Recent zbMATH articles in MSC 54https://zbmath.org/atom/cc/542021-01-08T12:24:00+00:00Werkzeug\(s\)-\(R\)-continuous functions.https://zbmath.org/1449.540212021-01-08T12:24:00+00:00"Etebar, Masoumeh"https://zbmath.org/authors/?q=ai:etebar.masoumehSummary: A new class of continuous functions, namely \(s\)-\(R\)-continuous functions, is introduced. The relations of \(s\)-\(R\)-continuity with continuity and other variants of continuity are discussed. Properties of \(s\)-\(R\)-continuous functions are studied.Common fixed point theorems in metric space using new CLR property.https://zbmath.org/1449.540532021-01-08T12:24:00+00:00"Bhutia, Jigmi Dorjee"https://zbmath.org/authors/?q=ai:bhutia.jigmi-dorjee"Tiwary, Kalishankar"https://zbmath.org/authors/?q=ai:tiwary.kalishankarSummary: In this paper we present a new property of functions called weakly \(CLR^*_{(f,g),T}\) property which generalizes the class of mappings already defined in the literature [\textit{V. Popa}, ``Fixed point theorems for two pairs of mappings satisfying a new type of common limit range property'', Filomat 31, No. 11, 3181--3192 (2017; \url{doi:10.2298/FIL1711181P})]. We obtain a common fixed point theorem in metric space for mappings satisfying the weakly \(CLR^*_{(f,g),T}\) property. Some examples to justify our results are given.Some fixed point theorems of single-valued mapping under \(c\)-distance.https://zbmath.org/1449.540642021-01-08T12:24:00+00:00"Han, Yan"https://zbmath.org/authors/?q=ai:han.yan"Xu, Shaoyuan"https://zbmath.org/authors/?q=ai:xu.shaoyuan"Duan, Jiangmei"https://zbmath.org/authors/?q=ai:duan.jiangmei"Dai, Tingting"https://zbmath.org/authors/?q=ai:dai.tingtingSummary: Many existing well-known theorems for cone metric spaces under \(c\)-distance in the literature rely strongly either on the assumption of the continuity of the mappings or the normality of the cone. In this paper, some new fixed point theorems for one single-valued mapping under \(c\)-distance in cone metric spaces are presented. The results do not require these two conditions. Furthermore, the existence and uniqueness of the fixed point are obtained. Finally, some supporting examples are given to show that our results improve and generalize some well-known comparable results.On gluing of quasi-pseudometric spaces.https://zbmath.org/1449.510082021-01-08T12:24:00+00:00"Mutemwa, Yolanda"https://zbmath.org/authors/?q=ai:mutemwa.yolanda"Otafudu, Olivier Olela"https://zbmath.org/authors/?q=ai:otafudu.olivier-olela"Sabao, Hope"https://zbmath.org/authors/?q=ai:sabao.hopeSummary: The concept of gluing a family of \(T_0\)-quasi-metric spaces along subsets was introduced by the second author [Topology Appl. 263, 159--171 (2019; Zbl 1419.51010)]. In this article, we continue the study of externally Isbell-convex and weakly externally Isbell-convex subsets of a \(T_0\)-quasi-metric space. We finally investigate some properties of the resulting \(T_0\)-quasi-metric space obtained by gluing a family of Isbell-convex \(T_0\)-quasi-metric spaces attached
along isometric subspaces.Some new cyclic admissibility type with uni-dimensional and multidimensional fixed point theorems and its applications.https://zbmath.org/1449.540822021-01-08T12:24:00+00:00"Mongkolkeha, Chirasak"https://zbmath.org/authors/?q=ai:mongkolkeha.chirasak"Sintunavarat, Wutiphol"https://zbmath.org/authors/?q=ai:sintunavarat.wutipholSummary: In this paper, we introduce the concept of a cyclic \((\alpha,\beta)\)-admissible mapping type \(S\) and the notion of an \((\alpha,\beta)\)-\((\psi,\varphi)\)-contraction type \(S\). We also establish fixed point results for such contractions along with the cyclic \((\alpha,\beta)\)-admissibility type \(S\) in complete \(b\)-metric spaces and provide some examples for supporting our result. Applying our new results, we obtain fixed point results for cyclic mappings and multidimensional fixed point results. As application, the existence of the solution of a nonlinear integral equation is discussed.An improvement in ordered cone \(b\)-metric spaces over Banach algebras.https://zbmath.org/1449.540632021-01-08T12:24:00+00:00"Han, Yan"https://zbmath.org/authors/?q=ai:han.yan"Xu, Shaoyuan"https://zbmath.org/authors/?q=ai:xu.shaoyuan"Dong, Yanshou"https://zbmath.org/authors/?q=ai:dong.yanshouSummary: The purpose of this paper is to improve some famous theorems for contractive mapping from \(\rho \left({\alpha + \beta} \right) \in \left[ {0, \frac{1}{s}} \right)\) to \(\rho \left({\alpha + \beta} \right) \in \left[ {0, 1} \right)\) in ordered cone \(b\)-metric spaces over Banach algebras with coefficient \(s \ge 1\) (\(\left({\rho \left(x \right)} \right. \) is the spectral radius of the generalized Lipschitz constant \(\left. x \right)\). Moreover, some similar improvements in ordered cone \(b\)-metric spaces are also obtained, which are from \(\left({\alpha + \beta} \right) \in \left[ {0, \frac{1}{s}} \right)\) to \(\left({\alpha + \beta} \right) \in \left[ {0, 1} \right)\). Some examples are given to support that our new results are genuine improvements and generalizations.Some coincidence and common fixed point theorems concerning \(F\)-contraction and applications.https://zbmath.org/1449.541012021-01-08T12:24:00+00:00"Tomar, Anita"https://zbmath.org/authors/?q=ai:tomar.anita"Sharma, Ritu"https://zbmath.org/authors/?q=ai:sharma.rituSummary: The aim of this paper is to establish coincidence and common fixed point theorems for a discontinuous noncompatible pair of self-maps in noncomplete metric space without containment requirement of range space of involved maps acknowledging the notion of \(F\)-contraction introduced by \textit{D. Wardowski} [Fixed Point Theory Appl. 2012, Paper No. 94, 6 p. (2012; Zbl 1310.54074)]. Our results generalize, extend and improve analogous results existing in the literature and are supported with the help of illustrative examples associated with pictographic validations to demonstrate the authenticity of the postulates. Solutions of two-point boundary value problem of a second order differential equation arising in electric circuit and a Volterra type integral equation using Ćirić type as well as Hardy-Rogers-type \(F\)-crontactions are also given to exhibit the usability of obtained results.Fixed point theorems for generalized \(\beta\)-\(\psi\)-Geraghty contraction type maps in \(S\)-metric space.https://zbmath.org/1449.540962021-01-08T12:24:00+00:00"Singh, K. Anthony"https://zbmath.org/authors/?q=ai:singh.k-anthony"Singh, M. R."https://zbmath.org/authors/?q=ai:singh.manas-ranjan|singh.mahi-r|singh.medini-r|singh.maibam-ranjitSummary: In this paper, we introduce the notion of generalized \(\beta\)-\(\psi\)-Geraghty contraction type maps and \(\beta\)-\(\psi\)-Geraghty contraction type maps in the context of \(S\)-metric space and establish some fixed point theorems for such maps. Our results (with some modifications) extend the fixed point results of \textit{E. Karapinar} [Filomat 28, No. 1, 37--48 (2014; Zbl 06704732)] to complete \(S\)-metric space. An example is also given to illustrate our result.Common fixed points for \(S\)-weakly \(B\)-contraction mappings.https://zbmath.org/1449.540832021-01-08T12:24:00+00:00"Murugesu, Menaka"https://zbmath.org/authors/?q=ai:murugesu.menaka"Muthiah, Marudai"https://zbmath.org/authors/?q=ai:muthiah.marudai"Khan, Mohammad Saeed"https://zbmath.org/authors/?q=ai:khan.mohammad-saeedSummary: The purpose of this paper is to prove common fixed point theorems for \(S\)-weakly \(B\)-contraction mappings in complete metric space which generalize and unify many fixed point theorems available in the literature.The research of periodic shadowing property and equicontinuity in the product \(G\)-space.https://zbmath.org/1449.370142021-01-08T12:24:00+00:00"Ji, Zhanjiang"https://zbmath.org/authors/?q=ai:ji.zhanjiang"Zhang, Gengrong"https://zbmath.org/authors/?q=ai:zhang.gengrong"Tu, Jingxian"https://zbmath.org/authors/?q=ai:tu.jingxianSummary: The concept of \(G\)-periodic shadowing property and \(G\)-equicontinuity is introduced in the product space under the action of topological group. By using the property of the product map, we study the relationship of these dynamical properties between product mapping \(f \times g\) and sub mapping \(f, g\). We obtain the following conclusions. (1) The product map \(f \times g\) has the \(G\)-periodic shadowing property if and only if the map \(f\) has the \({G_1}\)-periodic shadowing property and the map \(g\) has the \({G_2}\)-periodic shadowing property. (2) The product map \(f \times g\) has the \(G\)-equicontinuity if and only if the map \(f\) has the \({G_1}\)-equicontinuity and the map \(g\) has the \({G_2}\)-equicontinuity. These results enrich the theory of \(G\)-periodic shadowing property and \(G\)-equicontinuity in the product space under the action of topological group.A note on lattice-valued filters.https://zbmath.org/1449.540102021-01-08T12:24:00+00:00"Zhao, Hu"https://zbmath.org/authors/?q=ai:zhao.hu"Zhang, Hongying"https://zbmath.org/authors/?q=ai:zhang.hongyingSummary: We first pointed out that the product of two lattice-valued filters was not necessarily lattice-valued filter by a counterexample. Then we defined inverse operation and multiplication for lattice-valued filters on groups, and characterized the results of inverse operation and multiplication.Endpoints of generalized \(\phi\)-contractive multivalued mappings of integral type.https://zbmath.org/1449.540802021-01-08T12:24:00+00:00"Mohammadi, Babak"https://zbmath.org/authors/?q=ai:mohammadi.babak"Alizadeh, Esmaeil"https://zbmath.org/authors/?q=ai:alizadeh.esmaeilSummary: Recently, some researchers have established some results on existence of endpoints for multivalued mappings. In particular, \textit{B. Mohammadi} and \textit{Sh. Rezapour} [Sci. Bull., Ser. A, Appl. Math. Phys., Politeh. Univ. Buchar. 77, No. 1, 17--20 (2015; Zbl 1349.54113)] used the technique of \(\alpha\)-\(\psi\)-contractive mappings, due to \textit{B. Samet} et al. [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 4, 2154--2165 (2012; Zbl 1242.54027)], to give some results about endpoints of Suzuki type quasi-contractive multifunctions satisyfing property (BS). In this paper, we prove existence and uniqueness of endpoint for multivalued mappings satisfing the weaker conditions generalized \(\phi\)-contractivity of integral type and property (HS). This result generalize and improve Mohammadi and Rezapour's result. Also, we give an example to illustrate the usability of the result.Expansion theorem of Deng's metric.https://zbmath.org/1449.540122021-01-08T12:24:00+00:00"Chen, Peng"https://zbmath.org/authors/?q=ai:chen.peng"Qiu, Xin"https://zbmath.org/authors/?q=ai:qiu.xinSummary: In this paper, we give the pointwise extension theorems of the Deng's metric on a completely distributive lattice and its several other equivalent axiom forms, and point out that the topology and uniform structure induced by the Deng's metric are the topology induced by the Erceg's metric and the Hutton's uniform structure respectively.C-class function on some common fixed point theorems for weakly subsequentially continuous mappings in Menger spaces.https://zbmath.org/1449.540512021-01-08T12:24:00+00:00"Beloul, Said"https://zbmath.org/authors/?q=ai:beloul.said"Ansari, Arslan Hojet"https://zbmath.org/authors/?q=ai:ansari.arslan-hojetSummary: The aim of this paper is to prove some common fixed point theorems for two weakly subsequentially continuous and compatible of type (E) pairs of self mappings in Menger spaces. Two examples are given to illustrate our results.Approaching simultaneous Fredholm integral equations using common fixed point theorems in complex valued metric spaces.https://zbmath.org/1449.540472021-01-08T12:24:00+00:00"Alfaqih, Waleed M."https://zbmath.org/authors/?q=ai:alfaqih.waleed-mohd"Imdad, Mohammad"https://zbmath.org/authors/?q=ai:imdad.mohammad"Gubran, Rqeeb"https://zbmath.org/authors/?q=ai:gubran.rqeebSummary: The aim of this paper is to discuss the existence and uniqueness of a common solution for the following system of linear Fredholm integral equations (of the second kind): \[u(t)=f_i(t)+\beta\int^b_aK_i(t,s)F_i(u(s)\,ds,\quad t,s\in[a,b],\] where \(\beta\in\mathbb{R}\), \(f_i,K_i\) and \(F_i\) are given continuous functions, \(i=1,2\), while \(u\) is an unknown function to be determined. To establish this, we prove a common fixed point theorem for two self-mappings defined on a complex metric space. Moreover, we prove coincidence and common fixed point theorems for two weakly compatible self-mappings defined on a complex metric space.Fixed point theorem for integral type contraction quasi \(b\)-metric space.https://zbmath.org/1449.540932021-01-08T12:24:00+00:00"Rahman, Mujeeb Ur"https://zbmath.org/authors/?q=ai:rahman.mujeeb-ur"Sarwar, Mohammad"https://zbmath.org/authors/?q=ai:sarwar.mohammadSummary: In this paper, we introduce contractive conditions of integral type in the setting of dislocated quasi \(b\)-metric spaces and present a fixed point theorem in the framework of dislocated quasi \(b\)-metric