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 $$\underset{x\in M}{Min}\ f(x).$$
Ahmadi, P., & Khatibzadeh, H. (2015). On the convergence of solutions to a difference inclusion on Hadamard manifolds. Bulletin of the Iranian Mathematical Society, 41(4), 1045-1059.
MLA
P. Ahmadi; H. Khatibzadeh. "On the convergence of solutions to a difference inclusion on Hadamard manifolds". Bulletin of the Iranian Mathematical Society, 41, 4, 2015, 1045-1059.
HARVARD
Ahmadi, P., Khatibzadeh, H. (2015). 'On the convergence of solutions to a difference inclusion on Hadamard manifolds', Bulletin of the Iranian Mathematical Society, 41(4), pp. 1045-1059.
VANCOUVER
Ahmadi, P., Khatibzadeh, H. On the convergence of solutions to a difference inclusion on Hadamard manifolds. Bulletin of the Iranian Mathematical Society, 2015; 41(4): 1045-1059.