A Monotonicity Condition for Strong Convergence of the Mann Iterative Sequence for Demicontractive Maps in 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 ![]()