Existence and stability results of relaxation fractional differential equations with Hilfer--Katugampola fractional derivative.

Year 2020, Volume: 4 Issue: 4, 299 - 315, 30.12.2020

Mohammed Almalahı , Satish K. Panchal

https://doi.org/10.31197/atnaa.686693

Cited By: 1

Abstract

In this work, we present the existence, uniqueness, and stability result of solution to the nonlinear fractional
differential equations involving Hilfer-Katugampola derivative subject to nonlocal fractional integral bound-
ary conditions. The reasoning is mainly based upon properties of Mittag-Leffler functions, and fixed-point
methods such as Banach contraction principle and Krasnoselskii's fixed point theorem. Moreover, the gener-
alized Gornwall inequality lemma is used to analyze different types of stability. Finally, one example is given
to illustrate our theoretical results.

Keywords

Hilfer--Katugampola fractional derivative, Existence, Mittag-Leffler functions, Ulam stability

References

• [1] O.P. Agrawal, S.I. Muslih, D. Baleanu, Generalized variational calculus in terms of multi-parameters fractional derivatives, Communications in Nonlinear Science and Numerical Simulation. 16(12) (2011) 4756-4767.
• [2] O.P. Agrawal, Some generalized fractional calculus operators and their applications in integral equations, Fract. Calc. Appl. Anal. 15 (2012) 700-711.
• [3] M.A. Almalahi, M.S. Abdo, S.K. Panchal, ψ-Hilfer Fractional functional di?erential equation by Picard operator method. Journal of Nonlinear Dynamics (2020)
• [4] M.A. Almalahi, S.K. Panchal, E α -Ulam-Hyers stability result for ψ-Hilfer Nonlocal Fractional Differential Equation. Dis- continuity, Nonlinearity, and Complexity (2020)
• [5] M.A. Almalahi, M.S. Abdo, S.K. Panchal, Existence and Ulam-Hyers-Mittag-Lefller stability results of ψ-Hilfer nonlocal Cauchy problem. Rend. Circ. Mat. Palermo, II. Ser (2020). https://doi.org/10.1007/s12215-020-00484-8
• [6] M.A. Almalahi, M.S. Abdo, S.K. Panchal, Periodic boundary value problems for fractional implicit differential equations involving Hilfer fractional derivative. 9(2) (2020)
• [7] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44 (2017) 460-481.
• [8] D. Baleanu, O.P. Agrawal, S. I. Muslih, Lagrangians with linear velocities within Hilfer fractional derivative. In ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, American Society of Mechanical Engineers Digital Collection, (2011) 335-338).
• [9] Z. Gao, Yu, X, Existence results for BVP of a class of Hilfer fractional differential equations. Journal of Applied Mathematics and Computing, 56(1-2) (2018) 217-233.
• [10] R. Hilfer, Applications of Fractional Calculus in Physics, World scientific, Singapore, 1999.
• [11] D.H. Hyers, G. Isac, Th.M. Rassias, Stability of Functional Equations in Several Variables, Progr. Nonlinear Differential Equations Appl., Birkh 646user, Boston, 34 (1998).
• [12] U.N. Katugampola, New approach to a genaralized fractional integral, Appl. Math.Comput., 218(2011), no. 3, 860-865.
• [13] U.N. Katugampola, Existence and uniqueness results for a class of generalized fractional differential equations, Bull. Math. Anal. Appl., 1(2014).
• [14] U.N. Katugampola, New fractional integral unifying six existing fractional integrals, epint arxiv: 1612.08596, 6 pages. (2016).
• [15] A.A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Elsevier, Amsterdam, 207 (2006).
• [16] D.S. Oliveira , de oliveira E. Capelas, Hilfer-Katugampola fractional derivative. Comp Appl Math, (2017), 37: 3672-3690.
• [17] I. Podlubny, Fractional Di?erential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Math. Sci. Eng. 198, Elsevier, Amsterdam, 1999.
• [18] T.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72(2) (1978), 297-300.
• [19] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon (1987).
• [20] S.M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, 8, Inter-science, New York-London(1960).
• [21] da C Sousa J. Vanterler, de Oliveira E. Capelas, Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation. Appl Math Lett, (2018), 81: 50-56.
• [22] J.R. Wang, M. Feckan, Y. Zhou, Presentation of solutions of impulsive fractional Langevin equations and existence results. Eur. Phys. J. Spec. Top. 222 (2013) 1857-1874.
• [23] J.R. Wang, L Lv, Y Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative. Electron J Qual Theory Di?er Equ, (2011), 63: 1-10.
• [24] H. Ye, J. Gao, Y. Ding, A generalized Gronwall inequality and its application to a fractional diffequation. J Math Anal Appl, (2007), 328: 1075-1081.