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http://dx.doi.org/10.5666/KMJ.2020.60.3.647

Global Existence and Ulam-Hyers Stability of Ψ-Hilfer Fractional Differential Equations  

Kucche, Kishor Deoman (Department of Mathematics, Shivaji University)
Kharade, Jyoti Pramod (Department of Mathematics, Shivaji University)
Publication Information
Kyungpook Mathematical Journal / v.60, no.3, 2020 , pp. 647-671 More about this Journal
Abstract
In this paper, we consider the Cauchy-type problem for a nonlinear differential equation involving a Ψ-Hilfer fractional derivative and prove the existence and uniqueness of solutions in the weighted space of functions. The Ulam-Hyers and Ulam-Hyers-Rassias stabilities of the Cauchy-type problem is investigated via the successive approximation method. Further, we investigate the dependence of solutions on the initial conditions and their uniqueness using 𝜖-approximated solutions. Finally, we present examples to illustrate our main results.
Keywords
${\Psi}$-Hilfer fractional derivative; existence and uniqueness; Ulam-Hyers stability; successive approximations; ${\epsilon}$-approximate solution; dependence of solution;
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1 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. Difference Equ., (2017), Paper No. 180, 14 pp.
2 S. Abbas, M. Benchohra and A. Petrusel, Ulam stability for Hilfer type fractional differential inclusions via the weakly Picard operators theory, Fract. Calc. Appl. Anal., 20(2017), 384-398.   DOI
3 M. Benchohra and J. E. Lazreg, Existence and Ulam stability for nonlinear implicit fractional differential equations with Hadamard derivative, Stud. Univ. Babes-Bolyai Math., 62(1)(2017), 27-38.   DOI
4 E. Capelas de Oliveira and J. Vanterler da C. Sousa, Ulam-Hyers-Rassias stability for a class of fractional integro-differential equations, Results Math., 73(2018), Paper No. 111, 16 pp.
5 K. M. Furati, M. D. Kassim and N. E. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64(2012), 1616-1626.   DOI
6 K. M. Furati, M. D. Kassim and N. E. Tatar, Non-existence of global solutions for a differential equation involving Hilfer fractional derivative, Electron. J. Differential Equations, (2013), No. 235, 10 pp.
7 R. Hilfer, Applications of fractional calculus in Physics, World Scientific, Singapore, 2000.
8 J. Huang and Y. Li, Hyers-Ulam stability of delay differential equations of first order, Math. Nachr., 289(1)(2016), 60-66.   DOI
9 K. D. Kucche and S. T. Sutar, On existence and stability results for nonlinear fractional delay differential equations, Bol. Soc. Parana. Mat., 36(2018), 55-75.   DOI
10 I. A. Rus, Ulam stability of ordinary differential equations, Stud. Univ. Babes-Bolyai Math., 54(4)(2009), 125-133.
11 S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives: theory and applications, Gordon and Breach, Amsterdam, 1983.
12 A. H. Siddiqi, Functional analysis with applications, Tata McGraw-Hill Publishing Ltd, New Delhi, 1986.
13 S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publishers, New York, 1960.
14 J. Vanterler da C. Sousa and E. Capelas de. Oliveira, On the $\psi$-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60(2018), 72-91.   DOI
15 J. Vanterler da C. Sousa and E. Capelas de Oliveira, A Gronwall inequality and the Cauchy-type problem by means of $\Psi$-Hilfer operator, Diff. Equ. Appl., 11(2019), 87-106.
16 J. Vanterler da C. Sousa and E. Capelas de Oliveira, On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equations using the $\Psi$-Hilfer operator, J. Fixed Point Theory Appl., 20(3)(2018), Paper No. 96, 21 pp.   DOI
17 J. Wang, L. Lv and Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electronic Electron. J. Qual. Theory Differ. Equ., (2011), No. 63, 10 pp.
18 A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies 204, Elsevier Science B. V., Amsterdam, 2006.