On some midpoint-type algorithms.

Giuseppe Mari̇no,Roberta Zaccone

doi:10.31197/atnaa.407069

Abstract

We introduce iterative methods approximating fixed points for nonlinear operators defined on infinite-dimensional spaces. The starting points are the Implicit and Explicit Midpoint Rules, which generate polygonal functions approximating a solution for an ordinary differential equation infinite-dimensional spaces. The purpose is to determine suitable conditions on the mapping and the underlying space, in order to get strong convergence of the generated sequence to a common solution of a fixed point problem and a variational inequality.

Highlights

Let (X, · ) be an infinite dimensional Banach space, C ⊂ X a nonempty and closed set, T : C → C a nonlinear operator with F ix(T ) = {z ∈ C : T z = z} = ∅.A classical problem in Metric Fixed Point Theory can be formulated as: Examine the conditions under which the equation x = T x may be solved by successive approximations: x0 ∈ C, xn+1 = T xn, n ≥ 0. (1.1)Recall that a mapping T : C → C is said L-Lipschitzian if there exists a constant L ≥ 0 such thatT x − T y ≤ L x − y ∀x, y ∈ E.In particular, G
If the midpoint xn +xn+1 2 in the evaluation of is replaced with any convex combination between xn and xn+1, scheme (1.8) is named General Explicit Midpoint Rule for nonexpansive mappings. We provide for the latter scheme the same formal modification as for the Implicit Midpoint Rule (IMR) for nonexpansive mappings, following [43]
We show that the sequence generated by (1.9) strongly converges to the fixed point of

Summary

Introduction

In case u = 0, under the same assumptions of Theorem 3.5, we obtain strong convergence of the sequence (xn)n∈N, generated by xn+1 = βnxn + (1 − βn)T xn, n ≥ 0 xn+1 = αnxn + (1 − αn)T (snxn + (1 − sn)xn+1) − αnμnxn, n ≥ 0, to the point x∗ ∈ F ix(T ) nearest to 0 ∈ H, that is the fixed point of T with minimum norm x∗ = minx∈F ix(T ) x. A strong convergence result of the sequence (xn)n∈N generated by (1.10) to a fixed point of a quasinonexpansive operator is proved in the framework of Hilbert spaces: Theorem 3.9.

It results that

Thus it must be q

The mapping defined as

Let us choose

Let f

We observe that