×

zbMATH — the first resource for mathematics

A common fixed point theorem for expansive mappings under strict implicit conditions on b-metric spaces. (English) Zbl 1246.54035
Let \((X,d)\) be a \(b\)-metric space with parameter \(s\), in the sense of S. Czerwik [Acta Math. Inform. Univ. Ostrav. 1, 5–11 (1993; Zbl 0849.54036)]. Let \(S\) and \(T\) be two weakly compatible self-mappings of \(X\) such that: (1) \(S\) and \(T\) satisfy property (E.A) of M. Aamri and D. El Moutawakil [J. Math. Anal. Appl. 270, No. 1, 181–188 (2002; Zbl 1008.54030)]; (2) \(T(X)\subset S(X)\); and (3)  \[ G(d(Tx,Ty),d(Sx,Sy),d(Sx,Tx),\break d(Sy,Ty),d(Sx,Ty),d(Sy,Tx))>0 \] for all \(x,y\in X\) such that \(x\neq y\), where \(G:\mathbb{R}_+^6\to\mathbb{R}\) is continuous and satisfies: (a) \(G\) is nondecreasing in variable \(t_1\) and nonincreasing in variable \(t_2\); (b) \(G(st,0,0,t,\frac1st,0)<0\) for all \(t>0\); and (c) \(G(t,t,0,0,t,t)\leq0\) for all \(t>0\). The author proves that if \(S(X)\) or \(T(X)\) is a closed subspace of \(X\), then \(T\) and \(S\) have a unique common fixed point. If the \(b\)-metric \(d\) is weakly continuous (i.e., if \(\lim_{n\to\infty}d(x_n,x)=0\) implies \(\lim_{n\to\infty}d(x_n,y)=d(x,y)\) for every sequence \(\{x_n\}\) in \(X\) and all \(x,y\in X\)), then the same conclusion holds with weaker assumptions for the function \(G\).

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] Aamri, M., El Moutawakil, D.: Some new common fixed point theorems under strict contractive conditions. Math. Anal. Appl. 270 (2002), 181-188. · Zbl 1008.54030
[2] Czerwik, S.: Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostraviensis 1 (1993), 5-11. · Zbl 0849.54036
[3] Czerwik, S.: Nonlinear set-valued contraction mappings in b-metric spaces. Atti Sem. Mat. Fis. Univ. Modena 46, 2 (1998), 263-276. · Zbl 0920.47050
[4] Djoudi, A.: General fixed point theorems for weakly compatible maps. Demonstratio Math. 38 (2005), 195-205. · Zbl 1065.54020
[5] Imdad, M., Khan, T. I.: Fixed point theorems for some expansive mapping via implicit relations. Nonlinear Analysis Forum 9, 2 (2004), 209-218. · Zbl 1066.54038
[6] Imdad, M., Ali, J.: Jungck’s Common Fixed Point Theorem and E.A Property. Acta Mathematica Sinica, English Series 24, 1 (2008), 87-94. · Zbl 1158.54021
[7] Jungck, G.: Compatible mappings and common fixed points. Internat. J. Math. Math. Sci. 9 (1986), 771-779. · Zbl 0613.54029
[8] Jungck, G.: Common fixed points for noncontinuous nonself mappings on nonnumeric spaces. Far. East, J. Math.Sci. 4 (1996), 199-215. · Zbl 0928.54043
[9] Pant, R. P.: Common fixed points of contractive maps. J. Math. Anal. Appl. 226 (1998), 251-258. · Zbl 0916.54027
[10] Pant, R. P.: R-weak commutativity and common fixed points of noncompatible maps. Ganita 49, 1 (1998), 19-27. · Zbl 0977.54039
[11] Pant, R. P.: R-weak commutativity and common fixed points. Soochow J. Math. 25 (1999), 37-42. · Zbl 0918.54038
[12] Popa, V.: Fixed point theorems for implicit contractive mappings. Stud. Cerc. St. Ser. Mat. Univ. Bacău 7 (1997), 127-133. · Zbl 0967.54041
[13] Popa, V.: Some fixed point theorems for compatible mappings satisfying an implicit relation. Demonstratio Math. 32 (1999), 157-163. · Zbl 0926.54030
[14] Popa, V.: A General fixed point theorem for expansive mappings under strict implicit conditions. Stud. Cerc. St. Ser. Mat. Univ. Bacău 17 (2007), 197-200. · Zbl 1199.54242
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.