spaces. An example is given in the support of our main results.Applications of partial belong and total non-belong relations on soft separation axioms and decision-making problem.https://zbmath.org/1449.540132021-01-08T12:24:00+00:00"El-Shafei, M. E."https://zbmath.org/authors/?q=ai:el-shafei.mohammed-e"Al-shami, T. M."https://zbmath.org/authors/?q=ai:al-shami.tareq-mohammedSummary: This study introduces a new family of soft separation axioms and a real-life application utilizing partial belong and natural non-belong relations. First, we initiate the concepts of w-soft \(T_i\)-spaces \((i=0, 1, 2, 3, 4)\) with respect to distinct ordinary points. These concepts generate a wider family of soft spaces compared with soft \(T_i\)-spaces, p-soft \(T_i\)-spaces and e-soft \(T_i\)-spaces. We illustrate the relationships between w-soft \(T_i\)-spaces with the help of examples and discuss some sufficient conditions of soft topological spaces to be w-soft \(T_i\)-spaces. Additionally, we point out that stable or soft regular spaces are sufficient conditions for the equivalence among the concepts of soft \(T_i\), p-soft \(T_i\) and w-soft \(T_i\). We highlight on explaining the links between w-soft \(T_i\)-spaces and their parametric topological spaces and studying the role of enriched spaces in these links. Furthermore, we prove that w-soft \(T_i\)-spaces are hereditary and topological properties, and they are preserved under finite product soft spaces. Finally, we propose an algorithm to bring out the optimal choices. This algorithm is based on dividing the whole parameters set into parameter sets and then apply a partial belong relation in the favorite soft sets. This application is supported with an interesting example to show how to implement this algorithm.Fixed point belonging to the zero-set of a given function.https://zbmath.org/1449.541042021-01-08T12:24:00+00:00"Vetro, Francesca"https://zbmath.org/authors/?q=ai:vetro.francescaSummary: We prove the existence and uniqueness of a fixed point belonging to the zero-set of a given function. The results are established in the setting of metric spaces and partial metric spaces. Our approach combines the recent notions of \((F,\varphi)\)-contraction and \(\mathcal{Z}\)-contraction. The main result allows to deduce, as a particular case, some of the known results in the literature. An example supports the theory.Weakly \((s,r)\)-contractive multi-valued operators on \(b\)-metric space.https://zbmath.org/1449.541052021-01-08T12:24:00+00:00"Ye, Lingjuan"https://zbmath.org/authors/?q=ai:ye.lingjuan"Shen, Congcong"https://zbmath.org/authors/?q=ai:shen.congcongSummary: In this paper we introduce the notion of weakly \((s,r)\)-contractive multi-valued operator on \(b\)-metric space and establish some fixed point theorems for this operator. In addition, an application to the differential equation is given to illustrate the usability of the obtained results.Fuzzy structure space of semirings and \(\Gamma \)-semirings.https://zbmath.org/1449.161022021-01-08T12:24:00+00:00"Goswami, Sarbani Mukherjee"https://zbmath.org/authors/?q=ai:goswami.sarbani-mukherjee"Mukhopadhyay, Arup"https://zbmath.org/authors/?q=ai:mukhopadhyay.arup-ranjan|mukhopadhyay.arup-kumar"Sardar, Sujit Kumar"https://zbmath.org/authors/?q=ai:sardar.sujit-kumarSummary: The purpose of this paper is to study the fuzzy structure space of a semiring as well as of a \(\Gamma\)-semiring. We study separation axioms, compactness, etc. in the fuzzy structure space of a semiring. Similar study has also been accomplished in the setting of a \(\Gamma\)-semiring \(S\) by using the nice interplay between \(S\) and its left operator semiring \(L\).Improvements of fixed point theorems for mappings with \(\mathcal{A}\)-contractions on metric spaces.https://zbmath.org/1449.540912021-01-08T12:24:00+00:00"Piao, Yongjie"https://zbmath.org/authors/?q=ai:piao.yongjieSummary: In this paper, we introduce a new class \({\mathcal{A}^*}\) of 3-dimensional functions, which is a generalization of a known class \(\mathcal{A}\), obtain a fixed point theorem for a mapping and a common fixed point theorem for an infinite family of mappings, and discuss the existence problems of fixed points for a mapping on a nonempty set with two metrics under non-continuity and non-completeness. The obtained results generalize and improve many known conclusions.Cone \(D\)-metric spaces over Banach algebras and common fixed point theorems.https://zbmath.org/1449.541072021-01-08T12:24:00+00:00"Zhang, Xuezhi"https://zbmath.org/authors/?q=ai:zhang.xuezhi"Xue, Xifeng"https://zbmath.org/authors/?q=ai:xue.xifengSummary: In order to expand fixed point theory, the concept of cone \(D\)-metric spaces over Banach algebras is put forward in this manuscript. Then by establishing iterative sequence, the existence and uniqueness of common fixed point of compatible mappings are studied on cone \(D\)-metric spaces over Banach algebras without the normality of cone. Two new common fixed point theorems are proved, which improve and extend the corresponding results of some existing references.Quadruple random common fixed point results of generalized Lipschitz mappings in cone \(b\)-metric spaces over Banach algebras.https://zbmath.org/1449.540712021-01-08T12:24:00+00:00"Kongban, Chayut"https://zbmath.org/authors/?q=ai:kongban.chayut"Kumam, Poom"https://zbmath.org/authors/?q=ai:kumam.poomSummary: In this paper, we introduce the concept of cone \(b\)-metric spaces over Banach algebras and present some quadruple random coincidence points and quadruple random common fixed point theorems for nonlinear operators in such spaces.Some fixed point results for modified \(F\)-contractions in metric spaces via a new type of \((\alpha, \beta)\)-cyclic admissible mappings in metric space.https://zbmath.org/1449.540772021-01-08T12:24:00+00:00"Mebawondu, A. A."https://zbmath.org/authors/?q=ai:mebawondu.akindele-adebayo"Mewomo, O. T."https://zbmath.org/authors/?q=ai:mewomo.oluwatosin-temitopeSummary: The aim of this paper is to define the new type of mapping which is called modified Suzuki-Berinde \(F\)-contraction mapping in the framework of metric spaces. Fixed point theorems for such mappings in complete metric spaces are established. Furthermore, we present examples to support our main results and, using these examples, we establish that our main result is a generalization of the fixed point result of \textit{D. Wardowski} [Fixed Point Theory Appl. 2012, Paper No. 94, 6 p. (2012; Zbl 1310.54074)], \textit{H. Piri} and \textit{P. Kumam} [ibid. 2014, Paper No. 210, 11 p. (2014; Zbl 1371.54184)], and others in the literature.On retraction problem concerning inclusion of \(F\)-contractions in almost contractions.https://zbmath.org/1449.541032021-01-08T12:24:00+00:00"Udo-utun, Xavier Alexius"https://zbmath.org/authors/?q=ai:udo-utun.xavier-alexiusSummary: We have established, in the context of metric spaces, that an \(F\)-contraction restricted to an appropriate neighborhood of its fixed point is an almost contraction and obtain a definition of retractions on complete metric spaces. We have illustrated such an inclusion using an example studied in [\textit{D. Wardowski}, Fixed Point Theory Appl. 2012, Paper No. 94, 6 p. (2012; Zbl 1310.54074)] and in [\textit{G. Minak} et al., Bull. Belg. Math. Soc. - Simon Stevin 22, No. 3, 411--422 (2015; Zbl 1326.54046)].\(S_\alpha\)-connectedness in topological spaces.https://zbmath.org/1449.540062021-01-08T12:24:00+00:00"Tyagi, B. K."https://zbmath.org/authors/?q=ai:tyagi.brij-kishore"Bhardwaj, Manoj"https://zbmath.org/authors/?q=ai:bhardwaj.manoj"Singh, Sumit"https://zbmath.org/authors/?q=ai:singh.sumitSummary: In this paper, connectedness of a class of \(S_\alpha\)-open sets in a topological space \(X\) is introduced. The connectedness of this class on \(X\), called \(S_\alpha\)-connectedness, turns out to be equivalent to connectedness of \(X\) when \(X\) is locally indiscrete or with finite \(\alpha\)-topology. The \(S_\alpha\)-continuous and \(S_\alpha\)-irresolute mappings are defined and their relationship with other mappings such as continuous mappings and semi-continuous mappings are discussed. An intermediate value theorem is obtained. The hyperconnected spaces constitute a subclass of the class of \(S_\alpha\)-connected spaces.Integral type common fixed point theorems in modified intuitionistic fuzzy metric spaces.https://zbmath.org/1449.540522021-01-08T12:24:00+00:00"Beloul, Said"https://zbmath.org/authors/?q=ai:beloul.said"Tomar, Anita"https://zbmath.org/authors/?q=ai:tomar.anitaSummary: A-subsequential continuity, A-compatibility of type (E), compatibility of type (E), and weak subsequential continuity in a intuitionistic fuzzy metric space are introduced and the applicability of these notions in establishing the existence of a unique common fixed point is demonstrated. An example is given to outline our outcomes and a system of Fredholm equations is resolved as an application of our conclusions.Fuzzy uniform Scott topology.https://zbmath.org/1449.540142021-01-08T12:24:00+00:00"Li, Hui"https://zbmath.org/authors/?q=ai:li.hui.3|li.hui.2|li.hui.4|li.hui|li.hui.5|li.hui.1"Jiang, Guanghao"https://zbmath.org/authors/?q=ai:jiang.guanghao"Liu, Dongming"https://zbmath.org/authors/?q=ai:liu.dongmingSummary: The concepts of fuzzy uniform Scott open set and fuzzy uniform Scott topology on the fuzzy uniform posets are introduced and some properties are investigated. Also, the definition of fuzzy uniform Scott closed set as well as some equivalent characterizations are given. Furthermore, a property of US is presented and both the fuzzy uniform Scott open sets and fuzzy lower sets are proved to satisfy this property.Rationalized evaluation subgroups of mapping spaces between complex Grassmannians.https://zbmath.org/1449.550022021-01-08T12:24:00+00:00"Otieno, Paul Antony"https://zbmath.org/authors/?q=ai:otieno.paul-antony"Gatsinzi, Jean Baptiste"https://zbmath.org/authors/?q=ai:gatsinzi.jean-baptiste"Onyango-Otieno, Vitalis"https://zbmath.org/authors/?q=ai:onyango-otieno.vitalisSummary: We determine evaluation subgroups of the inclusion \(Gr(2,n)\hookrightarrow Gr(2,n+1)\) between complex Grassmannians.Ordinal spaces.https://zbmath.org/1449.540392021-01-08T12:24:00+00:00"Keller, K."https://zbmath.org/authors/?q=ai:keller.kara|keller.kirby|keller.kai-johannes|keller.klaus|keller.karsten|keller.kurt-h|keller.kurtis"Petrov, E."https://zbmath.org/authors/?q=ai:petrov.evgenii-aleksandrovichIn this paper, an axiomatic approach is proposed for ordinal spaces. According to the authors, an ordinal space is an ordered triplet \((X,L,\delta)\) where \(X\) is a nonempty set, \(L\) is a linearly ordered set with minimal element \(0\), and \(\delta:X\times X\to L\) is a surjective map with
i) \(\delta(x,y)=\delta(y,x)\), for all \(x,y\in X\)
ii) \(\delta(x,y)=0\) iff \(x=y\).
The paper consists of 9 sections, that may be described as follows. Section 1 is introductory. In Section 2, two approaches are provided for defining isomorphisms between ordinal spaces. Section 3 is devoted to the definition of sets of balls in ordinal spaces by starting from the cuts of linearly ordered sets. In Section 4, the topological properties of ordinal spaces are being investigated. Further, in Section 5, an isomorphism criterion is established for the Hasse diagrams \({\mathcal{H}}(B_X)\) and \({\mathcal{H}}(B_Y)\) of two finite ordinal spaces \(X\) and \(Y\). Section 6 is devoted to the formulation of two conjectures about the maximal and minimal number of balls in finite ordinal spaces. The objective of Section 7 and Section 8 is to study the embeddings of ordinal spaces into one dimensional and higher dimensional Euclidean spaces, respectively. Finally, in Section 9, an analog of Gromov-Hausdorff distance is proposed for ordinal spaces with a fixed finite number of points.
Reviewer: Mihai Turinici (Iaşi)Lindelöf domination versus \(\omega\)-domination of discrete subsets.https://zbmath.org/1449.540322021-01-08T12:24:00+00:00"Alas, O. T."https://zbmath.org/authors/?q=ai:alas.ofelia-teresa"Junqueira, L. R."https://zbmath.org/authors/?q=ai:junqueira.lucia-r"Wilson, R. G."https://zbmath.org/authors/?q=ai:wilson.richard-gFor a space \(X\) and families \(\mathcal{A}, \mathcal{B}\) of subsets of \(X\) say that \(\mathcal{A}\) is dominated by \(\mathcal{B}\) if each member of \(\mathcal{A}\) is a subset of the closure of some member of \(\mathcal{B}\). When \(\mathcal{B}\) consists of all countable (Lindelöf) subsets, say that all members of \(\mathcal{A}\) are \(\omega\)-(Lindelöf) dominated. If \(d(X)\le s(X)<\mathfrak{w}\) then all discrete subsets of \(X\times X\) are \(\omega\)-dominated if and only if \(X\) is separable. A perfect space in which all closed discrete subsets are Lindelöf dominated has countable cellularity. Examples are given of perfect regular (resp.\,normal) spaces in which all discrete subsets are Lindelöf dominated but there is a closed, discrete subset that it not \(\omega\)-dominated; the respective example requires the existence of a \(Q\)-set.
