The general solution of differential equations with Caputo-Hadamard fractional derivatives and impulsive effect

Advances in Difference Equations

Volumn 2015, Issue 1, 2015, Pages

  Zhang, Xianmin ^a  

^a JIUJIANG UNIVERSITY   (China)

Author keywords

Caputo Hadamard fractional derivative; fractional differential equations; general solution; impulse

Indexed keywords

EID: 84937955041     PISSN: 16871839     EISSN: 16871847     Source Type: Journal    
DOI: 10.1186/s13662-015-0552-1     Document Type: Article

References (32)

1. Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999)
2. Kilbas, AA, Srivastava, HH, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
3. Baleanu, D, Diethelm, K, Scalas, E, Trujillo, JJ: Fractional Calculus Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos. World Scientific, Singapore (2012)
4. Ye, H, Gao, J, Ding, Y: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075-1081 (2007)
5. Benchohra, M, Henderson, J, Ntouyas, SK, Ouahab, A: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338, 1340-1350 (2008)
6. Agarwal, RP, Benchohra, M, Hamani, S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973-1033 (2010)
7. Odibat, ZM: Analytic study on linear systems of fractional differential equations. Comput. Math. Appl. 59, 1171-1183 (2010)
8. Ahmad, B, Nieto, JJ: Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Topol. Methods Nonlinear Anal. 35(2), 295-304 (2010)
9. Bai, Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. TMA 72, 916-924 (2010)
10. Mophou, GM, N’Guérékata, GM: Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay. Appl. Math. Comput. 216, 61-69 (2010)
11. Deng, W: Smoothness and stability of the solutions for nonlinear fractional differential equations. Nonlinear Anal. TMA 72, 1768-1777 (2010)
12. Jarad, F, Abdeljawad, T, Baleanu, D: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2012, 142 (2012)
13. Kilbas, AA: Hadamard-type fractional calculus. J. Korean Math. Soc. 38(6), 1191-1204 (2001)
14. Butzer, PL, Kilbas, AA, Trujillo, JJ: Compositions of Hadamard-type fractional integration operators and the semigroup property. J. Math. Anal. Appl. 269, 387-400 (2002)
15. Butzer, PL, Kilbas, AA, Trujillo, JJ: Mellin transform analysis and integration by parts for Hadamard-type fractional integrals. J. Math. Anal. Appl. 270, 1-15 (2002)
16. Klimek, M: Sequential fractional differential equations with Hadamard derivative. Commun. Nonlinear Sci. Numer. Simul. 16, 4689-4697 (2011)
17. Ahmad, B, Ntouyas, SK: A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations. Fract. Calc. Appl. Anal. 17, 348-360 (2014)
18. Thiramanus, P, Ntouyas, SK, Tariboon, J: Existence and uniqueness results for Hadamard-type fractional differential equations with nonlocal fractional integral boundary conditions. Abstr. Appl. Anal. (2014). doi:10.1155/2014/902054
19. Gambo, YY, Jarad, F, Baleanu, D, Abdeljawad, T: On Caputo modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2014, 10 (2014)
20. Ahmad, B, Sivasundaram, S: Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations. Nonlinear Anal. Hybrid Syst. 3, 251-258 (2009)
21. Ahmad, B, Sivasundaram, S: Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Anal. Hybrid Syst. 4, 134-141 (2010)
22. Tian, Y, Bai, Z: Existence results for the three-point impulsive boundary value problem involving fractional differential equations. Comput. Math. Appl. 59, 2601-2609 (2010)
23. Cao, J, Chen, H: Some results on impulsive boundary value problem for fractional differential inclusions. Electron. J. Qual. Theory Differ. Equ. 2010, 11 (2010)
24. Wang, X: Impulsive boundary value problem for nonlinear differential equations of fractional order. Comput. Math. Appl. 62, 2383-2391 (2011)
25. Stamova, I, Stamov, G: Stability analysis of impulsive functional systems of fractional order. Commun. Nonlinear Sci. Numer. Simul. 19, 702-709 (2014)
26. Abbas, A, Benchohra, M: Impulsive hyperbolic functional differential equations of fractional order with state-dependent delay. Fract. Calc. Appl. Anal. 13, 225-242 (2010)
27. Abbas, A, Benchohra, M: Upper and lower solutions method for impulsive hyperbolic differential equations with fractional order. Nonlinear Anal. Hybrid Syst. 4, 406-413 (2010)
28. Abbas, A, Agarwal, RP, Benchohra, M: Darboux problem for impulsive partial hyperbolic differential equations of fractional order with variable times and infinite delay. Nonlinear Anal. Hybrid Syst. 4, 818-829 (2010)
29. Abbas, A, Benchohra, M, Gorniewicz, L: Existence theory for impulsive partial hyperbolic functional differential equations involving the Caputo fractional derivative. Sci. Math. Jpn. 72(1), 49-60 (2010)
30. Benchohra, M, Seba, D: Impulsive partial hyperbolic fractional order differential equations in Banach spaces. J. Fract. Calc. Appl. 1(4), 1-12 (2011)
31. Guo, T, Zhang, K: Impulsive fractional partial differential equations. Appl. Math. Comput. 257, 581-590 (2015)
32. Zhang, X, Zhang, X, Zhang, M: On the concept of general solution for impulsive differential equations of fractional order q∈(0,1)$\mathrm{q} \in (0, 1)$. Appl. Math. Comput. 247, 72-89 (2014)