Non-homogeneous continuous and discrete gradient systems‎: ‎the quasi-convex case

Document Type : Research Paper


Department of Mathematics‎, ‎University‎ ‎of Zanjan‎, ‎P‎. ‎O‎. ‎Box 45195-313‎, ‎Zanjan‎, ‎Iran.


‎In this paper‎, ‎first we study the weak and strong convergence of solutions to the‎ ‎following first order nonhomogeneous gradient system‎ ‎$$\begin{cases}-x'(t)=\nabla\phi(x(t))+f(t),\ \ \ \ \text{a.e.\ \ on}\ (0,\infty)\\‎‎x(0)=x_0\in H\end{cases}$$ to a critical point of $\phi$‎, ‎where‎ ‎$\phi$ is a $C^1$ quasi-convex function on a real Hilbert space‎ ‎$H$ with ${\rm Argmin}\phi\neq\varnothing$ and $f\in L^1(0,+\infty;H)$‎. ‎These results extend the‎ ‎results in the literature to non-homogeneous case‎. ‎Then the‎ ‎discrete version of the above system by backward Euler‎ ‎discretization has been studied‎. ‎Beside of the proof of the‎ ‎existence of the sequence given by the discrete system‎, ‎some‎‎results on‎ ‎the weak and strong convergence to the critical point of $\phi$ are also proved‎. ‎These results when $\phi$ is pseudo-convex (therefore the critical points‎ ‎are the same minimum points) may be applied in optimization for approximation of a‎ ‎minimum point of $\phi$‎.


Main Subjects

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