Existence, Uniqueness and Ulam-Hyers-Rassias Stability of Differential Coupled Systems with Riesz-Caputo Fractional Derivative

Salim, Abdelkrim; Lazreg, Jamal Eddine; Benchohra, Mouffak

doi:10.2478/tmmp-2023-0019

Abstract

This article deals with the existence, uniqueness and Ulam-Hyers--Rassias stability results for a class of coupled systems for implicit fractional differential equations with Riesz-Caputo fractional derivative and boundary conditions. We will employ the Banach’s contraction principle as well as Schauder’s fixed point theorem to demonstrate our existence results. We provide an example to illustrate the obtained results.

References

ABBAS, S.—BENCHOHRA, M.—GRAEF, J. R.—HENDERSON, J.: Implicit Differential and Integral Equations: Existence and stability. In: De Gruyter Series in Nonlinear Analysis and Applications, Vol. 26. Walter de Gruyter, Berlin, 2018.
Search in Google Scholar Back to article
ABBAS, S.—BENCHOHRA, M.—N’GUÉRÉKATA, G. M.: Topics in Fractional Differential Equations. In: Developments in Mathematics Vol. 27, Springer-Verlag, New York, 2012.
Search in Google Scholar Back to article
ABBAS, S.—BENCHOHRA, M.—N’GUÉRÉKATA, G M.: Advanced Fractional Differential and Integral Equations. Mathematics Research Developments. Nova Science Publishers, Inc., New York, 2015.
Search in Google Scholar Back to article
AHMAD, B.—ALSAEDI, A.—NTOUYAS, S.K.—TARIBOON, J.: Hadamard-type Fractional Differential Equations, Inclusions and Inequalities. Springer, Cham, 2017.
Search in Google Scholar Back to article
ALI, A.—SAMET, B.—SHAH, K.—KHAN, R. A.: Existence and stability of solution to a toppled systems of differential equations of non–integer order, Bound. Value Prob. (2017), Paper no. 16, 13 pp.
Search in Google Scholar Back to article
ALI, A.—SHAH, K.—JARAD, F.—GUPTA, V.—ABDELJAWAD, T.: Existence and stability analysis to a coupled system of implicit type impulsive boundary value problems of fractional-order differential equations, Adv. Difference Equ. 2019 (2019), Paper no. 101, 21 pp.
Search in Google Scholar Back to article
BALEANU, D.—GÜVENÇ, Z. B.—MACHADO, J. A. T., EDS.: New Trends in Nanotechnology and Fractional Calculus Applications. In: Selected papers based on the presentations at the workshop new trends in science and technology (NTST 08), and the workshop fractional differentiation and its applications (FDA 09), Ankara, Türkei, November 2008. Springer, Dordrecht, 2010.
Search in Google Scholar Back to article
BENKHETTOU, N.—SALIM, A.—AISSANI, K. —BENCHOHRA, M.—KARAPINAR, E.: Non-instantaneous impulsive fractional integro-differential equations with state--dependent delay, Sahand Commun. Math. Anal. 19 (2022), 93–109.
Search in Google Scholar Back to article
BOURIAH, S.—SALIM, A.—BENCHOHRA, M.: On nonlinear implicit neutral generalized Hilfer fractional differential equations with terminal conditions and delay, Topol. Algebra Appl. 10 (2022), 77–93.
Search in Google Scholar Back to article
CHEN, F.—BALEANU, D.—WU G.: Existence results of fractional differential equations with Riesz-Caputo derivative, Eur. Phys. J. 226 (2017), 3411–3425.
Search in Google Scholar Back to article
CHEN, F.—CHEN, A.—WU, X.: Anti-periodic boundary value problems with Riesz-Caputo derivative, Adv. Difference Equ. 2019 (2019), Paper no. 119, 13 pp.
Search in Google Scholar Back to article
DERBAZI, C.—HAMMOUCHE, H.—SALIM, A.—BENCHOHRA, M.: Measure of non-compactness and fractional hybrid differential equations with hybrid conditions, Differ. Equ. Appl. 14 (2022), 145–161.
Search in Google Scholar Back to article
GU, C. Y.—WU, G. C.: Positive solutions of fractional differential equations with the Riesz space derivative, Appl. Math. Lett. 95 (2019), 59–64.
Search in Google Scholar Back to article
HERIS, A.—SALIM, A.—BENCHOHRA, M.—KARAPINAR, E.: Fractional partial random differential equations with infinite delay, Results in Physics 37 (2022). https://doi.org/10.1016/j.rinp.2022.105557
Search in Google Scholar Back to article
HYERS,D.H.: On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222–224.
Search in Google Scholar Back to article
