Convergence Of Iterative Scheme For Nonlinear Oporator

Nonlinear function analysis theory has been prominent in mathematics. Among many aspects of it, the most important and interesting is to develop effective numericalmethods to generate approximate solutions. This paper will discuss it in the following way:Firstly, the background and current state of Nonlinear function analysis theory will be discussed.Secondly, it will introduce the new Ishikawa iterative scheme with errors for finitefamilies of nonself asymptotically nonexpansive mappings in a uniformly convex Banach space and prove convergence of the sequence {x_n} given by this scheme.Thirdly, it will study the new Mann type implicit iterative scheme for uniformly Lipschitzian asymptotically pseudocontractive mapping in a Hilbert space. Suppose K be a nonempty closed convex subset of Hilbert space E, T : K→K be a uniformlyLipschitzian asymptotically pseudocontractive mapping, and {a_n} is chosen in appropriate conditions, then we can prove for arbitrary x₀∈K, the sequence {x_n} given by the Mann type implicit iteration process x_n = a_nx_n-1 + (1 - a_n)Tⁿx_n,n > 0, weakly converges to fixed point of T.Finally, it will propose the new Mann iterative scheme with errors for total asymptotically hemicontractive mappings in a real Banach space. Suppose K be a nonempty closed convex subset of real Banach space E, T : K→K be a uniformly Lipschitzian total asymptotically hemicontractive mapping, then we can prove for arbitrary x₁∈K, the sequence {x_n} given by the Mann type iteration process with errors x_n+1 = (1 - a_n)x_n + a_nTⁿx_n +ξ_n, n > 0, strongly converges to fixed point of T under certain condition.