@article {
author = {Khatibzadeh, H. and Mohebbi, V.},
title = {Non-homogeneous continuous and discrete gradient systems: the quasi-convex case},
journal = {Bulletin of the Iranian Mathematical Society},
volume = {43},
number = {6},
pages = {2099-2110},
year = {2017},
publisher = {Springer and the Iranian Mathematical Society (IMS)},
issn = {1017-060X},
eissn = {1735-8515},
doi = {},
abstract = {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 $\phi$.},
keywords = {Gradient system,quasi-convex,backward Euler discretization,weak convergence,strong convergence},
url = {http://bims.iranjournals.ir/article_1086.html},
eprint = {http://bims.iranjournals.ir/article_1086_c9f5c603ae165a46745899906352a83b.pdf}
}