Abstract inclusions in Banach spaces with boundary conditions of periodic type

Lahcene Guedda; Ahmed Hallouz

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2014)

  • Volume: 34, Issue: 2, page 229-253
  • ISSN: 1509-9407

Abstract

top
We study in the space of continuous functions defined on [0,T] with values in a real Banach space E the periodic boundary value problem for abstract inclusions of the form ⎧ x ∈ S ( x ( 0 ) , s e l F ( x ) ) ⎨ ⎩ x (T) = x(0), where, F : [ 0 , T ] × → 2 E is a multivalued map with convex compact values, ⊂ E, s e l F is the superposition operator generated by F, and S: × L¹([0,T];E) → C([0,T]; ) an abstract operator. As an application, some results are given to the periodic boundary value problem for nonlinear differential inclusions governed by m-accretive operators generating not necessarily a compact semigroups.

How to cite

top

Lahcene Guedda, and Ahmed Hallouz. "Abstract inclusions in Banach spaces with boundary conditions of periodic type." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 34.2 (2014): 229-253. <http://eudml.org/doc/270538>.

@article{LahceneGuedda2014,
abstract = {We study in the space of continuous functions defined on [0,T] with values in a real Banach space E the periodic boundary value problem for abstract inclusions of the form ⎧ $x ∈ S(x(0), sel_\{F\}(x))$ ⎨ ⎩ x (T) = x(0), where, $F:[0,T] × → 2^E $ is a multivalued map with convex compact values, ⊂ E, $sel_\{F\}$ is the superposition operator generated by F, and S: × L¹([0,T];E) → C([0,T]; ) an abstract operator. As an application, some results are given to the periodic boundary value problem for nonlinear differential inclusions governed by m-accretive operators generating not necessarily a compact semigroups.},
author = {Lahcene Guedda, Ahmed Hallouz},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {measure of noncompactness; condensing operator; nonlinear abstract inclusion; accretive operator; integral solution; nonlinear semigroup},
language = {eng},
number = {2},
pages = {229-253},
title = {Abstract inclusions in Banach spaces with boundary conditions of periodic type},
url = {http://eudml.org/doc/270538},
volume = {34},
year = {2014},
}

TY - JOUR
AU - Lahcene Guedda
AU - Ahmed Hallouz
TI - Abstract inclusions in Banach spaces with boundary conditions of periodic type
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2014
VL - 34
IS - 2
SP - 229
EP - 253
AB - We study in the space of continuous functions defined on [0,T] with values in a real Banach space E the periodic boundary value problem for abstract inclusions of the form ⎧ $x ∈ S(x(0), sel_{F}(x))$ ⎨ ⎩ x (T) = x(0), where, $F:[0,T] × → 2^E $ is a multivalued map with convex compact values, ⊂ E, $sel_{F}$ is the superposition operator generated by F, and S: × L¹([0,T];E) → C([0,T]; ) an abstract operator. As an application, some results are given to the periodic boundary value problem for nonlinear differential inclusions governed by m-accretive operators generating not necessarily a compact semigroups.
LA - eng
KW - measure of noncompactness; condensing operator; nonlinear abstract inclusion; accretive operator; integral solution; nonlinear semigroup
UR - http://eudml.org/doc/270538
ER -

