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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 \(G\).

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
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