Abstract
Abstract In this paper, we show the existence of a coupled fixed point theorem of nonlinear contraction mappings in complete metric spaces without the mixed monotone property and give some examples of a nonlinear contraction mapping, which is not applied to the existence of coupled fixed point by using the mixed monotone property. We also study the necessary condition for the uniqueness of a coupled fixed point of the given mapping. Further, we apply our results to the existence of a coupled fixed point of the given mapping in partially ordered metric spaces. Moreover, some applications to integral equations are presented. MSC:47H10, 54H25.
Highlights
Let X be an arbitrary nonempty set
The applications of fixed point theorems are very important in diverse disciplines of mathematics, statistics, chemistry, biology, computer science, engineering and economics in dealing with problems arising in approximation theory, potential theory, game theory, mathematical economics, theory of differential equations, theory of integral equations, theory of matrix equations etc
Fixed point theorems are helpful for proving the existence of weak periodic solutions for a model describing the electrical heating of a conductor taking into account the Joule-Thomson effect
Summary
Let X be an arbitrary nonempty set. A fixed point for a self mapping f : X → X is a point x ∈ X such that fx = x. It states that if (X, d) is a complete metric space and T : X → X is a contraction mapping (i.e., d(Tx, Ty) ≤ kd(x, y) for all x, y ∈ X, where k is a nonnegative number such that k < ), T has a unique fixed point.
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