Geodesic
Periodic solutions for nonlinear evolution inclusions
Archivum mathematicum, Tome 32 (1996) no. 3, pp. 195-209 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this paper we prove the existence of periodic solutions for a class of nonlinear evolution inclusions defined in an evolution triple of spaces $(X,H,X^{*})$ and driven by a demicontinuous pseudomonotone coercive operator and an upper semicontinuous multivalued perturbation defined on $T\times X$ with values in $H$. Our proof is based on a known result about the surjectivity of the sum of two operators of monotone type and on the fact that the property of pseudomonotonicity is lifted to the Nemitsky operator, which we prove in this paper.
In this paper we prove the existence of periodic solutions for a class of nonlinear evolution inclusions defined in an evolution triple of spaces $(X,H,X^{*})$ and driven by a demicontinuous pseudomonotone coercive operator and an upper semicontinuous multivalued perturbation defined on $T\times X$ with values in $H$. Our proof is based on a known result about the surjectivity of the sum of two operators of monotone type and on the fact that the property of pseudomonotonicity is lifted to the Nemitsky operator, which we prove in this paper.
Classification : 34A60, 34C25, 34G20, 47H15, 47N20
Keywords: evolution triple; compact embedding; pseudomonotone operator; demicontinuity; coercive operator; dominated convergence theorem 
@article{ARM_1996_32_3_a4,
     author = {Kandilakis, Dimitrios A. and Papageorgiou, Nikolaos S.},
     title = {Periodic solutions for nonlinear evolution inclusions},
     journal = {Archivum mathematicum},
     pages = {195--209},
     year = {1996},
     volume = {32},
     number = {3},
     mrnumber = {1421856},
     zbl = {0908.34043},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_1996_32_3_a4/}
}
TY  - JOUR
AU  - Kandilakis, Dimitrios A.
AU  - Papageorgiou, Nikolaos S.
TI  - Periodic solutions for nonlinear evolution inclusions
JO  - Archivum mathematicum
PY  - 1996
SP  - 195
EP  - 209
VL  - 32
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/ARM_1996_32_3_a4/
LA  - en
ID  - ARM_1996_32_3_a4
ER  - 
%0 Journal Article
%A Kandilakis, Dimitrios A.
%A Papageorgiou, Nikolaos S.
%T Periodic solutions for nonlinear evolution inclusions
%J Archivum mathematicum
%D 1996
%P 195-209
%V 32
%N 3
%U http://geodesic.mathdoc.fr/item/ARM_1996_32_3_a4/
%G en
%F ARM_1996_32_3_a4
Kandilakis, Dimitrios A.; Papageorgiou, Nikolaos S. Periodic solutions for nonlinear evolution inclusions. Archivum mathematicum, Tome 32 (1996) no. 3, pp. 195-209. http://geodesic.mathdoc.fr/item/ARM_1996_32_3_a4/

[1] Ash, R.: Real Analysis and Probability. Academic Press, New York (1972). | MR

[2] Becker, R. I.: Periodic solutions of semilinear equations of evolution of compact type. J. Math. Anal. Appl. 82 (1981), 33-48. | MR | Zbl

[3] Brezis, H.: Operateurs Maximaux Monotones. North Holland, Amsterdam (1973). | Zbl

[4] Browder, F.: Existence of periodic solutions for nonlinear equations of evolution. Proc. Nat. Acad. Sci. USA 53 (1965), 1100-1103. | MR | Zbl

[5] Browder, F.: Pseudomonotone operators and nonlinear elliptic boundary value problems on unbounded domains. Proc. Nat. Acad. Sci. USA 74 (1977), 2659-2661. | MR

[6] Chang, K.-C.: Variational methods for nondifferentiable functions and their applications to partial differential equations. J. Math. Anal. Appl. 80 (1981), 102-129. | MR

[7] Blasi, F. S., Myjak, J.: On continuous approximations for multifunctions. Pacific J. Math. 123 (1986), 9-31. | MR

[8] Diestel, J., Uhl, J. J.: Vector Measures. Math. Surveys, 15, AMS Providence, Rhode Island (1977). | MR

[9] Goebel, K., Kirk, W.: Topics in Metric Fixed Point Theory. Cambridge Univ. Press, Cambridge (1990). | MR

[10] Gossez, J. P., Mustonen, V.: Pseudomonotonicity and the Leray-Lions condition. Differential and Integral Equations 6 (1993), 37-45. | MR

[11] Hirano, N.: Existence of periodic solutions for nonlinear evolution equations in Hilbert spaces. Proc. AMS 120 (1994), 185-192. | MR | Zbl

[12] Hu, S., Papageorgiou, N. S.: On the existence of periodic solutions for a class of nonlinear evolution inclusions. Bolletino UMI 7-B (1993), 591-605. | MR

[13] Hu, S., Papageorgiou, N. S.: Galerkin approximations for nonlinear evolution inclusions. Comm. Math. Univ. Carolinae 35 (1994), 705-720. | MR

[14] Lions, J. L.: Quelques Methods de Resolution des Problemes aux Limites Nonlineaires. Dunod, Paris (1969).

[15] Papageorgiou, N. S.: Convergence theorems for Banach space valued integrable multifunctions. Inter. J. Math. and Math. Sci. 10 (1987), 433-464. | MR | Zbl

[16] Papageorgiou, N. S.: On measurable multifunctions with applications to random multivalued equations. Math. Japonica 32 (1987), 437-464. | MR | Zbl

[17] Prüss, J.: Periodic solutions for semilinear evolution equations. Nonl. Anal. TMA 3 (1979), 221-235.

[18] Ton, B.-A.: Nonlinear evolution equations in Banach spaces. Proc. AMS 109 (1990), 653-661.

[19] Vrabie, I.: Periodic solutions for nonlinear evolution equations in a Banach space. Proc. AMS 109 (1990), 653-661. | MR | Zbl

[20] Wagner, D.: Survey of measurable selection theorems. SIAM J. Control Opt. 15 (1977), 859-903. | MR | Zbl

[21] Zeidler, E.: Nonlinear Functinal Analysis and its Applications. Springer-Verlag, New York (1990).