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Convergence of Paths for Perturbed Maximal Monotone Mappings in Hilbert Spaces
Fixed Point Theory and Applications volume 2010, Article number: 547828 (2011)
Abstract
Let be a Hilbert space and
a nonempty closed convex subset of
. Let
be a maximal monotone mapping and
a bounded demicontinuous strong pseudocontraction. Let
be the unique solution to the equation
. Then
is bounded if and only if
converges strongly to a zero point of A as
which is the unique solution in
, where
denotes the zero set of
, to the following variational inequality
, for all
.
1. Introduction and Preliminaries
Throughout this work, we always assume that is a real Hilbert space, whose inner product and norm are denoted by
and
, respectively. Let
be a nonempty closed convex subset of
and
a nonlinear mapping. We use
and
to denote the domain and the range of the mapping
.
and
denote strong and weak convergence, respectively.
Recall the following well-known definitions.
(1)A mapping is said to be monotone if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ1_HTML.gif)
(2)The single-valued mapping is maximal if the graph
of
is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping
is maximal if and only if for
,
for every
implies
.
(3) is said to be pseudomonotone if for any sequence
in
which converges weakly to an element
in
with
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ2_HTML.gif)
(4) is said to be bounded if it carries bounded sets into bounded sets; it is coercive if
as
.
(5)Let be linear normed spaces.
is said to be demicontinuous if, for any
we have
as
.
(6)Let be a mapping of a linear normed space
into its dual space
.
is said to be hemicontinuous if it is continuous from each line segment in
to the weak topology in
.
(7)The mapping with the domain
and the range
in
is said to be pseudocontractive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ3_HTML.gif)
(8)The mapping with the domain
and the range
in
is said to be strongly pseudocontractive if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ4_HTML.gif)
Remark 1.1.
For the maximal monotone operator , we can defined the resolvent of
by
. It is well know that
is nonexpansive.
Remark 1.2.
It is well-known that if is demicontinuous, then
is hemicontinuous, however, the converse, in general, may not be true. In reflexive Banach spaces, for monotone mappings defined on the whole Banach space, demicontinuity is equivalent to hemicontinuity.
To find zeroes of maximal monotone operators is the central and important topics in nonlinear functional analysis. We observe that is a zero of the monotone mapping
if and only if it is a fixed point of the pseudocontractive mapping
. Consequently, considerable research works, especially, for the past 40 years or more, have been devoted to the existence and convergence of zero points for monotone mappings or fixed points of pseudocontractions, see, for instance, [1–23].
In 1965, Browder [1] proved the existence result of fixed point for demicontinuous pseudocontractions in Hilbert spaces. To be more precise, he proved the following theorem.
Theorem Bo
Let be a Hilbert space,
a nonempty bounded and closed convex subset of
and
a demicontinuous pseduo-contraction. Then
has a fixed point in
.
In 1968, Browder [4] proved the existence results of zero points for maximal monotone mappings in reflexive Banach spaces. To be more precise, he proved the following theorem.
Theorem Bt
Let be a reflexive Banach space,
a maximal monotone mapping and
a bounded, pseudomonotone and coercive mapping. Then, for any
, there exists
such that
, or
is all of
.
For the existence of continuous paths for continuous pseudocontractions in Banach spaces, Morales and Jung [15] proved the following theorem.
Theorem MJ.
Let be a Banach space. Suppose that
is a nonempty closed convex subset of
and
is a continuous pseudocontraction satisfying the weakly inward condition. Then for each
, there exists a unique continuous path
,
, which satisfies the following equation
.
In 2002, Lan and Wu [14] partially improved the result of Morales and Jung [15] from continuous pseudocontractions to demicontinuous pseudocontractions in the framework of Hilbert spaces. To be more precise, they proved the following theorem.
Theorem LW.
Let be a bounded closed convex set in
. Assume that
is a demicontinuous weakly inward pseudocontractive map. Then
has a fixed point in
. Moreover; for every
,
defined by
converges to a fixed point of
.
In this work, motivated by Browder [3], Lan and Wu [14], Morales and Jung [15], Song and Chen [19], and Zhou [22, 23], we consider the existence of convergence of paths for maximal monotone mappings in the framework of real Hilbert spaces.
2. Main Results
Lemma 2.1.
Let be a nonempty closed convex subset of a Hilbert space
and
a demicontinuous monotone mapping. Then
is pseudomonotone.
Proof.
For any sequence which converges weakly to an element
in
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ5_HTML.gif)
we see from the monotonicity of that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ6_HTML.gif)
Combining (2.1) with (2.2), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ7_HTML.gif)
By taking , we arrive at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ8_HTML.gif)
which yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ9_HTML.gif)
Noticing that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ10_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ11_HTML.gif)
Let , for all
and
. By taking
and
in (2.7), we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ12_HTML.gif)
Noting that ,
,
, and
is demicontinuous, we have
as
, and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ13_HTML.gif)
This completes the proof.
Lemma 2.2.
