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Existence of solutions and Ulam stability for Caputo type sequential fractional differential equations of order \(\alpha\in(2,3)\). (English) Zbl 1382.34004

Summary: We study initial value problems of sequential fractional differential equations and inclusions involving a Caputo type differential operator of the form: \(\left({}^CD_{a+}^\alpha+\lambda_1{}^CD_{a+}^{\alpha -1}+\lambda_2^CD_{a+}^{\alpha -2}\right)\), where \(\alpha \in (2,3)\) and \(\lambda_i (i=1, 2) \) are nonzero constants. Several existence and uniqueness results are accomplished by means of fixed point theorems. Sufficient conditions for Ulam stability of the given problem are also presented. Examples are constructed for the illustration of obtained results. Then we investigate the inclusions case of the problem at hand. An initial value problem for coupled sequential fractional differential equations is also discussed.

MSC:

34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
34D10 Perturbations of ordinary differential equations
34A60 Ordinary differential inclusions
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