# 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)
Existence of solutions for double perturbed neutral functional evolution equation. (English) Zbl 1197.34154

Summary: We discuss double perturbed neutral functional evolution equation with infinite delay

$\frac{d}{dt}\left(x\left(t\right)-h\left(t,{x}_{t}\right)\right)=A\left(t\right)x\left(t\right)+f\left(t,{x}_{t}\right)+g\left(t,{x}_{t}\right),\phantom{\rule{1.em}{0ex}}t\in J=\left[0,b\right]\phantom{\rule{2.em}{0ex}}\left(1·1\right)$
${x}_{0}=\phi \in ℬ\phantom{\rule{2.em}{0ex}}\left(1·2\right)$

where $\left\{A\left(t\right):t>0\right\}$ is a family of linear closed operators in a real Banach space $X$ that generates an evolution system $\left\{U\left(t,s\right):0 and $D\left(A\left(t\right)\right)\subseteq X$ is dense in $X$. The history ${x}_{t}:\left(-\infty ,0\right]\to X$, ${x}_{t}\left(\theta \right)=x\left(t+\theta \right)$, belongs to some abstract phase space $ℬ$ defined axiomatically; $g,f,h$ are appropriate functions.

The existence of mild solutions to such equations is obtained by using the theory of the Hausdorff measure of noncompactness and a fixed point theorem, without the compactness assumption on the associated evolution system. Our results improve and generalize some previous results.

##### MSC:
 34K30 Functional-differential equations in abstract spaces 34K40 Neutral functional-differential equations 47N20 Applications of operator theory to differential and integral equations