Author/Authors :
Ahmadi، P. نويسنده University of Zanjan , , Khatibzadeh، H. نويسنده University of Zanjan ,
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
$$\underset{x\in M}{Min}\ f(x).$$
