A sufficient condition of viability for fractional differential equations with the Caputo derivative

Journal of Mathematical Analysis and Applications

Volumn 381, Issue 1, 2011, Pages 146-154

  Girejko, Ewa ^a   Mozyrska, Dorota ^a   Wyrwas, Malgorzata ^a  

^a BIALYSTOK UNIVERSITY OF TECHNOLOGY   (Poland)

Author keywords

Fractional derivative; Fractional differential equation; Viability

Indexed keywords

EID: 79955481320     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2011.04.004     Document Type: Article

References (31)

1. Zhang S. Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives. Nonlinear Anal. 2009, 71:2087-2093.
2. Bonilla B., Rivero M., Rodriguez-Germá L., Trujillo J.J. Fractional differential equations as alternative models to nonlinear differential equations. Appl. Math. Comput. 2007, 187:79-88.
3. Kosmatov N. Integral equations and initial value problems for nonlinear differential equations of fractional order. Nonlinear Anal. 2009, 70:2521-2529.
4. Agarwal R.P., Lakshmikantham V., Nieto J.J. On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. 2010, 72:2859-2862.
5. Luchko Y.F., Rivero M., Trujillo J.J., Velasco M.P. Fractional models, non-locality, and complex systems. Comput. Math. Appl. 2010, 59:1048-1056.
6. Wei Z., Che J. Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative. J. Math. Anal. Appl. 2010, 367:260-272.
7. Diethelm K., Ford N.J. Analysis of fractional differential equations. J. Math. Anal. Appl. 2002, 265:229-248.
8. Diethelm K. The Analysis of Fractional Differential Equations 2010, Springer-Verlag, Berlin.
9. Ciotir J., Rascanu A. Viability for differential equations driven by fractional Brownian motion. J. Differential Equations 2009, 247:1505-1528.
10. Nualart D., Rascanu A. Differential equations driven by fractional Brownian motion. Collect. Math. 2002, 53:55-81.
11. Ibrahim R.W. Existence of convex and non convex local solutions for fractional differential inclusions. Electron. J. Differential Equations 2009, 18:1-13.
12. Momani S.M., Hadid S.B., Alawenh Z.M. Some analytical properties of solutions of differential equations of noninteger order. Int. J. Math. Math. Sci. 2004, 13:697-701.
13. Kaczorek T. Fractional positive linear systems. Kybernetes 2009, 38(7/8):1059-1078.
14. Al-Bassam M.A. Some existence theorems on differential equations of generalized order. J. Reine Angew. Math. 1965, 218:70-78.
15. D. Bugajewska, M. Zima, On positive solutions of nonlinear fractional differential equations, in: Proceedings of the 4th International Conference on Dynamical Systems and Differential Equations, Wilmington, NC, USA, 2002, pp. 141-146.
16. Zhang S. The existence of a positive solution for a nonlinear fractional differential equation. J. Math. Anal. Appl. 2000, 252:804-812.
17. El-Shahed M. Positive solutions for boundary value problem of nonlinear fractional differential equations. Abstr. Appl. Anal. 2007, 1:1-8.
18. Bai Z., Lü H. Positive solutions for boundary value problems of nonlinear fractional differential equations. J. Math. Anal. Appl. 2005, 311(2):495-505.
19. Nagumo M. Uber die lage der intergralkurven gewohnlicher differentialgleichung. Proc. Phys. Math. Soc. Japan 1942, 24:551-559.
20. Aubin J.P. Viability Theory 1991, Birkhäuser, Berlin.
21. Bothe D. Multivalued differential equations on graphs. J. Nonlinear Anal. TAMS 1992, 18:245-252.
22. Frankowska H., Plaskacz S., Rzezuchowski T. Measurable viability theorems and the Hamilton-Jacobi-Bellman equation. J. Differential Equations 1995, 116:265-305.
23. Cardaliaguet P., Plaskacz S. Viability and invariance for differential games with applications to Hamilton-Jacobi-Isaacs equations. Banach Center Publ. 1996, vol. 35:149-158. Polish Acad. Sci.
24. Quincampoix M. Frontieres de domaines d'invariance et de viabilité pour des inclusions différentielles avec contraintes. C. R. Math. Acad. Sci. Paris Sér. I 1990, 311(7):411-416.
25. Cañada A., Drábek P., Fonda A. Handbook of Differential Equations Ordinary Differential Equations 2005, Elsevier/North-Holland.
26. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations. North-Holland Math. Stud. 2006, vol. 204. Elsevier Science B.V., Amsterdam.
27. Podlubny I. Fractional Differential Equations. Math. Sci. Eng. 1999, vol. 198. Academic Press, San Diego.
28. Lakshmikantham V., Leeela S., Devi J.V. Theory of Fractional Dynamic Systems 2009, Cambridge Scientific Publishers, UK.
29. Aubin J.P., Frankowska H. Set-Valued Analysis 1990, Birkhäuser Boston Inc., Boston, MA.
30. Kaczorek T. Fractional positive continuous-time linear systems and their reachability. Int. J. Appl. Math. Comput. Sci. 2008, 18(2):223-228.
31. Kaczorek T. Reachability of cone fractional continuous-time linear systems. Int. J. Appl. Math. Comput. Sci. 2009, 19(1):89-93.