General decay for a viscoelastic equation of variable coefficients with a time-varying delay in the boundary feedback and acoustic boundary conditions

Boukhatem, Yamna; Benabderrahmane, Benyattou

doi:10.1016/s0252-9602(17)30084-x

Cited by 13 publications

(8 citation statements)

References 25 publications

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“…In this example, we can show that h not be necessarily of exponential or polynomial decay but under general assumption on the kernel function h which gives a much larger class of functions h , the uniform stability of the system () is established with an explicit formula of the decay rates of the energy. For more examples of kernel functions and the decay rates of the energy, see other studies 13,18,27 …”

Section: Stability Resultsmentioning

confidence: 99%

“…Recently, there is a different result according to the general decay for a several problems with internal or boundary feedback and for constant or variable delay. For instance, Boukhatem and Benabderrahmane 27 considered a variable coefficient viscoelastic equation with a time‐varying delay in the boundary feedback and acoustic boundary conditions and nonlinear source term. They established a general decay results where the kernel memory satisfies Equation () and

\int_{0}^{+ \infty} \frac{h (s)}{G^{- 1} (- h^{'} (s))} d s + \sup_{s \in ℝ_{+}} \frac{h (s)}{G^{- 1} (- h^{'} (s))} < + \infty,

where G is an increasing positive strictly convex function satisfying some additional properties.…”

Section: Introductionmentioning

confidence: 99%

“…For more examples of kernel functions and the decay rates of the energy, see other studies. 13,18,27…”

mentioning

confidence: 99%

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Optimal decay for second‐order abstract viscoelastic equation in Hilbert spaces with infinite memory and time delay

Chellaoua

Boukhatem

2020

Math Methods in App Sciences

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In this paper, we consider a second-order abstract viscoelastic equation in Hilbert spaces with infinite memory, time delay, and a kernel function h ∶ R + → R + satisfying, for all t ≥ 0, h ′ (t) ≤ − (t)G(h(t)) where and G are functions satisfying some specific properties. For this much larger class of kernel functions and under a suitable conditions, we prove well-posedness of solution by using semi-group theory. Then, we establish an explicit and general decay results of the energy solution by introducing a suitable Lyapunov functional and some properties of the convex functions. Finally, some applications are given. This work improves the previous results with finite memory to infinite memory and without time delay term to those with delay.

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Section: Stability Resultsmentioning

confidence: 99%

\int_{0}^{+ \infty} \frac{h (s)}{G^{- 1} (- h^{'} (s))} d s + \sup_{s \in ℝ_{+}} \frac{h (s)}{G^{- 1} (- h^{'} (s))} < + \infty,

where G is an increasing positive strictly convex function satisfying some additional properties.…”

Section: Introductionmentioning

confidence: 99%

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Optimal decay for second‐order abstract viscoelastic equation in Hilbert spaces with infinite memory and time delay

Chellaoua

Boukhatem

2020

Math Methods in App Sciences

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“…Just as differential equations are called continuous dynamical systems, difference equations are called discrete dynamical systems that play a very important role in describing many problems in the real world. For example, when describing continuous variables such as parameter t , where t T  , mathematical models for many practical problems can be established by differential equations (see, e.g., [1][2][3][4][5][6][7][8][9][10][11] and the references therein). However, many variables in the real world are not necessarily continuous in performancewe call them discrete variables-such as number of times, days, and so on.…”

Section: Introductionmentioning

confidence: 99%

Dynamic Behavior of a Fractional-Type Fuzzy Difference System

et al. 2022

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In this paper, our aim is to study the following fuzzy system: xn+1=Axn−1xn−2+Bxn−3D+Cxn−4, n=0,1,2,⋯, where {xn} is a sequence of positive fuzzy numbers and the initial conditions x−4,x−3,x−2,x−1,x0 and the parameters A,B,C,D are positive fuzzy numbers. Firstly, the existence and uniqueness of positive fuzzy solutions of the fuzzy system are proved. Secondly, the dynamic behavior of the equilibrium points for the fuzzy system are studied by means of the fuzzy sets theory, linearization method and mathematical induction. Finally, the results obtained in this paper are simulated by using the software package MATLAB 2016, and the numerical results not only show the dynamic behavior of the solutions for the fuzzy system, but also verify the effectiveness of the proposed results.

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“…Dai and Yang in [8] considered the same problem in [13] and solved the open problem proposed by Kirane and Said-Houari. Recently, Boukhatem and Benabderrahmane in [5] considered a variable coefficient viscoelastic equation with a time-varying delay in the boundary feedback and acoustic boundary conditions and nonlinear source term. They established a general decay results of the energy via suitable Lyapunov functionals and some properties of the convex functions where the kernel memory satisfies the equation (1.2).…”

Section: Introductionmentioning

confidence: 99%

Stability result for an abstract delayed evolution equation with arbitrary decay in viscoelasticity

Chellaoua¹,

Boukhatem²

2019

Third International Conference of Mathematical Sciences (Icms 2019)

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The paper is concerned with a second-order abstract semilinear evolution equation with infinite memory and time delay. With the help of the semigroup arguments and under suitable conditions on initial data and the kernel memory function, we state and prove the global existence of solution. Then, we establish the decay rates of the energy using the multiplier method by defining a suitable Lyapunov functional. This work extends previous works with time delay for a much wider class of kernels. We give also some applications to illustrate our results.

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General decay for a viscoelastic equation of variable coefficients with a time-varying delay in the boundary feedback and acoustic boundary conditions

Cited by 13 publications

References 25 publications

Optimal decay for second‐order abstract viscoelastic equation in Hilbert spaces with infinite memory and time delay

Optimal decay for second‐order abstract viscoelastic equation in Hilbert spaces with infinite memory and time delay

Dynamic Behavior of a Fractional-Type Fuzzy Difference System

Stability result for an abstract delayed evolution equation with arbitrary decay in viscoelasticity

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