Existence and uniqueness of solutions for delay evolution equations of second order in time. (Q1405312)

The paper deals with the following second-order delay evolution equation \[ u''(t) + A(t)u(t) + B(t, u'(t)) = F(t, u_t, u'_t) + f(t), \;\;t \in (0,T), \] \[ u(t)= \psi(t) , \;\;t \in [-h,0], \] where \ \(u\in C([-h,T];V)\),\ \(u'\in C([-h,T];H)\;\cap \;L^p([0,T];W)\), \ \(f\in L^{p'}([0,T];W^*) \cap L^1([0,T];H)\) \ and \(F:[0,T]\times C([-h,0];V) \times C([-h,0];H)\to H\) is a nonlinear operator, eventually depending on spatial derivatives of \(u\) and \(u'\), too. Here, \(V, H\) are real Hilbert spaces, with \(H\) separable, and \(W\) is a reflexive real Banach space, such that \(V\) and \(W\) are dense in \(H\) and the injections of \(V\) and \(W\) into \(H\) are continuous. The authors prove existence and uniqueness theorems and provide two examples of applications of their results, which extend previous ones by \textit{M. Artola} in [Ann. Sci. Éc. Norm. Supér., IV Sér., 2, 137-253 (1969; Zbl 0179.19303)]. The approach is of variational type.