Fractional Integrals Variable Exponent Spaces 1st Kind Integral Equations Singular Integral Equations Generalized Abel Equation Fredholmness Fractional Smoothness 45A05 45B05 45E05 45P05 46E30
Issue Date:
2007
Publisher:
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation:
Fractional Calculus and Applied Analysis, Vol. 10, No 4, (2007), 399p-421p
Abstract:
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.
