Strict solutions to nonlinear hyperbolic neutral differential equations (Q1809028)

By using the theory of integrated semigroups, conditions for existence, uniqueness, and regularity of solutions to the Cauchy problem \(x_0=\varphi\in C_E:=C([-r,0],E) \) for the neutral functional-differential equation \[ [x(t) - Lx_t]' = A_0 x(t) + F(t,x_t),\qquad t\geq 0,\tag \(*\) \] are obtained. Herein, \(E\) is a Banach space, \(F:[0,T]\times C_E \to E\), and \(x_t(\theta):=x(t+\theta)\) for all \(x\in C([-r,T],E)\), \(\theta\in[-r,0]\), and \(t\in[0,T]\). The operator \(A_0\) is assumed to satisfy the Hille-Yosida condition on \(E\), without being densely defined. The linear and continuous operator \(L:C_E\to E\) is supposed to have a range in \(D(A_0)\) and be defined as \[ Lx=L_0x + \sum_{j=1}^nB_jx(-h_j),\qquad(x\in C_E), \] where \(L: C_E\to E\) is linear and continuous, \(\|L_0\|<1\), \(0<h_1<\dots<h_n=r\) are given numbers, and \(B_j \in {\mathcal L} (E)\) for \(j=1,2,\dots,n\). Assuming the Lipschitz condition on the mapping \(F(t,\cdot):C_E \to E\) (with \(t\) fixed), the authors prove that, for \(\varphi\in C_E\) such that \(\varphi(0)\in \overline{D(A_0)}\), the above-stated Cauchy problem for Equation \((*)\) has a unique integral solution, which is representable in terms of the integrated semigroup generated by the operator \(A_0\).