A common fixed point theorem for expansive mappings under strict implicit conditions on b-metric spaces (Q2904099)

Let \((X,d)\) be a \(b\)-metric space with parameter \(s\), in the sense of \textit{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 \textit{M. Aamri} and \textit{D. El Moutawakil} [J. Math. Anal. Appl. 270, No.~1, 181--188 (2002; Zbl 1008.54030)]; (2)~\(T(X)\subset S(X)\); and (3)~NEWLINE\[NEWLINEG(d(Tx,Ty),d(Sx,Sy),d(Sx,Tx),\break d(Sy,Ty),d(Sx,Ty),d(Sy,Tx))>0NEWLINE\]NEWLINE 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~\(G\).