Abstract
In this paper, we consider a dynamical system for solving equilibrium problems in the framework of Hilbert spaces. First, we prove that under strong pseudo-monotonicity and Lipschitz-type continuity assumptions, the dynamical system has a unique equilibrium solution, which is also globally exponentially stable. Then, we derive the linear rate of convergence of a discrete version of the proposed dynamical system to the unique solution of the problem. Global error bounds are also provided to estimate the distance between any trajectory and this unique solution. Some numerical experiments are reported to confirm the theoretical results.
Highlights
The equilibrium problem (EP for short) is a very general mathematical model in the sense that it includes, as special cases, the optimization problem, the variational inequality, the saddle point problem, the Nash equilibrium problemin noncooperativeCommunicated by Hedy Attouch
Among them, fixed point-type methods play an important role, because they are simple in form and useful in practice. The idea of these methods comes from the proximal point and the projection methods for solving a variational inequality (VI for short) [1,2,13,20]
Namely strong pseudo-monotonicity and Lipschitz continuity, we prove that the proposed dynamical system has a unique equilibrium point
Summary
The equilibrium problem (EP for short) is a very general mathematical model in the sense that it includes, as special cases, the optimization problem, the variational inequality, the saddle point problem, the Nash equilibrium problemin noncooperative.
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