Darboux problem for fractional order neutral functional partial hyperbolic differential equations

International Journal of Dynamical Systems and Differential Equations

Volumn 2, Issue 3-4, 2009, Pages 301-312

  Abbas, Saïd ^a   Benchohra, Mouffak ^b   Zhou, Yong ^c  

^a UNIVERSITY OF SAIDA   (Algeria)

^b UNIVERSITY OF SIDI BEL ABBES   (Algeria)

^c XIANGTAN UNIVERSITY   (China)

Author keywords

Caputo fractional order derivative; Contraction; Delay; Fixed point; Fractional order; Functional hyperbolic neutral differential equations; Left sided mixed riemann liouville integral; Solution

Indexed keywords

DIFFERENTIATION (CALCULUS); INITIAL VALUE PROBLEMS; SHRINKAGE; SOLUTIONS;

CAPUTO FRACTIONAL ORDER DERIVATIVES; DELAY; FIXED POINTS; FRACTIONAL ORDER; LEFT-SIDED MIXED RIEMANN-LIOUVILLE INTEGRALS; NEUTRAL DIFFERENTIAL EQUATION;

FIXED POINT ARITHMETIC;

EID: 77249102599     PISSN: 17523583     EISSN: 17523591     Source Type: Journal    
DOI: 10.1504/IJDSDE.2009.031110     Document Type: Article

References (28)

1. Abbas, S. and Benchohra, M. (2009a) 'Partial hyperbolic differential equations with finite delay involving the Caputo fractional derivative', Commun. Math. Anal., Vol. 7, No. 2, pp.62-72.
2. Abbas, S. and Benchohra, M. (2009b) 'Darboux problem for perturbed partial differential equations of fractional order with finite delay', Nonlinear Anal.: Hybrid Systems, Vol. 3, pp.597-604.
3. Agarwal, R.P., Benchohra, M. and Hamani, S. (2009a) 'A survey on existence result for boundary value problems of nonlinear fractional differential equations and inclusions', Acta. Appl. Math., DOI 10.1007/s10440-008-9356-6.
4. Agarwal, R.P., Zhou, Y. and He, Y. (2009b) 'Existence of fractional neutral functional differential equations', Comput. Math. Appl., doi:10.1016/j.camwa.2009.05.010.
5. Belarbi, A., Benchohra, M. and Ouahab, A. (2006) 'Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces', Appl. Anal., Vol. 85, pp.1459-1470.
6. Benchohra, M., Graef, J.R. and Hamani, S. (2008a) 'Existence results for boundary value problems of nonlinear fractional differential equations with integral conditions', Appl. Anal., Vol. 87, No. 7, pp.851-863.
7. Benchohra, M., Hamani, S. and Ntouyas, S.K. (2008b) 'Boundary value problems for differential equations with fractional order', Surv. Math. Appl., Vol. 3, pp.1-12.
8. Benchohra, M., Henderson, J., Ntouyas, S.K. and Ouahab, A. (2008c) 'Existence results for functional differential equations of fractional order', J. Math. Anal. Appl., Vol. 338, pp.1340-1350.
9. x)', J. Appl. Math. Stochastic Anal., Vol. 3, pp.163-168.
10. xy ∈ F(x, y, z) in Banach space', Ann. Soc. Math. Pol. Ser. I, Comment. Math., Vol. 30, No. 1, pp.41-49.
11. Diethelm, K. and Freed, A.D. (1999) 'On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity', in Keil, F., Mackens, W., Voss, H. and Werther, J. (Eds.): Scientifice Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer-Verlag, Heidelberg, pp.217-224.
12. Diethelm, K. and Ford, N.J. (2002) 'Analysis of fractional differential equations', J. Math. Anal. Appl., Vol. 265, pp.229-248.
13. Kamont, Z. (1999) Hyperbolic Functional Differential Inequalities and Applications, Kluwer Academic Publishers, Dordrecht.
14. Kamont, Z. and Kropielnicka, K. (2005) 'Differential difference inequalities related to hyperbolic functional differential systems and applications', Math. Inequal. Appl., Vol. 8, No. 4, pp.655-674.
15. Kilbas, A.A., Bonilla, B. and Trujillo, J. (2000) 'Nonlinear differential equations of fractional order in a space of integrable functions', Dokl. Ross. Akad. Nauk, Vol. 374, No. 4, pp.445-449.
16. Kilbas, A.A. and Marzan, S.A. (2005) 'Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions', Differential Equations, Vol. 41, pp.84-89.
17. Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam.
18. Lakshmikantham, V., Leela, S. and Vasundhara, J. (2009) Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge.
19. Lakshmikantham, V. and Pandit, S.G. (1985) 'The method of upper, lower solutions and hyperbolic partial differential equations', J. Math. Anal. Appl., Vol. 105, pp.466-477.
20. Mainardi, F. (1997) 'Fractional calculus: some basic problems in continuum and statistical mechanics', in Carpinteri, A. and Mainardi, F. (Eds.): Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien, pp.291-348.
21. Pandit, S.G. (1997) 'Monotone methods for systems of nonlinear hyperbolic problems in two independent variables', Nonlinear Anal., Vol. 30, pp.235-272.
22. Vityuk, A.N. (1997) 'Existence of solutions of partial differential inclusions of fractional order', Izv. Vyssh. Uchebn., Ser. Mat., Vol. 8, pp.13-19.
23. Vityuk, A.N. and Golushkov, A.V. (2004) 'Existence of solutions of systems of partial differential equations of fractional order', Nonlinear Oscil., Vol. 7, No. 3, pp.318-325.
24. Yu, C. and Gao, G. (2005) 'Existence of fractional differential equations', J. Math. Anal. Appl., Vol. 310, pp.26-29.
25. Zhang, S. (2006) 'Positive solutions for boundary-value problems of nonlinear fractional diffrential equations', Electron. J. Differential Equations, Vol. 2006, No. 36, pp.1-12.
26. Zhou, Y. (2008) 'Existence and uniqueness of fractional functional differential equations with unbounded delay', Int. J. Dyn. Syst. Differ. Equ., Vol. 1, No. 4, pp.239-244.
27. Zhou, Y., Jiao, F. and Li, J. (2009a) 'Existence and uniqueness for p-type fractional neutral differential equations', Nonlinear Anal., Vol. 71, pp.2724-2733.
28. Zhou, Y., Jiao, F. and Li, J. (2009b) 'Existence and uniqueness for fractional neutral differential equations with infinite delay', Nonlinear Anal., Vol. 71, pp.3249-3256.