ON THE STEEPEST DESCENT APPROXIMATION TO SOLUTIONS OF NONLINEAR STRONGLY ACCRETIVE OPERATOR EQUATIONS

A new inequality of Banach spaces and the asymptotic stability theory on the equilibrium point of a certain type of initial value problem are used to establish the global convergence of the steepest descent approximation for accretive operator equations. Let X be a real uniformly smooth Banach space and A: X ➝ X be a demicontinuous, strongly accretive operator. It is proved under suitable assumptions on an that the iterative process xn+1 = xn - anAxn, x₀ ϵ X, n = 0, 1, 2,··· converges strongly to the unique solution of the equation Ax = 0. The theorem obtained generalises and improves several existing results.