Document Type : Research Paper
Authors
1 Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes, P.O. Box 89 Sidi Bel Abbes 22000, Algeria.
2 Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes, P.O. Box 89 Sidi Bel Abbes 22000, Algeria and Faculty of Technology, Hassiba Benbouali University of Chlef, P.O. Box 151 Chlef 02000, Algeria.
Abstract
This article deals with the existence, uniqueness and Ulam type stability results for a class of boundary value problems for fractional differential equations with Riesz-Caputo fractional derivative. The results are based on Banach contraction principle and Krasnoselskii's fixed point theorem. An illustrative example is given to validate our main results.
Keywords
- Riesz-Caputo fractional derivative
- Existence
- Measure of noncompactness
- Fixed point
- Ulam stability
- Non-instantaneous impulses
Main Subjects
[1] S. Abbas, M. Benchohra, J.J. Nieto, Caputo-Fabrizio fractional differential equations with non instantaneous impulses, Rend. Circ. Mat. Palermo., 71 (2022), pp. 131-144.
[2] S. Abbas, M. Benchohra, J.R. Graef and J. Henderson, Implicit Differential and Integral Equations: Existence and stability, Walter de Gruyter, London, 2018.
[3] S. Abbas, M. Benchohra and G.M. N'Gu\'er\'ekata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2014.
[4] S. Abbas, M. Benchohra and G.M. N'Gu\'er\'ekata, Topics in Fractional Differential Equations, Springer-Verlag, New York, 2012.
[5] R.S. Adiguzel, U. Aksoy, E. Karapinar and I.M. Erhan, Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM., 115 (3) (2021), 16 pp.
[6] R.S. Adiguzel, U. Aksoy, E. Karapinar and I.M. Erhan, On the solutions of fractional differential equations via Geraghty type hybrid contractions, Appl. Comput. Math., 20 (2021), pp. 313-333.
[7] R.S. Adiguzel, U. Aksoy, E. Karapinar and I.M. Erhan, On the solution of a boundary value problem associated with a fractional differential equation, Math. Meth. Appl. Sci., (2020).
[8] H. Afshari, H. Aydi and E. Karapinar, On generalized $\alpha-\psi-$Geraghty contractions on $b$-metric spaces, Georgian Math. J., 27 (1) (2020), pp. 9-21.
[9] H. Afshari, H. Hosseinpour and H.R. Marasi, Application of some new contractions for existence and uniqueness of differential equations involving Caputo-Fabrizio derivative, Adv. Difference Equ., 321 (2021), 13 pp.
[10] H. Afshari and E. Karapinar, A discussion on the existence of positive solutions of the boundary value problems via $\psi$-Hilfer fractional derivative on $b$-metric spaces, Adv. Difference Equ., 2020 (2020), pp. 1-11.
[11] B. Ahmad, A. Alsaedi, S.K. Ntouyas and J. Tariboon, Hadamard-type Fractional Differential Equations, Inclusions and Inequalities. Springer, Cham, 2017.
[12] L. Bai and J.J. Nieto and J.M. Uzal, On a delayed epidemic model with non-instantaneous impulses, Commun. Pure Appl. Anal., 19 (2020), pp. 1915-1930.
[13] L. Bai and J.J. Nieto, Variational approach to differential equations with not instantaneous impulses, Appl. Math. Lett., 73 (2017), pp. 44-48.
[14] F. Chen, D. Baleanu and G. Wu, Existence results of fractional differential equations with Riesz-Caputo derivative, Eur. Phys. J., 226 (2017), pp. 3411-3425.
[15] F. Chen, A. Chen and X. Wu, Anti-periodic boundary value problems with Riesz-Caputo derivative, Adv. Difference Equ., 2019 (2019).
[16] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
[17] C.Y. Gu and G.C. Wu, Positive solutions of fractional differential equations with the Riesz space derivative, Appl. Math. Lett., 95 (2019), pp. 59-64.
[18] E. Hernandez, D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 141 (5) (2013), pp. 1641-1649.
[19] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A., 27 (1941), pp. 222-224.
[20] A.A. Kilbas, H.M. Srivastava and Juan J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Amsterdam, 2006.
