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evolution triple; compact embedding; pseudomonotone operator; demicontinuity; coercive operator; dominated convergence theorem
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.
[1] Ash, R.: Real Analysis and Probability. Academic Press, New York (1972). MR 0435320
[2] Becker, R. I.: Periodic solutions of semilinear equations of evolution of compact type. J. Math. Anal. Appl. 82 (1981), 33-48. MR 0626739 | Zbl 0465.34014
[3] Brezis, H.: Operateurs Maximaux Monotones. North Holland, Amsterdam (1973). Zbl 0252.47055
[4] Browder, F.: Existence of periodic solutions for nonlinear equations of evolution. Proc. Nat. Acad. Sci. USA 53 (1965), 1100-1103. MR 0177295 | Zbl 0135.17601
[5] Browder, F.: Pseudomonotone operators and nonlinear elliptic boundary value problems on unbounded domains. Proc. Nat. Acad. Sci. USA 74 (1977), 2659-2661. MR 0445124
[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 0614246
[7] Blasi, F. S., Myjak, J.: On continuous approximations for multifunctions. Pacific J. Math. 123 (1986), 9-31. MR 0834135
[8] Diestel, J., Uhl, J. J.: Vector Measures. Math. Surveys, 15, AMS Providence, Rhode Island (1977). MR 0453964
[9] Goebel, K., Kirk, W.: Topics in Metric Fixed Point Theory. Cambridge Univ. Press, Cambridge (1990). MR 1074005
[10] Gossez, J. P., Mustonen, V.: Pseudomonotonicity and the Leray-Lions condition. Differential and Integral Equations 6 (1993), 37-45. MR 1190164
[11] Hirano, N.: Existence of periodic solutions for nonlinear evolution equations in Hilbert spaces. Proc. AMS 120 (1994), 185-192. MR 1174494 | Zbl 0795.34051
[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 1244409
[13] Hu, S., Papageorgiou, N. S.: Galerkin approximations for nonlinear evolution inclusions. Comm. Math. Univ. Carolinae 35 (1994), 705-720. MR 1321241
[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 0896595 | Zbl 0619.28009
[16] Papageorgiou, N. S.: On measurable multifunctions with applications to random multivalued equations. Math. Japonica 32 (1987), 437-464. MR 0914749 | Zbl 0634.28005
[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 1015686 | Zbl 0701.34074
[20] Wagner, D.: Survey of measurable selection theorems. SIAM J. Control Opt. 15 (1977), 859-903. MR 0486391 | Zbl 0427.28009
[21] Zeidler, E.: Nonlinear Functinal Analysis and its Applications. Springer-Verlag, New York (1990).
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