On the theory of fractional terminal value problem with <em>ψ</em>-Hilfer fractional derivative

Mohammed A. Almalahi; Mohammed S. Abdo; Satish K. Panchal; Mohammed A. Almalahi; Mohammed S. Abdo; Satish K. Panchal

doi:10.3934/math.2020312

AIMS Mathematics

2020, Volume 5, Issue 5: 4889-4908. doi: 10.3934/math.2020312

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Research article

On the theory of fractional terminal value problem with ψ-Hilfer fractional derivative

Department of Mathematics. Dr. Babasaheb Ambedkar Marathwada University Aurangabad, (M. S). 431001. India

Received: 30 December 2019 Accepted: 12 May 2020 Published: 03 June 2020
MSC : 34A08, 34B15, 34A12, 47H10

In this paper, we prove the existence and uniqueness of solutions of a new class of boundary value problems of terminal type for ψ-Hilfer fractional differential equations. The technique used in the analysis relies on the Banach contraction principle and Krasnosleskii fixed point theorem. Moreover, we use generalized Gronwall inequality with singularity to establish uniqueness and continuous dependence of the δ-approximate solution. Finally, we demonstrate some examples to illustrate our main results.
- fractional differential equations,
- Gronwall inequality,
- fixed point theorem,
- terminal value problem
Citation: Mohammed A. Almalahi, Mohammed S. Abdo, Satish K. Panchal. On the theory of fractional terminal value problem with ψ-Hilfer fractional derivative[J]. AIMS Mathematics, 2020, 5(5): 4889-4908. doi: 10.3934/math.2020312

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Abstract

In this paper, we prove the existence and uniqueness of solutions of a new class of boundary value problems of terminal type for ψ-Hilfer fractional differential equations. The technique used in the analysis relies on the Banach contraction principle and Krasnosleskii fixed point theorem. Moreover, we use generalized Gronwall inequality with singularity to establish uniqueness and continuous dependence of the δ-approximate solution. Finally, we demonstrate some examples to illustrate our main results.

