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 Mat. Sb. (N.S.), 1973, Volume 90(132), Number 2, Pages 214–228 (Mi msb3007)

Monotonicity in the theory of almost periodic solutions of nonlinear operator equations

V. V. Zhikov

Abstract: In a Banach space with a strictly convex norm we consider a nonlinear equation $u'+A(t)u=0$ of general form. Suppose that a “monotonicity” condition is satisfied: for any two solutions $u_1(t)$ and $u_2(t)$ the function $g(t)=\|u_1(t)-u_2(t)\|$ is nonincreasing with respect to $t$; suppose $A(t)$ is almost periodic (in some sense) with respect to $t$.
The basic theorem reads as follows: given strong (weak) continuity of the solutions with respect to the initial conditions and the coefficients, there exists at least one almost periodic solution if there exists a compact (weakly compact) solution on $t\geqslant0$.
Bibliography: 26 titles.

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English version:
Mathematics of the USSR-Sbornik, 1973, 19:2, 209–223

Bibliographic databases:

UDC: 519.4+517+513.88
MSC: Primary 47H15, 34C25, 34G05; Secondary 34H05, 47H10

Citation: V. V. Zhikov, “Monotonicity in the theory of almost periodic solutions of nonlinear operator equations”, Mat. Sb. (N.S.), 90(132):2 (1973), 214–228; Math. USSR-Sb., 19:2 (1973), 209–223

Citation in format AMSBIB
\Bibitem{Zhi73} \by V.~V.~Zhikov \paper Monotonicity in the theory of almost periodic solutions of nonlinear operator equations \jour Mat. Sb. (N.S.) \yr 1973 \vol 90(132) \issue 2 \pages 214--228 \mathnet{http://mi.mathnet.ru/msb3007} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=341221} \zmath{https://zbmath.org/?q=an:0259.34071} \transl \jour Math. USSR-Sb. \yr 1973 \vol 19 \issue 2 \pages 209--223 \crossref{https://doi.org/10.1070/SM1973v019n02ABEH001746} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. V. Zhikov, B. M. Levitan, “Favard theory”, Russian Math. Surveys, 32:1 (1977), 129–180
2. A. A. Pankov, “Bounded and almost periodic solutions of evolutionary variational inequalities”, Math. USSR-Sb., 36:4 (1980), 519–533
3. A. A. Pankov, “Boundedness and almost periodicity in time of solutions of evolutionary variational inequalities”, Math. USSR-Izv., 20:2 (1983), 303–332
4. D. N. Cheban, “Bounded solutions of linear almost periodic differential equations”, Izv. Math., 62:3 (1998), 581–600
5. David N. Cheban, Peter E. Kloeden, Björn Schmalfuß, “Global attractors for $V$-monotone nonautonomous dynamical systems”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2003, no. 1, 47–57
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