Convergence and Rate of Convergence of a Non-Autonomous Gradient System on Hadamard Manifolds

In this paper we consider the following nonhomogeneous gradient system on a Hadamard manifold M { (- x '( t))(x(0)=x0 ,) (- grad) (phi(x( t)) + e( t),) where phi: M -> R is a geodesically convex function of class C-2 with argmin phi not equal circle divide . We prove global convergence of solutions of the gradient systemto a minimum point of phi We also discuss on the rate of convergence of phi(x(t)) to the minimum value of. as well as the rate of convergence of ||x '(t)|| and d(x(t), p) to zero, where p is the minimum point of phi. Finally, we present some problems and future directions to study.