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The decompositional structure of certain fractional integral operators. (English) Zbl 1429.26009
Summary: The aim of this paper is to investigate the decompositional structure of generalized fractional integral operators whose kernels are the generalized hypergeometric functions of certain type. By using the Mellin transform theory proposed by P. L. Butzer and S. Jansche [J. Fourier Anal. Appl. 3, No. 4, 325–376 (1997; Zbl 0885.44004)], we prove that these operators can be decomposed in terms of Laplace and inverse Laplace transforms. As applications, we derive two very general results involving the \(H\)-function. We also show that these fractional integral operators when being understood as integral equations possess the \(\mathcal{L}\) and \(\mathcal{L}^{-1}\) solutions. We also consider the applications of the decompositional structures of the fractional integral operators to some specific integral equations and one of such integral equations is shown to possess a solution in terms of an Aleph \((\aleph)\)-function.
MSC:
26A33 Fractional derivatives and integrals
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
44A10 Laplace transform
44A15 Special integral transforms (Legendre, Hilbert, etc.)
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
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