On a Class of Fractional Type Integral Equations in Variable Exponent Spaces

2000 Mathematics Subject Classification: 45A05, 45B05, 45E05,45P05, 46E30

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.

∗ This work was made under the project “Variable Exponent Analysis” supported by INTAS grant Nr.06-1000017-8792. * Supported by Fundação para a Ciência ea Tecnologia (FCT) (Grant No. SFRH / BD / 22977 / 2005), through Programa Operacional Ciência e Inovação 2010 (POCI2010) of the Portuguese Government, co-financed by the European Community Fund FSE.