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Zbl 1197.34154
Ye, Runping; Dong, Qixiang; Li, Gang
Existence of solutions for double perturbed neutral functional evolution equation. (English)
[J] Int. J. Nonlinear Sci. 8, No. 3, 360-367 (2009). ISSN 1749-3889; ISSN 1749-3897/e

Summary: We discuss double perturbed neutral functional evolution equation with infinite delay $$\tfrac{d}{dt}(x(t)-h(t,x_t))=A(t)x(t)+f(t,x_t)+g(t,x_t),\quad t\in J=[0,b]\tag{1.1}$$ $$x_0=\varphi\in{\cal B}\tag{1.2}$$ where $\{A(t):t > 0\}$ is a family of linear closed operators in a real Banach space $X$ that generates an evolution system $\{U(t,s):0 < s \le t <\infty\}$ and $D(A(t))\subseteq X$ is dense in $X$. The history $x_t : (-\infty,0]\to X$, $x_t(\theta)=x(t+\theta)$, belongs to some abstract phase space $\cal B$ 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 2000:: *34K30 Functional-differential equations in abstract spaces
34K40 Neutral equations
47N20 Appl. of operator theory to differential and integral equations

Keywords: evolution equations with infinite delay; evolution system; mild solution; Hausdorff measure of noncompactness; phase space

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Zentralblatt MATH Berlin [Germany]