Wafaa Rahou, Abdelkrim Salim, Mouffak Benchohra, Jamal Eddine Lazreg

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In this paper, we investigate the existence and Ulam stability results for a class of boundary value problems for implicit Riesz-Caputo fractional differential equations with non-instantaneous impulses involving both retarded and advanced arguments. The result are based on Monch fixed point theorem associated with the technique of measure of noncompactness. An illustrative example is given to validate our main results.


Riesz-Caputo fractional derivative, existence, measure of noncompactness, xed point, Ulam stability, non-instantaneous impulses, retarded arguments, delay, implicit, anticipation

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bibitem{8} {sc S. Abbas, M. Benchohra, J.R. Graef {rm and} J. Henderson}: textit{ Implicit Differential and Integral Equations: Existence and stability}. Walter de Gruyter, London, 2018.

bibitem{9} {sc S. Abbas, M. Benchohra {rm and} G M. N'Gu'{e}r'{e}kata}: textit{ Topics in Fractional Differential Equations}. Springer-Verlag, New York, 2012.

bibitem{10} {sc S. Abbas, M. Benchohra {rm and} G M. N'Gu'{e}r'{e}kata}: textit{Advanced Fractional Differential and Integral Equations}. Nova Science Publishers, New York, 2014.

bibitem{11} {sc B. Ahmad, A. Alsaedi, S.K. Ntouyas {rm and} J. Tariboon}: textit{Hadamard-type Fractional Differential Equations, Inclusions and Inequalities}. Springer, Cham, 2017.

bibitem{Appell} {sc J. Appell}: textit{Implicit Functions, Nonlinear Integral Equations, and the Measure of Noncompactness of the superposition Operator}. { J. Math. Anal. Appl.} {bf 83}, (1981), 251--263.

bibitem{BaNi} {sc L. Bai, J. J. Nieto}: textit{Variational approach to differential equations with not instantaneous impulses}, { Appl. Math. Lett.} {bf 73} (2017), 44--48.

bibitem{2} {sc J. Banas {rm and} K. Goebel}: textit{ Measures of noncompactness in Banach spaces}. Marcel Dekker, New York, 1980.

bibitem{Praveen2} {sc A. Benkerrouche, M. S. Souid, S. Etemad, A. Hakem, P. Agarwal, S. Rezapour, S.K. Ntouyas, J. Tariboon}: textit{Qualitative study on solutions of a Hadamard variable order boundary problem via the Ulam–Hyers–Rassias stability}. { Fractal Fract.} {bf 2021} (2021), 5, 108. https://doi.org/10.3390/fractalfract5030108

bibitem{13} {sc F. Chen, D. Baleanu, {rm and} G. Wu}: textit{Existence results of fractional differential equations with Riesz-Caputo derivative}, { Eur. Phys. J.} {bf226} (2017), 3411--3425.

bibitem{14} {sc F. Chen, A. Chen, {rm and} X. Wu}: textit{Anti-periodic boundary value problems with Riesz-Caputo derivative}, { Adv. Difference Equ.} {bf 2019} (2019). https://doi.org/10.1186/s13662-019-2001-z

bibitem{16} {sc C. Y. Gu, G. C. Wu}: textit{Positive solutions of fractional differential equations with the Riesz space derivative}. { Appl. Math. Lett.} {bf 95} (2019), 59--64.

bibitem{HeOr} {sc E. Hern`{a}ndez, D. O'Regan}: textit{On a new class of abstract impulsive differential equations}, { Proc. Amer. Math. Soc}. {bf 141} (2013), no. 5, 1641--1649.

bibitem{Hye} {sc D. H. Hyers}: textit{On the stability of the linear functional equation}, { Proc. Nat. Acad. Sci. U. S. A.} {bf 27} (1941), 222--224.

bibitem{1} {sc A. A. Kilbas, H. M. Srivastava, {rm and} J. J. Trujillo}: textit{ Theory and Applications of Fractional Differential Equations}. North-Holland Mathematics Studies, Amsterdam, 2006.

bibitem{LaBeSa} {sc J. E. Lazreg, M. Benchohra {rm and} A. Salim}: textit{Existence and Ulam stability of ${psi}$-Generalized $psi$-Hilfer Fractional Problem}. { J. Innov. Appl. Math. Comput. Sci.} {bf 2} (2022), 1--13.

bibitem{LiWang} {sc M. Li {rm and} Y. Wang}: textit{Existence and iteration of monotone positive solutions for fractional boundary value problems with Riesz-Caputo derivative}. { Engineering Letters.} {bf 29} (2021), 1--5.

bibitem{LuLuQi} {sc D. Luo, Z. Luo, H. Qiu}: textit{Existence and Hyers-Ulam stability of solutions for a mixed fractional-order nonlinear delay

difference equation with parameters}. { Math. Probl. Eng.} {bf 2020}, 9372406 (2020).

