This article is cited in 5 papers

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.

UDC: 519.4+517+513.88

MSC: Primary 47H15, 34C25, 34G05; Secondary 34H05, 47H10

Received: 21.06.1972

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

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