On solution of integral equation of Abel-Volterra type (Q1891553)

The authors study asymptotic expansions of the solution of the equation \[ \varphi (x) = a(x) \int_ 0^ x (x - t)^{\alpha - 1} \varphi (t) dt + f(x), \] as \(x \to 0 +\), where \(0 < \alpha < 1\). It is assumed that \(a\) and \(f\) have expansions of the form \(a(x) \sim \sum_{k = - 1}^ \infty a_ k x^{\alpha k}\), \(f(x) \sim \sum_{k = - 1}^ \infty f_ k x^{\alpha k}\), and it is shown that then the solution has a similar expansion \(\varphi (x) \sim \sum_{k = - 1}^ \infty \varphi_ k x^{\alpha k}\) when \(x \to 0 +\). Recursive formulas for calculating the coefficients \(\varphi_ k\) are given. The case where \(a(x) = ax^{\alpha m}\), \(m = - 1,0, \dots\), and where one can find an explicit expression for the solution is studied, too.