Reviewer: David B. Gauld (Auckland)Some generalized continuous maps via ideal.https://zbmath.org/1449.540222021-01-08T12:24:00+00:00"Tiwari, Rajesh Kumar"https://zbmath.org/authors/?q=ai:tiwari.rajesh-kumar"Maitra, J. K."https://zbmath.org/authors/?q=ai:maitra.j-k"Vishwakarma, Ravi"https://zbmath.org/authors/?q=ai:vishwakarma.raviSummary: The \(\mu^*\)-open sets are sets where the closure has been considered in topological space and interior in generalized topological space. In this paper, we have studied the properties of \(\mu^*\)-open sets and defined the \(\mu^*\)-continuity in generalized topology on topological space. The \(I_\mu\)-open sets are generalized open sets of \(\mu^*\)-open sets. Some properties of \(I_\mu\)-open sets have been proved and defined the \(I_\mu\)-continuity and weakly \(I_\mu\)-continuity in generalized topology on topological spaces via ideal. Further, we have developed some classical properties on \(\mu^*\)-continuity, \(I_\mu\)-continuity and weakly \(I_\mu\)-continuity.Multidimensional \(p\)-adic metric and genetic code.https://zbmath.org/1449.540402021-01-08T12:24:00+00:00"Kozyrev, Sergeĭ Vladimirovich"https://zbmath.org/authors/?q=ai:kozyrev.sergei-v"Khrennikov, Andreĭ Yur'evich"https://zbmath.org/authors/?q=ai:khrennikov.andrei-yuSummary: We discuss the family of metrics in multimensional \(p\)-adic spaces. For a metric under consideration the set of balls differs from the set of balls for the standard multidimensional metric. Moreover, it is possible to consider the metrics for which the form of the sets of balls depends on the scale and position. As the example of the introduced metric spaces we study the 2-adic parametrization of the genetic code. We show that the degeneracy of the genetic code is described by the metric space from the considered family, i.e. the map of the genetic code is constant for the balls with respect to the metric from the introduced class.On the intermediate multivalued functions.https://zbmath.org/1449.540282021-01-08T12:24:00+00:00"Maslyuchenko, V. K."https://zbmath.org/authors/?q=ai:maslyuchenko.volodymyr-k"Mel'nyk, V. S."https://zbmath.org/authors/?q=ai:melnyk.v-sSummary: If \(X\) is a topological space, \(Y =\mathbb R\), then we say that maps \( g : X\to \mathbb R\), and \(h: X\to \mathbb R\), form a Hahn's pair (resp., strict Hahn's pair), if \(g\) is upper semicontinuous, \(h\) is lower semicontinuous and \(g (x)\leq h (x)\) (\(g(x)<h(x)\)) on \(X\). The austrian mathematician H. Hahn proved, that every Hahn's pair \((g, h)\) on metric space \(X\) has continuous intermediate function \(f: X\to \mathbb R\). For topological spaces \(X\) and \(Y\) we consider conditions, by which for arbitrary multivalued maps \(G: X\to Y\) and \(H: X\to Y\), such, that \(G (x)\subseteq H (x)\) for each \(x\in X\) and \(G\) and \(H\) are respectively upper and lower semicontinuous, there is \(F: X\to Y\) continuous, such, that \(G (x) \subseteq F (x)\subseteq H (x)\). We also consider conditions on topological spaces, by which the Hahn's theorem on the intermediate function has a multivalued analogue. The set \(\mathcal{P} (Y)\) is equipped with natural partial order, which is the relation of inclusion \(\subseteq\) of \(Y\) subsets, that allows to transfer the concept of Hahn's pair to the case of multivalued maps. We say that two multivalued maps form a Hahn (Hahn strict) pair, if \(G\) is upper semicontinuous, \(H\) is lower semicontinuous and \(G (x)\subseteq H (x)\) (\(G (x)\subset H (x))\) for each \(x\in X\). We show that for a normal \(T_1\)-space \(X\) and any Hahn's pair \((G,H)\) of multivalued maps \(G: X\to Y\) and \(H: X\to Y\), which values are segments, there is an intermediate continuous map \(F : X \to Y\), that has segments as values in \(\mathbb R\). Then we show, that the existence of an intermediate continuous map \(F: X\to Y\) for a Hahn's pair \((G, H)\) of maps \(G : X \to\mathbb R\), \(G (x) = (-\infty, g (x)]\), and \(H : X \to\mathbb R\), \(H (x) = (-\infty, h (x)]\), implies that \((g, h)\) is a Hahn's pair on \(X\) and has a continuous intermediate function \(f: X \to\mathbb R\) on \(X\).On lower and upper semi-continuous functions.https://zbmath.org/1449.540042021-01-08T12:24:00+00:00"Renukadevi, V."https://zbmath.org/authors/?q=ai:renukadevi.vellapandi"Vadakasi, S."https://zbmath.org/authors/?q=ai:vadakasi.subramanianThe authors study semi-continuous functions and Baire spaces in the context of generalized topological spaces. In particular, they give conditions which guarantee that a real-valued semi-continuous function is cliquish. (If \(\lambda\) and \(\mu\) are generalized topologies on a set \(X\), a real-valued function \(f\) on \(X\) is called \((\lambda, \mu)\)-cliquish if the set of all points in which \(f\) is \(\mu\)-continuous, is \(\lambda\)-dense.)
Reviewer: Heinz-Peter Butzmann (Mannheim)Sequences of contractions on cone metric spaces over Banach algebras and applications to nonlinear systems of equations and systems of differential equations.https://zbmath.org/1449.540462021-01-08T12:24:00+00:00"Alecsa, Cristian Daniel"https://zbmath.org/authors/?q=ai:alecsa.cristian-danielSummary: It is well known that fixed point problems of contractive-type mappings defined on cone metric spaces over Banach algebras are not equivalent to those in usual metric spaces (see [\textit{H. Huang} et al., Positivity 23, No. 1, 21--34 (2019; Zbl 07053419)] and [\textit{H. Liu} and \textit{S. Xu}, Fixed Point Theory Appl. 2013, Paper No. 320, 10 p. (2013; Zbl 1295.54062)]). In this framework, the novelty of the present paper represents the development of some fixed point results regarding sequences of contractions in the setting of cone metric spaces over Banach algebras. Furthermore, some examples are given in order to strengthen our new concepts. Also, based on the powerful notion of a cone metric space over a Banach algebra, we present important applications to systems of differential equations and coupled functional equations, respectively, that are linked to the concept of sequences of contractions.Some types of soft paracompactness via soft ideals.https://zbmath.org/1449.540192021-01-08T12:24:00+00:00"Turanli, Elif"https://zbmath.org/authors/?q=ai:turanli.elif"Demir, İzzettin"https://zbmath.org/authors/?q=ai:demir.izzettin"Özbakir, Oya Bedre"https://zbmath.org/authors/?q=ai:bedre-ozbakir.oyaSummary: In this paper, we introduce the soft \(\mathcal{I}\)-paracompact spaces and the soft \(\mathcal{I}\)-S-paracompact spaces. First, we investigate the relationships between these spaces and soft paracompact spaces. Also, we give some fundamental properties of these spaces. Finally, we prove that soft \(\mathcal{I}\)-S-paracompact spaces are invariant under perfect mappings.The strong \(G\)-shadowing property of the inverse limit spaces and the product spaces of group action.https://zbmath.org/1449.370162021-01-08T12:24:00+00:00"Ji, Zhanjiang"https://zbmath.org/authors/?q=ai:ji.zhanjiang"Zhang, Gengrong"https://zbmath.org/authors/?q=ai:zhang.gengrong"Tu, Jingxian"https://zbmath.org/authors/?q=ai:tu.jingxianSummary: The concept of the strong \(G\)-shadowing property is given in the metric spaces under the action of topological group. Then the dynamical properties of the strong \(G\)-shadowing property in the inverse limit spaces and the product spaces under the action of topological group are studied. The following conclusions are obtained. Let the system \( ({X_f}, \bar G, \bar d, \sigma)\) be the inverse limit spaces of the system \( (X, G, D, f)\). Then \(f\) has the \(G\)-shadowing property if and only if \(\sigma\) has the \({\bar G}\)-shadowing property. The product map \({f_1} \times {f_2}\) has the strong \(G\)-shadowing property if and only if the map \({f_1}\) has the strong \({G_1}\)-shadowing property and the map \({f_2}\) has the strong \({G_2}\)-shadowing property. These results enrich the theory of strong \(G\)-shadowing property in the inverse limit spaces and the product spaces under the action of topological group.Best proximity point for proximal Berinde nonexpansive mappings on starshaped sets.https://zbmath.org/1449.470972021-01-08T12:24:00+00:00"Bunlue, Nuttawut"https://zbmath.org/authors/?q=ai:bunlue.nuttawut"Suantai, Suthep"https://zbmath.org/authors/?q=ai:suantai.suthepSummary: In this paper, we introduce the new concept of proximal mapping, namely proximal weak contractions and proximal Berinde nonexpansive mappings. We prove the existence of best proximity points for proximal weak contractions in metric spaces, and for proximal Berinde nonexpansive mappings on starshaped sets in Banach spaces. Examples supporting our main results are also given. Our main results extend and generalize some of the well-known best proximity point theorems for proximal nonexpansive mappings in the literature.On generalized geometric difference of six dimensional rough ideal convergent of triple sequence defined by Musielak-Orlicz function.https://zbmath.org/1449.400012021-01-08T12:24:00+00:00"Esi, Ayhan"https://zbmath.org/authors/?q=ai:esi.ayhan"Subramanian, N."https://zbmath.org/authors/?q=ai:subramanian.nagarajanSummary: We introduce a rough ideal convergent of triple sequence spaces defined by Musielak-Orlicz function, using an six dimensional infinite matrix, and a generalized geometric difference Zweier six dimensional matrix operator \(B^p_{(abc)}\) of order \(p\). We obtain some topological and algebraic properties of these spaces.Coincidence points and common fixed points for mappings with \(\phi \)-contractive conditions on metric spaces.https://zbmath.org/1449.540922021-01-08T12:24:00+00:00"Piao, Yongjie"https://zbmath.org/authors/?q=ai:piao.yongjieSummary: In this paper, we obtain existence theorems of common fixed points and coincidence points for multi-valued mappings and single-valued mappings satisfying \(\phi \)-contractive type conditions on metric spaces, and also give several fixed point theorems.Notes on common coincidence and fixed points of an operator and a multivalued weakly commutative map.https://zbmath.org/1449.540852021-01-08T12:24:00+00:00"Negoescu, Nicoleta"https://zbmath.org/authors/?q=ai:negoescu.nicoleta(no abstract)Some fixed point theorems in Menger probabilistic partial metric spaces with application to Volterra type integral equation.https://zbmath.org/1449.540602021-01-08T12:24:00+00:00"Ghanenia, Amir"https://zbmath.org/authors/?q=ai:ghanenia.amir"Khanehgir, Mahnaz"https://zbmath.org/authors/?q=ai:khanehgir.mahnaz"Allahyari, Reza"https://zbmath.org/authors/?q=ai:allahyari.reza"Mehrabinezhad, Mohammad"https://zbmath.org/authors/?q=ai:mehrabinezhad.mohammadSummary: In this paper, we introduce the notion of Menger probabilistic partial metric space and prove some fixed point theorems in the framework of such spaces. Some examples and an application to Volterra type integral equations are given to support the obtained results. Finally, we apply successive approximations method to find a solution for a Volterra type integral equation with high accuracy.Fixed point theorems for generalized \(F\)-contractions and generalized \(F\)-Suzuki-contractions in complete dislocated \(S_b\)-metric spaces.https://zbmath.org/1449.540782021-01-08T12:24:00+00:00"Mehravaran, Hamid"https://zbmath.org/authors/?q=ai:mehravaran.hamid"Khanehgir, Mahnaz"https://zbmath.org/authors/?q=ai:khanehgir.mahnaz"Allahyari, Reza"https://zbmath.org/authors/?q=ai:allahyari.rezaSummary: In this paper, first we describe the notion of dislocated \(S_b\)-metric space and then we introduce the new notions of generalized \(F\)-contraction and generalized \(F\)-Suzuki-contraction in the setup of dislocated \(S_b\)-metric spaces. We establish some fixed point theorems involving these contractions in complete dislocated \(S_b\)-metric spaces. We also furnish some examples to verify the effectiveness and applicability of our results.\(\mathcal{T}^{\mathrm{min}}_{(\alpha,\beta)\text{-}\gamma}\) spaces.https://zbmath.org/1449.540072021-01-08T12:24:00+00:00"Wu, Yaoqiang"https://zbmath.org/authors/?q=ai:wu.yaoqiangSummary: First, this paper introduces the concepts of minimal \( (\alpha,\beta)\)-\(\gamma\)-open sets and \(\mathcal{T}^{\mathrm{min}}_{(\alpha,\beta)\text{-}\gamma}\) spaces. Then, we discuss their properties in topological spaces. Furthermore, we give the concept of minimal \( (\alpha,\beta)\)-\(\gamma\)-continuous and \( (\alpha,\beta)\)-\(\gamma\)-\(LF\) spaces. Also, we obtain their essential topological properties.Insertion of a contra-Baire-1 (Baire-.5) function between two comparable real-valued functions.https://zbmath.org/1449.260052021-01-08T12:24:00+00:00"Mirmiran, Majid"https://zbmath.org/authors/?q=ai:mirmiran.majid"Naderi, Binesh"https://zbmath.org/authors/?q=ai:naderi.bineshFor a topological space \(X\), the function \(f : X \to \mathbb{R}\) is called \textit{Baire-.5} if the preimage of every open subset of \(\mathbb{R}\) is a \(G_{\delta}\)-set in \(X.\)
A property \(P\) defined relative to real-valued function on a topological space is a \textit{B-.5-property} if any constant function has property \(P\) and the sum of a function with property \(P\) and any Baire-.5 function also has property \(P.\) Let \(P_{1}\) and \(P_{2}\) be two B-.5-properties. Then, a space \(X\) has the \textit{weak B-.5-insertion property for \((P_{1},P_{2})\)} (\textit{B-.5-insertion property for \((P_{1},P_{2})\))} if for any functions \(g, f : X \to \mathbb{R}\) such that \(g \leq f,\) (\(g < f,\)) \(g\) has property \(P_{1}\) and \(f\) has property \(P_{2},\) then there exists a Baire-.5 function \(h\) such that \(g \leq h \leq f\) (\(g < h < f\)).
The main results are the following: for a topological space such that \(F_{\sigma}\)-kernel of sets are \(F_{\sigma}\)-sets, a sufficient condition is given for the weak B-.5-insertion property. Also for a space with the weak B-.5-insertion property, a necessary and sufficient condition is given for the space to have the B-.5-insertion property.
Reviewer: Zoltán Finta (Cluj-Napoca)Almost contra-\(P_S\)-continuity in topological spaces.https://zbmath.org/1449.540232021-01-08T12:24:00+00:00"Tyagi, Brij K."https://zbmath.org/authors/?q=ai:tyagi.brij-kishore"Singh, Sumit"https://zbmath.org/authors/?q=ai:singh.sumit"Bhardwaj, Manoj"https://zbmath.org/authors/?q=ai:bhardwaj.manojThere is a part of general topology that deals with the study of different variants of contra-continuity. These studies were initiated by Dontchev, who in 1996 defined \textit{contra-continuous functions} as such \(f:X\to Y\) for which \(f^{-1}(V)\) is closed in \(X\) for every open set \(V\) in \(Y\), see \textit{J. Dontchev} [Int. J. Math. Math. Sci. 19, No. 2, 303--310 (1996; Zbl 0840.54015)]. In the paper under review the authors introduce a new property of such type, namely the \textit{almost contra-\(P_S\)-continuity}. Let \(X\) and \(Y\) be topological spaces. A function \(f:X\to Y\) is called almost contra-\(P_S\)-continuous if \(f^{-1}(V)\) is \(P_S\)-closed in \(X\) for each regular open set \(V\) of \(Y\). From the Introduction: ``The paper is organized as follows. Section 2 develops the necessary preliminaries. In Section 3, the concept of almost contra-\(P_S\)-continuity is introduced, and we obtain characterization and basic properties of this notion. In Section 4, separation axioms are studied in relation to almost contra-\(P_S\)-continuity. In Section 5, \(P_S\)-closed graph of almost contra-\(P_S\)-continuous functions are defined. Section 6 gives sufficient conditions for a space to be connected and hyperconnected.''
