Existence and uniqueness of fixed points for mixed monotone operators with applications (Q855474)

Let \(P\) be a cone of a Banach space \(E\). An operator \(A: P \times P \to E\) is said to be mixed monotone if \(A(\cdot,y)\) is increasing and \(A(x, \cdot)\) is decreasing. An element \(x \in P\) is called a fixed point of \(A\) if \(A(x,x)=x\). The study of fixed point theory of mixed monotone operators has been the focus of a large number of papers since it was introduced by \textit{D.\,Guo} and \textit{V.\,Lakshmikantham} [Nonlinear Anal., Theory Methods Appl.\ 11, 623--632 (1987; Zbl 0635.47045)]. In the paper under review, the authors study the existence and uniqueness of fixed points of a class of mixed monotone operators. An application to a class of integral equations is given.