Existence and stability of fractional differential equations involving generalized Katugampola derivative

Authors

  • Sandeep P. BHAIRAT “Institute of Chemical Technology” Mumbai, Faculty of Engineering Mathematics, Marathwada Campus, Jalna, (M.S) - 431 203, India, e-mail: sp.bhairat@marj.ictmumbai.edu.in https://orcid.org/0000-0002-8363-9621

DOI:

https://doi.org/10.24193/subbmath.2020.1.03

Keywords:

Fractional differential equations, fixed point theory, stability of solutions.

Abstract

The present article deals with the existence and stability results for a class of fractional differential equations involving generalized Katugampola derivative. Some fixed point theorems are used to obtain the results and enlightening examples of obtained result are also given.

Mathematics Subject Classification (2010): 26A33, 34K37, 35B35.

References

Abbas, S., Benchohra, M., N’Guerekata, G.M., Topics in Fractional Differential Equations, New York, Springer, 2012.

Abbas, S., Benchohra, M., Lagreg, J.E., Alsaedi, A., Zhou, Y., Existence and Ulam stability for fractional differential equations of Hilfer-Hadamard type, Adv. Difference Equat., 180(2017), no. 1, 14 pages, DOI: 10.1186/s13662-017-1231-1.

Abbas, S., Benchohra, M., Lagreg, J.E., Zhou, Y., A survey on Hadamard and Hilfer fractional differential equations: Analysis and stability, Chaos Solitons Fractals, 102(2017), 47-71.

Adjabi, Y., Jarad, F., Baleanu, D., Abdeljawad, T., On Cauchy problems with Caputo- Hadamard fractional derivatives, J. Comp. Anal. Appl., 21(2016), no. 1, 661-681.

Ahmad, B., Ntouyas, S.K., Initial value problem of fractional order Hadamard-type functional differential equations, Electron. J. Differential Equations, 77(2015), 1-9.

Almeida, R., Variational problems involving a Caputo-type fractional derivative, J. Optim. Theory Appl., 174(2017), 276-294.

Almeida, R., Malinowska, A.B., Odzijewicz, T., Fractional differential equations with dependence on the Caputo-Katugampola derivative, J. Comput. Nonlinear Dynam., 11(2016), no. 6, 11 pages.

Anderson, D.R., Ulness, D.J., Properties of the Katugampola fractional derivative with potential application in quantum mechanics, J. Math. Phys., 56(2015), 18 pages.

Bhairat, S.P., On stability of generalized Cauchy-type problem, Special issue: RAMSA-17, DCDIS: Series A - Mathematical Analysis (In Press), 9 pages. arXiv:1808.03079v1[math.CA], 2018.

Bhairat, S.P., New approach to existence of solution of weighted Cauchy-type problem, arXiv:1808.03067v1[math.CA], 10 pages, 2018.

Bhairat, S.P., Existence and continuation of solution of Hilfer fractional differential equations, J. Math. Modeling., 7(2019), no. 1, 1-20, DOI: 10.22124/jmm.2018.9220.1136.

Bhairat, S.P., Dhaigude, D.B., Ulam stability for system of nonlinear implicit fractional differential equations, Progress in Nonlinear Dynamics and Chaos, 6(2018), no. 1, 29-38.

Bhairat, S.P., Dhaigude, D.B., Existence of solution of generalized fractional differential equation with nonlocal initial conditions, Mathematica Bohemica, 144(2019), no. 1, 1-18.

Carduneanu, C., Integral Equations and Stability of Feedback Systems, New York, Academic Press, 1973.

Chitalkar-Dhaigude, C.P., Bhairat, S.P., Dhaigude, D.B., Solution of fractional differential equations involving Hilfer fractional derivatives: Method of successive approximations, Bull. Marathwada Math. Soc., 18(2017), no. 2, 1-13.

Dhaigude, D.B., Bhairat, S.P., Existence and uniqueness of solution of Cauchy-type problem for Hilfer fractional differential equations, Commun. Appl. Anal., 22(2018), no. 1, 121-134.

Dhaigude, D.B., Bhairat, S.P., On existence and approximation of solution of Hilfer fractional differential equations, (Under review) arXiv:1704.02464v2 [math.CA], 2018.

Dhaigude, D.B., Bhairat, S.P., Local existence and uniqueness of solution of Hilfer fractional differential equations, Nonlinear Dyn. Syst. Theory., 18(2018), no. 2, 144-153.

Diaz, J., Margolis, B., A fixed point theorem as the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., 74(1968), no. 2, 305-309.

Furati, K.M., Kassim, M.D., Tatar, N.-E., Existence and uniqueness for a problem in- volving Hilfer fractional derivative, Computers Math. Appl., 64(2012), no. 6, 1616-1626.

Granas, A., Dugundji, J., Fixed Point Theory, Springer, New York, 2003.

Jung, S.M., Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.

Kassim, M.D., Furati, K.M., Tatar, N.-E., On a differential equation involving Hilfer-Hadamard fractional derivative, Abstr. Appl. Anal., (2012), 17 pages, DOI: 10.1155/2012/391062.

Katugampola, U.N., New approach to a generalized fractional integral, Appl. Math. Comput., 218(2011), 860-865.

Katugampola, U.N., A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6(2014), 1-15.

Katugampola, U.N., Existence and uniqueness results for a class of generalized fractional differenital equations, eprint arXiv:1411.5229v2 [math.CA], 2016.

Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of the Fractional Differential Equations, 204. Elsevier, Amsterdam, 2006.

Kilbas, A.A., Hadamard-type fractional calculus, J. Korean Math. Soc., 38(2001), no.6, 1191-1204.

Oliveira, D.S., Oliveira, E.C., Hilfer-Katugampola fractional derivative, eprint arXiv:1705.07733v1 [math.CA], 2017.

Ulam, S.M., A Collection of Mathematical Problems, Interscience, New York, 1968. [31] Vivek, D., Kanagrajan, K., Elsayed, E.M., Some existence and stability results for Hilfer fractional implicit differential equations with nonlocal conditions, Mediterr. J. Math., 15:15(2018), DOI: 10.1007/s00009-017-1061-0.

Wang, J., Lu, L., Zhou, Y., New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci., 17(2012), 2530-2538.

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Published

2020-03-06

How to Cite

BHAIRAT, S. P. (2020). Existence and stability of fractional differential equations involving generalized Katugampola derivative. Studia Universitatis Babeș-Bolyai Mathematica, 65(1), 29–46. https://doi.org/10.24193/subbmath.2020.1.03

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