Reviewer: Tomasz Natkaniec (Gdańsk)Almost continuity on generalized topological and minimal structure spaces.https://zbmath.org/1449.540012021-01-08T12:24:00+00:00"Al-Saadi, H. S."https://zbmath.org/authors/?q=ai:al-saadi.hanan-saad"Omran, S."https://zbmath.org/authors/?q=ai:omran.s-a|omran.salehThe concepts of generalized topology and generalized continuity were introduced by \textit{Á. Császár} [Acta Math. Hung. 96, No. 4, 351--357 (2002; Zbl 1006.54003)]. The concept of generalized topological and minimal structure spaces was introduced by \textit{S. Buadong} et al. [Int. J. Math. Anal., Ruse 5, No. 29--32, 1507--1516 (2011; Zbl 1244.54002)]. In this paper, the authors introduce the notion of almost \(gm\)-continuity on generalized topological and minimal structure spaces and study some of its basic properties. Moreover, the authors discuss some notions of \(gm\)-open sets and give their properties. Finally, the authors introduce the notion of \(gm\)-closure continuous functions and investigate the relations between them, resp. their effects on some notions of sets.
Reviewer: Shou Lin (Ningde)Fixed point results for \(\phi\)-\((\gamma,\eta, n, m)\)-contractions with applications to nonlinear integral equations.https://zbmath.org/1449.540622021-01-08T12:24:00+00:00"Hammad, Hasanen A."https://zbmath.org/authors/?q=ai:hammad.hasanen-abuelmagd"La Sen, Manuel De"https://zbmath.org/authors/?q=ai:de-la-sen.manuelSummary: The aim of this paper is to introduce a new class of pair of contraction mappings, called \(\phi\)-\((\gamma,\eta, n, m)\)-contraction pairs, and obtain common fixed point theorems for a pair of mappings in this class, satisfying a weakly compatible condition. As an application, we use mappings of this class to find the existence of solutions for nonlinear integral equations on the space of continuous functions and in some of its subspaces. Moreover, some examples are given here to illustrate the applicability of these results.Point-star networks and images of metric spaces.https://zbmath.org/1449.540412021-01-08T12:24:00+00:00"Lin, Shou"https://zbmath.org/authors/?q=ai:lin.shou"Huang, Yanhui"https://zbmath.org/authors/?q=ai:huang.yanhui"Zhang, Jing"https://zbmath.org/authors/?q=ai:zhang.jing.12|zhang.jing.3|zhang.jing.2|zhang.jing.1|zhang.jing.6|zhang.jing.10|zhang.jing.8|zhang.jing.9|zhang.jing.7|zhang.jing.11|zhang.jing.5Summary: A sequence \(\{\mathcal{P}_i\}_{i\in \mathbb{N}}\) of covers of a topological space \(X\) is called a point-star network for \(X\) if the family \(\{{\mathrm{st}} (x, \mathcal{P}_i)\}_{i\in \mathbb{N}}\) is a network at \(x\) in \(X\) for each \(x \in X\). The main purpose of this paper is to characterize the spaces which have a point-star network consisting of \(cs\)-finite \(cs\)-coverings and express them as certain images of metric spaces. It is proved that the following are equivalent for a topological space \(X\) when the property \(\mathfrak{P}\) of set families satisfies some suitable conditions: (1) \(X\) has a point-star network consisting of \(cs\)-coverings with property \(\mathfrak{P}\). (2) \(X\) has a point-star network consisting of \(sn\)-coverings with property \(\mathfrak{P}\). (3) \(X\) is a Cauchy \(sn\)-symmetric space with a \(\sigma\)-\(\mathfrak{P}\) \(cs\)-network. (4) \(X\) is a Cauchy \(sn\)-symmetric space with a \(\sigma\)-\(\mathfrak{P}\) \(sn\)-network. (5) \(X\) is a sequence-covering, \(\pi\) and \(\sigma\)-\(\mathfrak{P}\)-image of a metrizable space. (6) \(X\) is a 1-sequence-covering, compact and \(\sigma\)-\(\mathfrak{P}\)-image of a metrizable space. The above and some related works contain the study of locally finite or point-finite families as a special case, expand the research from bases to \(cs\)-networks, and enrich the idea of mutual classification on mappings and spaces.Coincidence point results in B-metric spaces via \(C_F\)-\(s\)-simulation function.https://zbmath.org/1449.540612021-01-08T12:24:00+00:00"Gupta, Anuradha"https://zbmath.org/authors/?q=ai:gupta.anuradha"Rohilla, Manu"https://zbmath.org/authors/?q=ai:rohilla.manuSummary: The notion of \(C_F\)-\(s\)-simulation function is introduced and the existence and uniqueness of coincidence point of two self mappings in the framework of b-metric spaces is investigated. An example with a corresponding numerical simulation is also provided to support the obtained result.Common best proximity points theorems.https://zbmath.org/1449.540562021-01-08T12:24:00+00:00"Chen, Lijun"https://zbmath.org/authors/?q=ai:chen.lijunSummary: In this paper, an existence and uniqueness common best proximity point theorem for a pair of non-self mappings was proved. Moreover, an example was given to support our main result, which generalized some well-known results in literature.New separation axioms in soft topological space.https://zbmath.org/1449.540172021-01-08T12:24:00+00:00"Prasannan, A. R."https://zbmath.org/authors/?q=ai:prasannan.a-r"Biswas, Jayanta"https://zbmath.org/authors/?q=ai:biswas.jayantaSummary: The more general form of soft separation axioms are defined in soft topological spaces and its interrelationship with existing soft separation axioms are studied. It was interesting to go through separation axiom as in [\textit{M. Shabir} and \textit{M. Naz}, Comput. Math. Appl. 61, No. 7, 1786--1799 (2011; Zbl 1219.54016)] shown that there are limited relation between \(T_i\) axioms (\(i= 0,1,2,3\)). In this paper, it is shown that these axioms are stronger than the existing separation axioms in soft topological spaces.Fixed point theorems for generalized contractive and expansive type mappings over a \(C^*\)-algebra valued metric space.https://zbmath.org/1449.540942021-01-08T12:24:00+00:00"Roy, Kushal"https://zbmath.org/authors/?q=ai:roy.kushal"Saha, Mantu"https://zbmath.org/authors/?q=ai:saha.mantuSummary: In this paper, fixed points of generalized contractive mappings and \(n\)-times reasonable expansive mappings over a \(C^*\)-algebra valued metric space have been investigated. The results obtained so far are the existence of fixed points of generalized contractive mappings via the notion of \(d\)-point of a lower semi-continuous function on the underlying space. Also, a result on coincidence points of two mappings has been established. Some examples are given in support of fixed points of expansive mappings.Existence of Picard-Jungck operator using \(C_G\)-simulation functions in generalized metric spaces.https://zbmath.org/1449.540552021-01-08T12:24:00+00:00"Chandok, Sumit"https://zbmath.org/authors/?q=ai:chandok.sumitSummary: We provide some new results with short proofs for the existence of Picard-Jungck operators in the framework of generalized metric spaces using \(C_G\)-simulation functions. An example is also provided to illustrate the usability of the results.Asymptotic average and Lipschitz shadowing property of the product map under group action.https://zbmath.org/1449.370152021-01-08T12:24:00+00:00"Ji, Zhanjiang"https://zbmath.org/authors/?q=ai:ji.zhanjiang"Zhang, Gengrong"https://zbmath.org/authors/?q=ai:zhang.gengrong"Tu, Jingxian"https://zbmath.org/authors/?q=ai:tu.jingxianSummary: The shadowing property is significant both in theory and application. In this paper, we introduce the concept of \(G\)-asymptotic average shadowing property and \(G\) Lipschitz shadowing property in the product space under the action of a topological group. By means of properties of the product map and zero density sets, we study the relationship of these shadowing propertes between product mapping \(f \times g\) and sub mapping \(f, g\). We obtain the following result: (1) the product map \(f \times g\) has the \(G\)-asymptotic average shadowing property if and only if the map \(f\) has the \({G_1}\)-asymptotic average shadowing property and the map \(g\) has the \({G_2}\)-asymptotic average shadowing property; (2) the product map \(f \times g\) has the \(G\)-Lipschitz shadowing property if and only if the map \(f\) has the \({G_1}\)-Lipschitz shadowing property and the map \(g\) has the \({G_2}\)-Lipschitz shadowing property. These results enrich the theory of asymptotic average shadowing property and Lipschitz shadowing property in the product space under the action of topological group.Fixed point theorems for certain contractive mappings of integral type.https://zbmath.org/1449.540742021-01-08T12:24:00+00:00"Liu, Zeqing"https://zbmath.org/authors/?q=ai:liu.zeqing"Liu, Xu"https://zbmath.org/authors/?q=ai:liu.xu"Guo, Yuchen"https://zbmath.org/authors/?q=ai:guo.yuchen"Jung, Chahn Yong"https://zbmath.org/authors/?q=ai:jung.chahn-yongSummary: Some fixed point theorems and properties of diminishing orbital diameters for a few contractive mappings of integral type in complete metric spaces are proved. Four nontrivial examples are included.Fixed point theorems for some contractive mappings of integral type with \(w\)-distance.https://zbmath.org/1449.540752021-01-08T12:24:00+00:00"Liu, Zeqing"https://zbmath.org/authors/?q=ai:liu.zeqing"Wang, Haoyue"https://zbmath.org/authors/?q=ai:wang.haoyue"Liu, Na"https://zbmath.org/authors/?q=ai:liu.na"Kang, Shin Min"https://zbmath.org/authors/?q=ai:kang.shin-minSummary: The existence, uniqueness and iterative approximations of fixed points for some contractive mappings of integral type defined in complete metric spaces with \(w\)-distance are proved. Four examples are given to demonstrate that the results in this paper extend and improve some well-known results in the literature.Common fixed point theorems in \(GP\)-metric space and applications.https://zbmath.org/1449.541022021-01-08T12:24:00+00:00"Tomar, Anita"https://zbmath.org/authors/?q=ai:tomar.anita"Sharma, Ritu"https://zbmath.org/authors/?q=ai:sharma.ritu"Upadhyay, Shivangi"https://zbmath.org/authors/?q=ai:upadhyay.shivangi"Beloul, Said"https://zbmath.org/authors/?q=ai:beloul.saidSummary: A generalized condition \((B)\) is introduced in the context of \(GP\)-metric spaces to establish coincidence and common fixed point results for discontinuous mappings and utilized to solve an integral equation and a functional equation arising in dynamic programming. Our results are absolutely novel and provide a new dimension in fixed point theory and can not be attained from the available results in the literature. Conclusively two explanatory examples are also furnished for the sake of clarity.Completeness and compactness in fuzzy cone metric spaces.https://zbmath.org/1449.540162021-01-08T12:24:00+00:00"Majumder, Arunima"https://zbmath.org/authors/?q=ai:majumder.arunima"Bag, Tarapada"https://zbmath.org/authors/?q=ai:bag.tarapadaSummary: In this paper, some basic results in complete fuzzy cone metric space are studied and Cantor's intersection theorem is established in fuzzy setting. On the other hand some results in compact fuzzy cone metric spaces are developed.\(\gamma_\mu\)-Lindelöf generalized topological spaces.https://zbmath.org/1449.540052021-01-08T12:24:00+00:00"Roy, B."https://zbmath.org/authors/?q=ai:roy.bishwambhar"Noiri, T."https://zbmath.org/authors/?q=ai:noiri.takashiSummary: In this paper we have introduced new types of sets termed as \(\omega_{\gamma_{\mu}}\)-open sets with the help of an operation and a generalized topology. We have also defined a notion of \(\gamma_\mu\)-Lindelöf spaces and discussed some of its basic properties.The topological pressure on an arbitrary topological space.https://zbmath.org/1449.370122021-01-08T12:24:00+00:00"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.30"Cao, Jie"https://zbmath.org/authors/?q=ai:cao.jieSummary: The purpose of this paper is to enrich the theory of topological pressure and to gain a wider range of topological structure. Using the compact method this paper deals with general topological space, and makes the spatial structure more concise. A new topological pressure is defined in new space. The properties of the new topological pressure are discussed and proved.Some convergence theorems for new iteration scheme in CAT(0) spaces.https://zbmath.org/1449.471172021-01-08T12:24:00+00:00"Uddin, Izhar"https://zbmath.org/authors/?q=ai:uddin.izhar"Ali, Javid"https://zbmath.org/authors/?q=ai:ali.javid"Rakočević, Vladimir"https://zbmath.org/authors/?q=ai:rakocevic.vladimirSummary: In this paper, we construct an iteration scheme involving a hybrid pair of nonexpansive mappings. For this scheme, we prove some convergence theorems in CAT(0) spaces. In process, we remove a restricted condition (also called end-point condition) in previous existing results. Thus, we generalize and improve several relevant results cited in the literature.A common fixed point theorem for weakly reciprocally continuous systems of maps satisfying a general contractive condition of integral type.https://zbmath.org/1449.540702021-01-08T12:24:00+00:00"Khantwal, Deepak"https://zbmath.org/authors/?q=ai:khantwal.deepak"Gairola, U. C."https://zbmath.org/authors/?q=ai:gairola.umesh-cSummary: In this paper, we prove a common fixed point theorem for weakly reciprocally continuous systems of maps satisfying a general contractive inequality of integral type on a finite product of metric spaces. Our result generalizes the results of \textit{A. Branciari} [Int. J. Math. Math. Sci. 29, No. 9, 531--536 (2002; Zbl 0993.54040)], \textit{B. E. Rhoades} [ibid. 2003, No. 63, 4007--4013 (2003; Zbl 1052.47052)], \textit{J. Matkowski} [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 21, 323--324 (1973; Zbl 0255.47063)] and \textit{U. C. Gairola} [``A fixed point theorem for system of transformation satisfying a general contractive condition of integral type on product spaces'', J. Mountain Res. 3, 87--93 (2008); \url{http://jmr.sharadpauri.org/papers/2008/87-93.pdf}].Some coupled fixed point theorems in convex metric spaces.https://zbmath.org/1449.540872021-01-08T12:24:00+00:00"Olatinwo, Memudu Olaposi"https://zbmath.org/authors/?q=ai:olatinwo.memudu-olaposiSummary: We establish some coupled fixed point theorems in convex cone metric spaces for \((k,\mu)\)-Lipschitzian and \((k,\mu,L)\)-Lipschitzian mappings. We assume that the cone has nonempty interior. Our results generalize and extend several known results in the existing literature.On relative \(\beta\)-normality.https://zbmath.org/1449.540312021-01-08T12:24:00+00:00"Das, A. K."https://zbmath.org/authors/?q=ai:das.ashis-kumar|kumar-das.ashish|das.arup-kumar|das.ajit-kumar|das.ananga-kumar|das.anjan-kr|das.apurva-kumar|das.ajoy-k-r|das.anil-kuman|das.abhik-kumar|das.ashok-kumar|das.aloke-k|das.amal-k|das.anjan-kumar|das.arabinda-k|das.asok-k|das.ashok-kr|das.asim-kumar|das.ajoy-kumar|das.amar-k|das.asit-kumar|das.ajoy-kanti|das.amit-kumar"Raina, S. S."https://zbmath.org/authors/?q=ai:raina.s-s\textit{A. V. Arhangel'skii} and \textit{L. Ludwig} [Commentat. Math. Univ. Carol. 42, No. 3, 507--519 (2001; Zbl 1053.54030)] defined a space \(X\) to be \(\alpha\)-normal (resp. \(\beta\)-normal) provided for any two disjoint closed set \(A\), \(B\) of \(X\) there exist open sets \(U, V\) of \(X\) such that \(A \cap U\) is dense in \(A\) and \(B \cap V\) is dense in \(B\) and \(U \cap V = \emptyset\) (resp. \(\overline{U}\cap\overline{V}=\emptyset\)). They assumed they were working in \(T_1\)-spaces. In this paper the authors, for \(Y \subset X\), define the subspace \(Y\) to be relatively \(\alpha\)-normal (resp. relatively \(\beta\)-normal) if given two disjoint closed subsets \(A,B\) of \(X\) there exist open subsets \( U,V\) of \(X\) such that \((A \cap Y) \cap U\) is dense in \(A \cap Y\) and \((B \cap Y) \cap V\) is dense in \(B \cap Y\) and \(U \cap V = \emptyset\) (\(\overline{U}\cap\overline{V}=\emptyset\)). The authors show how their results are related to other kinds of weak normality. For an example of weak normality, a set is called mildly normal by \textit{M. K. Singal} and \textit{A. R. Singal} [Kyungpook Math. J. 13, 27--31 (1973; Zbl 0266.54006)] if instead of two disjoint closed sets, one takes two disjoint regular closed sets (this notion was also defined under the name \(\kappa\)-normal by \textit{E. V. Shchepin} [Sib. Mat. Zh. 13, 1182--1196 (1972; Zbl 0256.54011)]). In the paper under review, except for the example of Fort's space, the examples are spaces on sets of three or four points, none of which are \(T_1\)-spaces.
