On the existence of solutions to wave integrodifferential equations with subdifferential operators (Q1802982)

Let \(H\) be a real Hilbert space, \(\psi,\varphi:H\to\mathbb{R}_ +\) lower semicontinuous proper convex functions and \(\partial\psi\), \(\partial\varphi\) their subdifferentials and let \(a:[0,T]\to\mathbb{R}\) and \(f:[0,T]\times H\to H\) be continuous functions. -- The author studies the initial value problem \[ u''(t)+\partial\psi u(t)+\partial\varphi u(t)+\int^ t_ 0a(t-s)\;\partial\varphi u(s)\;ds\ni f(t,u(t)),\quad t\in[0,T]; \tag{P} \] \[ u(0)=u_ 0,\quad u'(0)=u_ 1, \] and proves that under suitable assumptions on \(\psi,\varphi,a,f\) and \(u_ 0,u_ 1\) there exists at least a generalized solution of (P) in \([0,T]\). The result is obtained through energy estimates of the Yosida approximants of the problem (P). An application is given to a vibrating string problem which takes into account a unilateral constraint and a memory term. The paper generalizes the results proved by the author when \(a\equiv 0\) [ibid. 22, 21-30 (1985; Zbl 0573.35061)].