On an Initial Value Problem for Singular Integro-Differential Equations

Abstract

Abstract. The purpose of this paper is to study the existence and asymptotic behaviour of solutions of a nonlinear singular integro-differential equation. IntroductionIn the past two decades, several papers have been devoted to the study of singular initial value problems for differential and integro-differential equations under various conditions on the nonlinearity and the kernel (see e.g. [7]). Integro-differential equations have different properties from ordinary differential equations even in the simplest cases (see [2]). Therefore known qualitative methods of investigation of ordinary differential equations, e.g. Wazewki's topological method, cannot be applied to integrodifferential equations. The fundamental tools used in the existence proofs of all above mentioned works are essentially Schauder-Tychonoff's fixed point theorem and Banach contraction principle.In this paper we deal with the following problem t (1) g(t)y'(t) = a(t)y(t){l + f{t,y(t),\K(t,s,y(s))ds)), y(

Singular Initial Value Problem for a System of Integro‐Differential Equations

“…Singular initial value problem for ordinary differential and integro-differential equations is fairly well studied see, e.g., 1-16 , but the asymptotic properties of the solutions of such equations are only partially understood. Although the singular initial value problems were widely considered using various methods see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13]16 , our approach to this problem is essentially different from others known in the literature. In particular, we use a combination of the topological method of T. Ważewski 8 and Schauder's fixed point theorem 11 .…”

Singular Cauchy Initial Value Problem for Certain Classes of Integro-Differential Equations

“…The singular Cauchy problem for first-order differential and integro-differential equations resolved or unresolved with respect to the derivatives of unknowns is fairly well studied see, e.g., 1-16 , but the asymptotic properties of the solutions of such equations are only partially understood. Although the singular Cauchy problems were widely considered by using various methods see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][16][17][18] , the method used here is based on a different approach. In particular, we use a combination of the topological method of T. Ważewski see, e.g., 19, 20 and Schauder's fixed point theorem 21 .…”