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Baskakov, A. G.; Strukova, I. I.; Trishina, Irina A.

Solutions almost periodic at infinity to differential equations with unbounded operator coefficients. (English. Russian original) Zbl 1398.34086

Sib. Math. J. 59, No. 2, 231-242 (2018); translation from Sib. Mat. Zh. 59, No. 2, 293-308 (2018).
Let \(J\) be either \(\mathbb R\) or \(\mathbb R_+\), \(X\) be a complex Banach space and \(A\) be a linear operator on \(X\) which generates a strongly continuous semigroup \(U: \mathbb R_{+}\mapsto X\). Consider the differential equation \[ x'(t)-Ax(t)=\psi(t)\text{ for } t\in J.\eqno(1) \] A continuous function \(x:J\mapsto X\) is called a mild solution to (1) if \[ x(t)=U(t-s)x(s)+\int_s^tU(t-\tau)\psi(\tau)d\tau \] for all \(s,t \in J\) with \(s\leq t\). In this paper, the authors give several, but equivalent definitions of the space of functions almost periodic at infinity, a new class of functions in \(C_{b, u}(J, X)\), the space of bounded, uniformly continuous functions from \(J\) to \(X\).
As the main results, the authors obtain sufficient spectral conditions on \(A\) for the existence of mild solutions of (1) which are almost periodic at infinity.
Reviewer: Lan Nguyen (Bowling Green/KY)

Cited in 7 Documents

MSC:

34G10 Linear differential equations in abstract spaces
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
47D07 Markov semigroups and applications to diffusion processes

Keywords:

functions almost periodic at infinity; generator of semi-group; mild solutions to abstract differential equations
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References:

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