Existence of Solutions and Ulam Stability for Caputo Type Sequential Fractional Differential Equations of Order α ∈ (2,3)

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Bashir Ahmad, Mohammed M. Matar, Ola M. El-Salmy, Existence of Solutions and Ulam Stability for Caputo Type Sequential Fractional Differential Equations of Order α ∈ (2,3), Int. J. Anal. Appl., 15 (1) (2017), 86-101.

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Bashir Ahmad, Mohammed M. Matar, Ola M. El-Salmy

Abstract

We study initial value problems of sequential fractional differential equations and inclusions involving a Caputo type differential operator of the form: $\left(^{C}D_{a+}^{\alpha }+\lambda _{1}~^{C}D_{a+}^{\alpha -1}+\lambda _{2}~^{C}D_{a+}^{\alpha -2}\right),$ 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.

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