Existence and stability of square-mean almost automorphic solution for neutral stochastic evolution equations with Stepanov-like terms on time scales

Dhama, Soniya; Abbas, Syed

doi:10.1007/s13398-018-0547-3

Cited by 14 publications

(6 citation statements)

References 16 publications

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“…Now, we define the operators by where and are the Sobolev spaces (Wang and Agarwal 2014 ; Edmunds and Evans 2018 ). Clearly, it is well known that generate the evolution operators such that for all with and (please see (Dhama and Abbas 2019 ; Wang and Agarwal 2014 )).…”

Section: Examplesmentioning

confidence: 99%

“…( 2021a ), the authors examined the stability results for switched dynamic systems on arbitrary time domain by using the Lyapunov function and time scales theory. However, only a few researchers studied the stability, existence of almost periodic, periodic solutions, and controllability results of abstract equations on arbitrary time domain by using the time scale theory (Dhama and Abbas 2019 ; Kumar and Malik 2019 ; Kumar et al. 2021 ; Wang and Agarwal 2014 ).…”

Section: Introductionmentioning

confidence: 99%

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Existence, stability and controllability results for a class of switched evolution system with impulses over arbitrary time domain

2022

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We establish several qualitative properties of a neutral switched impulsive evolution system on an arbitrary time domain by using the theory of time scales. This is the first attempt for switched evolution systems with impulses in abstract spaces. First, we investigate the existence of a unique solution and Ulam’s type stability results. After that, we establish the total controllability results, i.e., controllability not only with respect to the endpoint of the interval but also on the impulsive points. We transform the controllability problem into a solvability problem of an operator equation. We used the Banach fixed point theory and evolution operator theory to establish these results. To illustrate the effectiveness and implications of the developed results, we provide theoretical and simulated numerical examples for different time domains.

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Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Existence, stability and controllability results for a class of switched evolution system with impulses over arbitrary time domain

2022

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“…Indeed, it is essential to analyze whether the presence of some random terms in the equations of the models may produce a very different behavior of their solutions. Although there exists a wide literature on this topic, see [1,21], [23]- [25].…”

Section: Introductionmentioning

confidence: 99%

A new result on stabilization analysis for stochastic nonlinear affine systems via Gamidov’s inequality

Caraballo

Ezzine²,

Hammami³

2022

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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The Lyapunov approach is one of the most effective and efficient methods for the investigation of the stability of stochastic systems. Several authors analyzed the stability and stabilization of stochastic differential equations via Lyapunov techniques. Nevertheless, few results are concerned with the stability of stochastic systems based on the knowledge of the solution of the system explicitly. The originality of our work is to investigate the problem of stabilization of stochastic perturbed control-bilinear systems based on the explicit solution of the system by using the integral inequalities of the Gronwall type in particular Gamidov's inequality. Namely, under some restrictions on the perturbed term, and based on the method of integral inequalities, we prove that the stochastic system can be stabilized by constant feedback. Further, we study the problem of stabilization of stochastic perturbed control affine systems based on the use of bilinear approximation. Different examples are provided to verify the effectiveness of the proposed results.

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“…With the implementation and development of time scales, since a large number of natural phenomena cannot be presented through classical differential equations and it is inevitably influenced by random factors, hence, the theory of stochastic differential equations has occupied the more and more important position and became an active topic of investigation; see previous studies [8][9][10][11][12] for details. Further, as a fusion of time scales and stochastic differential equations, stochastic dynamic equations driven by Brownian motion on time scales entered the broad view of researchers, and the stability, existence, and uniqueness of mild solutions have been wildly studied; see previous literature [13][14][15][16] and references therein. In recent years, although the research of stochastic dynamic equations have became a quite hot object and focused increasing attention on the topic of almost automorphic stochastic processes and its generalization of a variety of abstract ZHU nonautonomous stochastic nonlinear equations, there are still many equations that have not been explored and are far from enough.…”

Section: Introductionmentioning

confidence: 99%

The doubly weighted pseudo‐almost automorphic mild solutions to a class of stochastic dynamic equations

Zhu

2022

Math Methods in App Sciences

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This paper first obtains the necessary and sufficient condition betweenStepanov-like doubly weighted pseudo-almost automorphic stochastic processes and Stepanov-like asymptotically almost automorphic stochastic processes in a suitable set and, second, establishes the composition theorem based on Δ-measurability for nonequivalent weight functions; these new conclusions acquired perfect and extended some pre-existing related results, which enrich the dynamics of Stepanov-like doubly weighted pseudo-almost automorphic stochastic processes. Finally, for a class of nonautonomous stochastic equations with the coefficients of Stepanov-like doubly weighted pseudo-almost automorphic stochastic processes that satisfies non-Lipschitz conditions, we investigate the existence and uniqueness of the doubly weighted pseudo-almost automorphic mild solutions.

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Existence and stability of square-mean almost automorphic solution for neutral stochastic evolution equations with Stepanov-like terms on time scales

Cited by 14 publications

References 16 publications

Existence, stability and controllability results for a class of switched evolution system with impulses over arbitrary time domain

Existence, stability and controllability results for a class of switched evolution system with impulses over arbitrary time domain

A new result on stabilization analysis for stochastic nonlinear affine systems via Gamidov’s inequality

The doubly weighted pseudo‐almost automorphic mild solutions to a class of stochastic dynamic equations

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