## Leader type contractions, periodic and fixed points and new completivity in quasi-gauge spaces with generalized quasi-pseudodistances.(English)Zbl 1258.47075

Definition 1. Let $$X$$ be a nonempty set.
(i) A quasi-pseudometric on $$X$$ is a map $$p: X\times X\to [0,\infty)$$ such that: $$(P_1)$$: $$\forall x\in X\{p(x,x)= 0\}$$, and $$(P_2)$$: $$\forall x,y,z\in X\{p(x,z)\leq p(x,y)+ p(y,z)\}$$. For given quasi-pseudometric $$p$$ on $$X$$, a pair $$(X,p)$$ is called a quasi-pseudometric space. A quasi-pseudometric space $$(X,p)$$ is called Hausdorff if $$\forall x,y\in X\{x\neq y\Rightarrow p(x,y)> 0 \vee p(y,x)> 0\}$$.
(ii) Each family $${\mathcal P}= \{p_\alpha: \alpha\in{\mathcal A}\}$$ of quasi-pseudometrics $$p_\alpha: X\times X\to [0,\infty)$$, $$\alpha\in{\mathcal A}$$, is called a quasi-gauge space.
(iii) Let the family $${\mathcal P}= \{p_\alpha: \alpha\in{\mathcal A}\}$$ be a quasi-gauge on $$X$$. The topology $$\tau({\mathcal P})$$, having as a subbase the family $${\mathcal B}(D)= \{B(x,\varepsilon_\alpha) : x\in X$$, $$\varepsilon_\alpha> 0$$, $$\alpha\in{\mathcal A}\}$$ of all balls $$B(x,\varepsilon_\alpha)= \{y\in X: p_\alpha(x,y)< \varepsilon_\alpha$$, $$x\in X$$, $$\alpha> 0$$, $$\alpha\in{\mathcal A}\}$$, is called the topology induced by $${\mathcal P}$$ on $$X$$.
(iv) A topological space $$(X,\tau)$$ such that there is a quasi-gauge $${\mathcal P}$$ on $$X$$ with $$\tau= \tau({\mathcal P})$$ is called a quasi-gauge space and is denoted by $$(X,{\mathcal P})$$.
(v) A quasi-gauge space $$(X,{\mathcal P})$$ is called Hausdorff if the quasi-gauge $${\mathcal P}$$ has the property: $$\forall x,y\in X\{x\neq y\Rightarrow\exists\alpha \in{\mathcal A}\{ p_\alpha(x,y)> 0 \vee\,p_\alpha(y,x)> 0\}\}$$.
Recently, the concept of generalized pseudodistance was introduced by the authors [Appl. Math. Lett. 24, No. 3, 325-328 (2011; Zbl 1206.54068); Fixed Point Theory Appl. 2011, Article ID 712706 (2011; Zbl 1213.81161)].
In the present paper, the authors introduce the notions of generalized quasipseudodistances in quasi-gauge spaces.
Definition 2. Let $$(X,{\mathcal P})$$ be a quasi-gauge space. The family $${\mathcal I}= \{J_\alpha: \alpha\in{\mathcal A}\}$$ of maps $$J_\alpha: X\times X\to [0,\infty)$$, $$\alpha\in{\mathcal A}$$, is said to be of left (right) $${\mathcal I}$$-family of generalized quasipseudodistances on $$X$$ if the following conditions hold:
$$({\mathcal I}_1)$$: $$\forall\alpha\in{\mathcal A}$$, $$\forall x,y,z\in X\{J_\alpha(x,z)\leq J_\alpha(x,y)+ J_\alpha(y,z)\}$$; and
$$({\mathcal I}_2)$$ For any sequences $$(u_m: m\in \mathbb {N})$$ and $$(v_m: m\in \mathbb {N})$$ in $$X$$ satisfying $\forall\alpha\in{\mathcal A}\Biggl\{\lim_{m\to\infty}\;\sup_{n> m}\,J_\alpha(u_m,u_n)= 0\Biggr\}\;\Biggl(\forall\alpha\in{\mathcal A}\Biggl\{\lim_{m\to\infty}\;\sup_{n> m}\, J_\alpha(u_n, u_m)= 0\Biggr\}\Biggr)$ and $\forall\alpha\in{\mathcal A}\Biggl\{\lim_{m\to\infty}\, (J_\alpha(v_m,u_m)= 0\Biggr\}\;\Biggl(\forall\alpha\in{\mathcal A}\Biggl\{\lim_{m\to\infty}\, J_\alpha(u_m, v_m)= 0\Biggr\}\Biggr),$ the following holds: $\forall\alpha\in{\mathcal A}\Biggl\{\lim_{m\to\infty}\, p_\alpha(v_m,u_m)= 0\Biggr\}\;\Biggl(\forall\alpha\in{\mathcal A}\Biggl\{\lim_{m\to\infty}\, p_\alpha(u_m, v_m)= 0\Biggr\}\Biggr).$ In this paper, the notion of generalized quasipseudodistances in quasi-gauge sapces is used in natural way to define contractions of leader type and to obtain for these contractions periodic and fixed point theorems without Hausdorff and sequentially complete assumptions about these spaces and without graph assumptions about these contractions.

