Existence of Solutions for a Class of Nonlinear Convolution Integral Equations

Abstract

The existence, uniqueness, boundedness and monotonicity of solutions for a class of nonlinear convolutional integral equations are discussed. First, the problem of solving the equation is transformed into a fixed-point problem of an operator. Then, Arzela – Ascoli theorem and Schauder fixed point theorem are used to prove the existence of the solution. Then, Gronwall inequality and its related lemma are used to prove the uniqueness of the solution. Secondly, the sufficient and necessary conditions for boundedness of nonnegative solutions are given by using the supremum principle and Weierstrass aggregation point theorem. Finally, under given conditions, the monotonicity of the nonnegative solution is discussed, which extends the existing results.