Some results on multi-valued weakly Jungck mappings in \(b\)-metric space
Let \((X, d)\) be a \(b\)-metric space, \(S, T : X \to CB(X)\). The pair \((S, T)\) is called a multivalued \((\theta,\varphi)\)-weak \(J\)-contraction if there exists a constant \(\theta \in (0, 1)\) and a continuous monotone increasing function \(\varphi : \mathbb{R}^+ \to \mathbb{R}^+\) satisfying \(\varphi(0) = 0\), such that \(H(Tx, Ty) \leq \theta d(Sx, Sy) + \varphi(D(Sy, Tx))\), for all \(x, y \in X\). (This definition is a generalization of the corresponding definition in \textit{M.\,Berinde} and \textit{V.\,Berinde} [J.~Math.\ Anal.\ Appl.\ 326, No.\,2, 772--782 (2007; Zbl 1117.47039)]. Under the additional hypotheses that \(T(X) \subset S(X)\), with \(S(X)\) a complete subspace of \(CB(X)\), and \(S\) is continuous, the author proves that \(S\) and \(T\) have a common coincidence point \(z\) and obtains an error estimate for \(d(Sx_n, Sz)\). The second theorem in the paper is a similar result using a different contractive condition.
scientific article; zbMATH DE number 5485404
scientific article