Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions. (English) Zbl 1171.34052

The authors present the existence of integral solutions and extremal integral solutions for the following problem
\[ \begin{aligned} y'(t)\in Ay(t)+F(t,y_t), &\quad t\in [0,T],\\ \Delta y|_{t=t_k}\in I_k(y(t_k)), &\quad k=1,\ldots,m, \\ y(t)=\varphi(t),&\quad t\in[-r,0], \end{aligned} \]
where \(F: [0,T]\times {\mathcal D}\to {\mathcal P}(E)\) are a multivalued maps, \({\mathcal P}(E)\) is the family of all nonempty subsets of \(E\), \(A: D(A)\subset E\to E\) is a nondensely defined closed linear operator on \(E\), \(0<r<\infty\), \(0=t_{0}<t_1<\cdots<t_m<t_{m+1}=T\),
\[ \begin{split} {\mathcal D}= \{\psi:[-r,0]\to E: \psi\text{ is continuous everywhere except for a finite number}\\ \text{of points } \bar{t}\text{ at which }\psi(\bar{t}^-)\text{ and }\psi(\bar{t}^+)\text{ exist and satisfy }\psi(\bar{t}^-)=\psi(\bar{t})\}, \end{split} \]
\(\varphi\in {\mathcal D},\) \(I_k\in E\to {\mathcal P}(E)\) \((k=1,\ldots,m)\), \(\Delta y|_{t=t_k}= y(t_k^+)- y(t_k^-)\), \(y(t_k^+)= \lim_{h\to 0^+}y(t_k+h)\) and \(y(t_k^-)= \lim_{h\to 0^+} y(t_k-h)\) stand for the right and the left limits of \(y(t)\) at \(t=t_k\), respectively.
For any function \(y\) defined on \([-r,b]\) and any \(t\in J\), \(y_t\) refers to the element of \({\mathcal D}\) such that
\[ y_t(\theta)=y(t+\theta),\quad \theta\in[-r,0]; \]
thus the function \(y_t\) represents the history of the state from time \(t-r\) up to the present time \(t\). Also the controllability of above problem are investigated. An examples is presented.


34K45 Functional-differential equations with impulses
34K30 Functional-differential equations in abstract spaces
34K35 Control problems for functional-differential equations
93B05 Controllability
Full Text: DOI


[1] Abada, N.; Benchohra, M.; Hammouche, H., Existence and controllability results for impulsive partial functional differential inclusions, Nonlinear anal., 69, 2892-2909, (2008) · Zbl 1160.34068
[2] Abada, N.; Benchohra, M.; Hammouche, H.; Ouahab, A., Controllability of impulsive semilinear functional differential inclusions with finite delay in Fréchet spaces, Discuss. math. differ. incl. control optim., 27, 2, 329-347, (2007) · Zbl 1145.34047
[3] Adimy, M.; Bouzahir, H.; Ezzinbi, K., Local existence and stability for some partial functional differential equations with unbounded delay, Nonlinear anal., 48, 323-348, (2002) · Zbl 0996.35080
[4] Adimy, M.; Ezzinbi, K., A class of linear partial neutral functional-differential equations with nondense domain, J. differential equations, 147, 285-332, (1998) · Zbl 0915.35109
[5] Ahmed, N.U., Semigroup theory with applications to systems and control, Pitman res. notes math. ser., vol. 246, (1991), Longman Scientific & Technical/John Wiley & Sons Harlow/New York · Zbl 0727.47026
[6] Ahmed, N.U., Dynamic systems and control with applications, (2006), World Scientific Hackensack, NJ · Zbl 1214.93117
[7] Ahmed, N.U., Systems governed by impulsive differential inclusions on Hilbert spaces, Nonlinear anal., 45, 693-706, (2001) · Zbl 0995.34053
[8] Ahmed, N.U., Optimal control for impulsive systems in Banach spaces, Int. J. differ. equ. appl., 1, 1, 37-52, (2000) · Zbl 0959.49023
[9] Arendt, W., Vector valued Laplace transforms and Cauchy problems, Israel J. math., 59, 327-352, (1987) · Zbl 0637.44001
[10] Arendt, W., Resolvent positive operators and integrated semigroup, Proc. London math. soc., 3, 54, 321-349, (1987) · Zbl 0617.47029
