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http://dx.doi.org/10.22771/nfaa.2021.26.01.14

QUALITATIVE ANALYSIS OF A PROPORTIONAL CAPUTO FRACTIONAL PANTOGRAPH DIFFERENTIAL EQUATION WITH MIXED NONLOCAL CONDITIONS  

Khaminsou, Bounmy (Department of Mathematics, Faculty of Science, Burapha University)
Thaiprayoon, Chatthai (Department of Mathematics, Faculty of Science, Burapha University)
Sudsutad, Weerawat (Department of General Education, Faculty of Science and Health Technology Navamindradhiraj University)
Jose, Sayooj Aby (Ramanujan Centre for Higher Mathematics, Alagappa University)
Publication Information
Nonlinear Functional Analysis and Applications / v.26, no.1, 2021 , pp. 197-223 More about this Journal
Abstract
In this paper, we investigate existence, uniqueness and four different types of Ulam's stability, that is, Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability of the solution for a class of nonlinear fractional Pantograph differential equation in term of a proportional Caputo fractional derivative with mixed nonlocal conditions. We construct sufficient conditions for the existence and uniqueness of solutions by utilizing well-known classical fixed point theorems such as Banach contraction principle, Leray-Schauder nonlinear alternative and $Krasnosel^{\prime}ski{\breve{i}}{^{\prime}}s$ fixed point theorem. Finally, two examples are also given to point out the applicability of our main results.
Keywords
Existence; uniqueness; fixed point theorem; generalized proportional fractional derivative; nonlocal boundary condition; Ulam-Hyers stability;
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1 K. Liu, M. Feckan, D. O'Regan and J. Wang, Hyers-Ulam stability and existence of solutions for differential equations with Caputo-Fabrizio fractional derivative, Mathematics, 2019, 7, 333.   DOI
2 K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, NewYork, 1993.
3 T. Miura, Go Hirasawa and Sin-El Takahasi, Note on the Hyers-Ulam-Rassias stability of the first order linear differential equation, Nonlinear Funct. Anal. Appl., 11(5) (2006), 851-858.
4 M. Obloza, Hyers stability of the linear differential equation, Rocznik Nauk-Dydakt. Prace Mat., 13 (1993), 259-270.
5 I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
6 G. Rahman, A. Khan, T. Abdeljawad and K.S. Nisar, The Minkowski inequalities via generalized proportional fractional integral operators, Adv. Differ. Equ. 2019(287) (2019).
7 T.M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300.   DOI
8 T.M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl., 2003(158) (2003), 106-113.
9 I.A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math., 26 (2010), 103-107.
10 S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, 1993.
11 W. Shammakh and H.Z. Alzumi, Existence results for nonlinear boundary value problem involving generalized proportional derivative, Adv. Differ. Equ., (2019), 2019:94.   DOI
12 W. Sudsutad, J. Alzabut, C. Tearnbucha and C. Thaiprayoon, On the oscillation of differential equations in frame of generalized proportional fractional derivatives, AIMS Mathematics, 5(2) (2020), 856-871, doi:10.3934/math.2020058.   DOI
13 S.M. Ulam, Problem in Modern Mathematics, John Wiley and Sons, New York, NY, USA, 1940.
14 S.M. Ulam, A collection of mathematical Problems, Interscience, New York, NY, USA, 1968.
15 J. Vanterler da C. Sousa, and E. Capelas de Oliveira, On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equations using the ψ-Hilfer operator. J. Fixed Point Theory Appl., 2018, 20, 96.   DOI
16 T. Abdeljawad, F. Jarad, S. Mallak and J. Alzabut, Lyapunov type inequalities via fractional proportional derivatives and application on the free zero disc of Kilbas-Saigo generalized Mittag- Leffler functions, European Physical J. Plus, (2019), 134:247.   DOI
17 J. Wang, L. Lv and Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 2011(63) (2011), 1-10.
18 J. Wang, Y. Zhou and M. Medved, Existence and stability of fractional differential equations with Hadamard derivative, Topol. Methods Nonlinear Anal., 41 (2013), 113-133.
19 Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.
20 S. Abbsa, M. Benchohra, J. Lagreg, A. Alsaedi and Y. Zhou, Existence and Ulam stability results for fractional differential equations of Hilfer-Hadamard type, Adv. Differ. Equ., 2017(180) (2017).
21 M. Adam,, On the Hyers-Ulam-Rassias stability of a generalized quadratic and additive functional equation Nonlinear Funct. Anal. Appl., 14(5) (2009), 699-705.
