Abstract inclusions in Banach spaces with boundary conditions of periodic type

Guedda, Lahcene ; Hallouz, Ahmed

Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 34 (2014) no. 2, pp. 229-253.

Résumé

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

Détail

Keywords: measure of noncompactness, condensing operator, nonlinear abstract inclusion, accretive operator, integral solution, nonlinear semigroup

Bibliographie

[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.

[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] 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

[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] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces (Marcel Dekker, 1980)

[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

[7] V. Barbu, Analysis and control of nonlinear infinite-dimensional systems (Academic Press Inc., Boston, 1993)

[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

[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.

[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

[11] D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Israel J. Math 108 (1998) 109-138. doi: 10.1007/BF02783044

[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] 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.

[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

[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] J. Diestel and J.J. Uhl, Jr., Vector measures, American Mathematical Society (Providence, R.I., 1977). doi: 10.1090/surv/015.

[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] L. Górniewicz, Topological fixed point theory of multivalued mappings (Kluwer Academic Publishers, 1999). doi: 10.1007/978-94-015-9195-9

[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

[20] T. Kato, Nonlinear evolution equations, Proc. Sympos. Appl. Math 17 (1965) 50-67. doi: 10.1090/psapm/017/0184099

[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

[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).

[23] N.S. Papageorgiou, On multivalued evolution equations and differential inclusions in Banach spaces, Comment. Math. Univ. St. Paul 36 (1) (1987) 21-39.

[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] 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

[26] Jan Prüss, On semilinear evolution equations in Banach spaces, J. Reine Angew. Math 303/304 (1978) 144-158

[27] A. Tolstonogov, Differential inclusions in a Banach space (Kluwer Academic Publishers, 2000). doi: 10.1007/978-94-015-9490-5

[28] I.I. Vrabie, Compactness methods for nonlinear evolutions (Longman, Harlow, 1987)

[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