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Controllability of mixed Volterra-Fredholm-type integro-differential inclusions in Banach spaces. (English) Zbl 1167.93007

Summary: The paper establishes a sufficient condition for the controllability of semilinear mixed Volterra-Fredholm-type integro-differential inclusions in Banach spaces. We use Bohnenblust-Karlin’s fixed point theorem combined with a strongly continuous operator semigroup. Our main condition

for each positive number r and xC(J,X) with x r, there exists a function l r L 1 (J, + ) such that

sup|f|:f(t) F t , x (t) , 0 t g (t,s,x(s)) d x , 0 b h (t,s,x(s)) d sl r (t)

for a.e. tJ,

only depends upon the local properties of multivalued map on a bounded set. An example is also given to illustrate our main results.

MSC:
93B05Controllability
93C25Control systems in abstract spaces
47H10Fixed point theorems for nonlinear operators on topological linear spaces
34A60Differential inclusions
34A37Differential equations with impulses
93B28Operator-theoretic methods in systems theory
References:
[1]Balachandran, K.; Dauer, J. P.: Controllability of nonlinear systems in Banach spaces: a survey, J. optim. Theory appl. 115, 7-28 (2002) · Zbl 1023.93010 · doi:10.1023/A:1019668728098
[2]Balachandran, K.; Sakthivel, R.: Controllability of integrodifferential system in Banach spaces, Appl. math. Comput. 118, 63-71 (2001) · Zbl 1034.93005 · doi:10.1016/S0096-3003(00)00040-0
[3]Benchohra, M.; Ntouyas, S. K.: Controllability for functional differential and integrodifferential inclusions in Banach spaces, J. optim. Theory appl. 113, 449-472 (2002) · Zbl 1020.93002 · doi:10.1023/A:1015352503233
[4]Benchohra, M.; Gatsori, E. P.; Ntouyas, S. K.: Controllability results for semilinear evolution inclusions with nonlocal conditions, J. optim. Theory appl. 118, 493-513 (2003) · Zbl 1046.93005 · doi:10.1023/B:JOTA.0000004868.61288.8e
[5]Benchohra, M.; Górniewicz, L.; Ntouyas, S. K.; Ouahab, A.: Controllability results for impulsive functional differential inclusions, Rep. math. Phys. 54, 211-228 (2004) · Zbl 1130.93310 · doi:10.1016/S0034-4877(04)80015-6
[6]D.N. Chalishajar, Controllability of damped second order initial value problem for a class of differential inclusions with nonlocal conditions on noncompact intervals, in: Proceedings of the International Conference on Applied Analysis and Differential Equations (ICAADE), Iasi, Romania, World Science Publications, Singapore, September 2006, pp. 55 – 69 (January 2007). · Zbl 1156.93008
[7]Chang, Y. K.: Controllability of impulsive functional differential systems with infinite delay in Banach spaces, Chaos, solitons and fractals 33, 1601-1609 (2007) · Zbl 1136.93006 · doi:10.1016/j.chaos.2006.03.006
[8]Chang, Y. K.; Li, W. T.; Nieto, J.: Controllability of evolution differential inclusions in Banach spaces, Nonlinear anal. Theory methods anal. 67, 623-632 (2007) · Zbl 1128.93005 · doi:10.1016/j.na.2006.06.018
[9]Fu, X.: Controllability of abstract neutral functional differential systems with unbounded delay, Appl. math. Comput. 151, 299-314 (2004) · Zbl 1044.93008 · doi:10.1016/S0096-3003(03)00342-4
[10]Chalishajar, D. N.: Controllability of mixed Volterra – Fredholm-type integro-differential systems in Banach spaces, J. franklin inst. 344, 12-21 (2007) · Zbl 1119.93016 · doi:10.1016/j.jfranklin.2006.04.002
[11]Yosida, K.: Functional analysis, (1980)
[12]Deimling, K.: Multivalued differential equations, (1992) · Zbl 0760.34002
[13]Hu, S.; Papageorgiou, N.: Handbook of multivalued analysis, (1997)
[14]Bohnenblust, H. F.; Karlin, S.: On a theorem of ville, Contributions to the theory of games, 155-160 (1950) · Zbl 0041.25701
[15]Quinn, M. D.; Carmichael, N.: An approach to nonlinear control problem using fixed point methods, degree theory and pseudo-inverses, Numer. funct. Anal. optim. 23, 109-154 (1991)
[16]Lasota, A.; Opial, Z.: An application of the Kakutani – Ky Fan theorem in the theory of ordinary differential equations, Bull. acad. Polon. sci. Ser. sci. Math. astronom. Phys. 13, 781-786 (1965) · Zbl 0151.10703