Existence of positive almost automorphic solutions to nonlinear delay integral equations

This paper is concerned with the existence of positive almost automorphic solutions to some nonlinear delay integral equations. We first establish a new fixed point theorem for mixed monotone operator in a cone, and then, with its help, we obtain existence theorems of positive almost automorphic solutions. Some examples are given to illustrate our results. As one will see, even in the case of almost periodicity, our theorems extend some earlier results, and moreover, the approach dealing with the integral equation arising in an epidemic problem in this paper is also new.