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Continuous selections of solution sets to second order evolution equations. (English) Zbl 1114.35312

Summary: We prove the existence of a continuous selection of the multivalued map \((\xi,\eta)\to A_F(\xi,\eta)\), where \(A_F(\xi,\eta)\) is the set of all mild solutions of the Cauchy problem \[ x''\in Ax+F(t,x),\;x(0)=\xi,\;x(0)=\eta \] assuming that \(F\) is Lipschitzian with respect to \(x\) and \(A\) is the infinitesimal generator of a strongly cosine family of linear operators on a Banach space \(E\).

MSC:

35G25 Initial value problems for nonlinear higher-order PDEs
47D09 Operator sine and cosine functions and higher-order Cauchy problems
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