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Time-dependent perturbation for cosine families in Banach spaces. (English) Zbl 0619.47037

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.

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
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