Quadruple Fixed Point Theorems under Nonlinear Contractive Conditions in Partially Ordered Metric Spaces

The notion of coupled fixed point was initiated by Gnana Bhaskar and Lakshmikantham 1 in 2006. In this paper, they proved some fixed point theorems under a set of conditions and utilized their theorems to prove the existence of solutions to some ordinary differential equations. Recently, Berinde and Borcut 2 introduced the notion of tripled fixed point and extended the results of Gnana Bhaskar and Lakshmikantham 1 to the case of contractive operator F : X × X × X → X, where X is a complete ordered metric space. For some related works in coupled and tripled fixed point, we refer readers to 3–32 . For simplicity we will denote the cross product of k ∈ N copies of the space X by X.