$\psi$--Caputo fractional differential equations with multi-point boundary conditions by Topological Degree Theory

Abstract

In this article, we discuss the existence and uniqueness of solutions to some nonlinear fractional differential equations involving the $\psi$--Caputo fractional derivative with multi-point boundary conditions. Our results rely on the technique of topological degree theory for condensing maps and the Banach contraction principle. Also, two illustrative examples are presented to illustrate our main results.

Keywords

• Fractional differential equations
• $\psi$--Caputo fractional derivative
• multi-point boundary conditions
• existence, uniqueness
• topological degree theory
• condensing maps

Kaynakça

1. [1] S. Abbas, M. Benchohra, J.R. Graef, J. Henderson, Implicit Fractional Di?erential and Integral Equations, Existence and Stability, de Gruyter, Berlin 2018.
2. [2] S. Abbas, M. Benchohra, G.M.N' Guérékata, Topics in Fractional Di?erential Equations, Springer, New York 2012.
3. [3] S. Abbas, M. Benchohra and G M. N'Guérékata, Advanced Fractional Di?erential and Integral Equations, Nova Science Publishers, New York, 2015.
4. [4] S. Abbas, M. Benchohra, J. Henderson, J. E. Lazreg, Weak solutions for a coupled system of partial Pettis Hadamard fractional integral equations, Adv. Theory Nonl. Anal. Appl. 1 (2017) No. 2, 136-146.
5. [5] S. Abbas, M. Benchohra, B. Samet, Y. Zhou, Coupled implicit Caputo fractional q-di?erence systems, Adv. Difference Equ. 2019, Paper No. 527, 19 pp.
6. [6] S. Abbas, M. Benchohra, N. Hamidi and J. Henderson, Caputo-Hadamard fractional differential equations in Banach spaces, Fract. Calc. Appl. Anal. 21 (2018), 1027-1045.
7. [7] M.S. Abdo, S.K. Panchal, A.M. Saeed, Fractional boundary value problem with ψ-Caputo fractional derivative, Proc. Indian Acad. Sci. (Math. Sci), 129:65 (2019).
8. [8] R.P. Agarwal, D. O'Regan, Toplogical degree theory and its applications, Taylor and Francis, 2006.

9. 9] A. Aghajani, E. Pourhadi, J.J. Trujillo, Application of measure of noncompactness to a Cauchy problem for fractional di?erential equations in Banach spaces. Fract. Calc. Appl. Anal. 16 (4) (2013), 962-977.
10. [10] A. Ali, B. Samet, K. Shah and R. A. Khan, Existence and stability of solution to a coppled systems of differential equations of non-integer order, Bound. Value Probl., 2017 (2017), Paper No. 16, 13 pp.
11. [11] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Communications in Nonlinear Science and Numerical Simulation., 2017, 44, 460-481.
12. [12] R. Almeida, Fractional Di?erential Equations with Mixed Boundary Conditions, Bull. Malays. Math. Sci. Soc. 42 (2019), 1687?1697. [13] R. Almeida, A.B. Malinowska, M.T.T. Monteiro, Fractional di?erential equations with a Caputo derivative with respect to a kernel function and their applications, Math. Meth. Appl. Sci. 41 (2018), 336-352
13. [14] R. Almeida, A.B. Malinowska, T. Odzijewicz, Optimal Leader-Follower Control for the Fractional Opinion Formation Model, J. Optim. Theory Appl. 182 (2019), 1171-1185.
14. [15] R. Almeida, M. Jleli, B. Samet, A numerical study of fractional relaxation-oscillation equations involving ψ-Caputo fractional derivative. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113 (2019), 1873-1891.
15. [16] M. Bahadur Zada, K. Shah and R. A. Khan, Existence theory to a coupled system of higher order fractional hybrid di?erential equations by topological degree theory, Int. J. Appl. Comput. Math., 4 (2018), Art. 102, 19 pp.
16. [17] M. Benchohra, S. Hamani, S.K. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal., 71 (2009), 2391?2396.
17. [18] M. Benchohra, S. Bouriah, J.J. Nieto, Existence of periodic solutions for nonlinear implicit Hadamard's fractional differ- ential equations, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 112 (2018), 25-35.
18. [19] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
19. [20] R. Hilfer, Application of fractional calculus in physics, New Jersey: World Scientific, 2001.
20. [21] F. Isaia, On a nonlinear integral equation without compactness, Acta Math. Univ. Comenian. (N.S.) 75 (2006), 233-240.
21. [22] R.A. Khan and K. Shah, Existence and uniqueness of solutions to fractional order multi-point boundary value problems, Commun. Appl. Anal. 19 (2015), 515-526.
22. [23] A. Khan, K. Shah, Y. Li, T.S. Khan, Ulam type stability for a coupled system of boundary value problems of nonlinear fractional di?erential equations. J. Funct. Spaces 2017, Art. ID 3046013, 8 pp.
23. [24] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Sudies Elsevier Science B.V. Amsterdam the Netherlands, 2006.
24. [25] K.D. Kucche, A.D. Mali, J.V.C. Sousa, On the nonlinear Ψ-Hilfer fractional differential equations, Comput. Appl. Math. 38 (2) (2019), Art. 73, 25 pp
25. [26] K.B. Oldham, Fractional differential equations in electrochemistry. Adv. Eng. Softw., 41 (1) (2010), 9-12.
26. [27] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1993.
27. [28] J. Sabatier, O.P. Agrawal, J.A.T. Machado, Advances in Fractional Calculus-Theoretical Developments and Applications in Physics and Engineering. Dordrecht: Springer, 2007.
28. [29] B. Samet, H. Aydi, Lyapunov-type inequalities for an anti-periodic fractional boundary value problem involving ψ-Caputo fractional derivative, J. Inequal. Appl. 2018, Paper No. 286, 11 pp.
29. [30] K. Shah, A. Ali and R. A. Khan, Degree theory and existence of positive solutions to coupled systems of multi-point boundary value problems, Bound. Value Probl., 2016, Paper No. 43, 12 pp.
30. [31] K. Shah and R. A. Khan, Existence and uniqueness results to a coupled system of fractional order boundary value problems by topological degree theory, Numer. Funct. Anal. Optim., 37(7) (2016), 887-899.
31. [32] K. Shah and W. Hussain, Investigating a class of nonlinear fractional differential equations and its Hyers-Ulam stability by means of topological degree theory, Numer. Funct. Anal. Optim., 40 (2019), no. 12, 1355-1372.
32. [33] M. Shoaib, K. Shah, R. Ali Khan, Existence and uniqueness of solutions for coupled system of fractional diferential equation by means of topological degree method, Journal Nonlinear Analysis and Application., 2018 no. 2 (2018) 124-135.
33. [34] S. Smirnov, Green's function and existence of a unique solution for a third-order three-point boundary value problem, Math. Model. Anal. 24 (2019), no. 2, 171-178.
34. [35] V.E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Heidelberg & Higher Education Press, Beijing, 2010.
35. [36] J. Wang, Y. Zhou and W. Wei, Study in fractional differential equations by means of topological degree methods, Numer. Funct. Anal. Optim., 33(2) (2012), 216-238.