Fixed point theorems for a class of nonlinear sum-type operators and its application to fractional q-difference equations*

Abstract

In this paper, we investigate the operator equation $A(x,\ x)+B(x,\ x)+C(x,\ x)+Dx+Ex=x$, in which A, B, C are three mixed monotone operators with different characters, D is an increasing operator, and E is a decreasing operator. The main difficulty in dealing with the sum of five operators is the proving of concavity of the sum-type operator, which produces more complexities than the sum of two or three operators. Our goal is to obtain the unique existence of positive solutions under some appropriate conditions by virtue of a fixed point theorem for mixed monotone operators. At last, we give an application to demonstrate the efficiency of our abstract result.