### MSC:

 47H99 Nonlinear operators and their properties 54A05 Topological spaces and generalizations (closure spaces, etc.) 54E15 Uniform structures and generalizations 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics

### Citations:

Zbl 1206.54068; Zbl 1213.81161
Full Text:

### References:

 [1] Alemany, E.; Romaguera, S., On right $$K$$-sequentially complete quasi-metric spaces, Acta Math. Hungar., 75, 3, 267-278 (1997) · Zbl 0924.54037 [2] Banach, S., Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales, Fund. Math., 3, 133-181 (1922) [3] Boyd, D. W.; Wong, J. S.W., On nonlinear contractions, Proc. Amer. Math. Soc., 20, 458-464 (1969) · Zbl 0175.44903 [4] Browder, F. E., On the convergence of successive approximations for nonlinear equations, Indag. Math., 30, 27-35 (1968) · Zbl 0155.19401 [5] Burton, T. A., Integral equations, implicit functions, and fixed points, Proc. Amer. Math. Soc., 124, 2383-2390 (1996) · Zbl 0873.45003 [6] Caccioppoli, R., Un teorema generale sullʼesistenza di elementi uniti in una transformazione funzionale, Rend. Accad. dei Lincei, 11, 794-799 (1930) [7] Dugundji, J., Topology (1966), Allyn and Bacon: Allyn and Bacon Boston · Zbl 0144.21501 [8] Dugundji, J.; Granas, A., Weakly contractive maps and elementary domain invariance theorems, Bull. Greek Math. Soc., 19, 141-151 (1978) · Zbl 0417.54010 [9] Geraghty, M. A., An improved criterion for fixed points of contractions mappings, J. Math. Anal. Appl., 48, 811-817 (1974) · Zbl 0308.54033 [10] Geraghty, M. A., On contractive mappings, Proc. Amer. Math. Soc., 40, 604-608 (1973) · Zbl 0245.54027 [11] Jachymski, J., Equivalence of some contractivity properties over metrical structures, Proc. Amer. Math. Soc., 125, 2327-2335 (1997) · Zbl 0887.47039 [12] Jachymski, J., On iterative equivalence of some classes of mappings, Ann. Math. Sil., 13, 149-165 (1999) · Zbl 0955.47036 [13] J. Jachymski, I. Jóźwik, Nonlinear contractive conditions: a comparison and related problems, in: Fixed Point Theory and Its Applications, in: Banach Center Publ., vol. 77, Warsaw, 2007, pp. 123-146.; J. Jachymski, I. Jóźwik, Nonlinear contractive conditions: a comparison and related problems, in: Fixed Point Theory and Its Applications, in: Banach Center Publ., vol. 77, Warsaw, 2007, pp. 123-146. · Zbl 1149.47044 [14] Kada, O.; Suzuki, T.; Takahashi, W., Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japon., 44, 381-391 (1996) · Zbl 0897.54029 [15] Kelly, J. C., Bitopological spaces, Proc. London Math. Soc., 13, 71-89 (1963) · Zbl 0107.16401 [16] Krasnoselʼskiĭ, M. A.; Vaĭnikko, G. M.; Zabreĭko, P. P.; Rutitskiĭi, Ya. B.; Stetsenko, V. Ya., Approximate Solution of Operator Equations (1972), Wolters-Noordhoof Publishing: Wolters-Noordhoof Publishing Groningen · Zbl 0231.41024 [17] Leader, S., Equivalent Cauchy sequences and contractive fixed points in metric spaces, Studia Math., 66, 63-67 (1983) · Zbl 0469.54028 [18] Lin, L.-J.; Du, W.