Abstract
This paper is devoted to the study of global attractors for problems with state-dependent impulses and possible nonuniqueness of solutions. We provide the criteria under which there exists the global attractor, being on one hand an invariant set, and on the other hand given by the difference of the minimal compact attracting set and the impulsive set M. The new condition (T) used to get the global attractor invariance is discussed and compared with other conditions used in literature for impulsive problems. The theory is illustrated by several examples.
Highlights
In the last 20 years, a lot of interest has been attracted to dynamical systems approach to evolutionary problems where the uniqueness of solutions is unknown, or the solution is known to be nonunique for a given initial data
The results are complemented by the discussion on the continuity of the impact time function, the comparison of our condition (T ) with other conditions used for invariance of the attractor for impulsive systems, the criteria for asymptotic compactness, and several examples of impulsive dynamical systems without uniqueness
In addition to them, we obtain the result that the set A = Ac\M is attracting, and second, similar as [20,30,31] we provide the criteria for the global attractor invariance but we show that the condition (24) of [20] is not needed for this invariance which significantly broadens the class of problems, for which our theory works with respect to that of [20,30,31], cf
Summary
In the last 20 years, a lot of interest has been attracted to dynamical systems approach to evolutionary problems where the uniqueness of solutions is unknown, or the solution is known to be nonunique for a given initial data. The results are complemented by the discussion on the continuity of the impact time function, the comparison of our condition (T ) with other conditions used for invariance of the attractor for impulsive systems, the criteria for asymptotic compactness, and several examples of impulsive dynamical systems without uniqueness. In addition to them, we obtain the result that the set A = Ac\M is attracting, and second, similar as [20,30,31] we provide the criteria for the global attractor invariance but we show that the condition (24) of [20] is not needed for this invariance which significantly broadens the class of problems, for which our theory works with respect to that of [20,30,31], cf Example 7.8 in Sect. The comparison of our condition (T ) used to obtain the invariance with the other conditions used in literature is contained in Sect. 7, and some examples which illustrate the constructed theory are presented in last Sect. 8
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