Reviewer: Jerry E. Vaughan (Greensboro)Open remote neighborhoods of topological systems and their applications.https://zbmath.org/1449.540252021-01-08T12:24:00+00:00"Feng, Dandan"https://zbmath.org/authors/?q=ai:feng.dandan"Wu, Hongbo"https://zbmath.org/authors/?q=ai:wu.hongboSummary: The concept of open remote neighborhood is proposed in topological system, and its properties with applications are studied. At first, the concept of open remote neighborhood is proposed in topological system, and its basic properties are discussed. Furthermore, a method of determining topological system is given by open remote neighborhood systems. The definition of continuous of mapping at a fixed point between topological systems is defined by open remote neighborhood systems, by which the equivalent form of continuous mapping between topological systems is given. At last, the equivalent forms of some separations of topological systems are given by using the open remote neighborhood systems.On cardinality bounds for \(\theta^n\)-Urysohn spaces.https://zbmath.org/1449.540112021-01-08T12:24:00+00:00"Basile, F. A."https://zbmath.org/authors/?q=ai:basile.fortunata-aurora"Carlson, N."https://zbmath.org/authors/?q=ai:carlson.nathan-a"Porter, J."https://zbmath.org/authors/?q=ai:porter.jack-rayThe authors define the notion of \(n\)-\(\theta\)-closure: for a subset \(A\) of a space \(X\), \(cl^n_{\theta}(A) =cl_{\theta}\cdots cl_{\theta}(A)\) (the operator \(cl_{\theta}\) taken \(n\)-times). Using this operator, they then define \(\theta^n\)-Urysohn spaces as spaces in which any two distinct points have neighbourhoods with disjoint \(n\)-\(\theta\)-closures. A few cardinal functions (as the \(n\)-\(\theta\)-almost Lindelöf degree \(\theta^n\)-\(aL(X)\) and the \(\theta^n\)-Urysohn cellularity \(\theta^n\)-\(Uc(X)\)) of a space \(X\) are defined and used to extend to \(\theta^n\)-Urysohn spaces some cardinal inequalities known for Urysohn spaces. Sample results are: (1) If \(X\) is a \(\theta^n\)-Urysohn space, then \(\vert X\vert \le 2^{\theta^n\!\!-Uc(X)\chi(X)}\) and \(\vert X\vert \le 2^{\theta^n\!\!-aL(X)\chi(X)}\); (2) if \(X\) is a homogeneous \(\theta^n\)-Urysohn space, then \(\vert X\vert \le 2^{\theta^n\!\!-Uc(X)\pi\chi(X)}\).
Reviewer: Ljubiša D. Kočinac (Niš)The colouring existence theorem revisited.https://zbmath.org/1449.030142021-01-08T12:24:00+00:00"Shelah, S."https://zbmath.org/authors/?q=ai:shelah.saharonA the main result of this paper is devoted to a proof of a statement, abbreviated as \(\mathbf{Pr}_1(\lambda,\lambda,\lambda,(\theta_0,\theta_1))\), for a successor \(\lambda\) of a regular cardinal with \(\lambda\ge\theta_1^+\) and \(\theta_1>\theta_0\ge\aleph_0\). The general statement \(\mathbf{Pr}_1(\lambda,\mu,\sigma,(\theta_0,\theta_1))\) asserts that there is a colouring \(c:[\lambda]^2\to\sigma\) with the property that whenever two matrices \(\langle \zeta(\varepsilon,\alpha,i):\alpha<\mu, i<\mathbf{i}_\varepsilon\rangle\) (\(\varepsilon=0,1\)) of ordinals in \(\lambda\), with no repeated entries and with disjoint ranges, and \(\gamma<\sigma\) are given there are \(\alpha_0<\alpha_1<\mu\) such that \(c\{\zeta(0,\alpha_0,i),\zeta(1,\alpha_1,j)\}=\gamma\) for all \(i<\mathbf{i}_0\) and \(j<\mathbf{i}_1\).
The author also discusses some variants of this statement and their relations to earlier work on negative partition relations. There are also applications in general topology relating to the existence of regular (zero-dimensional) spaces with various compactness properties and discretely untouchable points.
Reviewer: K. P. Hart (Delft)New common fixed point results for pairs of self-maps satisfying common \(\left({E.A} \right)\) property in \(G\)-metric space.https://zbmath.org/1449.540662021-01-08T12:24:00+00:00"Hu, Pin"https://zbmath.org/authors/?q=ai:hu.pin"Gu, Feng"https://zbmath.org/authors/?q=ai:gu.feng|gu.feng.1Summary: In the framework of \(G\)-metric space, some new fixed point theorems are proved in the condition of mapping pair satisfying common \(\left({E. A} \right)\) property and weak compatibility. In the meantime, a specific example is provided to support the new result.A new common fixed point theorem for Suzuki type contractions via generalized \(\Psi\)-simulation functions.https://zbmath.org/1449.540692021-01-08T12:24:00+00:00"Joonaghany, Gholamreza Heidary"https://zbmath.org/authors/?q=ai:joonaghany.gholamreza-heidary"Farajzadeh, Ali"https://zbmath.org/authors/?q=ai:farajzadeh.ali-p"Azhini, Mahdi"https://zbmath.org/authors/?q=ai:azhini.mahdi"Khojasteh, Farshid"https://zbmath.org/authors/?q=ai:khojasteh.farshidSummary: In this paper, a new stratification of mappings, which is called \(\Psi\)-simulation functions, is introduced to enhance the study of the Suzuki type weak-contractions. Some well-known results in weak-contractions fixed point theory are generalized by our research. The methods used in proving the main results are new and different from the usual methods. Some suitable examples are furnished to demonstrate the validity of the hypothesis of our results and reality of our generalizations.A note on resolvability.https://zbmath.org/1449.540082021-01-08T12:24:00+00:00"Bhaskara Rao, K. P. S."https://zbmath.org/authors/?q=ai:bhaskara-rao.k-p-sSummary: A topological space is said to be resolvable if it has two disjoint dense subsets. If \(\aleph\) is a cardinal number (finite or infinite), a topological space is said to be \(\aleph\)-resolvable if it has a paiwise disjoint family of \(\aleph\) many dense subsets. \textit{A. Illanes} [Proc. Am. Math. Soc. 124, No. 4, 1243--1246 (1996; Zbl 0856.54010)] showed that a topological space which is \(\kappa\) resolvable for every finite integer \(\kappa\) is necessarily \(\aleph_0\)-resolvable. We generalize this result to infinite cardinals. We show that if a topological space \(X\) is \(\kappa\)-resolvable for every \(\kappa< \aleph\) and if cofinality of \(\aleph\) is \(\aleph_0\), then, \(X\) is \(\aleph\)-resolvable.Fixed point theory in \(\varepsilon\)-connected orthogonal metric space.https://zbmath.org/1449.540592021-01-08T12:24:00+00:00"Eshaghi, Gordji Madjid"https://zbmath.org/authors/?q=ai:eshaghi-gordji.madjid"Habibi, Hasti"https://zbmath.org/authors/?q=ai:habibi.hastiSummary: The existence of a fixed point in orthogonal metric spaces has been initiated by \textit{M. E. Gordji} et al. [Fixed Point Theory 18, No. 2, 569--578 (2017; Zbl 1443.47051)]. In this paper, we prove existence and uniqueness of a fixed point for mappings on \(\varepsilon\)-connected orthogonal metric space. As a consequence, we obtain the existence and uniqueness of a fixed point for analytic functions of one complex variable. The paper concludes with some illustrating examples.Fixed point theorems on locally T-convex spaces.https://zbmath.org/1449.540572021-01-08T12:24:00+00:00"Chen, Zhiyou"https://zbmath.org/authors/?q=ai:chen.zhiyouSummary: In this paper, in locally T-convex spaces, two new fixed point theorems are established without using the KKM technique, which respectively take the famous Schauder's and Browder's fixed point theorems on H-spaces as their particular cases, so that these two important theorems are generalized to T-convex spaces.Common fixed points in complex valued \(A_b\)-metric space.https://zbmath.org/1449.540952021-01-08T12:24:00+00:00"Singh, K. Anthony"https://zbmath.org/authors/?q=ai:singh.k-anthony"Singh, M. R."https://zbmath.org/authors/?q=ai:singh.mahi-r|singh.medini-r|singh.maibam-ranjit|singh.manas-ranjanSummary: In this paper, we prove two common fixed point theorems for two self mappings in complex valued \(A_b\)-metric space. Our results generalize the common fixed point results in complex valued \(b\)-metric space by \textit{A. A. Mukheimer} [``Some common fixed point theorems in complex valued \(b\)-metric spaces'', Sci. World J. 2014, Article ID 587825 (2014; \url{doi:10.1155/2014/5878250})] which are already generalizations of the results of \textit{A. Azam} et al. [Numer. Funct. Anal. Optim. 32, No. 3, 243--253 (2011; Zbl 1245.54036)] and \textit{S. Bhatt} et al. [Int. J. Pure Appl. Math. 73, No. 2, 159--164 (2011; Zbl 1246.54036)].Proximity point properties for admitting center maps.https://zbmath.org/1449.540732021-01-08T12:24:00+00:00"Labbaf Ghasemi, Mohammad Hosein"https://zbmath.org/authors/?q=ai:labbaf-ghasemi.mohammad-hussein"Haddadi, Mohammad Reza"https://zbmath.org/authors/?q=ai:haddadi.mohammad-reza"Eftekhari, Noha"https://zbmath.org/authors/?q=ai:eftekhari.nohaSummary: In this work, we investigate a class of admitting center maps on a metric space. We state and prove some fixed point and best proximity point theorems for them. We obtain some results and relevant examples. In particular, we show that, if \(X\) is a reflexive Banach space with the Opial condition and \(T:C\rightarrow X\) is a continuous admitting center map, then \(T\) has a fixed point in \(X\). Also, we show that, under some conditions, the set of all best proximity points is nonempty and compact.Domain representable Lindelöf spaces are cofinally Polish.https://zbmath.org/1449.540352021-01-08T12:24:00+00:00"Tkachuk, Vladimir V."https://zbmath.org/authors/?q=ai:tkachuk.vladimir-vSummary: We prove that, for any cofinally Polish space \(X\), every locally finite family of non-empty open subsets of \(X\) is countable. It is also established that Lindelöf domain representable spaces are cofinally Polish and domain representability coincides with subcompactness in the class of \(\sigma\)-compact spaces. It turns out that, for a topological group \(G\) whose space has the Lindelöf \(\sum\)-property, the space \(G\) is domain representable if and only if it is Čech-complete. Our results solve several published open questions.On weak quasicontractions in $b$-metric spaces.https://zbmath.org/1449.540792021-01-08T12:24:00+00:00"Mitrović, Zoran D."https://zbmath.org/authors/?q=ai:mitrovic.zoran-d"Hussain, Nawab"https://zbmath.org/authors/?q=ai:hussain.nawabThe authors obtain a generalization of a result of \textit{M. Bessenyei} [Publ. Math. 89, No. 3, 287--295 (2016; Zbl 1399.47139)] in $b$-metric spaces using weak quasicontraction under a comparison function, i.e., satisfying ad hoc requirements. They derive known results as corollaries, such as those, among others, due to \textit{Nguyen Van Dung} and \textit{Vo Thi Le Hang} [J. Fixed Point Theory Appl. 18, No. 2, 267--284 (2016; Zbl 1398.54080)].