[11] Balachandran, K.; Dauer, J.P., Controllability of nonlinear systems in Banach spaces: A survey, J. optim. theory appl., 115, 7-28, (2002), Dedicated to Professor Wolfram Stadler · Zbl 1023.93010
[12] Bainov, D.D.; Simeonov, P.S., Systems with impulsive effect, (1989), Horwood Chichester · Zbl 0671.34052
[13] Benchohra, M.; Gatsori, E.P.; Górniewicz, L.; Ntouyas, S.K., Controllability results for evolution inclusions with nonlocal conditions, Z. anal. anwend., 22, 411-431, (2003) · Zbl 1052.34073
[14] Benchohra, M.; Gorniewicz, L.; Ntouyas, S.K., Controllability of some nonlinear systems in Banach spaces (the fixed point theory approach), (2003), Wydawnictwo Naukowe NOVUM Płock, Pawel Wlodkowic University College · Zbl 1059.49001
[15] Benchohra, M.; Gorniewicz, L.; Ntouyas, S.K.; Ouahab, A., Controllability results for nondensely defined semilinear functional differential equations, Z. anal. anwend., 25, 311-325, (2006) · Zbl 1101.93007
[16] Benchohra, M.; Henderson, J.; Ntouyas, S.K., Impulsive differential equations and inclusions, vol. 2, (2006), Hindawi Publishing Corporation New York · Zbl 1146.34055
[17] Benchohra, M.; Ntouyas, S.K., Existence and controllability results for multivalued semilinear differential equations with nonlocal conditions, Soochow J. math., 29, 157-170, (2003) · Zbl 1033.34068
[18] Benedetti, I., An existence result for impulsive functional differential inclusions in Banach spaces, Discuss. math. differ. incl. control optim., 24, 13-30, (2004) · Zbl 1071.34087
[19] Byszewski, L., Theorems about existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. math. anal. appl., 162, 494-505, (1991) · Zbl 0748.34040
[20] Byszewski, L., Existence and uniqueness of mild and classical solutions of semilinear functional-differential evolution nonlocal Cauchy problem, (), 25-33, 50th Anniv. Cracow Univ. Technol., Anniv. Issue 6
[21] Byszewski, L.; Akca, H., On a mild solution of a semilinear functional-differential evolution nonlocal problem, J. appl. math. stoch. anal., 10, 265-271, (1997) · Zbl 1043.34504
[22] Byszewski, L.; Lakshmikantham, V., Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. anal., 40, 11-19, (1991) · Zbl 0694.34001
[23] Carmichael, N.; Quinn, M.D., An approach to nonlinear control problems using the fixed point methods, degree theory and pseudo-inverses, Numer. funct. anal. optim., 7, 197-219, (1984-1985) · Zbl 0563.93013
[24] Da Prato, G.; Grisvard, E., On extrapolation spaces, Rend. accad. naz. lincei, 72, 330-332, (1982) · Zbl 0527.46055
[25] Da Prato, G.; Sinestrari, E., Differential operators with non-dense domains, Ann. sc. norm. super. Pisa cl. sci., 14, 285-344, (1987) · Zbl 0652.34069
[26] Deimling, K., Multivalued differential equations, (1992), de Gruyter Berlin · Zbl 0760.34002
[27] Dhage, B.C., Fixed-point theorems for discontinuous multivalued operators on ordered spaces with applications, Comput. math. appl., 51, 589-604, (2006) · Zbl 1110.47043
[28] Dhage, B.C.; Gastori, E.; Ntouyas, S.K., Existence theory for perturbed functional differential inclusions, Comm. appl. nonlinear anal., 13, 1-14, (2006)
[29] Engel, K.J.; Nagel, R., One-parameter semigroups for linear evolution equations, (2000), Springer-Verlag New York · Zbl 0952.47036
[30] Ezzinbi, K.; Liu, J., Nondensely defined evolution equations with nonlocal conditions, Math. comput. modelling, 36, 1027-1038, (2002) · Zbl 1035.34063
[31] Fu, X., Controllability of neutral functional differential systems in abstract space, Appl. math. comput., 141, 281-296, (2003) · Zbl 1175.93029
