Fractional Integrals Variable Exponent Spaces 1st Kind Integral Equations Singular Integral Equations Generalized Abel Equation Fredholmness Fractional Smoothness 45A05 45B05 45E05 45P05 46E30
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Fractional Calculus and Applied Analysis, Vol. 10, No 4, (2007), 399p-421p
We obtain a criterion of Fredholmness and formula for the Fredholm index of a certain class of one-dimensional integral operators M with a weak singularity in the kernel, from the variable exponent Lebesgue space L^p(·) ([a, b], ?) to the
Sobolev type space L^α,p(·) ([a, b], ?) of fractional smoothness. We also give formulas of closed form solutions ϕ ∈ L^p(·)
of the 1st kind integral equation M0ϕ = f, known as the generalized Abel equation, with f ∈ L^α,p(·), in dependence on the
values of the variable exponent p(x) at the endpoints x = a and x = b.