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Existence and asymptotic behaviour for solutions of dynamical equilibrium systems

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  • 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.
    Mathematics Subject Classification: Primary: 37N40, 46N10, 49J40, 90C33.


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