| Title:
|
Impulsive semilinear neutral functional differential inclusions with multivalued jumps (English) |
| Author:
|
Abada, Nadjet |
| Author:
|
Agarwal, Ravi P. |
| Author:
|
Benchohra, Mouffak |
| Author:
|
Hammouche, Hadda |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
56 |
| Issue:
|
2 |
| Year:
|
2011 |
| Pages:
|
227-250 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we establish sufficient conditions for the existence of mild solutions and extremal mild solutions for some densely defined impulsive semilinear neutral functional differential inclusions in separable Banach spaces. We rely on a fixed point theorem for the sum of completely continuous and contraction operators. (English) |
| Keyword:
|
impulsive semilinear neutral functional differential equation |
| Keyword:
|
densely defined operator |
| Keyword:
|
infinite delay |
| Keyword:
|
phase space |
| Keyword:
|
fixed point |
| Keyword:
|
mild solutions |
| Keyword:
|
extremal mild solution |
| MSC:
|
34A37 |
| MSC:
|
34G25 |
| MSC:
|
34K09 |
| MSC:
|
34K30 |
| MSC:
|
34K35 |
| MSC:
|
34K45 |
| MSC:
|
47N20 |
| idZBL:
|
Zbl 1224.34207 |
| idMR:
|
MR2810245 |
| DOI:
|
10.1007/s10492-011-0004-5 |
| . |
| Date available:
|
2011-03-26T21:05:20Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141440 |
| . |
| Reference:
|
[1] Abada, N., Benchohra, M., Hammouche, H.: Existence and controllability results for impulsive partial functional differential inclusions.Nonlinear Anal., Theory Methods Appl. 69 (2008), 2892-2909. Zbl 1160.34068, MR 2452100, 10.1016/j.na.2007.08.060 |
| Reference:
|
[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 (2007), 329-347. Zbl 1145.34047, MR 2413817, 10.7151/dmdico.1088 |
| Reference:
|
[3] Ahmed, N. U.: Semigroup Theory with Applications to Systems and Control. Pitman Research Notes in Mathematics Series, 246.Longman Scientific & Technical/John Wiley & Sons Harlow/New York (1991). MR 1100706 |
| Reference:
|
[4] Ahmed, N. U.: Dynamic Systems and Control with Applications.World Scientific Publishing Hackensack (2006). Zbl 1127.93001, MR 2257896 |
| Reference:
|
[5] Ahmed, N. U.: Systems governed by impulsive differential inclusions on Hilbert spaces.Nonlinear Anal., Theory Methods Appl. 45 (2001), 693-706. Zbl 0995.34053, MR 1841203, 10.1016/S0362-546X(99)00417-4 |
| Reference:
|
[6] Ahmed, N. U.: Optimal impulse control for impulsive systems in Banach spaces.Int. J. Differ. Equ. Appl. 1 (2000), 37-52. Zbl 0959.49023, MR 1734517 |
| Reference:
|
[7] Bajnov, D. D., Simeonov, P. S.: Systems with Impulse Effect. Stability, Theory and Applications.Ellis Horwood Chichester (1989). Zbl 0683.34032, MR 1010418 |
| Reference:
|
[8] Belmekki, M., Benchohra, M., Ezzinbi, K., Ntouyas, S. K.: Existence results for some partial functional differential equations with infinite delay.Nonlinear Stud. 15 (2008), 373-385. Zbl 1182.34102, MR 2483148 |
| Reference:
|
[9] Benchohra, M., Górniewicz, L., Ntouyas, S. K.: Controllability of Some Nonlinear Systems in Banach Spaces (The Fixed Point Theory Approach).Pawel Wlodkowic University College, Wydawnictwo Naukowe NOVUM Plock (2003). Zbl 1059.49001 |
| Reference:
|
[10] Benchohra, M., Henderson, J., Ntouyas, S. K.: Impulsive Differential Equations and Inclusions, Vol. 2.Hindawi Publishing Corporation New York (2006). MR 2322133 |
