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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 $$G$$.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems
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##### References:
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