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 u1(t) and u2(t) the function g(t)=‖u1(t)−u2(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⩾0.
Bibliography: 26 titles.
Citation:
V. V. Zhikov, “Monotonicity in the theory of almost periodic solutions of nonlinear operator equations”, Math. USSR-Sb., 19:2 (1973), 209–223
\Bibitem{Zhi73}
\by V.~V.~Zhikov
\paper Monotonicity in the theory of almost periodic solutions of nonlinear operator equations
\jour Math. USSR-Sb.
\yr 1973
\vol 19
\issue 2
\pages 209--223
\mathnet{http://mi.mathnet.ru/eng/sm3007}
\crossref{https://doi.org/10.1070/SM1973v019n02ABEH001746}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=341221}
\zmath{https://zbmath.org/?q=an:0259.34071}
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https://www.mathnet.ru/eng/sm3007
https://doi.org/10.1070/SM1973v019n02ABEH001746
https://www.mathnet.ru/eng/sm/v132/i2/p214
This publication is cited in the following 5 articles:
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
D. N. Cheban, “Bounded solutions of linear almost periodic differential equations”, Izv. Math., 62:3 (1998), 581–600
A. A. Pankov, “Boundedness and almost periodicity in time of solutions of evolutionary variational inequalities”, Math. USSR-Izv., 20:2 (1983), 303–332
A. A. Pankov, “Bounded and almost periodic solutions of evolutionary variational inequalities”, Math. USSR-Sb., 36:4 (1980), 519–533
V. V. Zhikov, B. M. Levitan, “Favard theory”, Russian Math. Surveys, 32:2 (1977), 129–180
Citation in format AMSBIB
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