Existence and approximate Lp and continuous solutions of nonlinear integral equations of the Hammerstein and Volterra types

In this paper, we consider a compact interval [a,b], a positive real number p⩾1 and give different existence results for Lp([a,b]) and C([a,b])-solutions of some nonlinear integral equations of the Hammerstein and Volterra types. The main ingredients of our existence results are the Shaefer's and Schauder's fixed point theorems combined with a general version of Gronwall's inequality. Moreover, we give a numerical method for the approximation of the solutions of the Volterra-Hammerstein integral equations. Finally, to illustrate the results of this work, we provide the reader with some numerical examples.