Reviewer: Salvatore Sessa (Napoli)On Nadler's multi-valued contraction principle in complete metric spaces.https://zbmath.org/1449.540902021-01-08T12:24:00+00:00"Petruşel, Adrian"https://zbmath.org/authors/?q=ai:petrusel.adrianSummary: The aim of this paper to present an extended variant of the multi-valued contraction principle. Under the classical assumptions considered by \textit{S. B. Nadler jun.} [Pac. J. Math. 30, 475--488 (1969; Zbl 0187.45002)] and [\textit{H. Covitz} and \textit{S. B. Nadler jun.}, Isr. J. Math. 8, 5--11 (1970; Zbl 0192.59802)] (i.e., the completeness of the metric space \((X,d)\) and the contraction assumption on a self multi-valued operator on \(X\) having nonempty and closed values), several other conclusions with respect to the fixed point problem are presented.On products and diagonals of mappings in generalized topological spaces.https://zbmath.org/1449.540022021-01-08T12:24:00+00:00"Cao, Chunfang"https://zbmath.org/authors/?q=ai:cao.chunfang"Shen, Rongxin"https://zbmath.org/authors/?q=ai:shen.rongxinSummary: Based on the theory of products of generalized topologies, we introduce the product mappings and the diagonal mappings in generalized topological spaces in this paper. We investigate some basic properties (especially, the continuity, openness and closedness) of the product mappings and the diagonal mappings in generalized topological spaces.Characterization of a b-metric space completeness via the existence of a fixed point of Ciric-Suzuki type quasi-contractive multivalued operators and applications.https://zbmath.org/1449.540482021-01-08T12:24:00+00:00"Alolaiyan, Hanan"https://zbmath.org/authors/?q=ai:alolaiyan.hanan-abdulaziz"Ali, Basit"https://zbmath.org/authors/?q=ai:ali.basit"Abbas, Mujahid"https://zbmath.org/authors/?q=ai:abbas.mujahidSummary: The aim of this paper is to introduce Ćirić-Suzuki type quasi-contractive multivalued operators and to obtain the existence of fixed points of such mappings in the framework of b-metric spaces. Some examples are presented to support the results proved herein. We establish a characterization of strong b-metric and b-metric spaces completeness. An asymptotic estimate of a Hausdorff distance between the fixed point sets of two Ćirić-Suzuki type quasi-contractive multivalued operators is obtained. As an application of our results, existence and uniqueness of multivalued fractals in the framework of b-metric spaces is proved.Some results on an equivalence relation on the set of closed and bounded valued multifunctions.https://zbmath.org/1449.540272021-01-08T12:24:00+00:00"Aydoğan, S. Melike"https://zbmath.org/authors/?q=ai:aydogan.s-melike"Rezapour, Sh."https://zbmath.org/authors/?q=ai:rezapour.shahram"Sakar, F. Müge"https://zbmath.org/authors/?q=ai:sakar.fethiye-mugeSummary: By using the notion of the fixed point set of multi-valued mappings, we introduce an equivalence relation on the set of all closed and bounded valued multifunctions on a metric space. By using the notion we provide some related results.Signed topological measures on locally compact spaces.https://zbmath.org/1449.280142021-01-08T12:24:00+00:00"Butler, S. V."https://zbmath.org/authors/?q=ai:butler.svetlana-vIn this paper a (signed) topological measure on a locally compact space \(X\) is a function \(\mu\) defined on the union of the families of open sets, \(\mathcal{O}(X)\), and closed sets, \(\mathcal{C}(X)\), with values in \([0,\infty]\) (in \([-\infty,\infty]\)) that is finitely additive on \(\mathcal{O}(X)\cup\mathcal{K}(X)\), where \(\mathcal{K}(X)\) is the family of compact sets. It is also required to satisfy two regularity conditions: if \(U\) is open then \(\mu(U)=\lim\{\mu(K):K\in\mathcal{K}(X), K\subseteq U\}\) and if \(F\) is closed then \(\mu(F)=\lim\{\mu(O):O\in\mathcal{O}(X), F\subseteq O\}\), where the limit is taken along the family on the right hand side, directed by (reverse) inclusion.
If \(\mu\) is only required to be additive on \(\mathcal{K}(X)\) then it is called a (signed) deficient topological measure.
The author proves some structural results on these measures: they are the difference of their positive and negative variations; the latter add up to the total variation. In special cases a signed topological measure can be written as the difference of two topological easures: if \(X\) is connected, locally connected and its one-point compactification has genus \(0\).
Reviewer: K. P. Hart (Delft)On a generalized soft metric space.https://zbmath.org/1449.541002021-01-08T12:24:00+00:00"Taş, N."https://zbmath.org/authors/?q=ai:tas.nihal-arabacioglu"Özgür, N. Y."https://zbmath.org/authors/?q=ai:yilmaz-ozgur.nihalSummary: In this paper our aim is to obtain new generalized fixed-point results. To do this, we introduce a new generalized soft metric space called a soft \(S\)-metric space. We investigate some basic facts, relations and topological properties of this space. Also we define a soft \(S\)-contraction condition and study some fixed-point theorems on a complete soft \(S\)-metric space with necessary examples.Common fixed point theorem in multiplicative metric spaces.https://zbmath.org/1449.540682021-01-08T12:24:00+00:00"Jiang, Yun"https://zbmath.org/authors/?q=ai:jiang.yun"Gu, Feng"https://zbmath.org/authors/?q=ai:gu.fengSummary: In the framework of multiplicative metric spaces, some new fixed point theorems for four mappings were proved using weakly compatible mappings under noncontinuous mappings. The results extended and improved some well-known comparable results in the literature.On \(\mathcal{I}\)-covering mappings and 1-\(\mathcal{I}\)-covering mappings.https://zbmath.org/1449.400062021-01-08T12:24:00+00:00"Zhou, Xiangeng"https://zbmath.org/authors/?q=ai:zhou.xiangeng"Liu, Li"https://zbmath.org/authors/?q=ai:liu.li.3|liu.li.2|liu.li.6|liu.li|liu.li.4|liu.li.1|liu.li.7|liu.li.5Summary: In this paper, we introduce the concepts of \(\mathcal{I}\)-covering mappings and 1-\(\mathcal{I}\)-covering mappings, discuss the difference between sequence-covering and \(\mathcal{I}\)-covering mappings by some examples. With those concepts, we get some interesting properties of \(\mathcal{I}\)-covering (1-\(\mathcal{I}\)-covering) mappings and some characterizations of \(\mathcal{I}\)-covering (1-\(\mathcal{I}\)-covering) and compact mapping images of metric spaces.Fixed point theorems under generalized \(F\)-contractive conditions in \({D^*}\)-metric spaces.https://zbmath.org/1449.541062021-01-08T12:24:00+00:00"Zhang, Xuezhi"https://zbmath.org/authors/?q=ai:zhang.xuezhi"Xue, Xifeng"https://zbmath.org/authors/?q=ai:xue.xifengSummary: In the framework of complete \({D^*}\)-metric spaces, the notions of \(F\)-contractive mappings and generalized \(F\)-contractive mappings are firstly put forward. Then the existence and uniqueness of fixed point for self-mappings are discussed respectively under the two contractive conditions in diverse methods. Some new fixed point theorems are proved, which improve several relative results.A natural selection of a graphic contraction transformation in fuzzy metric spaces.https://zbmath.org/1449.540492021-01-08T12:24:00+00:00"Alolaiyan, Hanan"https://zbmath.org/authors/?q=ai:alolaiyan.hanan-abdulaziz"Saleem, Naeem"https://zbmath.org/authors/?q=ai:saleem.naeem"Abbas, Mujahid"https://zbmath.org/authors/?q=ai:abbas.mujahidSummary: In this paper, we study sufficient conditions to find a vertex \(v\) of a graph such that \(Tv\) is a terminal vertex of a path which starts from \(v\), where \(T\) is a self graphic contraction transformation defined on the set of vertices. Some examples are presented to support the results proved herein. Our results widen the scope of various results in the existing literature.Generalized common fixed point results via greatest lower bound property.https://zbmath.org/1449.540722021-01-08T12:24:00+00:00"Kutbi, Marwan Amin"https://zbmath.org/authors/?q=ai:kutbi.marwan-amin"Ahmad, Jamshaid"https://zbmath.org/authors/?q=ai:ahmad.jamshaid"Azam, Akbar"https://zbmath.org/authors/?q=ai:azam.akbar"Al-Rawashdeh, Ahmed Saleh"https://zbmath.org/authors/?q=ai:saleh-al-rawashdeh.ahmedSummary: The aim of this paper is to unify the concept of greatest lower bound (g.l.b) property and establish some generalized common fixed results. We support our results by a nontrivial example.On topological spaces determined by subset systems.https://zbmath.org/1449.540092021-01-08T12:24:00+00:00"Chen, Yao"https://zbmath.org/authors/?q=ai:chen.yao"Xu, Xiaoquan"https://zbmath.org/authors/?q=ai:xu.xiaoquan.1|xu.xiaoquanSummary: For a subset system \(Z\), the induced operator \({d_Z}\) is introduced, and the topological spaces determined by \(Z\) are discussed. The main results are: (1) the topological spaces determined by the single point subset system are the same with the ones determined by the finite subset system; (2) if \(Z\) is one of the single subset system, the finite subset system, the power subset system and the chain system, then \({d_Z}\) is a topological operator and \(d_Z^2 = {d_Z}\). Therefore, for each \({T_0}\) space \( (X, \tau)\), \({d_Z} (\tau)\) is the coarsest \(ZD\) topology which is finer than \(\tau\).On irreducible determined spaces.https://zbmath.org/1449.540032021-01-08T12:24:00+00:00"Luo, Shuzhen"https://zbmath.org/authors/?q=ai:luo.shuzhen"Xu, Xiaoquan"https://zbmath.org/authors/?q=ai:xu.xiaoquanSummary: In this paper, the concept of irreducible determined spaces (RD-spaces for short) is introduced and some basic properties of RD-spaces are proved. It is shown that there exists RD-space that is not an MD-space by an example.\(G\)-sequentially compact spaces.https://zbmath.org/1449.540342021-01-08T12:24:00+00:00"Liu, Li"https://zbmath.org/authors/?q=ai:liu.li.7|liu.li.4|liu.li.1|liu.li.3|liu.li.5|liu.li.2|liu.li.6|liu.li"Zhou, Xiangeng"https://zbmath.org/authors/?q=ai:zhou.xiangeng"Liu, Fang"https://zbmath.org/authors/?q=ai:liu.fang|liu.fang.1Summary: In this paper, the notion of \(G\)-sequentially compact in general sets is introduced based on \(G\)-methods. After studying the basic properties, the relationship among \(G\)-sequentially compact spaces, sequentially compact spaces and countable compact spaces in topological spaces is discussed. Also the application of \(G\)-sequentially compact spaces when the method is defined as ideal convergence is demonstrated, and some classical results of sequentially compact spaces and countable compact spaces are extend.Proinov contractions and discontinuity at fixed point.https://zbmath.org/1449.540542021-01-08T12:24:00+00:00"Bisht, Ravindra K."https://zbmath.org/authors/?q=ai:bisht.ravindra-kishor"Pant, R. P."https://zbmath.org/authors/?q=ai:pant.r-p"Rakočević, Vladimir"https://zbmath.org/authors/?q=ai:rakocevic.vladimirSummary: In this paper, we show that the contractive definition considered by \textit{P. D. Proinov} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 64, No. 3, 546--557 (2006; Zbl 1101.54046)] is strong enough to generate a fixed point but does not force the mapping to be continuous at the fixed point. Thus we provide more answers to the open question posed by \textit{B. E. Rhoades} [Contemp. Math. 72, 233--245 (1988; Zbl 0649.54024)].Fixed point theorems for a class of mapping in fuzzy metric spaces.https://zbmath.org/1449.540862021-01-08T12:24:00+00:00"Nie, Hui"https://zbmath.org/authors/?q=ai:nie.hui"Zhang, Xinyu"https://zbmath.org/authors/?q=ai:zhang.xinyu"Zhang, Shuyi"https://zbmath.org/authors/?q=ai:zhang.shuyiSummary: The existence of fixed points of Ćirić-Altman type mappings in complete fuzzy metric spaces is studied by using the basic concept of fuzzy metric satisfying triangular inequalities and analysis method. Two new nonunique fixed point theorems are established in fuzzy metric spaces, and the theorems extend previous results.Uniform topological spaces based on normal fuzzy ideals in negative non-involutive residuated lattices.https://zbmath.org/1449.540382021-01-08T12:24:00+00:00"Liu, Chunhui"https://zbmath.org/authors/?q=ai:liu.chunhuiSummary: Topological structure is one of important research contents in the field of logical algebra. In order to describe the topological structure of negative non-involutive residuated lattices, based on the congruences induced by normal fuzzy ideals, uniform topological spaces are established and some of their properties are discussed. The following conclusions are proved: (1) every uniform topological space is first-countable, zero-dimensional, disconnected, locally compact and completely regular; (2) a uniform topological space is a \({T_1}\) space iff it is a \({T_2}\) space; (3) the lattice and adjoint operations in a negative non-involutive residuated lattice are continuous under the uniform topology, which make the negative non-involutive residuated lattice be topological negative non-involutive residuated lattice. Meanwhile, some necessary and sufficient conditions for the uniform topological spaces to be compact and discrete are obtained. Finally, the relationships between algebraic isomorphism and topological homeomorphism in topological negative non-involutive residuated lattice are discussed. The results of this paper have a positive role to reveal internal features of negative non-involutive residuated lattices on a topological level.Metrical fixed point theorems via locally finitely $T$-transitive binary relations under certain control functions.https://zbmath.org/1449.540442021-01-08T12:24:00+00:00"Alam, Aftab"https://zbmath.org/authors/?q=ai:alam.aftab"Arif, Mohammad"https://zbmath.org/authors/?q=ai:arif.mohammad"Imdad, Mohammad"https://zbmath.org/authors/?q=ai:imdad.mohammadSummary: In this paper, we extend a relation-theoretic contraction principle due to \textit{A. Alam} and \textit{M. Imdad} [J. Fixed Point Theory Appl. 17, No. 4, 693--702 (2015; Zbl 1335.54040)] to a nonlinear contraction using a relatively weaker class of continuous control functions employing a locally finitely $T$-transitive binary relation, which improves the corresponding fixed point theorems.New fixed point results on \(c\)-distance in cone metric spaces over Banach algebras.https://zbmath.org/1449.540652021-01-08T12:24:00+00:00"Han, Yan"https://zbmath.org/authors/?q=ai:han.yan"Xu, Shaoyuan"https://zbmath.org/authors/?q=ai:xu.shaoyuan"Liu, Xiu"https://zbmath.org/authors/?q=ai:liu.xiuSummary: The purpose of this paper is to obtain several fixed point and common fixed point results for generalized Lipschitz mappings on \(c\)-distance in cone metric spaces over Banach algebras, without the assumption that the underlying cone is normal or the mappings are continuous. These results greatly improve and extend several well-known