References

[1]	R. Hilfer, Applications of fractional calculus in physics, Singapore: World., 35 (2000), 87-130.
[2]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional di fferential equations, Elsevier Science Limited., 204, 2006.
[3]	I. Podlubny, Fractional di fferential equations: an introduction to fractional derivatives, fractional di fferential equations, to methods of their solution and some of their applications, Elsevier., 198, 1998.
[4]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Theory and applications, Yverdon: Gordon and Breach., 1, 1993.
[5]	R. P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional di fferential equations and inclusions, Acta. Appl. Math., 109 (2010), 973-1033. Available from: https://doi.org/10.1007/s10440-008-9356-6. doi: 10.1007/s10440-008-9356-6
[6]	B. Ahmad, J. J. Nieto, Riemann-Liouville fractional di fferential equations with fractional boundary conditions, Fixed Point Theory., 13 (2012), 329-336. Available from: https://doi.org/10.1186/1687-2770-2011-36.
[7]	M. Benchohra, R. Graef, J. S. Hamani, Existence results for boundary value problems with nonlinear fractional di fferential equations, Appl. Anal., 87 (2008), 851-863. Available from: https://doi.org/10.1080/00036810802307579. doi: 10.1080/00036810802307579
[8]	I. Podlubny, Fractional Differential equation, Academic Press, San Diego, 1999.
[9]	S. Zhang, Existence of solution for a boundary value problem of fractional order. Acta Math. Sc. Ser. B (Engl. Ed.), 26 (2006), 220-228. Available from: https://doi.org/10.1016/S0252-9602(06)60044-1.
[10]	M. A. Almalahi, S. K. Panchal, E_α-Ulam-Hyers stability result for ψ-Hilfer Nonlocal Fractional Differential Equation, Discontinuity Nonlinearity and Complexity, 10, 2021.
[11]	M. S. Abdo, S. K. Panchal, Fractional integro-di fferential equations involvinψ-Hilfer fractional derivative, Adv. Appl. Math. Mech., 11 (2019), 338-359. doi: 10.4208/aamm.OA-2018-0143
[12]	M. A. Almalahi, M. S. Abdo, S. K. Panchal, Existence and Ulam-Hyers-Mittag-Leffler stability results of ψ-Hilfer nonlocal Cauchy problem. Rend. Circ. Mat. Palermo, II. Ser (2020). Available from: https://doi.org/10.1007/s12215-020-00484-8.
[13]	M. A. Almalahi, M. S. Abdo, S. K. Panchal, ψ-Hilfer Fractional functional di fferential equation by Picard operator method, J. Appl. Nonlinear Dyn., 9 (2020), 685-702. Available from: https: //doiI:10.5890/JAND.2020.12.011.
[14]	M. Benchohra, S. Bouriah, J. J. Nieto, Terminal value problem for di fferential equations with Hilfer-Katugampola fractional derivative, Symmetry, 11 (2019), 672. Available from: https://doi.org/10.3390/sym11050672.
[15]	M. Benchohra, S. Bouriah, Existence and stability results for nonlinear boundary value problem for implicit di fferential equations of fractional order. Moroccan J. Pure Appl. Anal., 1 (2015), 22-37. Available from: https://doi.org/10.7603/s40956-015-0002-9.
[16]	N. J. Ford, M. L. Morgado, M. Rebelo, A nonpolynomial collocation method for fractional terminal value problems, J. Comput. Appl. Math., 275 (2015), 392-402. Available from: https://doi.org/10.1016/j.cam.2014.06.013. doi: 10.1016/j.cam.2014.06.013
[17]	S. H. Shah, M. ur Rehman, A note on terminal value problems for fractional di fferential equations on infinite interval, Appl. Math. Lett., 52 (2016), 118-125. Available from: https://doi.org/10.1016/j.aml.2015.08.008. doi: 10.1016/j.aml.2015.08.008
[18]	W. E. Shreve, Boundary Value Problems for y" = f (x, y, λ) on (a,∞), SIAM. J. Appl. Math., 17 (1969), 84-97. Available from: https://doi.org/10.1137/0117009.
[19]	M. A. Zaky, Existence, uniqueness and numerical analysis of solutions of tempered fractional boundary value problems, Appl. Numer. Math., 2019. Available from: https://doi.org/10.1016/j.apnum.2019.05.008.
[20]	M. A. Zaky, Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non-smooth solutions, J. Comput. Appl. Math., 357 (2019), 103-122. Available from: https://doi.org/10.1016/j.cam.2019.01.046. doi: 10.1016/j.cam.2019.01.046
[21]	A. R. Aftabizadeh, V. Lakshmikantham, On the theory of terminal value problems for ordinary di fferential equations, Nonlinear Anal.: Theory Methods Appl., 5 (1981), 1173-1180. doi: 10.1016/0362-546X(81)90011-0
[22]	W. E. Shreve, Terminal value problems for second order nonlinear di fferential equations, SIAM. J. Appl. Math., 18 (1970): 783-791. Available from: https://doi.org/10.1137/0118071.
[23]	V. Lakshmikantham, S. Leela, Di fferential and Integral Inequalities; Academic Press: New York, NY, USA, 1969; Volume I.
[24]	T. G. Hallam, A comparison principle for terminal value problems in ordinary di fferential equations, Trans. Am. Math. Soc,. 169 (1972), 49-57. Available from: https://doi.org/10.1090/S0002-9947-1972-0306611-3. doi: 10.1090/S0002-9947-1972-0306611-3
[25]	J. V. D. C. Sousa, E. C. de Oliveira, On the ψ-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul,. 60 (2018), 72-91. Available from: https://doi.org/10.1016/j.cnsns.2018.01.005. doi: 10.1016/j.cnsns.2018.01.005
[26]	K. M. Furati, M. D. Kassim, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64 (2012), 1616-1626. Available from: https://doi.org/10.1016/j.camwa.2012.01.009. doi: 10.1016/j.camwa.2012.01.009
[27]	R. Almeida, A Caputo fractional derivative of a function with respect to another function, Communications Commun Nonlinear Sci Numer Simulat., 44 (2017), 460-481. Available from: https://doi.org/10.1016/j.cnsns.2016.09.006. doi: 10.1016/j.cnsns.2016.09.006
[28]	T. A. Burton, A fixed-point theorem of Krasnoselskii, Appl. Math. Lett., 11 (1998), 85-88.
[29]	A. G. Dugundji, J. Fixed Point Theory, Springer-Verlag: New York, NY, USA, 2003.
[30]	J. V. D. C. Sousa, E. C. de Oliveira, A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator, Differ. Equ. Appl,. 11 (2019), 87-106.

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