Controllability of impulsive semilinear functional differential inclusions with finite delay in Fréchet spaces

N. Abada; M. Benchohra; H. Hammouche; A. Ouahab

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2007)

  • Volume: 27, Issue: 2, page 329-347
  • ISSN: 1509-9407

Abstract

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In this paper, we use the extrapolation method combined with a recent nonlinear alternative of Leray-Schauder type for multivalued admissible contractions in Fréchet spaces to study the existence of a mild solution for a class of first order semilinear impulsive functional differential inclusions with finite delay, and with operator of nondense domain in original space.

How to cite

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N. Abada, et al. "Controllability of impulsive semilinear functional differential inclusions with finite delay in Fréchet spaces." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 27.2 (2007): 329-347. <http://eudml.org/doc/271146>.

@article{N2007,
abstract = {In this paper, we use the extrapolation method combined with a recent nonlinear alternative of Leray-Schauder type for multivalued admissible contractions in Fréchet spaces to study the existence of a mild solution for a class of first order semilinear impulsive functional differential inclusions with finite delay, and with operator of nondense domain in original space.},
author = {N. Abada, M. Benchohra, H. Hammouche, A. Ouahab},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {semilinear functional differential inclusions; impulses; mild solution; fixed point; controllability; extrapolation space; nondensely defined operator; differential equations; differential inclusions},
language = {eng},
number = {2},
pages = {329-347},
title = {Controllability of impulsive semilinear functional differential inclusions with finite delay in Fréchet spaces},
url = {http://eudml.org/doc/271146},
volume = {27},
year = {2007},
}

TY - JOUR
AU - N. Abada
AU - M. Benchohra
AU - H. Hammouche
AU - A. Ouahab
TI - Controllability of impulsive semilinear functional differential inclusions with finite delay in Fréchet spaces
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2007
VL - 27
IS - 2
SP - 329
EP - 347
AB - In this paper, we use the extrapolation method combined with a recent nonlinear alternative of Leray-Schauder type for multivalued admissible contractions in Fréchet spaces to study the existence of a mild solution for a class of first order semilinear impulsive functional differential inclusions with finite delay, and with operator of nondense domain in original space.
LA - eng
KW - semilinear functional differential inclusions; impulses; mild solution; fixed point; controllability; extrapolation space; nondensely defined operator; differential equations; differential inclusions
UR - http://eudml.org/doc/271146
ER -

References

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  1. [1] N.U. Ahmed, Semigroup Theory with Applications to Systems and Control, Pitman Research Notes in Mathematics Series, 246. Longman Scientific & Technical, Harlow John Wiley & Sons, Inc., New York, 1991. 
  2. [2] H. Amann, Linear and Quasilinear Parabolic Problems, Birkhäuser, Berlin, 1995. 
  3. [3] B. Amir and L. Maniar, Application de la théorie d'extrapolation pour la résolution des équations différentielles à retard homogènes, Extracta Math. 13 (1998), 95-105. 
  4. [4] B. Amir and L. Maniar, Composition of pseudo almost periodic functions and Cauchy problems with operator of nondense domain, Ann. Math. Blaise Pascal 6 (1999), 1-11. Zbl0941.34059
  5. [5] W. Arendt, C.J.K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplacian Transforms and Cauchy Problems, Monographs Math, Vol. 96, Birkhäuser Verlag, 2001. Zbl0978.34001
  6. [6] M. Benchohra, L. Górniewicz, S.K. Ntouyas and A. Ouahab, Controllability results for impulsive functional differential inclusions, Rep. Math. Phys. 54 (2004), 211-227. Zbl1130.93310
  7. [7] M. Benchohra, L. Górniewicz, S.K. Ntouyas and A. Ouahab, Existence results for nondensely defined impulsive semilinear functional differential equations, Nonlinear Analysis and Applications, edited by R.P. Agarwal and D. O'Regan, Kluwer, 2003. Zbl1051.34068
  8. [8] M. Benchohra, J. Henderson and S.K. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, Vol. 2, New York, 2006. Zbl1130.34003
  9. [9] M. Benchohra and A. Ouahab, Controllability results for functional semilinear differential inclusions in Fréchet spaces, Nonlinear Anal. 61 (2005), 405-423. Zbl1086.34062
  10. [10] G. Da Prato and E. Grisvard, On extrapolation spaces, Rend. Accad. Naz. Lincei. 72 (1982), 330-332. Zbl0527.46055
  11. [11] G. Da Prato and E. Sinestrari, Differential operators with non-dense domains, Ann. Scuola Norm. Sup. Pisa Sci. 14 (1987), 285-344. Zbl0652.34069
  12. [12] K.J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. Zbl0952.47036
  13. [13] M. Frigon, Fixed Point Results for Multivalued Contractions on Gauge Spaces, Set Valued Mappings with Applications in Nonlinear Analysis, 175-181, Ser. Math. Anal. Appl. 4, Taylor & Francis, London, 2002. 
  14. [14] E.P. Gatsori, L. Górniewicz and S.K. Ntouyas, Controllability results for nondensely defined evolution impulsive differential inclusions with nonlocal conditions, Panamer. Math. J. 15 (2) (2005), 1-27. Zbl1075.93003
  15. [15] G. Guhring, F. Rabiger and W. Ruess, Linearized stability for semilinear non-autonomous evolution equations to retarded differential equations, Differential Integral Equations 13 (2000), 503-527. Zbl0990.34068
  16. [16] J. Henderson and A. Ouahab, Existence results for nondensely defined semilinear functional differential inclusions in Fréchet spaces, Electron. J. Qual. Theory Differ. Equ. (2005), (11), 1-17. Zbl1111.34045
  17. [17] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer, Dordrecht, The Netherlands, 1991. 
  18. [18] V. Lakshmikantham, D.D. Bainov and P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. Zbl0719.34002
  19. [19] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser Verlag, 1995. Zbl0816.35001
  20. [20] L. Maniar and A. Rhandi, Extrapolation and inhomogeneous retarded differential equations on infinite-dimensional spaces, Circ. Mat. Palermo 47 (2) (1998), 331-346. Zbl0916.34065
  21. [21] R. Nagel and E. Sinestrari, Inhomogeneous Volterra Integrodifferential Equations for Hille-Yosida operators, In Functional Analysis, edited by K.D. Bierstedt, A. Pietsch, W.M. Ruess and D. Voigt, 51-70, Marcel Dekker, 1998. Zbl0790.45011
  22. [22] J. Neerven, The Adjoint of a Semigroup of Linear Operators, Lecture Notes in Math. 1529, Springer-Verlag, New York, 1992. Zbl0780.47026
  23. [23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. 
  24. [24] M.D. Quinn and N. Carmichael, An approach to nonlinear control problem using fixed point methods, degree theory, pseudo-inverses, Numer. Funct. Anal. Optim. 7 (1984-85), 197-219. 
  25. [25] A.M. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. 

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