A Monotonicity Condition for Strong Convergence of the Mann Iterative Sequence for Demicontractive Maps in Hilbert Spaces

Akuchu Besheng George, Celestin Akwumbuom Nse

Department of Mathematics, Federal University of Technology, Owerri, Nigeria.
Department of Mathematics, University of Nigeria, Nsukka, Nigeria.

DOI: 10.4236/am.2014.515212                

Abstract

Let be a real Hilbert space and C be a nonempty closed convex subset of H. Let T : C → C be a demicontractive map satisfying 〈Tx, x〉 ≥ ‖x‖² for all x ∈ D (T). Then the Mann iterative sequence given by x_n + 1 = (1 - a_n) x_n + a_nT x_n, where a_n ∈ (0, 1) n ≥ 0, converges strongly to an element of F (T):= {x ∈ C : Tx = x}. This strong convergence is obtained without the compactness-type assumptions on C, which many previous results (see e.g. [1]) employed.

Keywords

Demicontractive Maps, Mann Iterative Sequence, Strong Convergence, Monotonicity, Hilbert Spaces

1. Introduction

Let be a real Hilbert space. A mapping is said to be demicontractive if there exists a constant such that

(1.1)

for all, where More often than not, is assumed to be in the interval However, this is a restriction of convenience. If then is called a hemicontractive map.

On the otherhand, is said to satisfy condition (A) if there exists such that

(1.2)

for all Inequality (1.2) is equivalent to

(1.3)

The above classes of maps were studied independently by Hicks and Kubicek [2] and Maruster [3] . It is

however shown in [4] that the two classes of maps coincide if and

The class of demicontractive maps includes the class of quasi-nonexpansive and the class of strictly pseudocontractive maps. Any strictly pseudocontractive mapping with a nonempty fixed point set is demicontractive.

If is a closed convex subset of any Banach space and is any map, then the Mann itera-

tion sequence [5] is given by where satisfying certain condi-

tions. Several authors (see e.g. [1] - [3] [5] [6] ) have studied the convergence of the Mann iteration sequence to fixed points of certain mappings in certain Banach spaces. However, the Mann iteration sequence is very suitable for the study of convergence to fixed points of demicontractive mappings. It is well known (see e.g. [4] ) that demicontractivity alone is not sufficient for the convergence of the Mann iteration sequence. Some additional smoothness properties of such as continuity and demiclosedness are necessary.

A map is said to be demiclosed at a point if whenever is a sequence in the domain of such that converges weakly to and converges strongly to then

In [7] , Maruster studied the convergence of the Mann iteration sequence for demicontractive maps, in finite dimensional spaces, with an application to the study of the so-called relaxation algorithm for the solution of a particular convex feasibility problem. More precisely, he proved the following:

Theorem 1 [7] : Let be a nonlinear mapping, where is the m-Euclidean space. Suppose the following are satisfied:

1) is demiclosed at 0.

2) is demicontractive with constant or equivalently satisfies condition with

Then the Mann iteration sequence converges to a point of for any starting

Maruster [4] noted that in infinite dimensional spaces, demicontractivity and demiclosedness of are not sufficient for strong convergence. However, the two conditions ensure weak convergence. More precisely, he proved the following:

Theorem 2 [3] : Let be a nonlinear mapping with where is a closed convex subset of a real Hilbert space Suppose the following conditions are satisfied:

1) is demiclosed at 0.

2) is demicontractive with constant or equivalently satisfies condition with.

3).

Then the Mann iteration sequence converges weakly to a fixed point of, for any starting

2. Strong Convergence

As noted above, demicontractivity and demiclosedness of T are not sufficient for strong convergence of the Mann iteration sequence in infinite dimensional spaces. Some additional conditions on T, or some modifications of the Mann iteration sequence are required for strong convergence to fixed points of demicontractive maps. Such additional conditions or modifications have been studied by several authors (see e.g. [1] [2] [6] [8] [9] ).

There is however an interesting connection between the strong convergence of the Mann iteration sequence to a fixed point of a demicontractive map, T, and the existence of a non-zero solution of a certain variational inequality. This connection was observed by Maruster [3] , and has been studied by several authors. More precisely, Maruster proved the following theorem:

Theorem 3 [3] : Suppose satisfies the conditions of Theorem 2. If in addition there exists such that

(1.4)

for all then starting from a suitable the Mann iteration sequence converges strongly to an element of

The conditions of/and the variational inequality in Theorem 3 have been used and generalized by several authors (see e.g. [8] [9] ). The existence of a non-zero solution to the variational inequality is sometimes gotten under very stringent conditions. In [4] remark 4, Maruster and Maruster made the following observation “It would therefore be interesting to study more closely the existence of a non-zero solution of the variational inequality”.

The purpose of this paper is to provide a monotonicity condition under which the Mann iteration sequence converges strongly to a fixed point of a demicontractive map. The convergence does not need to pass through the variational inequality (1.4). The condition is embodied in the following theorem:

Before we state and prove our theorem, we give the following definition which will be useful in the sequel.

Definition 1: Let be a real Hilbert space with inner product and norm and let be a nonempty closed convex subset of. The orthogonal projection of onto is defined by and has the following properties:

1)

2).

Theorem 4: Suppose satisfies:

1) The conditions of Theorem 2.

2) for all Then starting from a suitable the Mann iteration sequence converges strongly to an element of

Proof. Choose such that This implies there exists such that Suppose Then using (1.3) and condition (ii) of Theorem 4, we have

Since from Theorem 2, then

Example: Let (reals) and be a nonempty closed convex subset of Define by

Then it is easily verifiable that is demicontractive and satisfies condition (ii) of our theorem for

Remark 1: In [4] Maruster and Maruster noted that if satisfies the positivity type condition then it is sufficient to find a non-zero solution of the variational inequality This motivates the condition in our theorem. As a matter of fact, our theorem is a necessity result.

Remark 2: We note that one of the ways of choosing is as follows: For any choose where and is the metric projection from into This follows since it is well known (see Definition 1) that is firmly nonexpansive (i.e. satisfies condition (ii) of Definition 1), so that

This implies where

Conflicts of Interest

The authors declare no conflicts of interest.

References

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