22 B. Ahmad, A. Alsaedi, S.K. Ntouyas and J. Tariboon, Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, Springer International Publishing AG, 2017.
23 B. Ahmad, M. M. Matar and O. El-Salmy, Existence of solutions and Ulam stability for Caputo type sequential fractional differential equations of order α ∈ (2, 3), Inter. J. Anal. Appl., 15(1) (2017), 86-101.
24 I. Ahmed, P. Kuman, K. Shah, P. Borisut, K. Sitthithakerngkiet and M.A. Demba, Stability results for implicit fractional Pantograph differential equations via ϕ-Hilfer fractional derivative with a nonlocal Riemann-Liouville fractional integral condition. Mathematics, 2020, 8, 94; doi:10.3390/math8010094.   DOI
25 J. Alzabut, T. Abdeljawad, F. Jarad and W. Sudsutad, A Gronwall inequality via the generalized proportional fractional derivative with applications, J. Inequ. Appl., (2019) 2019:101.   DOI
26 J. Alzabut, W. Sudsutad, Z. Kayar and H. Baghani, A new Gronwall-Bellman inequality in a frame of generalized proportional fractional proportional derivative, Mathematics, 2019, 7, 747, 1-15,; doi:10.3390/math7080747.   DOI
27 S. Asawasamrit, W. Nithiarayaphaks, S.K. Ntouyas and J. Tariboon, Existence and stability analysis for fractional differential equations with mixed nonlocal conditions, Mathematics, 2019, 7, 117.   DOI
28 D.R. Anderson, Second-order self-adjoint differential equations using a proportional-derivative controller, Commun. Appl. Nonlinear Anal., 24 (2017), 17-48.
29 T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66.   DOI
30 A. Aphithana, S.K. Ntouyas and J. Tariboon, Existence and Ulam-Hyers stability for Caputo conformable differential equations with four-point integral conditions, Adv. Differ. Equ., 2019, 2019, 139.   DOI
31 K. Balachandran and K. Uchiyama, Existence of solutions of nonlinear integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces, Proc. Indian Acad. Sci. (Math. Sci.) 110 (2000), 225-232.   DOI
32 M. Benchohra and S. Bouriah, Existence and stability results for nonlinear boundary value problem for implicit differential equations of fractional order, Moroccan J. Pure Appl. Anal., 1(1) (2015), 22-36.   DOI
33 A.V. Bitsadze and A.A. Samarskii, Some elementary generalizations of linear elliptic boundary value problems, Soviet Math. Dokl. 10 (1969), 398-400.
34 L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162 (1991), 494-505.   DOI
35 E. Capelas de Oliveira and J. Vanterler da C. Sousa, Ulam-Hyers-Rassias stability for a class of fractional integro-differential equations, Results Math., 2018, 73, 111.   DOI
36 K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer, New York, 2010.
37 M. Eshaghi Gordji and F. Habibian, Hyers-Ulam-Rassias stability of quadratic derivations on Banach algebras, Nonlinear Funct. Anal. Appl., 14(5) (2009), 759-766.
38 E. Hilb, Zur Theorie der Entwicklungen willkurlicher Funktionen nach Eigenfunktionen, Math. Z., 58 (1918), 1-9.   DOI
39 A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, USA 2003.
40 S. Harikrishman, E. Elsayed and K. Kanagarajan, Existence and uniqueness results for fractional Pantograph equations involving ψ-Hilfer fractional derivative. Dyn. Contin. Discret. Impuls. Syst., 2018(25) (2018), 319-328.
41 D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA,, 27(4) (1941), 222-224.   DOI
42 J. Jarad, T. Abdeljawad and J. Alzabut, Generalized fractional derivatives generated by a class of local proportional derivatives, Eur. Phys. J. Spec. Top., 226 (2017), 3457-3471.   DOI
43 B. Khaminsou, C. Thaiprayoon, J. Alzabut and W. Sudsutad, Nonlocal boundary value problems for integro-differential Langevin equation via the generalized Caputo proportional fractional derivative, Bound. Value Prob., (2020), 2020:176.   DOI
44 A. Khan, M.I. Syam, A. Zada and H. Khan, Stability analysis of nonlinear fractional differential equations with Caputo and Riemann-Liouville derivatives, Eur. Phys. J. Plus, 2018 (2018) 133: 264.
45 A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of the Fractional Differential Equations, North-Holland Mathematics Studies, 204, (2006).
46 M.A. Krasnoselskii, Two remarks on the method of successive approximations. Usp. Mat. Nauk 10 (1955), 123-127.
47 V. Lakshmikantham, S. Leela and J.V. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, 2009.