comparable results in the literature. Moreover, some examples are given to support our new results. Furthermore, an application to the existence and uniqueness of a solution to a class of integral equation is also given.Fixed point for mappings satisfying Kannan type inequality in fuzzy metric spaces involving \(\mathrm{t}\)-norms with equi-continuous iterates.https://zbmath.org/1449.540582021-01-08T12:24:00+00:00"Dutta, P. N."https://zbmath.org/authors/?q=ai:dutta.p-n"Choudhury, Binayak S."https://zbmath.org/authors/?q=ai:choudhury.binayak-samadder"Das, Pradyut"https://zbmath.org/authors/?q=ai:das.pradyutSummary: In this paper, we define a coupled weak compatible condition and use it to derive certain coupled coincidence point theorems for four mappings in fuzzy metric spaces. We use here a \(\mathrm{t}\)-norm which has equicontinuous iterates at 1. Some coupled fixed point results in metric spaces are obtained by applications of the results. Our results are obtained without any assumption of continuity on the mappings. Our main result is supported by an illustrative example. Some corollaries are also obtained.On open almost \(s\)-images of metric spaces.https://zbmath.org/1449.540262021-01-08T12:24:00+00:00"Ling, Xuewei"https://zbmath.org/authors/?q=ai:ling.xuewei"Lin, Shou"https://zbmath.org/authors/?q=ai:lin.shouSummary: The notion of an almost \(s\)-mapping was previously introduced as follows: a mapping \(f\) from a topological space \(X\) onto a topological space \(Y\) is called an almost \(s\)-mapping provided that \(f^{-1} (y)\) is a separable set in \(X\) for each non-isolated point \(y\) in \(Y\). In this paper, we study basic relations among almost \(s\)-mappings, near \(s\)-mappings and boundary-\(s\)-mappings, obtain internal characterizations of open almost \(s\)-images of metric spaces, and discuss the properties of countably bi-quotient boundary-\(s\)-mappings on metric spaces.Some coincidence point results of selfmappings satisfying implicit \(\phi\)-contractive conditions on Menger PM spaces.https://zbmath.org/1449.540972021-01-08T12:24:00+00:00"Song, Mingliang"https://zbmath.org/authors/?q=ai:song.mingliangSummary: By using \(\Gamma\)-type real functions, some coincidence point and common fixed point theorems of selfmappings satisfying implicit \(\phi\)-contractive conditions on Menger PM spaces are obtained. As applications, we establish some coincidence point and common fixed point theorems of selfmappings satisfying implicit \(\phi\)-contractive conditions on metric spaces.Bernstein Stancu operator of rough \(I\)-core of triple sequences.https://zbmath.org/1449.400052021-01-08T12:24:00+00:00"Subramanian, N."https://zbmath.org/authors/?q=ai:subramanian.nagarajan"Esi, A."https://zbmath.org/authors/?q=ai:esi.ayhan"Ozdemir, M. K."https://zbmath.org/authors/?q=ai:ozdemir.mustafa-kemalSummary: We introduce and study some basic properties of Bernstein-Stancu polynomials of rough $I$-convergent of triple sequences and also study the set of all Bernstein-Stancu polynomials of rough \(I\)-limits of a triple sequence and relation between analytic ness and Bernstein-Stancu polynomials of rough \(I\)-core of a triple sequence.Properties of bioperation-semiseparated sets.https://zbmath.org/1449.540292021-01-08T12:24:00+00:00"Nirmala, R."https://zbmath.org/authors/?q=ai:nirmala.r-joice"Shanthi, S."https://zbmath.org/authors/?q=ai:shanthi.s-anita|shanthi.s-d"Rajesh, N."https://zbmath.org/authors/?q=ai:rajesh.neelamegarajan|rajesh.n-r|rajesh.namegaleshSummary: In this paper, we introduce the notion \( (\gamma, \gamma')\)-semiseparated sets and study some of their basic properties.\( ({{\Lambda_\pi},m})\)-closed sets and decompositions of \(m\)-continuity.https://zbmath.org/1449.540202021-01-08T12:24:00+00:00"Al-Omari, Ahmad"https://zbmath.org/authors/?q=ai:al-omari.ahmad-abdullah"Noiri, Takashi"https://zbmath.org/authors/?q=ai:noiri.takashi"Al-Saadi, Hanan"https://zbmath.org/authors/?q=ai:al-saadi.hanan-saadSummary: We introduce and investigate the notions of \( ({{\Lambda_\pi}, m})\)-closed sets and \(\pi gm\)-locally closed sets in a topological space \( ({X, \tau})\) with a minimal structure \({m_X}\) on \(X\). We obtain decompositions of \({m_X}\)-closed sets by combining these sets and \(\pi gm\)-closed sets. As the consequence, we obtain decompositions of \(m\)-continuity which is a unified form of generalizations of continuity.Common fixed point theorem for generalized nonexpansive mappings on ordered orbitally complete metric spaces and application.https://zbmath.org/1449.540842021-01-08T12:24:00+00:00"Nashine, Hemant Kumar"https://zbmath.org/authors/?q=ai:nashine.hemant-kumar"Agarwal, Ravi P."https://zbmath.org/authors/?q=ai:agarwal.ravi-pSummary: We propose a common fixed point theorem for a new notion of generalized nonexpansive mappings for two pairs of maps in an ordered orbitally complete metric space. To illustrate our result, we give throughout the paper two examples. Existence of solutions for a certain system of functional equations arising in dynamic programming is also presented as application.Fixed point theorems for generalized contraction mappings on b-rectangular metric spaces.https://zbmath.org/1449.540452021-01-08T12:24:00+00:00"Alecsa, Cristian Daniel"https://zbmath.org/authors/?q=ai:alecsa.cristian-danielSummary: In the present article, we study some fixed point theorems for a hybrid class of generalized contractive operators in the context of b-rectangular metric spaces. Examples justifying theorems and an open problem regarding to further generalizations for this type of operators are also given.Hardy-Rogers type mappings on dislocated quasi metric spaces.https://zbmath.org/1449.540882021-01-08T12:24:00+00:00"Padcharoen, Anantachai"https://zbmath.org/authors/?q=ai:padcharoen.anantachai"Gopal, Dhananjay"https://zbmath.org/authors/?q=ai:gopal.dhananjay"Akkasriworn, Naknimit"https://zbmath.org/authors/?q=ai:akkasriworn.naknimitSummary: We prove some common fixed point results for two \(\alpha\)-dominated mappings of Hardy-Rogers type on a closed ball of a left (right) \(K\)-sequentially complete dislocated quasi-metric space and give some example to support our result.Improvements on Bogin-type fixed point theorems in complete metric spaces.https://zbmath.org/1449.540992021-01-08T12:24:00+00:00"Suzuki, Tomonari"https://zbmath.org/authors/?q=ai:suzuki.tomonariSummary: We improve Bogin-type fixed point theorems in complete metric spaces. We also compare these theorems with a Ćirić-type fixed point theorem and others.Topological entropy of free semigroup actions.https://zbmath.org/1449.370132021-01-08T12:24:00+00:00"Zhang, Wenda"https://zbmath.org/authors/?q=ai:zhang.wenda"Xue, Licui"https://zbmath.org/authors/?q=ai:xue.licuiSummary: In this paper, we define the entropy and preimage entropy of free semigroup actions in a new method. Based on these definitions, we get some relations between topological entropy and measure entropy, and the relations among kinds of preimage entropies. The main results of this paper are as follows: (1) the topological entropy is invariant under equi-conjugacy; (2) the power rule for the measure-theoretic entropy holds.A related fixed point theorem for three pairs of mappings on complete metric spaces.https://zbmath.org/1449.540672021-01-08T12:24:00+00:00"Jain, R. K."https://zbmath.org/authors/?q=ai:jain.raj-kumar|jain.rajendra-k|jain.rakesh-kumar|jain.rajinder-kumar|jain.ratnesh-k|jain.raj-krishan|jain.rajeev-kumar"Bhupendra"https://zbmath.org/authors/?q=ai:bhupendra."Fisher, Brian"https://zbmath.org/authors/?q=ai:fisher.brianSummary: In this paper we prove a related fixed point theorem for three pairs of mappings, on three complete metric spaces, satisfying rational type contractive conditions.\(\mathcal{I}_2\)-asymptotically lacunary statistical equivalence of weight \(g\) of double sequences of sets.https://zbmath.org/1449.400042021-01-08T12:24:00+00:00"Kişi, Ömer"https://zbmath.org/authors/?q=ai:kisi.omerSummary: In this paper, our aim is to introduce new notions, namely, Wijsman asymptotically \(\mathcal{I}_2\)-statistical equivalence of weight \(g\), Wijsman strongly asymptotically \(\mathcal{I}_2\)-lacunary equivalence of weight \(g\) and Wijsman asymptotically \(\mathcal{I}_2\)-lacunary statistical equivalence of weight \(g\) of double set sequences. We mainly investigate their relationship and also make some observations about these classes.Fixed point problems concerning contractive type operators on KST-Spaces.https://zbmath.org/1449.470942021-01-08T12:24:00+00:00"Ansari, Arslan H."https://zbmath.org/authors/?q=ai:ansari.arslan-hojat"Guran, Liliana"https://zbmath.org/authors/?q=ai:guran.liliana"Latif, Abdul"https://zbmath.org/authors/?q=ai:latif.abdulSummary: Using the concept of \(w\)-distance, we prove some results on the existence of fixed points for contractive type operators, namely, \((\alpha,\mu)\)-\(\psi\)-contractive operators. Applications are also presented. Our results improve and generalize a number of known results of fixed point theory, including the recent results of \textit{L. Guran} and \textit{M.-F. Bota} [``Ulam-Hyers stability problems and fixed point theorems concerning \(\alpha\)-\(\psi\)-type contractive operators on KST-spaces'' (submitted), see: Linear Nonlinear Anal. 5, No. 3, 379--390 (2019), \url{http://yokohamapublishers.jp/online2/oplna/vol5/p379.html}] and \textit{A. H. Ansari} and \textit{S. Shukla} [J. Adv. Math. Stud. 9, No. 1, 37--53 (2016; Zbl 1353.54030)].Fixed point theorems for contractions in semicomplete semimetric spaces.https://zbmath.org/1449.540982021-01-08T12:24:00+00:00"Suzuki, Tomonari"https://zbmath.org/authors/?q=ai:suzuki.tomonariSummary: We introduce the concept of semicompleteness on semimetric space, which is weaker than completeness. We prove fixed point theorems for contractions in semicomplete semimetric spaces. Also, we generalize Jachymski-Matkowski-Świątkowski's fixed point theorem in semimetric spaces.Existence of solutions of implicit integral equations via \(Z\)-contraction.https://zbmath.org/1449.540892021-01-08T12:24:00+00:00"Patle, Pradip R."https://zbmath.org/authors/?q=ai:patle.pradip-ramesh"Patel, Deepesh Kumar"https://zbmath.org/authors/?q=ai:patel.deepesh-kumarSummary: The main focus of this work is to assure that the sum of a compact operator with a \(Z\)-contraction admits a fixed point. The concept of condensing mapping (in the sense of Hausdorff non-compactness measure) is used to establish the concerned result which generalizes some of the existing state-of-art in the literature. Presented result is used to verify the actuality of solutions of implicit integral equations.Proximal point algorithms involving fixed point iteration for nonexpansive mappings in CAT\((\kappa)\) spaces.https://zbmath.org/1449.471132021-01-08T12:24:00+00:00"Pakkaranang, Nuttapol"https://zbmath.org/authors/?q=ai:pakkaranang.nuttapol"Kumam, Poom"https://zbmath.org/authors/?q=ai:kumam.poom"Cholamjiak, Prasit"https://zbmath.org/authors/?q=ai:cholamjiak.prasit"Suparatulatorn, Raweerote"https://zbmath.org/authors/?q=ai:suparatulatorn.raweerote"Chaipunya, Parin"https://zbmath.org/authors/?q=ai:chaipunya.parinSummary: In this paper, we propose a new modified proximal point algorithm involving fixed point iteration for nonexpansive mappings in CAT(1) spaces. Under some mild conditions, we prove that the sequence generated by our iterative algorithm \(\Delta\)-converges to a common solution of certain convex optimization and fixed point problems.Some coincidence point theorems in ordered metric spaces via \(w\)-distances.https://zbmath.org/1449.540812021-01-08T12:24:00+00:00"Mongkolkeha, Chirasak"https://zbmath.org/authors/?q=ai:mongkolkeha.chirasak"Cho, Yeol Je"https://zbmath.org/authors/?q=ai:cho.yeol-jeSummary: The purpose of this paper is to prove some existence theorems of coincidence points for generalized weak contractions in the setting of partially ordered sets with a metric via \(w\)-distances and give some example to illustrate our main results.Topological properties of category Fb-PTOP.https://zbmath.org/1449.540152021-01-08T12:24:00+00:00"Li, Hui"https://zbmath.org/authors/?q=ai:li.hui|li.hui.5|li.hui.2|li.hui.1|li.hui.4|li.hui.3"Wang, Ruiying"https://zbmath.org/authors/?q=ai:wang.ruiyingSummary: In this paper, we introduced the category Fb-PTOP, which is composed of fuzzifying \(b\)-pre-topological spaces and \(b\)-irresolute mappings. Furthermore, we proved that Fb-PTOP is the topological category on Set.On some generalized countably compact spaces.https://zbmath.org/1449.540242021-01-08T12:24:00+00:00"Yang, Erguang"https://zbmath.org/authors/?q=ai:yang.erguangSummary: We first give alternative expressions of some generalized countably compact spaces such as quasi-\(\gamma\) spaces, quasi-Nagata spaces, \({M^\#}\)-spaces and \(wM\)-spaces with \(g\)-functions. Then by means of these expressions, we present some characterizations of the corresponding spaces with real-valued functions.Cellular-compact spaces and their applications.https://zbmath.org/1449.540332021-01-08T12:24:00+00:00"Tkachuk, V. V."https://zbmath.org/authors/?q=ai:tkachuk.vladimir-v"Wilson, R. G."https://zbmath.org/authors/?q=ai:wilson.richard-gIn the following all spaces are Hausdorff. The authors define a space \(X\) to be \textit{cellular-compact} if for every family \(\mathcal{U}\) of disjoint nonempty open subsets of \(X\) there exists a compact subspace \(K\subset X\) such that \(K\cap U\neq\emptyset\) for every \(U\in\mathcal{U}\), and they recall that in [Monatsh. Math. 186, No. 2, 345--353 (2018; Zbl 1398.54008)], \textit{A. Bella} and \textit{S. Spadaro} defined a space \(X\) to be \textit{cellular-Lindelöf} if for every family \(\mathcal{U}\) of disjoint nonempty open subsets of \(X\) there exists a Lindelöf subspace \(L\subset X\) such that \(L\cap U\neq\emptyset\) for every \(U\in\mathcal{U}\). They note that every cellular-compact space is cellular-Lindelöf, and as was shown by others with the property cellular-Lindelöf, it is straightforward to verify that the property cellular-compact is preserved by continuous maps, inherited by regular-closed subspaces, and possessed by extension spaces, and furthermore, the closure of the set of isolated points of any cellular-compact space is compact. The authors next recall that a space \(X\) is said to be \textit{linearly \(H\)-closed} if every nested open cover of \(X\) has a dense member. This concept was studied in several articles, where it was shown that every linearly \(H\)-closed space \(X\) is \textit{feebly compact} (i.e., every locally finite family of open subsets of \(X\) is finite).
