Common fixed point theorems of compatible and weakly compatible maps in Menger spaces. (English) Zbl 1172.54027

Fang, Jin-Xuan

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 5-6, 1833-1843 (2009).

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 B. Singh and 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.

Reviewer: Dorel Miheţ (Timişoara)

MSC:

54H25	Fixed-point and coincidence theorems (topological aspects)
54E70	Probabilistic metric spaces

Keywords:

Menger space; common fixed points; compatible maps; weakly compatible maps

Citations:

Zbl 1068.54044

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