Existence and stability results for fractional differential equations involving generalized Katugampola derivative

Sandeep P Bhairat

Abstract


Present article deals with 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.

Keywords


Fractional differential equations; Fixed point theory; Stability of solutions.

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References

S. Abbas, M. Benchohra, G. M. N'Guerekata, Topics in fractional differential equations, New York, Springer, (2012).

S. Abbas, M. Benchohra, J. E. Lagreg, A. Alsaedi and Y. Zhou, Existence and Ulam stability for fractional differential equations of Hilfer-Hadamard type, Adv. Diff. Equat., 180, 2017.

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

B.Ahmad, S.K.Ntouyas, Initial value problem of fractional order Hadamard-type functional differential equations, Electron J Differ Equ., 77, 1{9, 2015.

Y.Adjabi, F.Jarad, D.Baleanu, and T.Abdeljawad, On Cauchy problems with Caputo-Hadamard fractional derivatives, J. Comp. Anal. App., 21 (1), 661-681, 2016.

R. Almeida, A. B. Malinowska and T. Odzijewicz, Fractional differential equations with dependence on the Caputo-Katugampola derivative, J. Comput. Nonlinear Dynam., 11 (6), 11 pages, 2016.

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

C. Carduneanu, Integral equations and stability of feedback systems, New York, Academic Press, 1973.

Sandeep P. Bhairat and D. B. Dhaigude, Ulam stability for system of nonlinear implicit fractional differential equations, Progress in Nonlinear Dynamics and Chaos, 6 (1), 29-38, 2018.

Sandeep P. Bhairat, On stability of generalized Cauchy-type problem,

arXiv:1808.03079v1[math.CA], 9 Aug, 2018. 13 pages.

Sandeep P. Bhairat, New approach to existence of solution of weighted Cauchy-type problem, arXiv:1808.03067v1[math.CA], 9 Aug, 2018. 10 pages.

Sandeep P. Bhairat and D. B. Dhaigude, Existence of solution of generalized fractional differential equation with nonlocal initial conditions, Mathematica Bohemica (In Press), 15 pages, 2018.

C. P. Chitalkar-Dhaigude, Sandeep P. Bhairat and D. B. Dhaigude, Solution of fractional differential equations involving Hilfer fractional derivatives: Method of successive approximations, Bull. Marathwada Math. Soc., 18 (2) 2017, 1-13.

D. B. Dhaigude and Sandeep P. Bhairat, Existence and uniqueness of solution of Cauchy-type problem for Hilfer fractional differential equations, Communications in Applied Analysis, 22 (1), 121-134, 2018.

D. B. Dhaigude and Sandeep P. Bhairat, Existence and continuation of solution of Hilfer fractional differential equations, arXiv:1704.02462v1 [math.CA], 2017.

D. B. Dhaigude and Sandeep P. Bhairat, On existence and approximation of solution of Hilfer fractional differential equations, arXiv:1704.02464v2 [math.CA], 2017. (accepted in IJPAM).

D. B. Dhaigude and Sandeep P. Bhairat, Local existence and uniqueness of solution of Hilfer fractional differential equations, Nonlinear Dyn. Syst. Theory., 18 (2), 144-153, 2018.

J. Diaz, B. Margolis, A fixed point theorem as the alternative for contractions on a generalized complete metric space, Bull Amer Math Soc., 74 (2), 305-9, 1968.

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

K. M. Furati, M. D. Kassim, and N.-E. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comp. Math. Appl., 64 (6), 1616-1626, 2012.

A. Granas, J. Dugundji, Fixed point theory, Springer, New York, (2003).

S. M. Jung, Hyers-Ulam-Rassias stability of functional equations in mathematical analysis, Hadronic Press, Palm Harbor (2001).

M. D. Kassim, K. M. Furati and N.-E. Tatar, On a differential equation involving Hilfer-Hadamard fractional derivative, Abst. Appl. Anal., 17 pages, 2012.

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

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

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

A. A. Kilbas, H.M.Srivastava and J. J. Trujillo, Theory and applications of the fractional differential equations, 204. Elsevier, Amsterdam, (2006).

A. A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc., 38 (6), 1191{1204, 2001.

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

S. M. Ulam, A collection of mathematical problems, Interscience, New York, (1968).

D Vivek, K Kanagrajan and E M Elsayed, Some existence and stability results for Hilfer fractional implicit dierential equations with nonlocal conditions, Mediterr. J. Math., 15:15, 2018. DOI: 10.1007/s00009-017-1061-0.

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




DOI: http://dx.doi.org/10.24193/subbmath.2020.1.03

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