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Forward Omega Limit Sets of Nonautonomous Dynamical Systems DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2020065 DYNAMICAL SYSTEMS SERIES S Volume 13, Number 4, April 2020 pp. 1103{1114 FORWARD OMEGA LIMIT SETS OF NONAUTONOMOUS DYNAMICAL SYSTEMS Hongyong Cui, Peter E. Kloeden and Meihua Yang School of Mathematics and Statistics, and Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074, China Dedicated to J¨urgen Scheurle on his 65th birthday. Abstract. The forward !-limit set !B of a nonautonomous dynamical system ' with a positively invariant absorbing family B = fB(t); t 2 Rg of closed and bounded subsets of a Banach space X which is asymptotically compact is shown to be asymptotically positive invariant in general and asymptotic negative invariant if ' is also strongly asymptotically compact and eventually continuous in its initial value uniformly on bounded time sets independently of the initial time. In addition, a necessary and sufficient condition for a '- invariant family A = fA(t); t 2 Rg in B of nonempty compact subsets of X to be a forward attractor is generalised to this context. 1. Introduction. Omega limit sets play a fundamental role in characterising the behaviour of dynamical systems. Indeed, the attractor of an autonomous dynamical system is an omega limit set. In such systems the omega limit sets are invariant. The situation is more complicated in nonautonomous dynamical systems because the behaviour now depends on the actual time and not on the the time that has elapsed since starting. This leads to new concepts of nonautonomous attractors that consist of families of compact subsets. Two kinds have been distinguished, pullback attractors which depend on information from the distant past and forward attractors which depend on the future behaviour of the system. The constituent subsets depend on time and are mapped into each other by the system, i.e., are invariant in a new nonautonomous sense. Vishik [15] also proposed another kind of nonautonomous attractor, which he called a uniform attractor. It is essentially a forward omega limit set which attracts bounded sets uniformly in initial times. No claim about its invariance was made by Vishik, but see [1,2,4,7] for a later discussion for skew product semiflows. In addition, the convergence uniformly in all initial times is far too strong, because forward attractor is concerned only about what happens in the distant future and not in the past. 2010 Mathematics Subject Classification. Primary: 35B40; Secondary: 35B41, 37L30. Key words and phrases. Nonautonomous dynamical system, forward attracting set, omega limit sets, uniform attractor, asymptotic positive invariance, asymptotic negative invariance, strong asymptotic compactness, upper semi continuous dependence on parameters. HC was partially funded by China Postdoctoral Science Foundation 2017M612430. PEK and MY were partially supported by the Chinese NSF grant 11571125. 1103 1104 HONGYONG CUI, PETER E. KLOEDEN AND MEIHUA YANG Recently, Kloeden & Yang [12] showed that the forward omega limit set !B of a nonautonomous dynamical system ' is asymptotic positive invariant for nonau- tonomous difference equations with a positively invariant compact absorbing set B. This concept has been long known in the differential equations literature [8, 13]. Also in [9] it was shown that !B is also asymptotic negative invariant provided ' is eventually continuous in its initial value inside the compact absorbing set B uniformly on bounded time sets independently of the initial time. These results were used to show the upper semi continuous dependence of !B on parameters. These results are generalised here to a nonautonomous dynamical system ' with a positively invariant absorbing family B = fB(t); t 2 Rg of closed and bounded subsets of X. This requires a successive strengthening of assumptions on com- pactness, such as asymptotic compactness and strong asymptotic compactness, and some eventual uniform properties. In addition, a necessary and sufficient condition in [10] for a '-invariant family A = fA(t); t 2 Rg in B of nonempty compact subsets of X to be a forward attractor is shown to hold when, amongst other things, the absorbing family is strongly asymptotically compact and the process is eventually continuous in its initial value uniformly on bounded time sets independently of the initial time. 2. Dissipative nonautonomous systems. Let distX (·; ·) be the