A study of fractional integro-differential equations via Hilfer-Hadamard fractional derivative

D. Vivek; K. Kanagarajan; E. M. Elsayed

doi:10.2478/gm-2019-0007

$A study of fractional integro-differential equations via Hilfer-Hadamard fractional derivative Cover$

.blurhash-client-img { display: none !important; }

A study of fractional integro-differential equations via Hilfer-Hadamard fractional derivative

General Mathematics

Volume 27 (2019): Issue 1 (June 2019)

By: D. Vivek, K. Kanagarajan and E. M. Elsayed

Open Access

|Dec 2019

Abstract

In this paper, we investigate the existence of solution of integro-differential equations (IDEs) with Hilfer-Hadamard fractional derivative. The main results are obtained by using Schaefer’s fixed point theorem. Some Ulam stability results are presented.

References

[1] S. Abbas, M. Benchohra, S. Sivasundaram, Dynamics and Ulam stability for Hilfer type fractional differential equations, Nonlinear Stud., vol. 23, no. 4, 2016, 627-637.
Search in Google Scholar Back to article
[2] S. Abbas, M. Benchohra, N. Hamidi, Hilfer and HilferHadamard Fractional Differential Equations with Random Effects, Libertas Mathematica (new series), vol. 37, 2017, 45-64.
Search in Google Scholar Back to article
[3] S. Abbas, M. Banchohra, J. E. Lazreg, J. J. Nieto, On a coupled system of Hilfer-Hadamard fractional differential equations in Banach spaces, J. Nonlinear Funct. Anal. 2018, 2018, Article ID 12.10.23952/jnfa.2018.12
Search in Google Scholar Back to article
[4] B. Ahmad, S. K. Ntouyas, Initial value problems for hybrid Hadamard fractional differential equations, Electron. J. Differential Equations, vol. 161, 2014, 1-8.
Search in Google Scholar Back to article
[5] B. Ahmad, S. Sivasundaram, Some existence results for fractional integrodifferential equations with nonlocal conditions, Commun. Appl. Anal., vol. 12, 2008, 107-112.
Search in Google Scholar Back to article
[6] B. Ahmad, J. J. Nieto, Riemann-Liouville fractional differential equations with fractional boundary conditions, Fixed point Theory, vol. 13, 2013, 329-336.
Search in Google Scholar Back to article
[7] S. Andras, J. J. Kolumban, On the Ulam-Hyers stability of first order differential systems with nonlocal initial conditions, Nonlinear Anal. Theory Methods Appl., vol. 82, 2013, 1-11.10.1016/j.na.2012.12.008
Search in Google Scholar Back to article
[8] K. Balachandran, S. Kiruthika, J. J. Trujillo, Existence results for fractional impulsive integrodifferential equations in Banach spaces, Commun Nonlinear Sci. Numer. Simul., vol. 16, 2011, 1970-1977.10.1016/j.cnsns.2010.08.005
Search in Google Scholar Back to article
[9] K. Balachandran, S. Kiruthika, J. Y. Park, Controllability of fractional integrodifferential systems in Banach spaces, Nonlinear Anal. Hybri., vol. 3, 2009, 363-367.10.1016/j.nahs.2009.01.014
Search in Google Scholar Back to article
[10] K. Balachandran, S. Kiruthika, Existence results for fractional integrodifferential equations with nonlocal condition via resolvent operators, Comput. Math. Appl., vol. 62, 2011, 1350-1358.10.1016/j.camwa.2011.05.001
Search in Google Scholar Back to article
[11] P. L. Butzer, A. A. Kilbas, J. J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl., vol. 269, 2002, 1-27.10.1016/S0022-247X(02)00049-5
Search in Google Scholar Back to article
[12] P. L. Butzer, A. A. Kilbas, J. J. Trujillo, Mellin transform analysis and integration by parts for Hadamard-type fractional integrals, J. Math. Anal. Appl., vol. 270, 2002, 1-15.10.1016/S0022-247X(02)00066-5
Search in Google Scholar Back to article
[13] K. M. Furati, M. D. Kassim, N. E. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., vol. 64, 2012, 1616-1626.10.1016/j.camwa.2012.01.009
Open DOI Search in Google Scholar Back to article
[14] K. M. Furati, M. D. Kassim, N. E. Tatar, Non-existence of global solutions for a differential equation involving Hilfer fractional derivative, Electron. J. Differential Equations, vol. 235, 2013, 1-10.
Search in Google Scholar Back to article
[15] H. Gu, J. J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput., vol. 257, 2014, 344-354.10.1016/j.amc.2014.10.083
Search in Google Scholar Back to article
[16] V. Gupta, J. Dabas, Existence of solution for fractional impulsive integrodifferential equation with integral boundary conditions, Func. Anal.-TMA, vol. 1, 2015, 56-68.
Search in Google Scholar Back to article
[17] V. Gupta, J. Dabas, Michal Feckan, Existence results of solutions for impulsive fractional differential equations, Nonauton. Dyn. Syst., vol. 5, 2018, 35-51.10.1515/msds-2018-0003
