A Fixed Point Theorem for Weakly Compatible Mappings Satisfying a General Contractive Condition of Operator Type

Abstract

In this paper, we prove a fixed point theorem for weakly compatible mappings satisfying a general contractive condition of operator type. In short, we are going to study mappings \( A, B, S, T: X \to X \) for which there exists a right continuous function \( \psi: \mathbb{R}^+ \to \mathbb{R}^+ \) such that \(\psi(0) = 0\) and \(\psi(s)\leq s\) for \(s > 0.\) Moreover, for each \( x, y \in X \), one has \(O(f; d(Sx, Ty)) \leq \psi(O(f; M(x,y))),\) where \( O(f; \cdot) \) and \( f \) are defined in the first section. Also in the first section, we give some examples for \( O(f; \cdot) \). The second section contains the main result. In the last section, we give some corollaries and remarks.