Some Properties for Exact Generalized Processes

Abstract

In this work, we define an exact generalized process and we establish some results such as monotonicity, compactness, and upper semicontinuity for the multivalued process defined by the exact generalized process. The main result is on compactness, invariance, and attraction properties of ω-limit sets.Keywords Nonautonomous dynamical systems · Pullback attraction · Generalized processes · Multivalued processes IntroductionThe concept of attraction is fundamental to analyze the asymptotic behavior of solutions of partial differential equations or inclusions. See, for example, [1][2][3][4][5][6][7][8][9][10][11]19] for autonomous, [12][13][14][15][16][17] for nonautonomous and [18,19] for stochastic problems. A concept which has been used a lot in the last years to deal with nonautonomous problems is pullback attraction, which attracts the solutions of the problem from −∞, i.e., the initial time goes to −∞ while the final time remains fixed [16,17,20]. The authors in [13] gave results which guarantee the existence of pullback attractors for processes, extending results of the autonomous case. In [21], the authors present abstract results that provide sufficient conditions to the existence of pullback attractors for multivalued processes. Observe that is also possible considering forward attractors for nonautonomous problems [22].

Asymptotic behavior of coupled inclusions with variable exponents

“…Let U G be the multivalued process defined by the generalized process G. We know from [23] that for all t ≥ s in R the map x → U G (t, s)x ∈ P (H × H) is closed, so we obtain from Theorem 18 in [4] the following result Theorem 3.3. If for any t ∈ R there exists a nonempty compact set D(t) which pullback attracts all bounded sets of H × H at time t, then the set A = {A(t)} t∈R with A(t) = B∈B(H×H) ω(t, B), is the unique compact, negatively invariant pullback attracting set which is minimal in the class of closed pullback attracting nonautonomous sets.…”

“…Moreover, we prove that the system (S) is in fact asymptotically autonomous. It makes use of a collection of ideas and results of some recent, distinct previous works [15,22,23,27] of the authors, which are applied here to a new problem to yield interesting new results. Regarding [13,14,15] where an equation and a single inclusion of this type of problems were considered, the coupled system can not be treated in the same way as the single case, the principal additional technical difficulty is to adjust the results considering two inclusions, in this sense, the main technical difficulty appears to prove dissipativity.…”

“…In order to study the asymptotic behavior of the solutions of the system (S) we will work with a multivalued process defined by a generalized process. We will review these concepts which had been considered in [22,23] and can be used in the study of infinite dimensional dynamical systems. Definition 2.6.…”

“…A multivalued process {U G (t, τ )} t≥τ defined by a generalized process G is a family of multivalued operators U G (t, τ ) : 22,23]). Let G be an exact generalized process.…”

On asymptotically autonomous dynamics for multivalued evolution problems

“…It is a well-known result that for each fixed par (t, τ ) with t ≥ τ , the multivalued evolution process associated with problem (2) U (t, τ ) : X → P (X) is an upper semicontinuous map if for example the multivalued process is defined by an Exact Generalized Process (see Theorem 12.3 in [13]).…”