Existence of solutions for a coupled system of Caputo-Hadamard type fractional di¤erential equations with Hadamard fractional integral conditions

Year 2021, , 316 - 329, 30.09.2021

Houas Mohamed

https://doi.org/10.31197/atnaa.683278

Cited By: 3

Abstract

In this work, we study existence and uniqueness of solutions for a coupled system of nonlinear fractional di¤erential equations involving two Caputo-Hadamard-type fractional derivatives. By applying the Banach's fixed point theorem and Shaefer's fixed point theorem, the existence of solutions is obtained.The results obtained in this work are well illustrated with the aid of examples.

Keywords

Caputo-Hadamard fractional calculs, Existence, Boundary conditions

References

• [1 ]S. Abbas, M. Benchohra, J. Lazreg and J.J Nieto. On a coupled system of Hilfer-Hadamard fractional differentialequations in Banach spaces. J. Nonlinear Funct. Anal.2018, Article ID 12 (2018).
• [2] M. Benchohra, S. Hamani and S. K. Ntouyas. Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal. 71 (2009), (7-8), 2391-2396.
• [3] W. Benhamida, S. Hamani and J. Henderson. Boundary value problems for Caputo-Hadamard fractional differential equations. Advances in the Theory of Nonlinear Analysis and its Applications. 2 (3), (2018), 138-145.
• [4] W. Benhamida, J. Henderson and S. Hamani. Boundary value problems for Hadamard fractional di¤erential equations with nonlocal multi-point boundary conditions. Fractional Di¤erential Calculus. 8(1), (2018), 165-176.
• [5] Z. Dahmani, S. Belarbi. New results for fractional evolution equations using Banach fixed point theorem. Int. J. Nonlinear Anal. Appl. 5(2), (2014), 22-30.
• [6] Z. Dahmani, L. Tabharit. Fractional order differential equations involving Caputo derivative. Theory Appl. Math. Comput. Sci. 4(1) (2014), 40-55.
• [7] M. Houas, Solvability of a system of fractional hybrid differential equations. Commun. Optim. Theory. Article ID12, (2018), 1-9.
• [8] M. Houas, Z. Dahmani and M. Benbachir, New results for a boundary value problem for di¤erential equations of arbitrary order. International Journal of Modern Mathematical Sciences. 7 (2013), 195-211.
• [9] M. Houas, Z. Dahmani. New results for a coupled system of fractional differential equations. Facta. Univ. Ser. Math. Inform. 28(2) (2013), 133-150.
• [10] M. Houas, M. Bezziou. Existence and stability results for fractional differential equations with two Caputo fractional derivatives. Facta. Univ. Ser. Math. Inform. 34(2) (2019), 341-357.
• [11] M. Houas, Z. Dahmani. On existence of solutions for fractional differential equations with nonlocal multi-point boundary conditions. Lobachevskii. J. Math. 37(2) (2016), 120-127.
• [12] M. Houas. Existence of solutions for fractional differential equations involving two Riemann-Liouville fractional orders.Anal. Theory Appl., Vol. 34(3), (2018), 253-274.
• [13] F. Jarad, T. Abdeljawad, D. Baleanu. Caputo-type modification of the Hadamard fractional derivatives. Adv. Di¤er. Equ. 2012, 142 (2012)
• [14] A.A.Kilbas, H.M. Srivastava, J.J.Trujillo. Theory and applications of fractional differential equations. North-Holland Mathematics Studies. Vol. 204. Elsevier Science B.V. Amsterdam. (2006).
• [15] S. K. Ntouyas and J.Tariboon. Fractional integral problems for Hadamard-Caputo fractional Langevin di¤erential inclusions. J. Appl. Math. Comput. 51, (2016), 13-33.
• [16] W. Shammakh. A study of Caputo-Hadamard-type fractional differential equations with nonlocal boundary conditions. Journal of Function Spaces. Vol. 2016, Article ID 7057910, 9 pages.
• [17] A.Taieb and Z. Dahmami. A coupled system of nonlinear differential equations involving m nonlinear terms. Georgian Math. J. 23(3), 2016, 447-458.
• [18] A.Taieb. Existence of solutions and the Ulam stability for a class of singular nonlinear fractional integro-di¤erential equations. Article ID 4, (2019), 1-22
• [19] P. Thiramanus, S.K. Ntouyas and J. Tariboon. Positive solutions for Hadamard fractional differential equations on infinite domain. Adv. Dif-ference. Equ. 83 (2016), 1-18.
• [20] W.Yang Y. Qinc. Positive solutions for nonlinear Hadamard fractional differential equations with integral boundary conditions. Science Asia. 43 (2017), 201-206.
• [21] G. Wang, W. Liu, C. Ren. Existence of solutions for multi-point nonlinear differential equations of fractional orders with integral boundary conditions. Elec. J. Diff. Equ. 54 (2012), 1-10.
• [22] W. Yang. Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions. Comput. Math. Appl. 63(1) (2012), 288-297.