A note on the existence of solutions to some nonlinear functional integral equations

Ioannis Purnaras

doi:10.14232/ejqtde.2006.1.17

Abstract

Substituting the usual growth condition by an assumption that a specific initial value problem has a maximal solution, we obtain existence results for functional nonlinear integral equations with variable delay. Appli- cation of the technique to initial value problems for differential equations as well as to integrodifferential equations are given. Nonlinear integral equations and nonlinear functional integral equations have been some topics of great interest in the field of nonlinear analysis for a long time. Since the pioneering work of Volterra up to our days, integral equations have attracted the interest of scientists not only because of their mathematical context but also because of their miscellaneous applications in various fields of science and technology. In particular, existence theory for nonlinear integral equations, strongly related with the evolution on fixed point theory, has been boosted ahead after the remarkable work of Krasnoselskii (6) which signaled a new era in the research of the subject. The present note is motivated by a recent paper by Dhage and Ntouyas (3) presenting some results on the existence of solutions to the nonlinear functional integral equation (E)

Highlights

Nonlinear integral equations and nonlinear functional integral equations have been some topics of great interest in the field of nonlinear analysis for a long time
The present note is motivated by a recent paper by Dhage and Ntouyas [3] presenting some results on the existence of solutions to the nonlinear functional integral equation (E)
On the basis of this observation, it is not difficult to see that application of the technique employed for the proof of Theorem 1 of this note can yield similar type results for integral equations as well as for differential equations that possess the same character as (E): their left hand side can be written as a sum of a contraction and a completely continuous operator

Summary

INTRODUCTION

Nonlinear integral equations and nonlinear functional integral equations have been some topics of great interest in the field of nonlinear analysis for a long time. Functional integral equation, existence theorem, Krasnoselskii’s type fixed point theorem, Nonlinear Alternative, maximal solution. On the basis of this observation, it is not difficult to see that application of the technique employed for the proof of Theorem 1 of this note can yield similar type results for integral equations as well as for differential equations that possess the same character as (E): their left hand side can be written as a sum of a contraction and a completely continuous operator.

PRELIMINARIES

MAIN RESULTS

APPLICATIONS