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Periodic solutions of second order differential equations in Banach spaces. (English) Zbl 1104.34041
The authors consider the maximal regularity for the second order periodic boundary value problem $$\gather u''(t)-aAu(t)-\alpha Au'(t)=f(t), \ 0\le t\le 2\pi,\\ u(0)=u(2\pi),\ u'(0)=u'(2\pi),\endgather$$ on a Banach space $X$. Necessary and sufficient conditions for the existence and uniqueness of periodic solutions in the spaces $L_{2\pi}^{p}({\Bbb R},X)$ ($1<p<\infty$) and $C_{2\pi}^{s}({\Bbb R},X)$ $(0<s<1)$ are given. Moreover, results on mild solutions are also presented. Two types of mild periodic solutions are considered. When the operator $A$ is the generator of a strongly continuous cosine function, characterizations are given in terms of Fourier multipliers and spectral properties of the cosine function. Throughout the paper, examples involving elliptic partial differential operators with Dirichlet boundary conditions are discussed.

34G10Linear ODE in abstract spaces
35L90Abstract hyperbolic equations
47D06One-parameter semigroups and linear evolution equations
47D09Operator sine and cosine functions and higher-order Cauchy problems
47N20Applications of operator theory to differential and integral equations
47F05Partial differential operators
35L70Nonlinear second-order hyperbolic equations
35G20General theory of nonlinear higher-order PDE
Full Text: DOI
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