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Liang, Jin; Xiao, Ti-Jun

Solvability of the Cauchy problem for infinite delay equations. (English) Zbl 1058.34106

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 58, No. 3-4, 271-297 (2004).
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 \(\varphi\in {\mathcal B}\). Here, \(\mathcal{B}\) is an admissible function space, i.e., a suitable chosen subspace of functions from \((-\infty,\sigma\,]\) to \(X\), with \(X\) a Banach space, \(\varphi\in \mathcal{B}\), \(x_t(s)=x(t+s)\), \(f\) is either in \(C([\,\sigma,T\,]\times [\,\sigma,T\,]\times X\times\mathcal{B};X)\) or in \(C([\,\sigma,T\,]\times X\times\mathcal{B};X)\) 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.
Reviewer: Ioan I. Vrabie (Iaşi)

Cited in 29 Documents

MSC:

34K30 Functional-differential equations in abstract spaces
47D62 Integrated semigroups
35Q55 NLS equations (nonlinear Schrödinger 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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\textit{J. Liang} and \textit{T.-J. Xiao}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 58, No. 3--4, 271--297 (2004; Zbl 1058.34106)
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References:

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