Global existence for second order functional semilinear equations
Global existence of solutions to the initial value problem \(x''(t) = Ax(t) + f(t, x_t)\), \(t \in [0,T]\); \(x_0 = \varphi\), \(x'(0) = \eta\) is proved using the Leray-Schauder principle, where \(A\) is the infinitesimal generator of a strongly continuous cosine family \(\{C(t) : t \in (- \infty, + \infty)\}\) in a Banach space \(X\) with norm \(|\cdot |\) and \(x_t = x(t + s)\), \(t \in [- r,0]\). The main assumptions are that \(f\) satisfies \(|f(t, \varphi) |\leq m(t) \omega (|\varphi |)\), where \(m, \omega\) are continuous nonnegative functions and \(MT \int^T_0 m(s) ds < \int^\infty_c (1/ \omega (s)) ds\), where \(M = \sup \{|C(t) |: t \in [0,T]\}\), \(c = M (|\varphi |+ T |\eta |)\).
scientific article; zbMATH DE number 892562