On the convergence of the trajectories of the dynamical Moudafi's viscosity approximation system (Q6611205)

In the present paper there is considered the continuous dynamical system (CDS)\N\[\N\left\{ \begin{aligned} x'(t)+x(t)=\theta(t)f(x(t))+(1-\theta(t))T(x(t))\\\Nx(0)=x_0,\\\N\end{aligned} \right.\N\]\Nwhere \(\theta \colon [0,+\infty) \rightarrow (0, 1]\) is a regular function.\N\NThe article consists of 4 sections. The first section is introduction.\N\NThe second section devoted to some classical notions and results from functional and convex analysis that are useful in the sequel of the paper.\N\NIn the third section the authors prove that if the function \(\theta(\cdot)\) satisfies some conditions, similar to the discrete ones, then, for any initial data \(x_0 \in C\), the CDS - system has a unique global solution \(x \in C_1([0,+\infty),H)\) which converges strongly in \(H\) as \(t \rightarrow+\infty\) to the unique solution \(q_*\) of the problem. There \(H\) is a real Hilbert space.\N\NIn the final section, the authors establish a general result that implies a precise estimation on the rate of the convergence of the residual term \(x(t)-T (x(t))\) to \(\theta\) in the particular case \( \theta(t)=\frac{K}{(1+t)^v}\) with \(K>0\) and \(v \in(0, 1]\). Then the authors deduce an estimate on the speed of convergence of \(x(t)\) at infinity to \(q_*\) in the particular case where \(T\) is a contraction mapping. Finally there have been shown the precision and in some cases the optimality of the obtained results through the study of some examples of the system (CDS) and its perturbed version.\N\NThe results of the work are new and actual.