SOME RESULTS IN FUNCTIONAL INTEGRAL EQUATIONS IN A BANACH SPACE

Abstract

This chapter presents some results in functional integral equations in a Banach space. The aim is to investigate some problems of a qualitative nature concerning functional integral equations. Lakshmikantham has highlighted that in the case of functional integral equations and functional differential equations in Banach spaces, there are some difficulties in imposing the assumptions because, in this case, the domain and the range of the functions involved in the functional equations are not in the same Banach space. Following the idea of Lakshmikantham, the chapter overcomes this difficulty by imposing conditions over a subset of the domain in a suitable way. Several results—namely, existence, uniqueness, boundedness, stability, and asymptotic behavior of the solutions for the equations—are thus obtained. The chapter also presents the results of Corduneanu and Miller as well as of Nohel and Wong on perturbation problems.