On solution of nonlinear Abel-Volterra integral equation (Q1276323)

The authors consider the integral equation \[ \varphi^m (x)= \frac{a(x)}{\Gamma (\alpha)}\int_0^x \frac{\varphi (t)}{(x-t)^{1-\alpha}} dt +f (x),\quad 0<x <d \leq \infty, \tag{1} \] with \(\alpha >0\) and \(m \in \mathbb{R}\), \(m \neq 0, -1, -2,\dots\) in some subclasses of continuous or bounded functions. Under some assumptions, the uniqueness of the solution is proved. The main part of the paper is devoted to an investigation of the asymptotic behaviour of the solution near the origin. Supposing that \(a(x)\) and \(f(x)\) have some special asymptotic expansions, the authors show that the solution of the equation (1) has the asymptotic expansion of the same form. In the particular case, when the equation (1) has the form \[ \varphi^m (x)= \frac{ax^{\alpha(pm-l)}}{\Gamma (\alpha)}\int_0^x \frac{\varphi (t)}{(x-t)^{1-\alpha}} dt -bx^{\alpha(pm-n)}f (x),\quad 0<x <d \leq \infty, \tag{2} \] with \(p,l,n \in \mathbb{Z}\) they obtain the solution in closed form via the solution of some transcendental equation. Solvability of the equation (2), and of more general form with kernel \(x^{\alpha -1}\) replaced by \(k(x)\), was also studied by \textit{N. Karapetyants} and the authors [J. Integral Equations Appl. 8, No. 4, 429-445 (1996; Zbl 0874.45002)], provided that \(a(x),\;k(x)\) and \(f(x)\) have power asymptotic behavior near the origin, but in that paper only the first asymptotic term was investigated.