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On stabilization of a solution of the Cauchy problem for a first-order differential equation in a Banach space. (English. Russian original) Zbl 0796.34040
Russ. Acad. Sci., Dokl., Math. 46, No. 2, 245-247 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 326, No. 2, 224-226 (1992).
The authors consider the initial value problem \(v(o)= v_ 0\) for the differential equation \(dv/dt= Av\), \(t>0\), where \(v\) belongs to a Banach space \(E\) and \(A\) is the generator of a strongly continuous cosine function. The necessary and sufficient conditions for existence of the limit \(\lim_{t\to+\infty} v(t)\) are formulated in this note.

34G10 Linear differential equations in abstract spaces
34C11 Growth and boundedness of solutions to ordinary differential equations
35K99 Parabolic equations and parabolic systems
35Q05 Euler-Poisson-Darboux equations