A new monotone iteration principle in the theory of nonlinear first order integro-differential equations

Abstract

In this paper the author proves the algorithms for the existence as well as approximations of the solutions for a initial value problem of nonlinear first order ordinary integro-differential equations using the operator theoretic techniques in a partially ordered metric space. The existence of maximal and minimal solutions as well as comparison theorems for the considered integro-differential equation are also obtained. The main results rely on the Dhage iteration principle embodied in the recent hybrid fixed point theorems of Dhage (2014) in a partially ordered normed linear space and the approximations of the solutions of the considered nonlinear integro-differential equations are obtained under weak mixed partial continuity and partial compactness or partial Lipschitz conditions. Our hypotheses and results are also illustrated by some concrete numerical examples. We claim that the approach to the results of this paper is new and include some known results for the nonlinear differential equations as special cases.