Global uniqueness results for partial functional and neutral functional evolution equations with infinite delay. (English) Zbl 1240.34378

Baghli, Selma; Benchohra, Mouffak

Differ. Integral Equ. 23, No. 1-2, 31-50 (2010).

The paper deals with functional differential evolution equations of the form \[ \frac {\roman d}{{\roman d}\,t}\big [y(t)-g(t,y_t)\big ]=A(t)\,y(t)+f(t,y_t),\quad y_0=\phi \in \mathcal {B}, \] where \(E\) is a Banach space, \(\mathcal {B}\) is an abstract phase space defined in an axiomatic way introduced by J. Hale and J. Kato [Funkc. Ekvacioj, Ser. Int. 21, 11–41 (1978; Zbl 0383.34055)], \(f\: [0,\infty )\times \mathcal B\to E,\) \(\phi \in \mathcal {B},\) \(\{A(t)\}_{0{\leq }t{<}\infty }\) is a family of linear closed operators \(E\to E\) that generates an evolution system of operators \(\{U(t,s)\}_{0{\leq }s{\leq }t{<}\infty }\), \(g\:[0,\infty )\times \mathcal {B}\to E\) and \(y_t(\theta )=y(t+\theta )\) for \(\theta \in (-\infty ,0].\) The authors provide sufficient conditions for the existence of a unique mild solution on \([0,\infty ).\) The main tool is the nonlinear alternative for contraction maps in Fréchet spaces due to M. Frigon and A. Granas [Ann. Sci. Math. Qué. 22, No. 2, 161–168 (1998; Zbl 1100.47514)].

Reviewer: Milan Tvrdý (Praha)

MSC:

34K30	Functional-differential equations in abstract spaces
34K40	Neutral functional-differential equations

Keywords:

functional differential evolution equation; neutral equation; continuation theorem; global uniqueness result; infinite delay

Citations:

Zbl 0383.34055; Zbl 1100.47514