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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 {\it 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 {\it M. Aamri} and {\it 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 $x\ne y$, where $G:\Bbb{R}_+^6\to\Bbb{R}$ 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 in topological spaces 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces
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##### References:
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