Existence and uniqueness of \(\xi\eta\)-multiple fixed points of mixed monotone operators (Q2853185)

Let \(E\) be a real Banach space and NEWLINE\[NEWLINEP_h=\{x\in E:\lambda(x)h\leq x\leq\mu(x)h,\text{ for some positive scalars }\lambda(x),\mu(x)\}.NEWLINE\]NEWLINE An operator \(A:P_h\times P_h\to P_h\) is said to be a mixed monotone operator if it is non-decreasing in the first component and non-increasing in the second component. A point \(x\in E\) is called \(\xi\eta\)-multiple fixed point of a mixed monotone operator \(A\) if \(x=A(\xi x,\eta x)=A(\eta x,\xi x)\). The authors give some necessary and sufficient conditions for a class of mixed monotone operators to have \(\xi\eta\)-multiple fixed points. They consider concavity and convexity properties along with monotonicities; see also \textit{Z.-D. Liang}, \textit{L.-L. Zhang} and \textit{S.-J. Li} [Z. Anal. Anwend. 22, No. 3, 529--542 (2003; Zbl 1065.47060)] for the existence and uniqueness problem of positive fixed points for a class of mixed monotone operators in an ordered Banach space. The uniqueness of the fixed points is also studied.