On the Asymptotic Behavior of Generalized Processes, with Applications to Nonlinear Evolution Equations
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چکیده
The invariance principle, introduced by LaSalle [40] and subsequentlygeneralized by Hale [34], gives information on the structure of w-limit sets in dynamical systems possessing a Liapunov function, and the principle and related methods have been used to determine the asymptotic behavior of solutions to a wide variety of evolution equations (see Kefs. [4, 10, 17-19, 23-29, 34, 48, 51, 55, 561). The principle has been extended by Dafermos [24] to compact processes, a special class of nonautonomous systems, including, in particular, dynamical and asymptotically dynamical systems, periodic, almost periodic, asymptotically periodic, and asymptotically almost periodic processes. In this paper we describe and apply some modified versions of the invariance principle for a class of nonautonomous systems which we call generulized processes. A generalized process is a natural extension of the concept of a process to evolutionary systems whose solutions for given initial data are not, or are not known to be, unique. ilside from treating nonuniqueness, this paper significantly weakens two hypotheses which are customarily made in connection with the invariance principle, namely, that the Liapunov function I’ be (i) continuous with respect to convergence in the phase space, and (ii) nonincreasing along solution paths. The need for weakening (i) ma! be seen from the problem of proving that all weak solutions zc(.v, t) of
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