| Reference:
|
[11] Benedetti, I.: An existence result for impulsive functional differential inclusions in Banach spaces.Discuss. Math., Differ. Incl. Control Optim. 24 (2004), 13-30. Zbl 1071.34087, MR 2118212, 10.7151/dmdico.1049 |
| Reference:
|
[12] Corduneanu, C., Lakshmikantham, V.: Equations with unbounded delay. A survey.Nonlinear Anal., Theory Methods Appl. 4 (1980), 831-877. Zbl 0449.34048, MR 0586852, 10.1016/0362-546X(80)90001-2 |
| Reference:
|
[13] Prato, G. Da, Grisvard, E.: On extrapolation spaces.Atti. Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 72 (1982), 330-332. Zbl 0527.46055, MR 0726298 |
| Reference:
|
[14] Dhage, B. C.: Multivalued maping and fixed point.Nonlinear Funct. Anal. Appl. 10 (2005), 359-378. MR 2194603 |
| Reference:
|
[15] Dhage, B. C.: A fixed point theorem for multi-valued mappings on ordered Banach spaces with applications II.Panam. Math. J. 15 (2005), 15-34. MR 2144192 |
| Reference:
|
[16] Dhage, B. C., Gatsori, E., Ntouyas, S. K.: Existence theory for perturbed functional differential inclusions.Commun. Appl. Nonlinear Anal. 13 (2006), 15-26. Zbl 1210.34084, MR 2226948 |
| Reference:
|
[17] Deimling, K.: Multivalued Differential Equations.Walter De Gruyter Berlin (1992). Zbl 0820.34009, MR 1189795 |
| Reference:
|
[18] Engel, K. J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations.Springer Berlin (2000). Zbl 0952.47036, MR 1721989 |
| Reference:
|
[19] Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones.Academic Press Boston (1988). Zbl 0661.47045, MR 0959889 |
| Reference:
|
[20] Górniewicz, L.: Topological Fixed Point Theory of Multivalued Mappings. Mathematics and Its Applications, 495.Kluwer Academic Publishers Dordrecht (1999). MR 1748378 |
| Reference:
|
[21] Hale, J. K.: Theory of Functional Differential Equations.Springer New York (1977). Zbl 0352.34001, MR 0508721 |
| Reference:
|
[22] Hale, J. K., Kato, J.: Phase space for retarded equations with infinite delay.Funkc. Ekvacioj, Ser. Int. 21 (1978), 11-41. Zbl 0383.34055, MR 0492721 |
| Reference:
|
[23] Hale, J. K., Lunel, S. H. Verduyn: Introduction to Functional Differential Equations. Applied Mathematical Sciences 99.Springer New York (1993). MR 1243878 |
| Reference:
|
[24] Heikkila, S., Lakshmikantham, V.: Monotone Iterative Technique for Nonlinear Discontinuous Differential Equations.Marcel Dekker Inc. New York (1994). MR 1280028 |
| Reference:
|
[25] Hernández, E., Henríquez, H. R.: Existence results for partial neutral functional differential equations with unbounded delay.J. Math. Anal. Appl. 221 (1998), 452-475. MR 1621730, 10.1006/jmaa.1997.5875 |
| Reference:
|
[26] Henríquez, H. R.: Existence of periodic solutions of partial neutral functional differential equations with unbounded delay.J. Math. Anal. Appl. 221 (1998), 499-522. MR 1621738, 10.1006/jmaa.1997.5899 |
| Reference:
|
[27] Hino, Y., Murakami, S., Naito, T.: Functional Differential Equations with Infinite Delay. Lecture Notes Math. Vol. 1473.Springer Berlin (1991). MR 1122588, 10.1007/BFb0084439 |
| Reference:
|
[28] Hu, Sh., Papageorgiou, N. S.: Handbook of Multivalued Analysis. Volume I: Theory.Kluwer Academic Publishers Dordrecht (1997). Zbl 0887.47001, MR 1485775 |
