Fixed point theorems of \(\phi\) convex \(-\psi\) concave mixed monotone operators and applications (Q2465857)

The existence and uniqueness of positive fixed points for \(\phi\) convex -- \(\psi\) concave mixed monotone operators is obtained without assuming the operators to be continuous and compact. The main difference from the known methods is that the author uses pre-images instead of images of the considered mappings. The obtained results are applied to the study of solvability of the integral equation \[ x(t)=\int_{\Omega}h(t)f(s)x^{\gamma}(s)\,ds,\;\gamma\in{\mathbb R}^1\setminus\{0,1\}. \]