Let be a nonempty closed convex subset of a Hilbert space
,
a maximal monotone mapping, and
a bounded, demicontinuous, and strongly monotone mapping. Then
has a unique zero in
.
Proof.
By using Lemma 2.1 and Theorem B2, we can obtain the desired conclusion easily.
Lemma 2.3.
Let be a nonempty closed convex subset of a Hilbert space
,
a maximal monotone mapping, and
a bounded, demicontinuous strong pseudocontraction with the coefficient
. For
, consider the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ14_HTML.gif)
where . Then, One has the following.
(i)Equation (2.10) has a unique solution for every
.
(ii)If is bounded, then
as
.
(iii)If , then
is bounded and satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ15_HTML.gif)
where denotes the zero set of
.
Proof.
-
(i)
From Lemma 2.2, one can obtain the desired conclusion easily.
-
(ii)
We use
to denote the unique solution of (2.10). That is,
. It follows that
. Notice that
(2.12)
From the boundedness of and
, one has
.
-
(iii)
For
, one obtains that
(2.13)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ18_HTML.gif)
That is, , for all
. This shows that
is bounded. Noticing that
, one arrives at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ19_HTML.gif)
This completes the proof.
Lemma 2.4.
Let be a nonempty closed convex subset of a Hilbert space
and
a maximal monotone mapping. Then
. If one defines
by
, for all
, then
is a nonexpansive mapping with
and
, where
denotes the set of fixed points of
.
Proof.
Noticing that is maximal monotone, one has
. It follows that
. For any
, one sees that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ20_HTML.gif)
which yields that is nonexpansive mapping. Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ21_HTML.gif)
That is, . On the other hand, for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ22_HTML.gif)
This completes the proof.
Set . Let
denote the Banach space of all bounded real value functions on
with the supremum norm,
a subspace of
, and
an element in
, where
denotes the dual space of
. Denote by
the value of
at
. If
, for all
, sometimes
will be denoted by
. When
contains constants, a linear functional
on
is called a mean on
if
. We also know that if
contains constants, then the following are equivalent.
(1).
(2), for all
.
To prove our main results, we also need the following lemma.
Lemma 2.5 (see [20, Lemma 4.5.4]).
Let be a nonempty and closed convex subset of a Banach space
. Suppose that norm of
is uniformly Gâteaux differentiable. Let
be a bounded set in
and
. Let
be a mean on
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ23_HTML.gif)
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ24_HTML.gif)
Now, we are in a position to prove the main results of this work.
Theorem 2.6.
Let be a Hilbert space and
a nonempty closed convex subset of
. Let
be a maximal monotone mapping and
a bounded demicontinuous strong pseudocontraction. Let
be as in Lemma 2.3. Then
is bounded if and only if
converges strongly to a zero point
of
as
which is the unique solution in
to the following variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ25_HTML.gif)
Proof.
The part is obvious and we only prove
. From Lemma 2.3, one sees that
as
. It follows from Lemma 2.4 that
as
. Define
,
, where
is a Banach limit. Then
is a convex and continuous function with
as
. Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ26_HTML.gif)
From the convexity and continuity of , we can get the convexity and continuity of the set
. Since
is continuous and
is a Hilbert space, we see that
attains its infimum over
; see [20] for more details. Then
is nonempty bounded and closed convex subset of
. Indeed,
contains one point only. Set
, where
. Notice that
is nonexpansive. Since every nonempty bounded and closed convex subset has the fixed point property for nonexpansive self-mapping in the framework of Hilbert spaces, then
has a fixed point
in
, that is,
. It follows from Lemma 2.4 that
. On the other hand, one has
. In view of Lemma 2.5, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ27_HTML.gif)
By taking in (2.23), we arrive at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ28_HTML.gif)
Combining (2.14) with (2.23) yields that . Hence, there exists a subnet
of
such that
. From (iii) of Lemma 2.3, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ29_HTML.gif)
Taking limit in (2.25), one gets that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ30_HTML.gif)
If there exists another subset of
such that
, then
is also a zero of
. It follows from (2.26) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ31_HTML.gif)
By using (iii) of Lemma 2.3 again, one arrives at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ32_HTML.gif)
Taking limit in (2.28), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ33_HTML.gif)
Adding (2.27) and (2.29), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ34_HTML.gif)
which yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ35_HTML.gif)
It follows that . That is,
converges strongly to
, which is the unique solution to the following variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F547828/MediaObjects/13663_2010_Article_1302_Equ36_HTML.gif)
Remark 2.7.
From Theorem 2.6, we can obtain the following interesting fixed point theorem. The composition of bounded, demicontinuous, and strong pseudocontractions with the metric projection has a unique fixed point. That is, .
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Acknowledgment
The third author was supported by the National Natural Science Foundation of China (Grant no. 10771050).
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Qing, Y., Qin, X., Zhou, H. et al. Convergence of Paths for Perturbed Maximal Monotone Mappings in Hilbert Spaces. Fixed Point Theory Appl 2010, 547828 (2011). https://doi.org/10.1155/2010/547828
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DOI: https://doi.org/10.1155/2010/547828