Hilfer-Hadamard Fractional Differential Equations; Existence and Attractivity

Öz

This work deals with a class of Hilfer-Hadamard differential equations. Existence and stability of solutions are presented. We use an appropriate fixed point theorem.

Anahtar Kelimeler

• Hilfer-Hadamard fractional derivative
• Schauder fixed- point Theorem
• uniformly locally attracting

Kaynakça

1. [1] S. Abbas, W. Albarakati and M. Benchohra, Existence and attractivity results for Volterra type nonlinear multi-delay Hadamard-Stieltjes fractional integral equations, PanAmer. Math. J. 26 (2016), 1-17.
2. [2] S. Abbas and M. Benchohra, Existence and attractivity for fractional order integral equations in Frchet spaces, Discuss. Math. Differ. Incl. Control Optim. 33 (2013), 1-17.
3. [3] S. Abbas and M. Benchohra, Existence and stability of nonlinear fractional order Riemann-Liouville, Volterra-Stieltjes multi-delay integral equations, J. Integ. Equat. Appl. 25 ( 2013 ), 143-158.
4. [4] S. Abbas, M. Benchohra, and T. Diagana, Existence and attractivity results for some fractional order partial integrodifferential equations with delay, Afr. Diaspora J. Math. 15 (2013), 87-100.
5. [5] S. Abbas, M. Benchohra and J. Henderson, Asymptotic attractive nonlinear frac- tional order Riemann-Liouville integral equations in Banach algebras, Nonlin. Stud. 20 (2013), 1-10.
6. [6] S. Abbas, M. Benchohra and J. Henderson, Existence and attractivity results for Hilfer fractional differential equations, J. Math. Sci. 243 (2019), 347-357.
7. [7] S. Abbas, M. Benchohra, J.-E. Lagreg, A. Alsaedi, Y. Zhou, Existence and Ulam stability for fractional differential equations of Hilfer-Hadamard type, Adv. Difference Equ. 180 (2017), 1-14.
8. [8] S. Abbas, M. Benchohra and G. M. N’Gu´ er´ ekata, Advanced Fractional Differential and Integral Equations, Nova Sci. Publ., New York, 2014.

9. [9] S. Abbas, M. Benchohra, and G. M.N’ Gu´ er´ ekata, Topics in Fractional Differential Equations, Dev. Math., 27, Springer, New York, 2015.
10. [10] S. Abbas, M. Benchohra, and J. J. Nieto, Global attractivity of solutions for non-linear fractional order Riemann-Liouville Volterra-Stieltjes partial integral equations, Electron. J. Qual. Theory Differ. Equat 81 (2012), 1-15.
11. [11] H. Afshari, E. Karapinar, A discussion on the existence of positive solutions of the boundary value problems via ψ-Hilfer fractional derivative on b-metric spaces. Adv. Difference Equ. 2020, 616.
12. [12] M. Benchohra, J. Henderson, S. K. Ntouyas and A. Ouahab, Existence results for functional differential equations of fractional order, J. Math. Anal. Appl. 338 (2008), 1340-1350.
13. [13] N. Bouteraa, S. Benaicha, The uniqueness of positive solution for higher-order nonlinear fractional differential equation with fractional multi-point boundary conditions, Adv. Theory Nonl. Anal. Appl. 2 (2) (2018), 74-84.
14. [14] C. Corduneanu, Integral Equations and Stability of Feedback Systems, Acad. Press, New York, 1973.
15. [15] A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
16. [16] R. Hilfer, Applications of Fractional Calculus in Physics , World Scientific, Singa- pore, 2000.
17. [17] R. Hilfer, Threefold introduction to fractional derivatives, R. Klages et al. (editors), Anomalous Transp.: Found. Appl., WileyVCH, Weinheim , pp. (2008), 17-73.
18. [18] R. Kamocki and C. Obcznski, On fractional Cauchy-type problems containing Hilfer’s derivative, Electron. J. Qual. Theory Differ. Equ. (2016), No. 50, 1-12.
19. [19] E. Karapinar, T. Abdeljawad, F. Jarad, Applying new fixed point theorems on fractional and ordinary differential equations. Adv. Difference Equ. 2019, Paper No. 421, 25 pp.
20. [20] M.D. Kassim, K.M. Furati, N.-E. Tatar, On a differential equation involving Hilfer-Hadamard fractional derivative, Abstr. Appl. Anal. (2012), Article ID 391062.
21. [21] M.D. Kassim, N.E. Tatar, Well-posedness and stability for a differential problem with Hilfer-Hadamard fractional derivative, Abstr. Appl. Anal. 1 (2013), 1-12.
22. [22] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
23. [23] V. Lakshmikantham and J. Vasundhara Devi, Theory of fractional differential equations in a Banach space, Eur. J. Pure Appl. Math. 1 (2008), 38-45.
24. [24] V. Lakshmikantham and A.S. Vatsala, Basic theory of fractional differential equations, Nonlin. Anal. 69 (2008), 2677-2682.
25. [25] V. Lakshmikantham and A. S. Vatsala, General uniqueness and monotone iterative technique for fractional differential equations, Appl. Math. Lett. 21 (2008), 828-834.
26. [26] K. Oldham, J. Spanier, The Fractional Calculus, Acad. Press, New York, 1974.
27. [27] Podlubny, Fractional Differential Equations, in: Mathematics in Science and Engineering, 198, Acad. Press, New York, 1999.
28. [28] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications , Gordon Breach, Tokyo-Paris-Berlin, 1993.
29. [29] S. Muthaiah, M. Murugesan, N. G. Thangaraj, Existence of solutions for nonlocal boundary value problem of Hadamard fractional differential equations, Adv. Theory Nonl. Anal. Appl. 3 (3) (2019), 162-173.
30. [30] V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to the Dynamics of Particles, Fields and Media, Springer, Beijing-Heidelberg, 2010.
31. [31] Z. Tomovski, R. Hilfer and H. M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler-type functions, Integral Transforms Spec. Funct. 21 (2010), No. 11, 797-814.
32. [32] J.-R. Wang and Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput. 266 (2015), 850-859.
33. [33] Y. Zhou, J.-R. Wang, L. Zhang, Basic Theory of Fractional Differential Equations, Second edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.