Geodesic
On a Class of Fractional Type Integral Equations in Variable Exponent Spaces
Fractional calculus and applied analysis, Tome 10 (2007) no. 4, pp. 399-421.

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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.
Keywords: Fractional Integrals, Variable Exponent Spaces, 1st Kind Integral Equations, Singular Integral Equations, Generalized Abel Equation, Fredholmness, Fractional Smoothness, 45A05, 45B05, 45E05, 45P05, 46E30 
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Rafeiro, Humberto; Samko, Stefan. On a Class of Fractional Type Integral Equations in Variable Exponent Spaces. Fractional calculus and applied analysis, Tome 10 (2007) no. 4, pp. 399-421. http://geodesic.mathdoc.fr/item/FCAA_2007_10_4_a3/