Search in Google Scholar Back to article
[18] R. W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Math., vol. 23, 2012, 1250056.10.1142/S0129167X12500565
Search in Google Scholar Back to article
[19] S. M. Jung, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett., vol. 17, 2004, 1135-1140.10.1016/j.aml.2003.11.004
Open DOI Search in Google Scholar Back to article
[20] R. Kamocki, C. Obcznnski, On fractional Cauchy-type problems containing Hilfer derivative, Electron J. Qual. Theory Differ. Equ., vol. 50, 2016, 1-12.10.14232/ejqtde.2016.1.50
Search in Google Scholar Back to article
[21] K. Karthikeyan, B. Ahmad, Existence results for boundary value problems of arbitrary order integrodifferential equations in Banach spaces, An. St. Univ. Ovidius Constant a, vol. 21, no. 2, 2013, 155-171, DOI: 10.2478/auom-2013-0029.10.2478/auom-2013-0029
Open DOI Search in Google Scholar Back to article
[22] K. Karthikeyan, J. J. Trujillo, Existence and uniqueness results for fractional integrodifferential equations with boundary value conditions, Commun. Nonlinear Sci. Numer. Simul., vol. 17, 2012, 4037-4043.10.1016/j.cnsns.2011.11.036
Open DOI Search in Google Scholar Back to article
[23] M. D. Kassim, N. E. Tatar, Well-psedness and stability for a differential problem with Hilfer-Hadamard fractional derivative, Abstr. Appl. Anal., vol. 1, 2013, 1-12.10.1155/2013/605029
Search in Google Scholar Back to article
[24] P. Muniyappan, S. Rajan, Hyers-Ulam-Rassias stability of fractional differential equation, Int. J. Pure Appl. Math., vol. 102, 2015, 631-642.10.12732/ijpam.v102i4.4
Search in Google Scholar Back to article
[25] Y. Rian, S. Shurong, S. Ying, H. Zhenlai, Boundary value problems for fractional differential equations with nonlocal boundary conditions, Adv. Differ. Equ., vol. 176, 2013.10.1186/1687-1847-2013-176
Search in Google Scholar Back to article
[26] I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math., vol. 26, 2010, 103-107.
Search in Google Scholar Back to article
[27] R. Hilfer, Application of fractional Calculus in Physics, World Scientific, Singapore, 1999.10.1142/3779
Search in Google Scholar Back to article
[28] R. Hilfer, Y. Luchko, Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Lioville fractional derivative, Fract. Calc. Appl. Anal., vol. 12, 2009, 289-318.
Search in Google Scholar Back to article
[29] J. R. Wang, Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput., vol. 266, 2015, 850-859.10.1016/j.amc.2015.05.144
Search in Google Scholar Back to article
[30] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in: Mathematics Studies, Elsevier, vol. 204, 2006.
Search in Google Scholar Back to article
[31] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, Engl. Trans.from the Russian, 1987.
Search in Google Scholar Back to article
[32] I. Podlubny, Fractional Differential Equations, in: Mathematics in Science and Engineering, Acad. Press, vol. 198, 1999.
Search in Google Scholar Back to article
[33] D. Vivek, K. Kanagarajan, S. Sivasundaram, Dynamics and stability of panto-graph equations via Hilfer fractional derivative, Nonlinear Stud., vol. 23, no. 4, 2016, 685-698.
Search in Google Scholar Back to article
[34] J. Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron J. Qual. Theory Differ. Equ., vol. 63, 2011, 1-10.10.14232/ejqtde.2011.1.63
Search in Google Scholar Back to article
[35] J. Wang, Y. Zhou, New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simulat., vol. 17, 2012, 2530-2538.10.1016/j.cnsns.2011.09.030
Open DOI Search in Google Scholar Back to article
[36] J. Wang, Y. Zhou, M. Medved, Existence and stability of fractional differential equations with Hadamard derivative, Topol. Methods Nonlinear Anal., vol. 41, no. 1, 2013, 113-133.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/gm-2019-0007 | Journal eISSN: 1584-3289 | Journal ISSN: 1221-5023

Journal RSS Feed

Language: English

Page range: 71 - 84

Submitted on: Oct 1, 2017

Accepted on: May 8, 2019

Published on: Dec 21, 2019

Published by: Lucian Blaga University of Sibiu

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

Fractional integro-differential equations,

Hadamard fractional derivative,

Hilfer fractional derivative,

Solution,

Ulam-Hyers stablity

Related subjects:

Mathematics,

Differential equations and dynamical systems,

Probability and statistics,

Numerical and computational mathematics,

Applied mathematics

© 2019 D. Vivek, K. Kanagarajan, E. M. Elsayed, published by Lucian Blaga University of Sibiu
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 27 (2019): Issue 1 (June 2019)