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Existence and asymptotic stability of periodic solution for evolution equations with delays. (English) Zbl 1233.34028

The author studies the existence and asymptotic stability of a time periodic solution for the evolution equation

${u}^{\text{'}}\left(t\right)+Au\left(t\right)=F\left(t,u\left(t\right)\right),u\left(t-{\tau }_{1}\right),u\left(t-{\tau }_{2}\right),\cdots ,u\left(t-{\tau }_{n}\right)\right),\phantom{\rule{4pt}{0ex}}t\in ℝ,$

where $A:D\left(A\right)\subset H\to H$ is a positive-definite selfadjoint operator on a Hilbert space $H$, $F:ℝ×{H}^{n+1}\to H$ is a nonlinear mapping which is $\omega$-periodic in $t$, and ${\tau }_{1},{\tau }_{2},\cdots ,{\tau }_{n}$ are positive constants.

Essential conditions on the nonlinearity are given, which guarantee that the equation has co-periodic solutions or an asymptotically stable co-periodic solution. The author improves and extends some previous results on the given topic by a different method. In this way, some milder conditions can be used, and the specific condition $n\le 3$ can be deleted. The results are based on analytic semigroups and on an integral inequality of Bellman type with delays. At the end, periodic solutions of delay parabolic equations are treated.

MSC:
 34K13 Periodic solutions of functional differential equations 34K30 Functional-differential equations in abstract spaces 47D06 One-parameter semigroups and linear evolution equations 34K20 Stability theory of functional-differential equations 35R10 Partial functional-differential equations