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

Abstract

‎The aim of this paper is to study the convergence of solutions of the‎ <br />‎following second order difference inclusion‎ <br />‎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 igeqslant 1 u_0=xin M‎, ‎quad‎ <br />‎underset{igeqslant 0}{sup} d(u_i,x)<+infty‎ , <br />‎end{cases}end{equation*}‎ <br />‎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 $f:Mrightarrow ]-infty,+infty]$‎. ‎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 ‎$$underset{xin M}{Min} f(x).$$‎