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Volume 2018

Existence and stability results for Langevin equations with Hilfer fractional derivative -by S. Harikrishnan, K. Kanagarajan and E. M. Elsayed

Abstract: In this paper, we study two Hilfer fractional derivatives of the form
\begin{equation}D^{\alpha_1, \beta} \left(D^{\alpha_2, \beta} + \lambda \right) x(t) = f(t, x(t)), \ t\in J:=(a,b],
\label{ee}\end{equation}\begin{equation}I^{1-\gamma} x(a) = x_a, \gamma = (\alpha_1+\alpha_2)(1-\beta) + \beta,\end{equation}

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where $\ D^{\alpha_1, \beta}, D^{\alpha_2, \beta}$ is two Hilfer fractional derivatives, $\alpha_1$ and $\alpha_2$ are the order of derivatives, $\beta$ determines to the type of initial condition used in the problem. In this problem existence of solution, Ulam-Hyers stability and generalized Ulam-Hyers-Rassias stability are analyzed. 
Keywords:Langevin equation, Hilfer fractional derivative, Existence, Ulam-Hyers stability.

DOI: https://doi.org/10.30697/rfpta-2018-3
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How to cite this article: 
S. Harikrishnan, K. Kanagarajan and E. M. Elsayed, Existence and stability results for Langevin equations with Hilfer fractional derivative, Results in Fixed Point Theory and Applications, vol. 2018, Article ID 20183, 10 pages, 2018.
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Copyright © 2018 SUMA Publishing Group., This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.