References

top
  1. [1] S. Aizicovici, N.S. Papageorgiou and Staicu, Periodic solutions of nonlinear evolution inclusions in Banach spaces, J. Nonlinear Convex Anal. 7 (2) (2006) 163-177. Zbl1110.34037
  2. [2] R.R. Akhmerov, M.I. Kamenskii, A.S. Potapov, B.N. Rodkina and B.N. Sadovskiĭ, Measures of Noncompactness and Condensing Operators, Number 55 in Oper. Theory Adv. Appl. (Birkhäuser, Basel, Boston, Berlin, 1992). doi: 10.1007/978-3-0348-5727-7 
  3. [3] R. Bader, B.D. Gel'man, M.I. Kamenskii and V.V. Obukhovskii, On the topological dimension of the solutions sets for some classes of operator and differential inclusions, Discuss. Math. DICO 22 (1) (2002) 17-32. doi: 10.7151/dmdico.1030 Zbl1041.47031
  4. [4] R. Bader, M.I. Kamenskii and V.V. Obukhovskii, On some classes of operator inclusions with lower semicontinuous nonlinearities, Topol. Methods Nonlinear Anal. 17 (1) (2001) 143-156. 
  5. [5] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces (Marcel Dekker, 1980) Zbl0441.47056
  6. [6] V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România (Bucharest, 1976). Translated from the Romanian. doi: 10.1007/978-94-010-1537-0 Zbl0328.47035
  7. [7] V. Barbu, Analysis and control of nonlinear infinite-dimensional systems (Academic Press Inc., Boston, 1993) Zbl0776.49005
  8. [8] Yu. G. Borisovich, B.D. Gelman, A.D. Myshkis, and V.V. Obukhovskii, Multi-valued analysis and operator inclusions, J. Soviet Math 39 (1987) 2772-2811. doi: 10.1007/BF01127054 Zbl0725.54015
  9. [9] Ph. Bénilan, Solutions intégrales d'équations d'évolution dans un espace de Banach, C.R. Acad. Sci. Paris (A-B) 274 (1972) A47-A50. Zbl0246.47068
  10. [10] P. Benilan and H. Brezis, Solutions faibles d'équations d'évolution dans les espaces de Hilbert, Ann. Inst. Fourier (Grenoble) 22 (2) (1972) 311-329. doi: 10.5802/aif.421 Zbl0226.47034
  11. [11] D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Israel J. Math 108 (1998) 109-138. doi: 10.1007/BF02783044 Zbl0922.47048
  12. [12] D. Bothe, Nonlinear Evolutions in Banach Spaces - Existence and Qualitative Theory with Applications to Reaction-Diffusion Systems, Habilitation thesis (Univ. of Paderborn, 1999). 
  13. [13] J.F. Couchouron and M. Kamenskii, A unified topological point of view for integro-differential inclusions and optimal control. (J.Andres, L. Górniewicz and P. Nistri eds.), Lecture Notes in Nonlinear Anal. 2 (1998) 123-137. Zbl1105.45005
  14. [14] J.-F. Couchouron and M. Kamenskii, An abstract topological point of view and a general averaging principle in the theory of differential inclusions, Nonlinear Anal. (A) 42 (6) (2000) 1101-1129. doi: 10.1016/S0362-546X(99)00181-9 Zbl0972.34049
  15. [15] J. Diestel, W.M. Ruess, and W. Schachermayer, On weak compactness in L¹(μ,X), Proc. Amer. Math. Soc. 118 (2) (1993) 447-453. 
  16. [16] J. Diestel and J.J. Uhl, Jr., Vector measures, American Mathematical Society (Providence, R.I., 1977). doi: 10.1090/surv/015. Zbl0369.46039
  17. [17] L. Górniewicz, A. Granas and W. Kryszewski, Sur la méthode de l'homotopie dans la théoorie des point fixes pour les applications multivoques, Partie 2: L 'indiee dans les ANRs compaetes, Comptes Rendus de l'Aeadémie des Sciences, Paris 308 (1989) 449-452. 
  18. [18] L. Górniewicz, Topological fixed point theory of multivalued mappings (Kluwer Academic Publishers, 1999). doi: 10.1007/978-94-015-9195-9 Zbl0937.55001
  19. [19] S. Gutman, Evolutions governed by m-accretive plus compact operators, Nonlinear Anal. 7 (7) (1983) 707-715. doi: 10.1016/0362-546X(83)90027-5 Zbl0518.34055
  20. [20] T. Kato, Nonlinear evolution equations, Proc. Sympos. Appl. Math 17 (1965) 50-67. doi: 10.1090/psapm/017/0184099 
  21. [21] N. Halidias and N.S. Papageorgiou, Nonlinear boundary value problems with maximal monotone terms, Aequationes Math 59 (2000) 93-107. doi: 10.1007/PL00000131 Zbl0947.34009
  22. [22] M.I. Kamenskii, V.V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, Number 7 in de Gruyter Series in Nonlinear Analysis and Applications, de Gruyter (Berlin, 2001). Zbl0988.34001
  23. [23] N.S. Papageorgiou, On multivalued evolution equations and differential inclusions in Banach spaces, Comment. Math. Univ. St. Paul 36 (1) (1987) 21-39. Zbl0641.47052
  24. [24] A. Pazy, Initial value problems for nonlinear differential equations in Banach spaces, in: Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. IV (Paris, 1981/1982), volume 84 of Res. Notes in Math. (Pitman, Boston, MA, 1983), 154-172. 
  25. [25] A. Pazy, Semigroups of linear operators and applications to partial differential equations (Springer-Verlag, New York, 1983). doi: 10.1007/978-1-4612-5561-1 Zbl0516.47023
  26. [26] Jan Prüss, On semilinear evolution equations in Banach spaces, J. Reine Angew. Math 303/304 (1978) 144-158 Zbl0398.34057
  27. [27] A. Tolstonogov, Differential inclusions in a Banach space (Kluwer Academic Publishers, 2000). doi: 10.1007/978-94-015-9490-5 Zbl1021.34002
  28. [28] I.I. Vrabie, Compactness methods for nonlinear evolutions (Longman, Harlow, 1987) 
  29. [29] I.I. Vrabie, Periodic solutions for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc. 109 (3) (1990) 653-661. doi: 10.1090/S0002-9939-1990-1015686-4 Zbl0701.34074

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.