[32] Górniewicz, L., Topological fixed point theory of multivalued mappings, Math. appl., vol. 495, (1999), Kluwer Academic Dordrecht · Zbl 0937.55001
[33] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press New York · Zbl 0661.47045
[34] Hale, J.K., Ordinary differential equations, Pure appl. math., (1969), John Wiley & Sons New York · Zbl 0186.40901
[35] Hale, J.K., Theory of functional differential equations, (1977), Springer-Verlag New York · Zbl 0425.34048
[36] Hale, J.K.; Verduyn Lunel, S., Introduction to functional-differential equations, Appl. math. sci., vol. 99, (1993), Springer-Verlag New York · Zbl 0787.34002
[37] Heikkila, S.; Lakshmikantham, V., Monotone iterative technique for nonlinear discontinuous differential equations, (1994), Marcel Dekker New York · Zbl 0804.34001
[38] Hu, Sh.; Papageorgiou, N., Handbook of multivalued analysis, vol. I: theory, (1997), Kluwer Academic Dordrecht
[39] Kamenskii, M.; Obukhovskii, V.; Zecca, P., Condensing multivalued maps and semilinear differential inclusions in Banach spaces, De gruyter ser. nonlinear anal. appl., (2001), de Gruyter Berlin · Zbl 0988.34001
[40] Kellermann, H.; Hieber, M., Integrated semigroup, J. funct. anal., 84, 160-180, (1989)
[41] Kisielewicz, M., Differential inclusions and optimal control, (1991), Kluwer Academic Dordrecht · Zbl 0731.49001
[42] Kolmanovskii, V.; Myshkis, A., Introduction to the theory and applications of functional-differential equations, Math. appl., vol. 463, (1999), Kluwer Academic Dordrecht · Zbl 0917.34001
[43] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
[44] Lasota, A.; Opial, Z., An application of the kakutani – ky Fan theorem in the theory of ordinary differential equations, Bull. acad. Pol. sci. ser. sci. math. astronom. phys., 13, 781-786, (1965) · Zbl 0151.10703
[45] Li, G.; Xue, X., Controllability of evolution inclusions with nonlocal conditions, Appl. math. comput., 141, 375-384, (2003) · Zbl 1029.93003
[46] Liu, J.H., Nonlinear impulsive evolution equations, Dyn. contin. discrete impuls. syst., 6, 77-85, (1999) · Zbl 0932.34067
[47] Migorski, S.; Ochal, A., Nonlinear impulsive evolution inclusions of second order, Dynam. systems appl., 16, 155-173, (2007) · Zbl 1128.34038
[48] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer-Verlag New York · Zbl 0516.47023
[49] Rogovchenko, Yuri V., Impulsive evolution systems: main results and new trends, Dyn. contin. discrete impuls. syst., 3, 1, 57-88, (1997) · Zbl 0879.34014
[50] Rogovchenko, Yuri V., Nonlinear impulsive evolution systems and applications to population models, J. math. anal. appl., 207, 2, 300-315, (1997) · Zbl 0876.34011
[51] Sakthivel, R.; Mahmudov, N.I.; Kim, J.H., Approximate controllability of nonlinear impulsive differential systems, Rep. math. phys., 60, 1, 85-96, (2007) · Zbl 1141.93015
[52] Samoilenko, A.M.; Perestyuk, N.A., Impulsive differential equations, (1995), World Scientific Singapore · Zbl 0837.34003
[53] Sinestrari, E., Continuous interpolation spaces and spatial regularity in nonlinear Volterra integrodifferential equations, J. integral equations, 5, 287-308, (1983) · Zbl 0519.45013
[54] Tolstonogov, A.A., Differential inclusions in a Banach space, (2000), Kluwer Academic Dordrecht · Zbl 0990.49002
[55] Wu, J., Theory and applications of partial functional differential equations, Appl. math. sci., vol. 119, (1996), Springer-Verlag New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.