Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces

Abstract

In this paper we extend the coupled fixed point theorems for mixed monotone operators F : X × X → X obtained in [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA 65 (2006) 1379-1393] by significantly weakening the involved contractive condition. Our technique of proof is essentially different and more natural. An example as well an application to periodic BVP are also given in order to illustrate the effectiveness of our generalizations. 1

COUPLED FIXED POINTS FOR MIXED g-MONOTONE UNDER RATIONAL CONTRACTIVE EXPRESSIONS IN PARTIALLY ORDERED METRIC SPACES

“…NASHINE AND A. GUPTA and proved the existence and uniqueness of a coupled coincidence point for such mapping satisfying the mixed g-monotone property in partially ordered metric spaces. Following this result other coupled coincidence point results appeared in [1] and [20]. Subsequently, several authors obtained many results of this kind (see, e.g., [4,6,7,10,11,13,16,17,18,19]).…”

“…Subsequently, several authors obtained many results of this kind (see, e.g., [4,6,7,10,11,13,16,17,18,19]). These results have a lot of applications, e.g., in proving existence of solutions of periodic boundary value problems (e.g., [1,2]) as well as particular integral equations (e.g., [5,8,9]). …”

Shorter proofs of some recent even-tupled coincidence theorems for weak contractions in ordered metric spaces

“…We note the same idea here, but in the case of coupled and tripled fixed point theorems, we have been first used in ( [3,28,33]). …”

Coupled Fixed Point Theorems With C-Class Functions

“…Bhaskar and Lakshmikantham [5] obtained some coupled fixed point theorems for mixed monotone operators using certain contractive-type condition defined on a partially ordered metric space. Later, Berinde [4] extended the work of Bhaskar and Lakshmikantham [5] by significantly weakening the involved contractive condition.…”