| Reference:
|
[29] Kamenskii, M., Obukhovskii, V., Zecca, P.: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. De Gruyter Series in Nonlinear Analysis and Applications.De Gruyter Berlin (2001). MR 1831201 |
| Reference:
|
[30] Kappel, F., Schappacher, W.: Some considerations to the fundamental theory of infinite delay equations.J. Differ. Equations 37 (1980), 141-183. Zbl 0466.34036, MR 0587220, 10.1016/0022-0396(80)90093-5 |
| Reference:
|
[31] Kisielewicz, M.: Differential Inclusions and Optimal Control.Kluwer Academic Publishers Dordrecht (1990). Zbl 0731.49001, MR 1135796 |
| Reference:
|
[32] Kolmanovskii, V., Myshkis, A.: Introduction to the Theory and Applications of Functional-Differential Equations. Mathematics and Its Applications, 463.Kluwer Academic Publishers Dordrecht (1999). MR 1680144 |
| Reference:
|
[33] Kuang, Y.: Delay Differential Equations: with Applications in Population Dynamics.Academic Press Boston (1993). Zbl 0777.34002, MR 1218880 |
| Reference:
|
[34] Lakshmikantham, V., Bainov, D. D., Simeonov, P. S.: Theory of Impulsive Differential Equations.World Scientific Singapore (1989). Zbl 0719.34002, MR 1082551 |
| Reference:
|
[35] Lakshmikantham, V., Wen, L., Zhang, B.: Theory of Differential Equations with Unbounded Delay. Mathematics and Its Applications.Kluwer Academic Publishers Dordrecht (1994). MR 1319339 |
| Reference:
|
[36] Lasota, A., Opial, Z.: An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations.Bull. Acad. Pol. Sci., Sér. Sci. Math. Astronom. Phys. 13 (1965), 781-786. Zbl 0151.10703, MR 0196178 |
| Reference:
|
[37] Liu, J. H.: Nonlinear impulsive evolution equations.Dyn. Contin. Discrete Impulsive Syst. 6 (1999), 77-85. Zbl 0932.34067, MR 1679758 |
| Reference:
|
[38] Migorski, S., Ochal, A.: Nonlinear impulsive evolution inclusions of second order.Dyn. Syst. Appl. 16 (2007), 155-173. Zbl 1128.34038, MR 2305434 |
| Reference:
|
[39] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations.Springer New York (1983). Zbl 0516.47023, MR 0710486 |
| Reference:
|
[40] Rogovchenko, Y. V.: Impulsive evolution systems: Main results and new trends.Dyn. Contin. Discrete Impulsive Syst. 3 (1997), 57-88. Zbl 0879.34014, MR 1435816 |
| Reference:
|
[41] Rogovchenko, Y. V.: Nonlinear impulsive evolution systems and applications to population models.J. Math. Anal. Appl. 207 (1997), 300-315. MR 1438916, 10.1006/jmaa.1997.5245 |
| Reference:
|
[42] Samoilenko, A. M., Perestyuk, N. A.: Impulsive Differential Equations.World Scientific Singapore (1995). Zbl 0837.34003, MR 1355787 |
| Reference:
|
[43] Schumacher, K.: Existence and continuous dependence for functional-differential equations with unbounded delay.Arch. Ration. Mech. Anal. 67 (1978), 315-335. Zbl 0383.34052, MR 0477379, 10.1007/BF00247662 |
| Reference:
|
[44] Shin, J. S.: An existence of functional differential equations.Arch. Ration. Mech. Anal. 30 (1987), 19-29. MR 0915258 |
| Reference:
|
[45] Wu, J.: Theory and Applications of Partial Functional Differential Equations. Applied Mathematical Sciences 119.Springer New York (1996). MR 1415838, 10.1007/978-1-4612-4050-1 |
| Reference:
|
[46] Yosida, K.: Functional Analysis, 6th ed.Springer Berlin (1980). Zbl 0435.46002, MR 0617913 |
| . |
