Hadamard-Type Integral Equations and Fractional Calculus Operators

Abstract

The paper is devoted to the study of the integral equation

$$ \frac{1}{{\Gamma (\alpha )}}\smallint _a^x{\left( {\frac{u}{x}} \right)^\mu }{\left( {\log \frac{x}{u}} \right)^{\alpha - 1}}f\left( u \right)\frac{{du}}{u} = g\left( x \right) \left( {0 < a < x < b} \right) $$

with realμ and α > 0 on a finite segment [a, b] of the real line. We prove conditions for the existence of a solution f(x) of this equation in the space X _µ, (a, b) of Lebesgue measurable functions f on (a, b) such that $\smallint _a^b|{u^{ - \mu - 1}}f(u)|du < \infty$. Explicit formulas for the solution f(x) are established. We also describe properties of the Hadamard-type fractional integrals defined by the left-hand side of the above equation and of the corresponding fractional derivatives.