Solutions Of Nonlinear Equations In Abstract Spaces And Applications

This paper is divided into three sections.In Section 1, without assuming operators to be continuous and cones to be normal, we use the Monch's compactness condition to study the existence and iterative approximation theorem of solutions and coupled minimal and maximal quasi-solutions for a class of nonlinear operator equations x = A(x, x] under more extensive and weaker conditions than those used in [18] in ordered Banach spaces. Then, we offer two applications to nonlinear Volterra type integral equations and the initial value problems for ordinary differential equations in Banach spaces.In Section 2, by establishing a new comparison result, we obtain the existence of extremal solutions and the explicit iterative approximation of solutions of the following periodic boundary value problems (PBVP) for nonlinear integro-differential equations of mixed type in Banach spaces:when the upper and lower solutions of PBVP satisfy a reverse condition.In Section 3, by use of a new comparison result and the Monch's fixed point theorem, the existence of solutions of the following initial value problems (IVP) for nonlinear second order impulsive integro-differential equations of mixed type inBanach spaces:is investigated under the weaker conditions. We cross out and weaken some key conditions in [31]. The results in this section generalize and improve the main results in [29,31]. We apply the abstract results to the initial value problems of infinite system for nonlinear impulsive integro-differential equations of mixed type.