They then prove that every cellular-compact space is linearly \(H\)-closed, and hence, or as can easily be shown directly, every cellular-compact space is feebly compact. Examples showing these concepts are distinct are given, including the Isbell-Mrówka space \(\Psi\), which was previously shown by others to be linearly \(H\)-closed, for by the authors' results concerning isolated points, that space cannot be cellular-compact, although it is cellular-Lindelöf. As a way of constructing non-compact, cellular-compact spaces, they prove that for any regular cellular-compact space \(X\) and non-isolated point \(p\) of \(X\), the space \(X\setminus\{p\}\) is cellular-compact if and only if there exists no disjoint local \(\pi\)-base at the point \(p\) in \(X\). A number of interesting corollaries of this theorem are obtained such as: if \(\kappa\) is an uncountable cardinal and \(X=[0,1]^\kappa\), then \(X\setminus\{p\}\) is cellular-compact for any point \(p\in X\), and if \(S\) is the Corson \(\Sigma\)-product in \(X\) based at the origin, then \(S\) is cellular-compact, as well as collectionwise normal, Fréchet-Urysohn, non-compact, and countably compact, and for each point \(p\in S\), the space \(S\setminus\{p\}\) is a cellular-compact space whose extent is \(\kappa\) and which has no dense countably compact subspace.
The authors give an example which shows that under CH, there exists a Tychonoff first countable separable cellular-compact space which is not compact. Several results presented are of the form: If \(X\) is a cellular-compact space which has some particular property \(P\), then \(X\) is compact. Two especially nice theorems they obtain are: If \(X\) is a cellular-compact first countable regular space, then \(X\) is countably compact and \(\vert X\vert \leq\mathfrak{c}\). A regular sequential cellular-compact space is maximal cellular-compact if and only if it has a disjoint local \(\pi\)-base at every point, and thus every first countable compact space is maximal cellular-compact. They conclude by raising a number of open questions.
Reviewer: Robert M. Stephenson Jr. (Columbia)\textit{Ps}-normal and \textit{Ps}-Tychonoff spaces.https://zbmath.org/1449.540372021-01-08T12:24:00+00:00"Bag, Sagarmoy"https://zbmath.org/authors/?q=ai:bag.sagarmoy"Manna, Ram Chandra"https://zbmath.org/authors/?q=ai:manna.ram-chandra"Patra, Sourav Kanti"https://zbmath.org/authors/?q=ai:patra.sourav-kantiSummary: A space \(X\) is called a \textit{Ps}-normal (\textit{Ps}-Tychonoff) space if there exists a normal (Tychonoff) space \(Y\) and a bijection \(f: X\mapsto Y\) such that \(f|_K:K\mapsto f(K)\) is a homeomorphism for any pseudocompact subset \(K\) of \(X\). We establish a few relations between \(C\)-normal, \textit{CC}-normal, \(L\)-normal, \(C\)-Tychonoff, \textit{CC}-Tychonoff spaces with \textit{Ps}-normal and \textit{Ps}-Tychonoff spaces.Suzuki \(\phi F\)-contractions and some fixed point results.https://zbmath.org/1449.470932021-01-08T12:24:00+00:00"Secelean, Nicolae-Adrian"https://zbmath.org/authors/?q=ai:secelean.nicolae-adrianSummary: The purpose of this paper is to combine and extend some recent fixed point results of \textit{T. Suzuki} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 11, 5313--5317 (2009; Zbl 1179.54071)] and \textit{N.-A. Secelean} and \textit{D. Wardowski} [Result. Math. 70, No. 3--4, 415--431 (2016; Zbl 1442.54048)]. The continuity and the completeness conditions are replaced by orbital continuity and orbital completeness, respectively. It is given an illustrative example of a Picard operator on a noncomplete metric space which is neither nonexpansive nor expansive and has a unique continuity point.Meet uniform continuous posets.https://zbmath.org/1449.060012021-01-08T12:24:00+00:00"Mao, Xuxin"https://zbmath.org/authors/?q=ai:mao.xuxin"Xu, Luoshan"https://zbmath.org/authors/?q=ai:xu.luoshanSummary: In this paper, as a generalization of uniform continuous posets, the concept of meet uniform continuous posets via uniform Scott sets is introduced. Properties and characterizations of meet uniform continuous posets are presented. The main results are: (1) A uniform complete poset \(L\) is meet uniform continuous iff \(\uparrow (U \cap \downarrow x)\) is a uniform Scott set for each \(x \in L\) and each uniform Scott set \(U\); (2) A uniform complete poset \(L\) is meet uniform continuous iff for each \(x \in L\) and each uniform subset \(S\), one has \(x \wedge \bigvee S = \bigvee \{x \wedge s| {s \in S}\}\). In particular, a complete lattice \(L\) is meet uniform continuous iff \(L\) is a complete Heyting algebra; (3) A uniform complete poset is meet uniform continuous iff every principal ideal is meet uniform continuous iff all closed intervals are meet uniform continuous iff all principal filters are meet uniform continuous; (4) A uniform complete poset \(L\) is meet uniform continuous if \({L^1}\) obtained by adjoining a top element 1 to \(L\) is a complete Heyting algebra; (5) Finite products and images of uniform continuous projections of meet uniform continuous posets are still meet uniform continuous.Coset spaces and cardinal invariants.https://zbmath.org/1449.220012021-01-08T12:24:00+00:00"Fernández, M."https://zbmath.org/authors/?q=ai:fernandez.mauricio|fernandez.m-legua|fernandez.miguel-angel|fernandez.mario|fernandez.mirtha-lina|fernandez.maite|fernandez.moises|fernandez.m-p|fernandez.maximiliano|fernandez.maria-c|fernandez.max|fernandez.melchor|fernandez.manuel-jose|fernandez.m-v|fernandez.marcel|fernandez.michael|fernandez.mary-f|fernandez.m-angeles|fernandez.manel|fernandez.marcela|fernandez.manuel|fernandez.marcos|fernandez.maribel|fernandez.m-l-c|fernandez.marcelo-o|fernandez.margarita|fernandez.marisa|fernandez.mariela"Sánchez, I."https://zbmath.org/authors/?q=ai:sanchez.ivan"Tkachenko, M."https://zbmath.org/authors/?q=ai:tkachenko.mikhail-gThe authors extend various results about cardinal invariants of topological groups to homogeneous spaces \(G/H\) of topological groups. The notions used in the paper can be found for instance in \textit{A. Arhangel'skii} and \textit{M. Tkachenko} [Topological groups and related structures. Hackensack, NJ: World Scientific; Paris: Atlantis Press (2008; Zbl 1323.22001)].
A typical result: If \(H\) is a closed subgroup of a feathered topological group \(G\) then \(\pi\chi(G/H)=\chi(H)\) and \(\pi w(G/H)=w(G/H)\). (Here \(\pi\chi, \chi,\pi w\), and \(w\) are the \(\pi\)-character, character, \(\pi\)-weight and weight, respectively.)
The paper ends with a list of five open problems.
Reviewer: Mihail I. Ursul (Oradea)Baire categorical aspects of first passage percolation. II.https://zbmath.org/1449.540422021-01-08T12:24:00+00:00"Maga, B."https://zbmath.org/authors/?q=ai:maga.balazsSummary: In this paper we continue our earlier work [\textit{B. Maga}, Acta Math. Hung. 156, No. 1, 145--171 (2018; Zbl 1424.54065)] about topological first passage percolation and answer certain questions asked in our previous paper. Notably, we prove that apart from trivialities, in the generic configuration there exists exactly one geodesic ray, in the sense that we consider two geodesic rays distinct if they only share a finite number of edges. Moreover, we show that in the generic configuration any not too small and not too large convex set arises as the limit of a sequence \(B(t_n)/t_n\) for some \(t_n\to\infty\). Finally, we define topological Hilbert first passage percolation, and amongst others we prove that certain geometric properties of the percolation in the generic configuration guarantee that we consider a setting linearly isomorphic to the ordinary topological first passage percolation.Countably compact group topologies on the free abelian group of size continuum (and a Wallace semigroup) from a selective ultrafilter.https://zbmath.org/1449.540432021-01-08T12:24:00+00:00"Boero, A. C."https://zbmath.org/authors/?q=ai:boero.ana-carolina"Pereira, I. C."https://zbmath.org/authors/?q=ai:pereira.i-castro"Tomita, A. H."https://zbmath.org/authors/?q=ai:tomita.artur-hideyukiIt is well known that a non-trivial free abelian group does not admit a compact Hausdorff group topology. On the other hand, it was shown that the free abelian group generated by \(\mathfrak{c}\) elements can be endowed with a countably compact Hausdorff group topology under some set-theoretic assumptions. \textit{R. E. Madariaga-Garcia} and \textit{A. H. Tomita} [Topology Appl. 154, No. 7, 1470--1480 (2007; Zbl 1116.54004)] also obtained such a group assuming the existence of \(\mathfrak{c}\) many pairwise incomparable selective ultrafilters and asked whether the existence of one selective ultrafilter implies the existence of a countably compact group topology on the free abelian group of size \(\mathfrak{c}\).
It was known that compact both-sided cancellative semigroups are topological groups. In the 1950's, Wallace asked whether every countably compact topological semigroup with both-sided cancellation is a topological group. A counterexample to Wallace's question has been called a Wallace semigroup. The main example in the above cited paper of \textit{R. E. Madariaga-Garcia} and \textit{A. H. Tomita} yields a Wallace semigroup from the existence of \(\mathfrak{c}\) selective ultrafilters.
In this paper, the authors prove that the existence of a selective ultrafilter implies the existence of a countably compact Hausdorff group topology on the free abelian group of size \(\mathfrak{c}\) answering the above first question, and the existence of a Wallace semigroup.
Reviewer: Kohzo Yamada (Shizuoka)\(G\)-connectedness.https://zbmath.org/1449.540302021-01-08T12:24:00+00:00"Ping, Zheng"https://zbmath.org/authors/?q=ai:ping.zhengSummary: A \(G\)-method on a set \(X\) is a function defined on a subset of \(X\)-valued sequences to the set \(X\). This paper introduces the notion of \(G\)-separated sets through \(G\)-methods. It defines \(G\)-connected subsets, obtains some characterizations of \(G\)-connected sets and discusses the relationships among connectedness, sequential connectedness and \(G\)-connectedness. The author investigates several properties of \(G\)-connected subsets, and extends sequential connectedness and \(G\)-connectedness on Hausdorff topological groups which satisfy first countability axiom to \(G\)-connectedness on an arbitrary set.Weak sub-sequential continuous maps in non Archimedean Menger PM space via \(C\)-class functions.https://zbmath.org/1449.540502021-01-08T12:24:00+00:00"Ansari, Arslan Hojat"https://zbmath.org/authors/?q=ai:ansari.arslan-hojat"Sharma, Rajinder"https://zbmath.org/authors/?q=ai:sharma.rajinderSummary: This study deals with an establishment of some common fixed point theorems for weak sub sequential continuous and compatibility of type (E) maps via C-class functions in a non Archimedean Menger Probabilistic Metric space.A new approach to the fuzzification of convex structures.https://zbmath.org/1449.540182021-01-08T12:24:00+00:00"Shi, Fu-Gui"https://zbmath.org/authors/?q=ai:shi.fugui|shi.fu-gui"Xiu, Zhen-Yu"https://zbmath.org/authors/?q=ai:xiu.zhenyuSummary: A new approach to the fuzzification of convex structures is introduced. It is also called an \(M\)-fuzzifying convex structure. In the definition of \(M\)-fuzzifying convex structure, each subset can be regarded as a convex set to some degree. An \(M\)-fuzzifying convex structure can be characterized by means of its \(M\)-fuzzifying closure operator. An \(M\)-fuzzifying convex structure and its \(M\)-fuzzifying closure operator are one-to-one corresponding. The concepts of \(M\)-fuzzifying convexity preserving functions, substructures, disjoint sums, bases, subbases, joins, product, and quotient structures are presented and their fundamental properties are obtained in \(M\)-fuzzifying convex structure.Ordered weak \(\varphi\)-contractions in cone metric spaces over Banach algebras and fixed point theorems.https://zbmath.org/1449.540762021-01-08T12:24:00+00:00"Malhotra, S. K."https://zbmath.org/authors/?q=ai:malhotra.sandeep-kumar"Bhargava, P. K."https://zbmath.org/authors/?q=ai:bhargava.pradeep-kumar"Shukla, Satish"https://zbmath.org/authors/?q=ai:shukla.satishSummary: In this work, we introduce the class of ordered weak \(\varphi\)-contractions in cone metric spaces over Banach algebras and prove some fixed point results for the mappings belonging to this new class. Our results generalize and extend some known fixed point results in cone metric spaces to the spaces equipped with a partial order. Some examples are given which illustrate the results proved herein.On soft ultrafilters.https://zbmath.org/1449.540362021-01-08T12:24:00+00:00"Salec, Alireza Bagheri"https://zbmath.org/authors/?q=ai:bagheri-salec.alirezaSummary: In this paper, the concept of soft ultrafilters is introduced and some of the related structures such as soft Stone-Čech compactification, principal soft ultrafilters and basis for its topology are studied.