Common Fixed-point Theorems for Nonlinear Weakly Contractive Mappings

The Banach contraction principle is one of the pivotal results in the metric fixed-point theory. It is a popular tool for the solution of existence problems in various fields of mathematics. There are several generalizations of the Banach contraction principle in the related literature on the metric fixed-point theory. Ran and Reurings [15] extended the Banach contraction principle in partially ordered metric spaces with some applications to linear and nonlinear matrix equations. Nieto and López [14] extended the results of Ran and Reurings and used their main result to obtain a unique solution of the first-order ordinary differential equation with periodic boundary conditions. Bhaskar and Lakshmikantham [3] introduced a concept of mixed monotone mappings and obtained some coupled fixed-point results. Moreover, they applied their results to a first-order differential equation with periodic boundary conditions. Alber and Guerre-Delabriere [1] introduced a concept of weakly contractive mappings and proved the existence of fixed point for these mappings in Hilbert spaces. In 2001, Rhoades [17] proved the fixed-point theorem which is a generalization of the Banach contraction principle. Weakly contractive mappings are closely related to the mappings of the Boyd–Wong [4] and Reich types [16]. Recently, Doric [9] proved a common fixed-point theorem for generalized ( ,φ)-weakly contractive mappings. Fixed-point problems involving weak contractions and mappings satisfying the inequalities of the weak contractive type were studied by numerous authors (see [1, 5–10, 17] and the references therein). In the present paper, we generalize the Chatterjea-type contraction mappings to (μ, )-generalized Chatterjea-type contraction mappings and deduce some common fixed-point results for single-valued mappings on ordered metric spaces. First, we recall some basic definitions and notation. Let (X, d) be a metric space. A mapping T : X ! X is said to be: