Asymptotic properties of mild solutions of nonautonomousevolution equations with applications to retarded differential equations

Abstract

We investigate the asymptotic properties of the inhomogeneous nonautonomous evo-is a Hille-Yosida operator on a Banach space X, B(t), t ∈ R, is a family of operators in ᏸ(D(A), X) satisfying certain boundedness and measurability conditions and f ∈ L 1 loc (R, X). The solutions of the corresponding homogeneous equations are represented by an evolution family (U B (t, s)) t≥s . For various function spaces Ᏺ we show conditions on (U B (t, s)) t≥s and f which ensure the existence of a unique solution contained in Ᏺ. In particular, if (U B (t, s)) t≥s is p-periodic there exists a unique bounded solution u subject to certain spectral assumptions on U B (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 (U B (t, s)) t≥s .

Parabolic evolution equations with asymptotically autonomous delay

“…To that purpose we have to establish new representation formulas for the mild solution of the inhomogeneous equation, which are inspired by the evolution semigroup approach and by the papers [13], [14], [15], [26]. Finally, if f and L(·) are, e.g., Hölder continuous in time, then the mild solution u of (1.1) is a classical one and satisfies again (1.7) as we show in Section 5.…”

“…By a standard fix point argument one easily constructs a mild solution which is unique by Gronwall's inequality; see e.g. [12,Thm. 3.2].…”

“…In Section 4 we derive representation formulas for the mild solutions, whereas in Proposition 5.2 its regularity is investigated in the parabolic case. For more results in this context we refer to [10], [13], [14], [15], [25], [26], [27], [34], [36,Chap. 4].…”

Almost Periodicity of Parabolic Evolution Equations with Inhomogeneous Boundary Values

“…These results for autonomous problems were partly extended to the case of time periodic A(·), see e.g. [6], [14], [15], [28]. The case of almost periodic A(·) was studied for special classes of parabolic problems in e.g.…”

On solutions of differential inclusions in homogeneous spaces of functions

“…Let us mention also that for differential equations with unbounded operator coefficients, existence results in certain particular functional spaces were obtained by Yu. Latushkin and S. Montgomery-Smith [14] (see also the recent monograph [5]), T. Naito and Nguen Van Minh [15], G. Gūhring and F. Rābiger [11], the first author [1], and others.…”