Coupled system of $\psi$--Caputo fractional differential equations without and with delay in generalized Banach spaces

Year 2022, Volume: 5 Issue: 1, 42 - 61, 31.03.2022

Choukri Derbazi Zidane Baitichezidane Mouffak Benchohra

https://doi.org/10.53006/rna.1007501

Cited By: 3

Abstract

The main objective of this research manuscript is to establish various existence and uniqueness results as well as the Ulam--Hyers stability of solutions to a Coupled system of $\psi$--Caputo fractional differential equations without and with delay in generalized Banach spaces. Existence and uniqueness results are obtained by applying Krasnoselskii's type fixed point theorem, Schauder's fixed point theorem in generalized Banach spaces, and Perov's fixed point theorem combined with the Bielecki norm. While Urs's approach is used to analyze the Ulam--Hyers stability of solutions for the proposed problem. Finally, Some examples are given to illustrate the obtained results.

Keywords

$\psi $--Caputo fractional derivative, Coupled system, Existence, Uniqueness, Fixed point, Bielecki norm, Ulam stability, Generalized Banach spaces

References

• [1] S. Abbas, M. Benchohra and G.M. N'Guérékata, Topics in fractional differential equations, Developments in Mathematics, 27, Springer, New York, 2012.
• [2] S. Abbas, M. Benchohra and G.M. N'Guerekata, Advanced fractional differential and integral equations, Mathematics Research Developments, Nova Science Publishers, Inc., New York, 2015.
• [3] S. Abbas, M. Benchohra, J.R. Graef, J. Henderson, Implicit fractional differential and integral equations, De Gruyter Series in Nonlinear Analysis and Applications, 26, De Gruyter, Berlin, 2018.
• [4] S. Abbas, M. Benchohra, N. Hamidi, J. Henderson, Caputo-Hadamard fractional differential equations in Banach spaces, Fractional Calculus and Applied Analysis. 21(4) (2018) 1027-1045.
• [5] S. Abbas, M. Benchohra, J.E. Lazreg, Y. Zhou, A survey on Hadamard and Hilfer fractional differential equations: analysis and stability, Chaos Solitons Fractals. 102 (2017), 47-71.
• [6] S. Abbas, M. Benchohra, B. Samet, Y. Zhou, Coupled implicit Caputo fractional q-difference systems, Advances in Di?er- ence Equations. 2019, 527 (2019). https://doi.org/10.1186/s13662-019-2433-5
• [7] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Communications in Nonlinear Science and Numerical Simulation. 44 (2017) 460-481.
• [8] R. Almeida, Functional differential equations involving the ψ-Caputo fractional derivative. Fractal and Fractional, 4 (2) (2020).
• [9] A. Atangana, D. Baleanu, New fractional derivative without nonlocal and nonsingular kernel: theory and application to heat transfer model, Thermal Science. 20 (2016) 763-769.
• [10] Z. Baitiche, C. Derbazi and M. Benchohra, ψ-Caputo fractional differential equations with multi-point boundary conditions by topological degree theory, Results in Nonlinear Analysis. 3 (4) (2020) 167-178.
• [11] Z. Baitiche, C. Derbazi, MM. Matar, Ulam-stability results for a new form of nonlinear fractional Langevin dif- ferential equations involving two fractional orders in the ψ-Caputo sense, Applicable Analysis. (2021) 16 pp. https://doi.org/10.1080/00036811.2021.1873300
• [12] M. Benchohra, S. Bouriah, J.E. Lazreg, J.J. Nieto, Nonlinear implicit Hadamard's fractional differential equations with delay in Banach space, Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica. 55 (1) (2016) 15-26.
• [13] V. Berinde, H. Fukhar-ud-din, and M. Paacurar, On the global stability of some k-order difference equations, Results in Nonlinear Analysis. 1 (1) (2018) 13-18.
• [14] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differen- tiation and Applications. 1 (2015) 73-85.
• [15] Y. Chen and H.-L. An, Numerical solutions of coupled Burgers equations with time- and space-fractional derivatives, Applied Mathematics and Computation. 200 (1) (2008) 87-95.
• [16] N.D. Cong, H.T. Tuan, Existence, uniqueness, and exponential boundedness of global solutions to delay fractional differ- ential equations, Mediterranean Journal of Mathematics. 14 (5) (2017) 193.
• [17] C. Derbazi, Z. Baitiche and M. Benchohra, Cauchy problem with ψ-Caputo fractional derivative in Banach spaces, Advances in the Theory of Nonlinear Analysis and its Application. 4 (4) (2020) 349-360.
• [18] L. Fu, Y. Chen, H. Yang, Time-space fractional coupled generalized zakharov-kuznetsov equations set for rossby solitary waves in two-layer fluids, Mathematics. 7 (1) (2019) p.41.
• [19] V. Ga?ychuk, B. Datsko and V. Meleshko, Mathematical modeling of time fractional reaction-di?usion systems, Journal of Computational and Applied Mathematics. 220 (2008) 215-225.