bibitem{5} {sc H. Monch}: textit{BVP for nonlinear ordinary differential equations of second order in Banach spaces}. textit{ Nonlinear Anal}. {bf 4} (1980), 985--999.

bibitem{NassBour1} {sc A. Naas, M. Benbachir, M. S. Abdo {rm and} A. Boutiara}: textit{Analysis of a fractional boundary value problem involving Riesz-Caputo fractional derivative.} { ATNAA.} {bf 1} (2022), 14--27.

bibitem{Praveen3} {sc G. Rajchakit, R. Sriraman, N. Boonsatit textit{et al.}}: textit{Global exponential stability of Clifford-valued neural networks with time-varying delays and impulsive effects}. { Adv Differ Equ.} {bf 2021} (2021). https://doi.org/10.1186/s13662-021-03367-z

bibitem{Praveen4} {sc G. Rajchakit, R. Sriraman, N. Boonsatit textit{et al.}}: textit{Exponential stability in the Lagrange sense for Clifford-valued recurrent neural networks with time delays}. { Adv Differ Equ.} {bf 2021} (2021). https://doi.org/10.1186/s13662-021-03415-8

bibitem{Ras} {sc T. M. Rassias}: textit{On the stability of the linear mapping in Banach spaces}. { Proc. Amer. Math. Soc.} {bf 72} (1978), 297--300.

bibitem{6} {sc I. Rus}: textit{Ulam stability of ordinary differential equations in a Banach space}. {Carpathian J. Math.} {bf26} (2011), 103--107.

bibitem{SaBaBeLa} {sc A. Salim, B. Ahmad, M. Benchohra {rm and} J. E. Lazreg}: textit{Boundary value problem for hybrid generalized Hilfer fractional differential equations}. { Differ. Equ. Appl.} {bf14} (2022), 379--391. {http://dx.doi.org/10.7153/dea-2022-14-27}

bibitem{SaBeGrLa} {sc A. Salim, M. Benchohra, J. R. Graef {rm and} J. E. Lazreg}: textit{Boundary value problem for fractional generalized Hilfer-type fractional derivative with non-instantaneous impulses}. { Fractal Fract.} {bf5} (2021), 1--21. {https://dx.doi.org/10.3390/fractalfract5010001}

bibitem{SaBeKaLa} {sc A. Salim, M. Benchohra, E. Karapinar, J. E. Lazreg}: textit{Existence and Ulam stability for impulsive generalized Hilfer-type

fractional differential equations}. { Adv. Difference Equ.} {bf 2020}, 601 (2020).

bibitem{SaBeLaHe} {sc A. Salim, M. Benchohra, J. E. Lazreg {rm and} J. Henderson}: textit{Nonlinear implicit generalized Hilfer-type fractional differential equations with non-instantaneous impulses in Banach spaces}. { Adv. Theo. Nonli. Anal. Appl.} {bf 4} (2020), 332--348. {https://doi.org/10.31197/atnaa.825294}

bibitem{SaLaAhBeNi} {sc A. Salim, J. E. Lazreg, B. Ahmad, M. Benchohra {rm and} J. J. Nieto}: textit{A study on ${jmath}$-generalized $psi$-Hilfer derivative operator}. { Vietnam J. Math.} (2022). {https://doi.org/10.1007/s10013-022-00561-8}

bibitem{4} {sc D. R. Smart}: textit{Fixed point theory}, Combridge Uni. Press, Combridge, 1974.

bibitem{Ula} {sc S. M. Ulam}: {textit Problems in Modern Mathematics}, Science Editions John Wiley & Sons, Inc., New York, 1964.

bibitem{Praveen1} {sc B. Unyong, V. Govindan, S. Bowmiya, G. Rajchakit, N. Gunasekaran, R. Vadivel, C. P. Lim, P. Agarwal}: textit{Generalized linear differential equation using Hyers-Ulam stability approach}. { AIMS Math.} {bf 6} (2021), 1607-1623. https://doi.org/10.3934/math.2021096

bibitem{WaFe} {sc J. R. Wang, M. Feckan}: {textit Non-Instantaneous Impulsive Differential Equations.} Basic Theory And Computation, IOP Publishing Ltd. Bristol, UK, 2018.

bibitem{WaFe2} {sc J. Wang, L. Lv, Y. Zhou}: textit{Ulam stability and data dependence for fractional differential equations with Caputo derivative}. { Electr. J. Qual. Theory Differ. Equ.} {bf 63} (2011), 1--10.

bibitem{YaWa} {sc D. Yang, J. Wang}: textit{Integral boundary value problems fornonlinear non-instataneous impulsive differential equations}. { J. Appl. Math. Comput.} {bf 55} (2017), 59--78.

bibitem{ZaSh} {sc A. Zada, S. Shah}: textit{Hyers-Ulam stability of first-order non-linear delay differential equations with fractional

integrable impulses}. { J. Math. Stat.} {bf 47} (2018), 1196--1205.

DOI: https://doi.org/10.22190/FUMI221202035R


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