Hausdorff semi- distance between nonempty closed and bounded subsets of a Banach space X and define 2 2 R≥ := f(t; t0) 2 R : t ≥ t0g: Since forward asymptotic behaviour depends on the future of the system and ∗ ∗ not on its past, it can be restricted to initial times t0 ≥ T for some T . In 2 fact, the system need only be defined for such times, i.e., the time set R∗;≥ := 2 ∗ f(t; t0) 2 R : t ≥ t0 ≥ T g. Consider a nonautonomous dynamical system ' on X defined as a process or two-parameter semi-group, i.e., 2 Definition 2.1. A process on the state space X is a mapping ' : R≥ × X ! X, which satisfies the initial value, 2-parameter semigroup and continuity properties: (i) '(t0; t0; x0) = x0 for all t0 2 R and x0 2 X; (ii) '(t2; t0; x0) = ' (t2; t1;'(t1; t0; x0)) for all t0 ≤ t1 ≤ t2 in R and x0 2 X; 2 (iii) the mapping (t; t0; x0) 7! '(t; t0; x0) of R≥ × X into X is continuous. We assume that the nonautonomous system ' is dissipative. In particular, we assume that ' has an absorbing family B = fB(t); t 2 Rg of closed and bounded subsets of X which is '-positively invariant, i.e., '(t; t0;B(t0)) ⊂ B(t); 8t ≥ t0: Assumption 1. There exists a '-positive invariant family B = fB(t); t 2 Rg of closed and bounded subsets of X such that for any bounded subset D of X and every ∗ t0 ≥ T there exists a TD;t0 ≥ 0 for which '(t; t0; x0) 2 B(t) for all t ≥ t0 + TD;t0 ; x0 2 D: In addition, we assume that ' is asymptotically compact on B. FORWARD OMEGA LIMIT SETS 1105 Assumption 2. The process ' is asymptotically compact for the '-positive in- variant family B = fB(t); t 2 Rg of closed and bounded subsets of X if there is a compact subset K of X such that lim distX ('(t; t0;B(t0));K) = 0: t!1 2.1. Omega limit sets. Let B = fB(t); t 2 Rg be '-positive invariant family of closed and bounded subsets of X. The omega limit set starting in B(t0) at time ∗ t0 ≥ T is defined as \ [ !B(t0)(t0) := ' (s; t0;B(t0)): t≥t0 s≥t Clearly, y 2 !B(t0)(t0) if and only if there exist sequences tn ! 1 and xn 2 B(t0) 0 0 such that φ(t ; t ; x ) ! y. Note that ! (t ) ⊂ ! 0 (t ) if t ≤ t . n 0 n B(t0) 0 B(t0) 0 0 0 Lemma 2.2. Assume that ' is asymptotically compact for the family B = fB(t); t 2 Rg, i.e., satisfies Assumption2. Then the omega limit set !B(t0)(t0) starting in B(t0) at time t0 is nonempty and compact with !B(t0)(t0) ⊂ K for each t0 2 R. Moreover, lim distX '(t; t0;B(t0));!B(t )(t0) = 0: (2.1) t!1 0 Proof. Consider a sequence yn = '(tn; t0; bn) 2 '(tn; t0;B(t0)), where tn ! 1. Then distX (yn;K) ≤ distX ('(tn; t0;B(t0));K) ! 0 as n ! 1: Since K is compact there is a kn 2 K such that kyn − knk = distX (yn;K) ! 0 as n ! 1: ¯ Moreover, there is a convergent subsequence knj ! k 2 K. Hence ¯ kynj − kk ≤ distX ynj ;K ! 0 as nj ! 1: ¯ Clearly, k 2 !B(t0)(t0). Thus !B(t0)(t0) is nonempty. Now let !n be a sequence in !B(t0)(t0). Then there exist sequences '(tn; t0; bn) with bn 2 B(t0) and tn ! 1 such that 1 k'(t ; t ; b ) − ! k < ; n 2 : n 0 n n n N Hence, distX (!n;K) ≤ k'(tn; t0; bn) − !nk + distX ('(tn; t0;B(t0));K) 1 ≤ + dist ('(t ; t ;B(t ));K) ! 0 as n ! 1; n X n 0 0 so distX (!n;K) ! 0. Arguing as above, there is a subsequence !nj and a! ¯ 2 K such that !nj ! !¯ as nj ! 1. Moreover,! ¯ 2 !B(t0)(t0), since !B(t0)(t0) is closed. Hence, !B(t0)(t0) is compact. Similarly, let ! 2 !B(t0)(t0). Then there exist sequences '(tn; t0; bn) with bn 2 B(t0) and tn ! 1 such that k'(tn; t0; bn) − !k ! 0 as n ! 1. Moreover, distX (!; K) ≤ k! − '(tn; t0; bn)k + distX ('(tn; t0; bn);K) ≤ k! − '(tn; t0; bn)k + distX ('(tn; t0;B(t0));K) ! 0 as n ! 1; so distX (!; K) = 0. Hence, ! 2 K. Since the choice of ! 2 !B(t0)(t0) was arbitrary, this implies that !B(t0)(t0) ⊂ K for each t0 2 R. 1106 HONGYONG CUI, PETER E. KLOEDEN AND MEIHUA YANG Finally, suppose that the forward convergence (2.1) does not hold. Then there exist an "0 > 0 and a sequence tn ! 1 as n ! 1 such that distX '(tn; t0;B(t0));!B(t0)(t0) ≥ 2"0; n 2 N: The set '(tn; t0;B(t0)) may not be compact, so the supremum defining the distance distX '(tn; t0;B(t0));!B(t0)(t0) = sup distX y; !B(t0)(t0) y2'(tn;t0;B(t0)) need not be attained. However, for each n 2 N there is a point yn 2 '(tn; t0;B(t0)) such that distX '(tn; t0;B(t0));!B(t0)(t0) − "0 ≤ distX yn;!B(t0)(t0) ≤ distX '(tn; t0;B(t0));!B(t0)(t0) ; which means that distX yn;!B(t0)(t0) ≥ "0; n 2 N: Since yn 2 '(tn; t0;B(t0)) and distX (yn;K) ≤ distX ('(tn; t0;B(t0));K) ! 0 as n ! 1; there exists a convergent subsequence ynk ! y¯ 2 K.
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