• [20] R. Hilfer, Applications of fractional calculus in physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.
• [21] A. Hussain, F. Jarad, E. Karapinar, A study of symmetric contractions with an application to generalized fractional di?erential equations. Advances in Difference Equations. 2021, 300 (2021). https://doi.org/10.1186/s13662-021-03456-z.
• [22] F. Jarad, T. Abdeljawad and D. Baleanu, Caputo-type modification of the Hadamard fractional derivatives, Advances in Di?erence Equations. 2012, 142 (2012). https://doi.org/10.1186/1687-1847-2012-142
• [23] E. Karapinar, S. Moustafa, A. Shehata, R. P. Agarwal, Fractional hybrid di?erential equations and coupled fixed-point results for α-admissible F(ψ 1 ,ψ 2 )-contractions in M-metric spaces. Discrete Dynamics in Nature and Society. 2020, Art. ID 7126045, 13 pp. [24] 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.
• [25] K.D. Kucche and S.T. Sutar, On existence and stability results for nonlinear fractional delay differential equations, Boletim da Sociedade Paranaense de Matemática. 3rd Série. 36 (4) (2018) 55-75.
• [26] J. Liang, Z. Liu and X. Wang, Solvability for a coupled system of nonlinear fractional differential equations in a Banach space, Fractional Calculus and Applied Analysis. 16 (1) (2013) 51-63.
• [27] J.-G. Liu et al., New fractional derivative with sigmoid function as the kernel and its models, Chinese Journal of Physics. 68 (2020) 533-541.
• [28] N.H. Luc et al., Reconstructing the right-hand side of a fractional subdi?usion equation from the final data, Journal of Inequalities and Applications. 2020 (53) (2020) 15 pp.
• [29] F. Mainardi, Fractional calculus and waves in linear viscoelasticity, Imperial College Press, London, 2010.
• [30] S. Muthaiah, M. Murugesan and N. G. Thangaraj, Existence of solutions for nonlocal boundary value problem of Hadamard fractional di?erential equations, Advances in the Theory of Nonlinear Analysis and its Application. 3 (3) (2019) 162-173
• [31] I.-R. Petre and A. Petru³el, Krasnoselskii's theorem in generalized Banach spaces and applications, Electronic Journal of Qualitative Theory of Di?erential Equations (85) (2012) 20 pp.
• [32] A.I. Perov, On the Cauchy problem for a system of ordinary differential equations. Pviblizhen. Met. Reshen. Di?er. Uvavn. Vyp. 2 (1964) 115?134. [33] N.D. Phuong et al., Fractional order continuity of a time semi-linear fractional di?usion-wave system, Alexandria Engi- neering Journal. 59 (6) (2020) 4959-4968.
• [34] I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.
• [35] R. Precup, The role of matrices that are convergent to zero in the study of semilinear operator systems, Mathematical and Computer Modelling. 49 (2009) 703-708.
• [36] R. Precup and A. Viorel, Existence results for systems of nonlinear evolution equations, International Journal of Pure and Applied Mathematics. 47 (2) (2008) 199-206.
• [37] I.A. Rus, Generalized contractions and applications, Cluj University Press, Cluj, 2001.
• [38] C. Urs, Coupled fixed point theorems and applications to periodic boundary value problems, Miskolc Mathematical Notes. 14 (1) (2013) 323?333.
• [39] A. Salim, M. Benchohra, E. Karapinar and J. E. Lazreg, Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations, Advances in Di?erence Equations. 601 (2020) 21pp.
• [40] J. Vanterler da Costa Sousa, Existence results and continuity dependence of solutions for fractional equations, Differential Equations & Applications. 12 (4) (2020) 377-396.
• [41] J. Vanterler da C. Sousa and E. Capelas de Oliveira, On the ψ-Hilfer fractional derivative, Communications in Nonlinear Science and Numerical Simulation. 60 (2018) 72-91.
• [42] J. Vanterler da C. Sousa, E. Capelas de Oliveira, Existence, uniqueness, estimation and continuous dependence of the solutions of a nonlinear integral and an integrodi?erential equations of fractional order, arXiv:1806.01441, (2018).
• [43] V.E. Tarasov, Fractional dynamics, Nonlinear Physical Science, Springer, Heidelberg, 2010.
• [44] N.H. Tuan et al., Approximate solution for a 2-D fractional differential equation with discrete random noise, Chaos, Solitons & Fractals. 133 (2020) 13 pp.
• [45] R.S. Varga, Matrix iterative analysis, second revised and expanded edition, Springer Series in Computational Mathematics, 27, Springer-Verlag, Berlin, 2000.
• [46] Y. Zhang and J. Wang, Nonlocal Cauchy problems for a class of implicit impulsive fractional relaxation differential systems, Journal of Applied Mathematics and Computing. 52 (2016) 323-343.
• [47] Y. Zhou, Basic theory of fractional di?erential equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014.