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1

Asymptotic analysis for coupled parabolic problem with Dirichlet-Fourier boundary conditions in a thin domain

Dilmi Mohamed

Journal of Mathematics and Applications

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2023

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Vol. 46

163--185

EN

This paper concerns the asymptotic behaviour of the initial boundary value problem of a class of reaction-diffusion systems (coupled parabolic systems) posed in a thin domain with Dirichlet-Fourier boundary conditions. We first prove the existence and uniqueness of the solution to the problem for fixed ε >0 by the Galerkin method. Then, we give the characterization of the limiting behaviour of these solution as the thinness tends to zero.

2

Source term model for elasticity system with nonlinear dissipative term in a thin domain

Dilmi Mohamed, Dilmi Mourad, Boulaaras Salah, Benseridi Hamid

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

437--451

EN

This article establishes an asymptotic behavior for the elasticity systems with nonlinear source and dissipative terms in a three-dimensional thin domain, which generalizes some previous works. We consider the limit when the thickness tends to zero, and we prove that the limit solution u∗ is a solution of a two-dimensional boundary value problem with lower Tresca’s free-boundary conditions. Moreover, we obtain the weak Reynolds-type equation.

3

Nonhomogeneous nonlinear oscillator with damping: asymptotic analysis in continuous and discrete time

Rouhani Behzad Djafari, Piranfar Mohsen Rahimi

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

274--282

EN

We consider the following second order evolution equation modelling a nonlinear oscillator with damping ü(t) +𝛾 ů(t) + Au(t) = f(t), where A is a maximal monotone andα-inverse strongly monotone operator in a real Hilbert space H. With suitable assumptions on 𝛾 and f(t) we show that A-1(0) ≠ ∅, if and only if (SEE) has a bounded solution and in this case we provide approximation results for elements of A-1(0) by proving weak and strong convergence theorems for solutions to (SEE) showing that the limit belongs to A-1(0). As a discrete version of (SEE), we consider the following second order difference equation un+1-un-αn(un-un-1)+λnAun+1 ∋ f(t), where A is assumed to be only maximal monotone (possibly multivalued). By using the results in [Djafari Rouhani B., Khatibzadeh H., On the proximal point algorithm, J. Optim. Theory Appl., 2008, 137, 411-417], we prove ergodic, weak and strong convergence theorems for the sequence un, and show that the limit is the asymptotic center of un and belongs to A−1(0). This again shows that A−1(0) ≠ ∅ if and only if un is bounded. Also these results solve an open problem raised in [Alvarez F., Attouch H., An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping, Set Valued Anal., 2001, 9, 3-11], namely the study of the convergence results for the inexact inertial proximal algorithm. Our paper is motivated by the previous results in [Djafari Rouhani B., Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl., 1990, 147, 465-476; Djafari Rouhani B., Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl., 1990, 151, 226–235; Djafari Rouhani B., Khatibzadeh H., Asymptotic behavior of bounded solutions to some second order evolution systems, Rocky Mountain J. Math., 2010, 40, 1289-1311; Djafari Rouhani B., Khatibzadeh H., A strong convergence theorem for solutions to a nonhomogeneous second order evolution equation, J. Math. Anal. Appl., 2010, 363, 648-654; Djafari Rouhani B., Khatibzadeh H., Asymptotic behavior of bounded solutions to a class of second order nonhomogeneous evolution equations, Nonlinear Anal., 2009, 70, 4369-4376; Djafari Rouhani B., Khatibzadeh H., On the proximal point algorithm, J. Optim. Theory Appl., 2008, 137, 411-417] and significantly improves upon the results of [Attouch H., Maingé P. E., Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects, ESAIM Control Optim. Calc. Var., 2011, 17(3), 836-857], and [Alvarez F., Attouch H., An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping, Set Valued Anal., 2001, 9, 3-11].

4

Oscillatory behavior of higher order nonlinear homogeneous neutral delay dynamic equations with positive and negative coefficients

Panigrahi S., Reddy P. R.

Journal of Applied Analysis

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2018

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Vol. 24, nr 2

139--154

EN

In this paper, we derive some sufficient conditions for the oscillatory and asymptotic behavior of solutions of the higher order nonlinear neutral delay dynamic equation with positive and negative coefficients. The results of this paper extend and generalize the results of [S. Panigrahi and P. Rami Reddy, Oscillatory and asymptotic behavior of fourth order non-linear neutral delay dynamic equations, Dyn. Contin. Discrete Impuls. Syst. Ser. AMath. Anal. 20 (2013), 143-163] and [S. Panigrahi, J. R. Graef and P. Rami Reddy, Oscillation results for fourth order nonlinear neutral dynamic equations, Commun. Math. Anal. 15 (2013), 11-28]. Examples are included to illustrate the validation of the results.

5

On a linear difference equation with several infinite lags

Čermák J.

Fasciculi Mathematici

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2010

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Nr 44

19-28

EN

This paper deals with asymptotic properties of the solutions of a variable order linear difference equation. As the main result, we derive the effective asymptotic estimate valid for all solutions of this equation. Moreover, we are going to discuss some consequences of this theoretical result, especially with respect to the numerical analysis of the multi-pantograph differential equation.

6

Oscillation and non-oscillation criteria for a neutral delay difference equation of first order

Rath R., Rath S. K.

Fasciculi Mathematici

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2009

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Nr 42

109-120

EN

In this paper, necessary and sufficient condition are obtained so that every bounded solution of Δ(yn - yn-k) + qnG(yσ(n)) = 0 is oscillatory, under a condition weaker than ...[wzór]

7

Asymptotic of Lp extremal polynomials off the unit circle

Khaldi R.

Demonstratio Mathematica

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2005

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Vol. 38, nr 3

623--632

EN

Let sigma be a positive measure whose support is the unit circle gamma plus a denumerable set of mass points, which accumulate at gamma and satisfy Blaschke^s condition. We study the asymptotics of Lp extremal polynomials (0 < p < oo) in the region exterior to r under Szego's condition.

8

Asymptotic behaviour of a discrete dynamical system generated by a simple evolutionary process

Karcz-Dulęba I.

International Journal of Applied Mathematics and Computer Science

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2004

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Vol. 14, no 1

79-90

EN

A simple model of phenotypic evolution is introduced and analysed in a space of population states. The expected values of the population states generate a discrete dynamical system. The asymptotic behaviour of the system is studied with the use of classical tools of dynamical systems. The number, location and stability of fixed points of the system depend on parameters of a fitness function and the parameters of the evolutionary process itself. The influence of evolutionary process parameters on the stability of the fixed points is discussed. For large values of the standard deviation of mutation, fixed points become unstable and periodical orbits arise. An analysis of the periodical orbits is presented.