Serizawa, Hisamitsu; Watanabe, Michiaki Time-dependent perturbation for cosine families in Banach spaces. (English) Zbl 0619.47037 Houston J. Math. 12, 579-586 (1986). Let A be the infinitesimal generator of a strongly continuous cosine family in a Banach space X. A condition on a linear time-dependent operator B(t) in X is given under which the Cauchy problem in X for \[ (d/dt)^ 2u(t)=(A+B(t))u(t),\quad t\in R \] has a unique solution. The result is applied to the ordinary differential operator of second order \[ a(x)(d/dx)^ 2+b(t,x)(d/dx)+c(t,x),a(x)>0 \] with Neumann boundary condition. Cited in 26 Documents MSC: 47D03 Groups and semigroups of linear operators 47D99 Groups and semigroups of linear operators, their generalizations and applications 35L15 Initial value problems for second-order hyperbolic equations Keywords:time-dependent perturbation; strongly continuous cosine family in a Banach space; time-dependent operator; Cauchy problem; ordinary differential operator of second order; Neumann boundary condition PDFBibTeX XMLCite \textit{H. Serizawa} and \textit{M. Watanabe}, Houston J. Math. 12, 579--586 (1986; Zbl 0619.47037)