On the convergence of solutions to a difference inclusion on Hadamard manifolds

Document Type: Research Paper

Authors

1 Department of Mathematics‎, ‎University of Zanjan‎, ‎45371-38791‎, ‎Zanjan‎, ‎Iran‎

2 Department of Mathematics‎, ‎University of Zanjan‎, ‎45371-38791‎, ‎Zanjan‎, ‎Iran

Abstract

‎The aim of this paper is to study the convergence of solutions of the‎
‎following second order difference inclusion‎
‎\begin{equation*}\begin{cases}\exp^{-1}_{u_i}u_{i+1}+\theta_i \exp^{-1}_{u_i}u_{i-1} \in c_iA(u_i),\quad i\geqslant 1\\ u_0=x\in M‎, ‎\quad‎
‎\underset{i\geqslant 0}{sup}\ d(u_i,x)<+\infty‎ ,
‎\end{cases}\end{equation*}‎
‎to a singularity of a multi-valued maximal monotone vector field $A$‎ ‎on a Hadamard manifold $M$‎, ‎where $\{c_i\}$ and $\{\theta_i\}$ are‎ ‎sequences of positive real numbers and $x$ is an arbitrary fixed‎ ‎point in $M$‎. ‎The results of this paper extend previous results in‎ ‎the literature from Hilbert spaces to Hadamard manifolds for general‎ ‎maximal monotone‎, ‎strongly monotone multi-valued vector fields and‎ ‎subdifferentials of proper‎, ‎lower semicontinuous and geodesically‎ ‎convex functions $f:M\rightarrow ]-\infty,+\infty]$‎. ‎In the recent case‎, ‎when $A=\partial f$‎, ‎we show that the sequence $\{u_i\}$‎, ‎given by‎ ‎the equation‎, ‎converges to a point of the solution set of the‎ ‎following constraint minimization problem‎ ‎$$\underset{x\in M}{Min}\ f(x).$$‎

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