-S., Ekelandʼs variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces, J. Math. Anal. Appl., 323, 360-370 (2006) · Zbl 1101.49022 [19] Matkowski, J., Integrable solution of functional equations, Dissertationes Math., 127 (1975) · Zbl 0318.39005 [20] Matkowski, J., Fixed point theorems for mappings with a contractive iterate at a point, Proc. Amer. Math. Soc., 62, 344-348 (1977) · Zbl 0349.54032 [21] Matkowski, J., Nonlinear contractions in metrically convex space, Publ. Math. Debrecen, 45, 103-114 (1994) · Zbl 0824.47047 [22] Meir, A.; Keeler, E., A theorem on contraction mappings, J. Math. Anal. Appl., 28, 326-329 (1969) · Zbl 0194.44904 [23] Mukherjea, A., Contractions and completely continuous mappings, Nonlinear Anal., 1, 235-247 (1977) · Zbl 0359.47033 [24] Rakotch, E., A note on contractive mappings, Proc. Amer. Math. Soc., 13, 459-465 (1962) · Zbl 0105.35202 [25] Reilly, I. L., A generalized contraction principle, Bull. Austral. Math. Soc., 10, 349-363 (1974) · Zbl 0278.54046 [26] Reilly, I. L., Quasi-gauge spaces, J. London Math. Soc. (2), 6, 481-487 (1973) · Zbl 0257.54034 [27] Reilly, I. L.; Subrahmanyam, P. V.; Vamanamurthy, M. K., Cauchy sequences in quasi-pseudo-metric spaces, Monatsh. Math., 93, 127-140 (1982) · Zbl 0472.54018 [28] Subrahmanyam, P. V.; Reilly, I. L., Some fixed point theorems, J. Austral. Math. Soc. Ser. A, 53, 304-312 (1992) · Zbl 0772.54041 [29] Suzuki, T., Subrahmanyamʼs fixed point theorem, Nonlinear Anal., 71, 1678-1683 (2009) · Zbl 1170.54016 [30] Suzuki, T., A definitive result on asymptotic contractions, J. Math. Anal. Appl., 335, 707-715 (2007) · Zbl 1128.54025 [31] Suzuki, T., Generalized distance and existence theorems in complete metric spaces, J. Math. Anal. Appl., 253, 440-458 (2001) · Zbl 0983.54034 [32] Tasković, M. R., A generalization of Banachʼs contraction principle, Publ. Inst. Math. (Beograd) (N.S.), 23, 37, 171-191 (1978) · Zbl 0403.54042 [33] Tataru, D., Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear terms, J. Math. Anal. Appl., 163, 345-392 (1992) · Zbl 0757.35034 [34] Vályi, I., A general maximality principle and a fixed point theorem in uniform spaces, Period. Math. Hungar., 16, 127-134 (1985) · Zbl 0551.47025 [35] Walter, W., Remarks on a paper by F. Browder about contraction, Nonlinear Anal., 5, 21-25 (1981) · Zbl 0461.47032 [36] Włodarczyk, K.; Plebaniak, R., Contractivity of Leader type and fixed points in uniform spaces with generalized pseudodistances, J. Math. Anal. Appl., 387, 533-541 (2012) · Zbl 1233.54019 [37] Włodarczyk, K.; Plebaniak, R., Maximality principle and general results of Ekeland and Caristi types without lower semicontinuity assumptions in cone uniform spaces with generalized pseudodistances, Fixed Point Theory Appl., 2010 (2010), Article ID 175453, 35 pp · Zbl 1201.54039 [38] Włodarczyk, K.; Plebaniak, R., Periodic point, endpoint, and convergence theorems for dissipative set-valued dynamic systems with generalized pseudodistances in cone uniform and uniform spaces, Fixed Point Theory Appl., 2010 (2010), Article ID 864536, 32 pp · Zbl 1193.37101 [39] Włodarczyk, K.; Plebaniak, R.; Doliński, M., Cone uniform, cone locally convex and cone metric spaces, endpoints, set-valued dynamic systems and quasi-asymptotic contractions, Nonlinear Anal., 71, 5022-5031 (2009) · Zbl 1203.54051 [40] Włodarczyk, K.; Plebaniak, R., A fixed point theorem of Subrahmanyam type in uniform spaces with generalized pseudodistances, Appl. Math. Lett., 24, 325-328 (2011) · Zbl 1206.54068 [41] Włodarczyk, K.; Plebaniak, R., Quasi-gauge spaces with generalized quasi-pseudodistances and periodic points of dissipative set-valued dynamic systems, Fixed Point Theory Appl., 2011 (2011), Article ID 712706, 23 pp · Zbl 1213.81161
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.