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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
References:
 [1] Amann, H.: Periodic solutions of semilinear parabolic equations, Nonlinear analysis: A collection of papers in honor of erich H. Rothe, 1-29 (1978) · Zbl 0464.35050 [2] Becher, R. I.: Periodic solutions of semilinear equations of evolution of compact type, J. math. Anal. appl. 82, 33-48 (1982) · Zbl 0465.34014 · doi:10.1016/0022-247X(81)90223-7 [3] Burton, T. A.; Zhang, B.: Periodic solutions of abstract differential equations with infinite delay, J. differential equations 90, 357-396 (1991) · Zbl 0760.34060 · doi:10.1016/0022-0396(91)90153-Z [4] Hale, J. K.; Lunel, S. M. Verduyn: Introduction to functional differential equations, (1993) [5] Haraux, A.: Nonlinear evolution equations - global behavior of solutions, Lecture notes in math. 841 (1981) · Zbl 0461.35002 [6] Henry, D.: Geometric theory of semilinear parabolic equations, Lecture notes in math. 840 (1981) · Zbl 0456.35001 [7] Li, Y.: Periodic solutions of semilinear evolution equations in Banach spaces, Acta math. Sin. 41, 629-636 (1998) · Zbl 1027.34067 [8] Li, Y.: Existence and uniqueness of periodic solution for a class of semilinear evolution equations, J. math. Anal. appl. 349, 226-234 (2009) · Zbl 1162.35007 · doi:10.1016/j.jmaa.2008.08.019 [9] Liu, J.: Bounded and periodic solutions of finite delays evolution equations, Nonlinear anal. 34, 101-111 (1998) · Zbl 0934.34066 · doi:10.1016/S0362-546X(97)00606-8 [10] Liu, J.: Periodic solutions of infinite delay evolution equations, J. math. Anal. appl. 247, 644-727 (2000) · Zbl 1056.34085 · doi:10.1006/jmaa.2000.6896 [11] Liu, J.: Bounded and periodic solutions of infinite delay evolution equations, J. math. Anal. appl. 286, 705-712 (2003) [12] Liu, Y. C.; Li, Z. X.: Schaefer type theorem and periodic solutions of evolution equations, J. math. Anal. appl. 316, 237-255 (2006) · Zbl 1096.35007 · doi:10.1016/j.jmaa.2005.04.045 [13] Okochi, H.: On the existence of periodic solutions to nonlinear abstract parabolic equations, J. math. Soc. Japan 40, 541-553 (1988) · Zbl 0679.35046 · doi:10.2969/jmsj/04030541 [14] Pazy, A.: Semigroups of linear operators and applications to partial differential equations, (1983) [15] Teman, R.: Infinite-dimensional dynamical systems in mechanics and physics, (1997) [16] Wu, J.: Theory and application of partial functional differential equations, (1996) [17] Xiang, X.; Ahmed, N. U.: Existence of periodic solutions of semilinear evolution equations with time lags, Nonlinear anal. 18, 1063-1070 (1992) · Zbl 0765.34057 · doi:10.1016/0362-546X(92)90195-K [18] Zhu, J. M.; Liu, Y. C.; Li, Z. X.: The existence and attractivity of time periodic solutions for evolution equations with delays, Nonlinear anal. Real world appl. 9, 842-851 (2008) · Zbl 1146.35306 · doi:10.1016/j.nonrwa.2007.01.004