Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator |
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Authors: | Živorad Tomovski |
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Affiliation: | Faculty of Natural Sciences and Mathematics, Institute of Mathematics, Saints Cyril and Methodius University, 1000 Skopje, Macedonia |
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Abstract: | This paper is devoted to proving the existence and uniqueness of solutions to Cauchy type problems for fractional differential equations with composite fractional derivative operator on a finite interval of the real axis in spaces of summable functions. An approach based on the equivalence of the nonlinear Cauchy type problem to a nonlinear Volterra integral equation of the second kind and applying a variant of the Banach’s fixed point theorem to prove uniqueness and existence of the solution is presented. The Cauchy type problems for integro-differential equations of Volterra type with composite fractional derivative operator, which contain the generalized Mittag-Leffler function in the kernel, are considered. Using the method of successive approximation, and the Laplace transform method, explicit solutions of the open problem proposed by Srivastava and Tomovski (2009) [11] are established in terms of the multinomial Mittag-Leffler function. |
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Keywords: | 33E12 33C20 26A33 47B38 47G10 |
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