KHAN, A.—SHAH, K.—LI, Y.—KHAN, T. S.: Ulam type stability for a coupled systems of boundary value problems of nonlinear fractional differential equations, J. Funct. Spaces (2017), Art. ID 3046013, 8. pp.
Search in Google Scholar Back to article
KILBAS, A. A.—SRIVASTAVA, H. M.—TRUJILLO, J. J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Vol. 204, Elsevier Science B. V., Amsterdam, 2006.
Search in Google Scholar Back to article
KRIM, S.—SALIM, A.—ABBAS, S.—BENCHOHRA, M.: On implicit impulsive conformable fractional differential equations with infinite delay in b-metric spaces, Rend. Circ. Mat. Palermo Ser. 2 (2022), 1–14.
Search in Google Scholar Back to article
LAZREG, J. E.—BENCHOHRA, M.—SALIM, A.: Existence and Ulam stability of k--generalized ψ-Hilfer fractional problem, J. Innov. Appl. Math. Comput. Sci. 2 (2022), 1–13.
Search in Google Scholar Back to article
LI, M.—WANG, Y.: Existence and iteration of monotone positive solutions for fractional boundary value problems with Riesz-Caputo derivative, Engineering Letters 29 (2021), 327–331.
Search in Google Scholar Back to article
LUO, D.—LUO, Z.—QIU, H.: Existence and Hyers-Ulam stability of solutions for a mixed fractional-order nonlinear delay difference equation with parameters, Math. Probl. Eng. 2020 (2020), Art. ID 9372406, 12 pp.
Search in Google Scholar Back to article
MONCH, H.: Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal. 4 (1980), 985–999.
Search in Google Scholar Back to article
NAAS, A.—BENBACHIR, M.—ABDO, M. S.—BOUTIARA, A.: Analysis of a fractional boundary value problem involving Riesz-Caputo fractional derivative, Adv. Theory Nonlinear Anal. Appl. 6 (2022) no. 1, 14–27.
Search in Google Scholar Back to article
RASSIAS, T. M.: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 72 (1978), 297–300.
Search in Google Scholar Back to article
RUS, I.: Ulam stability of ordinary differential equations in a Banach space,Carpathian J. Math. 26 (2011), 103–107.
Search in Google Scholar Back to article
SALIM, A.—AHMAD, B. —BENCHOHRA, M.—LAZREG, J. E.: Boundary value problem for hybrid generalized Hilfer fractional differential equations, Differ. Equ.Appl. 14 (2022), 379–391.
Search in Google Scholar Back to article
SALIM, A.—BENCHOHRA, M.—GRAEF, J. R.—LAZREG, J. E.: Initial value problem for hybrid ψ-Hilfer fractional implicit differential equations, J. Fixed Point Theory Appl. 24 (2022), 14 pp.
Search in Google Scholar Back to article
SALIM, A.—BENCHOHRA, M.—LAZREG, J. E.: Nonlocal k-generalized ψ-Hilfer impulsive initial value problem with retarded and advanced arguments, Appl. Anal. Optim. 6 (2022), 21–47.
Search in Google Scholar Back to article
SALIM, A.—BENCHOHRA, M.—LAZREG, J. E.—HENDERSON, J.: On k-generalized ψ-Hilfer boundary value problems with retardation and anticipation, Adv. Theory of Nonlinear Anal. and Appl. 6 (2022), no. 2, 173–190.
Search in Google Scholar Back to article
SALIM, A.—BENCHOHRA, M.—LAZREG, J. E.—KARAPINAR, E.: On k-generalized ψ-Hilfer impulsive boundary value problem with retarded and advanced arguments, J. Math. Ext. 15 (2021), 1–39.
Search in Google Scholar Back to article
SALIM, A.—LAZREG, J. E.—AHMAD, B.—BENCHOHRA, M.—NIETO, J. J.: Astudy on k-generalized ψ-Hilfer derivative operator, Vietnam J. Math. (2022), 1–19.
Search in Google Scholar Back to article
SHAH, K.—TUNC, C.: Existence theory and stability analysis to a system of boundary value problem, J. Taibah Univ. Sci. 11 (2017), no. 6, 1330–1342
Search in Google Scholar Back to article
SMART, D. R.: Fixed Point Theory. Cambridge Univ. Press, Cambridge, 1974.
Search in Google Scholar Back to article
ULAM, S. M.: Problems in Modern Mathematics, Science Editions John Wiley & Sons, Inc., New York, 1964.
Search in Google Scholar Back to article
WANG, J.—ZADA, A.—WAHEED, H.: Stability analysis of a coupled system of nonlinear implicit fractional anti-periodic boundary value problem, Math Meth Appl Sci. 42 (2019), 6706–6732.
Search in Google Scholar Back to article

Existence, Uniqueness and Ulam-Hyers-Rassias Stability of Differential Coupled Systems with Riesz-Caputo Fractional Derivative

Abstract

Paradigm

My account