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Existence and uniqueness of solution for neutral differential equations with state-dependent delay. (English) Zbl 1492.34078

The authors study the local existence and uniqueness of mild and strict solutions of the abstract system \[ \frac{d}{dt} (x(t) + G(t, x_{\sigma}(t))) = A (x(t) + G(t, x_{\sigma}(t))) + F(t, x_{\gamma}(t)) \] in some Banach space \(X\), subject to the initial condition \(x|_{[-\tau, 0]} = \varphi\). In this equation, \(A\) is the infinitesimal generator of a strongly continuous semi-group on \(X\), the functions \(F, G\) belong to \([0, T] \times X \to X\) and \(x_{\sigma}\) is the delayed solution defined by \(x_{\sigma}(t) := x(t - \sigma(t, x(t)))\) associated to the (bounded) state-dependent delay function \(\sigma: [0, T] \times X \to [0, \tau]\) (and similarly for \(x_{\gamma}\)).
They provide a complex set of conditions under which such results holds, whose effective validation is illustrated on two examples. In the first one, \(X = C(K, \mathbb{R}^n)\) where \(K \subset \mathbb{R}^n\) is compact and \(A:X \to X\) is the bounded operator such that \((A x)(\cdot) = M x(\cdot)\) for some \(M \in \mathbb{R}^{n\times n}\). In the second one, \(X = L^2(\Omega)\) where \(\Omega \subset \mathbb{R}^n\) is an open set with a smooth boundary and \(A : D(A) \subset X \to X\) is unbounded.

MSC:

34K30 Functional-differential equations in abstract spaces
34K40 Neutral functional-differential equations
34K43 Functional-differential equations with state-dependent arguments
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