Existence of Common Coupled Fixed Points of a Pair of Operators

Abstract

In this paper, we continue on this path by investigating the existence of common coupled fixed points of a pair of operators $A,B: \Omega\times \Omega \rightarrow\Omega,$ where mixed monotone type properties of the operators are not assumed, while the pair of operators is assumed to satisfy a new two-dimensional order type property, extending a well-known one-dimensional equivalent property, and guaranteeing the use of monotone iterative technique. To prove the existence of a common coupled fixed point for such pair of operators, it is assumed for operators to satisfy a useful condensing and contractive conditions (conditions ($C_{1}$) and ($C_{2}$) hereafter) involving the two-dimensional setting.