A Coupled Fixed Point Problem Under a Finite Number of Equality Constraints

Abstract

Let $(E,\Vert \cdot \Vert )$ be a Banach space with a cone P. Let $F,\varphi _i: E\times E\rightarrow E$ ($i=1,2,\ldots ,r$) be a finite number of mappings. In this chapter, we provide sufficient conditions for the existence and uniqueness of solutions to the problem: Find $(x,y)\in E\times E$ such that

$$\begin{aligned} \left\{ \begin{array}{lll} F(x,y)&{}=&{}x,\\ F(y,x)&{}=&{}y,\\ \varphi _i(x,y)&{}=&{}0_E,\,\, i=1,2,\ldots ,r, \end{array} \right. \end{aligned}$$

where $0_E$ is the zero vector of E. The main reference for this chapter is the paper [4].