# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A new theorem on exponential stability of periodic evolution families on Banach spaces. (English) Zbl 1053.47034
Authors’ abstract: We consider a mild solution ${v}_{f}\left(·,0\right)$ of a well-posed inhomogeneous Cauchy problem $\stackrel{˙}{v}\left(t\right)=A\left(t\right)v\left(t\right)+f\left(t\right)$, $v\left(0\right)=0$, on a complex Banach space $X$, where $A\left(·\right)$ is a 1-periodic operator-valued function. We prove that if ${v}_{f}\left(·,0\right)$ belongs to $A{P}_{0}\left({ℝ}_{+},X\right)$, for each $f\in A{P}_{0}\left({ℝ}_{+},X\right)$ then for each $x\in X$ the solution of the well-posed Cauchy problem $\stackrel{˙}{u}\left(t\right)=A\left(t\right)v\left(t\right)$, $u\left(0\right)=x$, is uniformly exponentially stable. The converse statement is also true. [$A{P}_{0}\left({ℝ}_{+},X\right)$ denotes the smallest closed subspace (of the Banach space of $X$-valued, bounded and uniformly continuous functions on ${ℝ}_{+}$) which contains the almost periodic functions as well as those functins which are zero on some interval $\left[0,t\right]$ and identical with an almost periodic function on $\left[t,\infty \right)$]. Our approach is based on the spectral theory of evolution semigroups.
##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 26D10 Inequalities involving derivatives, differential and integral operators 34A35 ODE of infinite order 34D05 Asymptotic stability of ODE 34B15 Nonlinear boundary value problems for ODE 45M10 Stability theory of integral equations 47A06 Linear relations (multivalued linear operators)