Asymptotic properties of mild solutions of nonautonomous evolution equations with applications to retarded differential equations

Gabriele Gühring and Frank Räbiger

Abstract and Applied Analysis, 1999, vol. 4, issue 3, 169-194

Abstract: We investigate the asymptotic properties of the inhomogeneous nonautonomous evolution equation (d/dt)u(t) = Au(t) + B(t)u(t) + f(t), t ∈ ℝ, where (A, D(A)) is a Hille‐Yosida operator on a Banach space X, B(t), t ∈ ℝ, is a family of operators in ℒ(D(A)¯,X) satisfying certain boundedness and measurability conditions and f∈L loc 1(ℝ,X). The solutions of the corresponding homogeneous equations are represented by an evolution family (UB(t,s))t≥s. For various function spaces ℱ we show conditions on (UB(t,s))t≥s and f which ensure the existence of a unique solution contained in ℱ. In particular, if (UB(t,s))t≥s is p‐periodic there exists a unique bounded solution u subject to certain spectral assumptions on UB(p, 0), f and u. We apply the results to nonautonomous semilinear retarded differential equations. For certain p‐periodic retarded differential equations we derive a characteristic equation which is used to determine the spectrum of (UB(t,s))t≥s.

Date: 1999
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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:4:y:1999:i:3:p:169-194

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