Attracting sets for one class of asymptotically compact systems with pulsed perturbation

The authors consider the pulsed dynamical systems generated by evolutionary processes. The trajectories of these processes undergo the pulsed perturbation when the energy functional reaches some fixed limit value. The generalization of the classical theory of global attractors of infinite dimensional dynamical systems in case of systems with impulse actions is carried out. It is established that for the dissipative pulsed dynamical system generated by the asymptotically compact semigroup, there exists a uniform attractor, i.e., a compact uniformly attracting set, minimal among all such sets in the phase space of the system. The result is applied to the weakly nonlinear wave equation with dissipation, the trajectories of which are subjected to impulsive perturbations upon attainment of a certain fixed subset in the phase space, so called the impulse set.

Existence and Global Asymptotic Behavior of S -Asymptotically ω -Periodic Solutions for Evolution Equation with Delay

This paper is concerned with the abstract evolution equation with delay. Firstly, we establish some sufficient conditions to ensure the existence results for the S -asymptotically periodic solutions by means of the compact semigroup. Secondly, we consider the global asymptotic behavior of the delayed evolution equation by using the Gronwall-Bellman integral inequality involving delay. These results improve and generalize the recent conclusions on this topic. Finally, we give an example to exhibit the practicability of our abstract results.

Inverse source problem for the abstract fractional differential equation

In a Banach space, the inverse source problem for a fractional differential equation with Caputo–Dzhrbashyan derivative is considered. The initial and observation conditions are given by elements from D(A), and the operator function on the right side is sufficiently smooth. Two types of the observation operator are considered: integral and at the final point. Under the assumptions that operator 𝐴 is a generator of positive and compact semigroup the uniqueness, existence and stability of the solution are proved.

Results on Existence and Uniqueness of Solution Of Impulsive Neutral Integro-Differential System

In this work, the solution of the impulsive neutral integro-differential system is analysed. The Du Bois–Reymond’s assumptions on solution variation of piecewise smooth functions are used to establish the existence of only the impulsive term as a solution of the system at the points of discontinuity. The theories of infinitesimal generator of a strongly continuous compact semigroup is used to formulate theorems on existence and uniqueness of system solution, and proves are provided using an approximate piecewise continuous, compact operator and a continuous positive non decreasing function . Results obtained are improvement on the qualitative analysis of impulsive neutral integro-differential system

On the Initial Value Problem of Stochastic Evolution Equations in Hilbert Spaces

The present paper studies the initial value problem of stochastic evolution equations with compact semigroup in real separable Hilbert spaces. The existence of saturated mild solution and global mild solution is obtained under the situation that the nonlinear term satisfies some appropriate growth conditions. The results obtained in this paper improve and extend some related conclusions on this topic. An example is also given to illustrate that our results are valuable.

On the special semigroup “at infinity”

This paper presents an important new technique for studying a particular compact semigroup, N∪{∞}, the one-point compactification of positive integers with usual addition, which is an important semigroup. Indeed, the semigroup N ∪ {∞} is constructed as the quotient semigroup of a particular compact right topological semigroup. In the study of such a semigroup, a major role is played by the substructures called standard oids. For instance, some of the already known results on the structure of N ∪ {∞} are obtained as immediate consequences.

On an Open Problem of Hereditary Property of Green's $\mathcal{H}$-Relation on Compact Semigroups

We give a negative answer to an open problem concerning the hereditary property of Green's [Formula: see text]-relation on a compact semigroup proposed by Carruth and Clark in 1972. In addition, we give a positive answer to this open problem by adding some suitable conditions on the compact semigroup.

A Characterization of Semilinear Dense Range Operators and Applications

We characterize a broad class of semilinear dense range operators given by the following formula, , where , are Hilbert spaces, , and is a suitable nonlinear operator. First, we give a necessary and sufficient condition for the linear operator to have dense range. Second, under some condition on the nonlinear term , we prove the following statement: If , then and for all there exists a sequence given by , such that . Finally, we apply this result to prove the approximate controllability of the following semilinear evolution equation: , where , are Hilbert spaces, is the infinitesimal generator of strongly continuous compact semigroup in , the control function belongs to , and is a suitable function. As a particular case we consider the controlled semilinear heat equation.