Common fixed point theorems of compatible and weakly compatible maps in Menger spaces
The main result of the paper is the following common fixed point theorem for compatible and weakly compatible self-mappings on Menger spaces under \(\varphi\)-contractive conditions, which improves the result of \textit{B. Singh} and \textit{S. Jain} [J. Math. Anal. Appl. 301, No.~2, 439--448 (2005; Zbl 1068.54044)]: Let \(\Phi\) be the class of all functions \(\varphi: \mathbb R_+\rightarrow \mathbb R_+\), such that \(\varphi\) is non-decreasing, upper semi-continuous from the right and \[ \sum_{n=0}^{\infty}\varphi^n(t)<\infty \] for all \(t>0.\) Let \(P, Q, L, M\) be self-maps on a complete Menger space \((X, {\mathcal F},\Delta)\) with \(\Delta\) a continuous t-norm of Hadžić-type. If (i)\ \(L(X)\subset Q(X), M(X)\subset P(X)\); (ii) either \(P\) or \(L\) is continuous; (iii)\ \((L, P)\) is compatible and \((M, Q)\) is weakly compatible; (iv) there exists \(\phi\in \Phi\) such that \[ F_{Lx, My}(\phi(t))\geq \min\{F_{Px, Lx}(t), F_{Qy, My}(t), F_{Px, qy}(t), F_{Qy, Lx}(\beta t), [F_{Px, \xi }* F_{\xi, My}]((2-\beta)t)\} \] for all \(x, y, \xi \in X\), \(\beta \in (0, 2)\) and \(t>0\), where \[ F_1 * F_2 (t):=\sup_{t_1+t_2=t} \min\{F_1 (t_1), F_2 (t_2)\} \quad (t>0), \] then \(L, M, P\) and \(Q\) have a unique common fixed point.
scientific article; zbMATH DE number 5573704
scientific article