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

V. V. Zhikov

Sbornik. Mathematics, Tome 19 (1973) no. 2, pp. 209-223

Résumé

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.