On approximate solution of some types of Integral equations

Abstract

The theory of Hammerstein and Volterra integral equations are

directed as a generalization of the . theory of ordinary 4ifferential equations. ln particular, one can generalize the fundamental existence theorems of differential equations to Hammerstein and Volterra integral equations. Many authors have considered the problem of best . approximate solution of non-linear differential equations {5, 6, 7, 10, 25, 27, 29]. Many of these results have been extended to integral equations

[7, 10, 13, 14, 16, 17,29,32].

in the case of differential equations, a successive approximations method has been developed to find best approximate solutions. This method, being based on the Remes algorithm, is inefficient in the case of integral equations. Hence we use the successive approximations method to prove an existence theorem for solutions ofHammerstein and Volterra integral equations. One of the main problems in solving different classes of non-linear integral equations is the search for an initial approximation with which the widely known iterative methods converge. The difficulty of evaluating the integrals required for the power kernel functions k.. ,