[21] N. Laledj, A. Salim, J.E. Lazreg, S. Abbas, B. Ahmad and M. Benchohra, On implicit fractional $q$-difference equations: Analysis and stability, Math. Methods Appl. Sci., 45 (2022), pp. 10775-10797.
[22] J.E. Lazreg, M. Benchohra and A. Salim, Existence and Ulam stability of $k$-generalized $\psi$-Hilfer fractional problem, J. Innov. Appl. Math. Comput. Sci., 2 (2022), pp. 1-13.
[23] D. Luo, Z. Luo and H. Qiu, Existence and Hyers-Ulam stability of solutions for a mixed fractional-order nonlinear delay difference equation with parameters, Math. Probl. Eng., 2020, 9372406 (2020).
[24] R. Ma, J.R. Wang and M. Li, Almost periodic solutions for two non-instantaneous impulsive biological models, Qual. Theory Dyn. Syst., 21 (2022).
[25] R. Nesraoui, D. Abdelkader, J.J. Nieto and A. Ouahab, Variational approach to non-instantaneous impulsive system of differential equations, Nonlinear Stud., 28 (2021), pp. 563-573.
[26] I. Petras, Fractional Calculus NonLinear Systems, Modelling, Analysis and Simulation, Springer, 2011.
[27] T.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), pp. 297-300.
[28] IA. Rus, Ulam stability of ordinary differential equations in a Banach space, Carpathian J. Math., 26 (2011), pp. 103-107.
[29] A. Salim, S. Abbas, M. Benchohra and E. Karapinar, A Filippov's theorem and topological structure of solution sets for fractional q-difference inclusions, Dynam. Syst. Appl., 31 (2022), pp. 17-34.
[30] A. Salim, S. Abbas, M. Benchohra and E. Karapinar, Global stability results for Volterra-Hadamard random partial fractional integral equations, Rend. Circ. Mat. Palermo, 2 (2022), pp. 1-13.
[31] A. Salim, B. Ahmad, M. Benchohra and J.E. Lazreg, Boundary value problem for hybrid generalized Hilfer fractional differential equations, Differ. Equ. Appl., 14 (2022), pp. 379-391.
[32] A. Salim, M. Benchohra, J.R. Graef and J.E. Lazreg, Initial value problem for hybrid $\psi$-Hilfer fractional implicit differential equations, J. Fixed Point Theory Appl., 24 (2022), 14 pp.
[33] A. Salim, M. Benchohra, J.R. Graef and J.E. Lazreg, Boundary value problem for fractional generalized Hilfer-type fractional derivative with non-instantaneous impulses, Fractal Fract., 5 (2021), pp. 1-21.
[34] A. Salim, M. Benchohra, E. Karapinar and J.E Lazreg, Existence and Ulam stability for impulsive generalized Hilfer-type
fractional differential equations, Adv. Difference Equ., 2020, 601 (2020).
fractional differential equations, Adv. Difference Equ., 2020, 601 (2020).
[35] A. Salim, M. Benchohra, J.E. Lazreg and J. Henderson, Nonlinear implicit generalized Hilfer-type fractional differential equations with non-instantaneous impulses in Banach spaces, Adv. Theo. Nonli. Anal. Appl., 4 (2020), pp. 332-348.
[36] A. Salim, J.E. Lazreg, B. Ahmad, M. Benchohra and J.J. Nieto, A study on $k$-generalized $\psi$-Hilfer derivative operator, Vietnam J. Math., (2022).
[37] D.R. Smart, Fixed point theory, Combridge Uni. Press, Combridge, 1974.
[38] S.M. Ulam, Problems in Modern Mathematics, Science Editions John Wiley Sons, Inc., New York, 1964.
[39] J.R. Wang and M. Fev ckan, Non-Instantaneous Impulsive Differential Equations. Basic Theory And Computation, IOP Publishing Ltd. Bristol, UK, 2018.
[40] J. Wang, L. Lv and Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electr. J. Qual. Theory Differ. Equ., 63 (2011), pp. 1-10.
[41] D. Yang and J. Wang, Integral boundary value problems for nonlinear non-instataneous impulsive differential equations, J. Appl. Math. Comput., 55 (2017), pp. 59-78.
)
