ON THE SOLUTION OF
NONLINEAR VOLTERRA CONVOLUTION
EQUATION WITH POWER NONLINEARITY

NIKOLAI K. KARAPETYANTS, ANATOLY A. KILBAS
AND MEGUMI SAIGO

Abstract:

The Volterra nonlinear convolution integral equation

$$\varphi^m(x)=a(x)\int^x_0k(x-t)\varphi(t)\,dt+f(x)$$

$$0<x<d\le\infty$$

with m>0, $m\ne1$, and real functions a(x), k(u) and f(x) is studied. Local estimates and asymptotic properties near zero for its solution $\varphi(x)$ are given, provided that a(x), k(u) and f(x) have power asymptotic behaviors near zero. The integral equation with a power kernel being solvable in closed form is considered.