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Periodic solutions for nonlinear evolution equations in a Banach space. (English) Zbl 0701.34074
Let A: D(A)$$\subseteq X\to 2^ X$$ be an operator in a real Banach space X with the property that -A generates a compact semigroup, and A-aI is m- accretive for some $$a>0$$. The author proves existence of T-periodic solutions to the nonlinear evolution relation $$u'(t)+Au(t)\ni F(t,u(t))$$ $$(0\leq t<\infty)$$, where $$F: [0,\infty)\times \overline{D(A)}\to X$$ is Carathéodory and satisfies $\lim_{r\to \infty}(1/r)\sup \{\| F(t,u)\|:\;0\leq t<\infty,\quad \| v\| \leq r\}<a.$
Reviewer: J.Appell

MSC:
 34G20 Nonlinear differential equations in abstract spaces 34C25 Periodic solutions to ordinary differential equations 35K55 Nonlinear parabolic equations
Keywords:
nonlinear evolution relation
Full Text:
References:
 [1] Pierre Baras, Compacité de l’opérateur \?\mapsto\? solution d’une équation non linéaire (\?\?/\?\?)+\?\?∋\?, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 23, A1113 – A1116 (French, with English summary). · Zbl 0389.47030 [2] Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. · Zbl 0328.47035 [3] Ronald I. Becker, Periodic solutions of semilinear equations of evolution of compact type, J. Math. Anal. Appl. 82 (1981), no. 1, 33 – 48. · Zbl 0465.34014 [4] Tomas Dominguez Benavides, Generic existence of periodic solutions of differential equations in Banach spaces, Bull. Polish Acad. Sci. Math. 33 (1985), no. 3-4, 129 – 135 (English, with Russian summary). · Zbl 0576.34039 [5] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans un espace de Hilbert, North-Holland, 1973. · Zbl 0252.47055 [6] Felix E. Browder, Existence of periodic solutions for nonlinear equations of evolution, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1100 – 1103. · Zbl 0135.17601 [7] Michael G. Crandall, Nonlinear semigroups and evolution governed by accretive operators, Nonlinear functional analysis and its applications, Part 1 (Berkeley, Calif., 1983) Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc., Providence, RI, 1986, pp. 305 – 337. · Zbl 0637.47039 [8] Klaus Deimling, Periodic solutions of differential equations in Banach spaces, Manuscripta Math. 24 (1978), no. 1, 31 – 44. · Zbl 0373.34032 [9] James H. Lightbourne III, Periodic solutions for a differential equation in Banach space, Trans. Amer. Math. Soc. 238 (1978), 285 – 299. · Zbl 0389.34041 [10] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). · Zbl 0189.40603 [11] Nicolae H. Pavel, Nonlinear evolution operators and semigroups, Lecture Notes in Mathematics, vol. 1260, Springer-Verlag, Berlin, 1987. Applications to partial differential equations. · Zbl 0626.35003 [12] J. Prüss, Periodic solutions of semilinear evolution equations, Nonlinear Anal. T.M.A. 3 (1979), 601-612. · Zbl 0419.34061 [13] Ioan I. Vrabie, The nonlinear version of Pazy’s local existence theorem, Israel J. Math. 32 (1979), no. 2-3, 221 – 235. · Zbl 0406.34064 [14] Ioan I. Vrabie, Compactness methods and flow-invariance for perturbed nonlinear semigroups, An. Ştiinţ. Univ. ”Al. I. Cuza” Iaşi Secţ. I a Mat. (N.S.) 27 (1981), no. 1, 117 – 125. · Zbl 0463.34054 [15] I. I. Vrabie, Compactness methods for nonlinear evolutions, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 32, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. With a foreword by A. Pazy. · Zbl 0721.47050 [16] -, Periodic solutions of nonlinear evolution equations in a Hilbert space, in preparation.
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