On the Solvability and Optimal Controls of Fractional Integrodifferential Evolution Systems with Infinite Delay

Journal of Optimization Theory and Applications

Volumn 152, Issue 1, 2012, Pages 31-50

  Wang, JinRong ^a   Zhou, Yong ^b   Medved', Milan ^c  

^a GUIZHOU UNIVERSITY   (China)

^b XIANGTAN UNIVERSITY   (China)

^c Faculty of Natural Sciences   (Slovakia)

Author keywords

Continuous dependence; Fractional integrodifferential evolution systems; Infinite delay; Optimal controls; Solvability

Indexed keywords

NONLINEAR EQUATIONS;

CONTINUOUS DEPENDENCE; EVOLUTION SYSTEMS; FRACTIONAL INTEGRODIFFERENTIAL EVOLUTION SYSTEM; INFINITE DELAY; MILD SOLUTION; OPTIMAL CONTROLS; PRIORI ESTIMATION; SOLVABILITY;

BANACH SPACES;

EID: 84855519663     PISSN: 00223239     EISSN: 15732878     Source Type: Journal    
DOI: 10.1007/s10957-011-9892-5     Document Type: Article

References (35)

1. Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and applications of fractional differential equations. In: North-Holland Mathematics Studies, vol. 204. Elsevier Science, Amsterdam (2006).
2. Miller, K. S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993).
3. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999).
4. Lakshmikantham, V., Leela, S., Devi, J. V.: Theory of Fractional Dynamic Systems. Scientific Publishers Cambridge, Cambridge (2009).
5. Agarwal, R. P., 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).
6. Agarwal, R. P., Belmekki, M., Benchohra, M.: A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative. Adv. Differ. Equ. 2009, e1-e47 (2009).
7. Agarwal, R. P., Zhou, Y., He, Y.: Existence of fractional neutral functional differential equations. Comput. Math. Appl. 59, 1095-1100 (2010).
8. Belmekki, M., Benchohra, M.: Existence results for fractional order semilinear functional differential with nondense domain. Nonlinear Anal. TMA 72, 925-932 (2010).
9. Benchohra, M., Henderson, J., Ntouyas, S. K., Ouahab, A.: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338, 1340-1350 (2008).
10. Benchohra, M., Henderson, J., Ntouyas, S. K., Ouahab, A.: Existence results for fractional functional differential inclusions with infinite delay and application to control theory. Fract. Calc. Appl. Anal. 11, 35-56 (2008).
11. Henderson, J., Ouahab, A.: Fractional functional differential inclusions with finite delay. Nonlinear Anal. TMA 70, 2091-2105 (2009).
12. Hu, L., Ren, Y., Sakthivel, R.: Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays. Semigroup Forum 79, 507-514 (2009).
13. Mophou, G. M., N'Guérékata, G. M.: Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay. Appl. Math. Comput. 216, 61-69 (2010).
14. Ren, Y., Qin, Y., Sakthivel, R.: Existence results for fractional order semilinear integro-differential evolution equations with infinite delay. Integral Equ. Oper. Theory 67, 33-49 (2010).
15. Chang, Y. K., Kavitha, V., Arjunan, M. M.: Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order. Nonlinear Anal. TMA 71, 5551-5559 (2009).
16. Jaradat, O. K., Al-Omari, A., Momani, S.: Existence of the mild solution for fractional semilinear initial value problems. Nonlinear Anal. TMA 69, 3153-3159 (2008).
17. El-Borai, M. M.: Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fractals 14, 433-440 (2002).
18. El-Borai, M. M.: The fundamental solutions for fractional evolution equations of parabolic type. J. Appl. Math. Stoch. Anal. 3, 197-211 (2004).
19. Hernández, E., O'Regan, D., Balachandran, K.: On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal. TMA 73, 3462-3471 (2010).
20. Wang, J., Zhou, Y.: Study of an approximation process of time optimal control for fractional evolution systems in Banach spaces. Adv. Differ. Equ. 2011, e1-e16 (2011).
21. Wang, J., Yang, Y., Wei, W.: Nonlocal impulsive problems for fractional differential equations with time-varying generating operators in Banach spaces. Opusc. Math. 30, 361-381 (2010).
22. Wang, J., Wei, W., Yang, Y.: On some impulsive fractional differential equations in Banach spaces. Opusc. Math. 30, 507-525 (2010).
23. Wang, J., Wei, W., Yang, Y.: Fractional nonlocal integrodifferential equations and its optimal control in Banach spaces. J. SIAM 14, 79-91 (2010).
24. Wang, J., Zhou, Y., Wei, W.: A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces. Commun. Nonlinear Sci. Numer. Simul. 16, 4049-4059 (2011).
25. Wang, J., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal., Real World Appl. 12, 262-272 (2011).
26. Zhou, Y.: Existence and uniqueness of fractional functional differential equations with unbounded delay. Int. J. Dyn. Differ. Equ. 1, 239-244 (2008).
27. Zhou, Y., Jiao, F., Li, J.: Existence and uniqueness for fractional neutral differential equations with infinite delay. Nonlinear Anal. TMA 71, 3249-3256 (2009).
28. Zhou, Y., Jiao, F.: Existence of extremal solutions for discontinuous fractional functional differential equations. Int. J. Dyn. Differ. Equ. 2, 237-252 (2009).
29. Zhou, Y., Jiao, F.: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59, 1063-1077 (2010).
30. Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal., Real World Appl. 11, 4465-4475 (2010).
31. Hale, J., Kato, J.: Phase spaces for retarded equations with infinite delay. Funkc. Ekvacioj 21, 11-41 (1978).
32. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. LNM, vol. 840. Springer, Berlin (1981).
33. 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).
34. Zeidler, E.: Nonlinear Functional Analysis and Its Application II/A. Springer, New York (1990).
35. 1-strong-weak lower semicontinuity of integral functional. Nonlinear Anal., Real World Appl. 11, 1399-1404 (1987).