Some Classes of Nonlinear Mixed Volterra and Singular Integral Equations

Some years ago the author dealt with nonlinear singular integral equations of Cauchy type using methods of monotone operator theory [4] and a novel application of Schauder's fixed point theorem [5], respectively. In the present note these methods are applied to some classes of nonlinear mixed Volterra and singular integral or integro-differential equations. First, equations with an integral operator composed by a nonlinear Volterra operator and the sum of operators of fractional integration and logarithmic type are considered. By differentiation these equations are reduced to singular integral equations of Cauchy type which under suitable assumptions can be studied through a combination of the method of monotone operator theory in [4] and the iteration method. Second, a quasi-linear Volterra integro -differential equation additionally involving the Cauchy singular integral operator and a related class of nonlinear Volterra integral equations additionally involving the logarithmic integral operator are dealt with. Again by differentiation these equations are reduced to singular integro -differential equations of Cauchy type treated in [5].