^{1}

^{*}

^{2}

^{*}

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 〈*T*x, x〉 ≥ ‖x‖^{2} for all x ∈ *D* (*T*). Then the Mann iterative
sequence given by x_{n }+ 1 = (1 - a_{n}) x_{n} + a_{n}*T* x_{n}, where a_{n} ∈ (0, 1) *F *(*T*):= {x ∈ *C* : *T*x = x}.
This strong convergence is obtained without the compactness-type assumptions on *C*, which many previous results (see e.g. [1])
employed.

Let

for all

On the otherhand,

for all

The above classes of maps were studied independently by Hicks and Kubicek [

however shown in [

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

tion sequence [

tions. Several authors (see e.g. [

A map

In [

Theorem 1 [

1)

2)

Then the Mann iteration sequence converges to a point of

Maruster [

Theorem 2 [

1)

2)

3)

Then the Mann iteration sequence converges weakly to a fixed point of

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. [

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 [

Theorem 3 [

for all

The conditions of/and the variational inequality in Theorem 3 have been used and generalized by several authors (see e.g. [

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

1)

2)

Theorem 4: Suppose

1) The conditions of Theorem 2.

2)

Proof. Choose

Since

Example: Let

Then it is easily verifiable that

Remark 1: In [

Remark 2: We note that one of the ways of choosing

This implies