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Zbl 1058.34106
Liang, Jin; Xiao, Ti-Jun
Solvability of the Cauchy problem for infinite delay equations. (English)
[J] Nonlinear Anal., Theory Methods Appl. 58, No. 3-4, A, 271-297 (2004). ISSN 0362-546X

By using topological methods, expressed in terms of the Kuratowski measure of noncompactness, the authors prove several existence results on mild solutions for the infinite delay functional-integral equation $$u(t)=g(t)+\int_{\sigma}^tf(t,s,u(s),u_s)\,ds,\quad \sigma\leq t\leq T, \qquad u_\sigma=\varphi,$$ and semilinear functional-differential equations of the form $$u'(t)=Au(t)+f(t,u(t),u_t),\quad 0\leq t\leq T, \qquad u_0=\varphi,$$ with $\varphi\in {\cal B}$. Here, $\cal{B}$ is an admissible function space, i.e., a suitable chosen subspace of functions from $(-\infty,\sigma\,]$ to $X$, with $X$ a Banach space, $\varphi\in \cal{B}$, $x_t(s)=x(t+s)$, $f$ is either in $C([\,\sigma,T\,]\times [\,\sigma,T\,]\times X\times\cal{B};X)$ or in $C([\,\sigma,T\,]\times X\times\cal{B};X)$ and $A:D(A)\subset X\to X$ is a linear operator. An extension to the case when $A$ may depend on $t$ as well is considered, and an application to a functional integro-differential equation of Schrödinger type is included.
[Ioan I. Vrabie (Iaşi)]

MSC 2000:: *34K30 Functional-differential equations in abstract spaces
47D62 Integrated semigroups
35Q55 NLS-like (nonlinear Schroedinger) equations
35K05 Heat equation

Keywords: delay equation; Cauchy problem; mild solution; local E-existence family; integrated operator semigroup; evolution family; Schrödinger-type equation

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Zentralblatt MATH Berlin [Germany]