On Some Nonlinear Alternatives of Leray-schauder Type and Functional Integral Equations

Nonlinear functional integral equations have been discussed in the literature extensively, for a long time. See for example, Subramanyam and Sundersanam [15], Ntouyas and Tsamatos [14], Dhage and Regan [10] and the references therein, Recently, the present author, in a series of papers [4, 6] initiated the study of nonlinear integral equations in a Banach algebra via fixed point techniques. In this papers we study a new class of nonlinear functional integral equations for the existence theory via a new nonlinear alternative of Leray-Schauder-type to be developed in this paper. In particular, given a closed and bounded interval J = [0, 1] ⊂ R, R denotes the set of all real numbers, we study the existence of the nonlinear functional integral equation (in short FIE)