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New spectral criteria for almost periodic solutions of evolution equations. (English) Zbl 0982.34074
The authors consider the linear, inhomogeneous integral equation \[ x(t)= U(t,s) x(s)+ \int^t_s U(t,\xi) f(\xi) d\xi,\tag{1} \] where \(f\) is continuous and \((U(t, s))_{t\geq s}\) is a 1-periodic evolutionary process in a complex Banach space. This problem yields results for the problem \[ {dx\over dt}= A(t) x+ f(t),\tag{2} \] where \(A\) is the generator of a \(C_0\)-semigroup and \(f\) is bounded and uniformly continuous with precompact range. The authors prove a spectral decomposition result for bounded uniformly continuous solutions to (1). This theorem is then used to prove the existence of almost-periodic (in the sense of Bohr) solutions with specific spectral properties. A corollary specifies the assumptions, including that \(f\) be almost-periodic, so that if there exists a bounded uniformly continuous solution \(u\) to (1), then there exists an almost-periodic solution \(w\) to (1) such that \(\sigma(w)= \sigma(f)\), where \(\sigma(w)\) is defined to be the closure of \(\{e^{i\xi}: \xi\in \text{sp}(w)\}\) and sp represents the Carleman spectrum. Another corollary specifies the assumptions so that if there exists a bounded uniformly continuous mild solution to (2), then there exists a quasi-periodic mild solution \(w\) with \(\text{sp}(w)= \text{sp}(f)\). The authors close the paper with a number of examples.

34L30 Nonlinear ordinary differential operators
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34G10 Linear differential equations in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
35B15 Almost and pseudo-almost periodic solutions to PDEs
45M15 Periodic solutions of integral equations
45N05 Abstract integral equations, integral equations in abstract spaces
34L05 General spectral theory of ordinary differential operators
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