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

‎The aim of this paper is to study the convergence of solutions of the‎ ‎following second order difference inclusion‎ {exp −1 u i u i+1 +θ i exp −1 u i u i−1 ∈c i A(u i),i⩾1u 0 =x∈M‎,‎‎‎sup i⩾0 d(u i, x)<+∞‎,‎ ‎to a singularity of a multi-valued maximal monotone vector field A ‎on a Hadamard manifold M ‎, ‎where {c i} and {θ 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→]−∞,+∞] ‎. ‎In the recent case‎, ‎when A=∂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 ‎Min x∈M f(x).