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Evolution Equations and Control Theory (EECT)
 

Existence and asymptotic behaviour for solutions of dynamical equilibrium systems

Pages: 1 - 14, Volume 3, Issue 1, March 2014      doi:10.3934/eect.2014.3.1

 
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Zaki Chbani - Laboratory LIBMA Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, 40000 Marrakech, Morocco (email)
Hassan Riahi - Laboratory LIBMA Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, 40000 Marrakech, Morocco (email)

Abstract: In this paper, we give an existence result for the following dynamical equilibrium problem: $\langle \frac{du}{dt},v-u(t)\rangle+F(u(t),v)\geq 0 \;\; \forall v\in K $ and for $a.e. \;t \geq 0$, where $K$ is a closed convex set in a Hilbert space and $ F:K \times K \rightarrow \mathbb{R}$ is a monotone bifunction. We introduce a class of demipositive bifunctions and use it to study the asymptotic behaviour of the solution $ u(t) $ when $ t\rightarrow\infty $. We obtain weak convergence of $ u(t) $ to some solution $x\in K$ of the equilibrium problem $F(x,y)\geq 0 $ for every $y\in K$. Our applications deal with the asymptotic behaviour of the dynamical convex minimization and dynamical system associated to saddle convex-concave bifunctions. We then present a new neural model for solving a convex programming problem.

Keywords:  Monotone bifunction, Cauchy problem, demipositive bifunction, asymptotic behaviour, convex minimisation, saddle point problem.
Mathematics Subject Classification:  Primary: 37N40, 46N10, 49J40, 90C33.

Received: February 2013;      Revised: January 2014;      Available Online: February 2014.

 References