- Research Article
- Open access
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Fixed Points for Pseudocontractive Mappings on Unbounded Domains
Fixed Point Theory and Applications volume 2010, Article number: 769858 (2009)
Abstract
We give some fixed point results for pseudocontractive mappings on nonbounded domains which allow us to obtain generalizations of recent fixed point theorems of Penot, Isac, and Németh. An application to integral equations is given.
1. Introduction
Let be a nonempty subset of a Banach space
with norm
. Recall that a mapping
is said to be nonexpansive whenever
for every
.
is said to have the fixed point property ((FPP) for short) if every nonexpansive selfmapping of each nonempty bounded closed and convex subset of
has a fixed point. It has been known from the outset of the study of this property (around the early sixties of the last century) that it depends strongly on "nice'' geometrical properties of the space. For instance, a celebrated result due to Kirk [1] establishes that those reflexive Banach spaces with normal structure (NS) have the (FPP). In particular, uniformly convex Banach spaces have normal structure (see [2, 3] for more information).
If is a closed convex of a Banach space enjoying the (FPP), in general it is not true that
has a fixed point due to the possible unboundedness of
(it is enough to consider any translation map, with nonnull vector, in the Banach space
). In 2003 Penot [4] showed that if
is a closed convex subset of a uniformly convex Banach space
,
is a nonexpansive mapping, and for some
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ1_HTML.gif)
(in other words if is asymptotically contractive), then
has a fixed point.
A celebrated fixed point result due to Altman [5] is the following.
Let be a separable Hilbert space, with inner product
and induced norm
. Let
be a weakly closed mapping where
is the closed ball with center
and radius
. Suppose that
maps the sphere
into a bounded set in
. If the following condition is satisfied:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ2_HTML.gif)
for all , then
has a fixed point in
.
In 2006, Isac and Németh [6] gave some fixed point results for nonexpansive nonlinear mappings in Banach spaces inspired by Penot's results where the asymptotically contractiveness was stated in similar terms to condition (1.2).
In this paper we generalize some Penot, Isac, and Németh's fixed point results in several ways. First, we will be concerned with pseudocontractive mappings, a more general class of mappings than the nonexpansive ones. Second, we use an inwardness condition weaker than , and finally our Altmann type assumptions are more general than those required in [4, 6].
We prove our fixed point results as a consequence of some results on the existence of zeroes for accretive operators. Among the problems treated by accretive operators theory, one of the most studied is just this one (see, e.g., Kirk and Schöneberg's paper [7] as well as [3, 8, 9] and the references therein). We obtain here several results of this type, and in particular we give a characterization in the setting of the Banach spaces with (FPP) of those -accretive operators which have zeroes.
2. Preliminaries
Throughout this paper we suppose that is a real Banach space and that
is its topological dual. We use
to denote the closed ball centered at
with radius
. We also use the notation
,
.
If , we will denote by
the normalized duality mapping at
defined by
. We will often use the mapping
defined by
.
A mapping will be called an operator on
. The domain of
is denoted by
and its range by
. It is well known that an operator
is accretive if and only if
for all
.
If, in addition, is for one, hence for all,
, precisely
, then
is called
-accretive. We say that
satisfies the range condition if
for all
.
We now recall some important facts regarding accretive operators which will be used in our paper (see, e.g., [10]).
Proposition 2.1.
Let be an operator on
. The following conditions are equivalent:
(i) is an accretive operator,
(ii)the inequality holds for all
, and for every
,
(iii)for each the resolvent
is a single-valued nonexpansive mapping.
Let be a nonempty subset of
and let
be a mapping. Recall that a sequence
of elements of
is said to be an a.f.p sequence for
whenever
. It is well known that if
is a nonexpansive mapping which maps a closed convex bounded subset
of
into itself, then such a mapping always has a.f.p. sequences in
.
When the Banach space has the (FPP), Morales [9] gave a characterization of those
-accretive operators
such that
. Let us recall such result.
Theorem 2.2.
Let be a Banach space with the FPP, and let
be an
-accretive operator. Then
if and only if the set
is bounded.
A mapping is said to be pseudocontractive if for every
, and for all positive
,
. Pseudocontractive mappings are easily seen to be more general than nonexpansive mappings ones. The interest in these mappings also stems from the fact that they are firmly connected to the well-known class of accretive mappings. Specifically
is pseudocontractive if and only if
is accretive where
is the identity mapping.
We say that a mapping is demiclosed at
if for any sequence
in
weakly convergent to
with
norm convergent to
one has that
. It is well known that if
is weakly compact and convex,
is nonexpansive, and
is demiclosed at
, then
has a fixed point in
.
We say that the mapping is weakly inward on
if
for all
. Such condition is always weaker than the assumption of
mapping the boundary of
into
. Recall that if
is a continuous accretive mapping,
is convex and closed, and
is weakly inward on
, then
has the range condition (see [11]).
We say that a semi-inner-product is defined on , if to any
there corresponds a real number denoted by
satisfying the following properties:
(s1) for
,
(s2) for
, and
,
(s3) for
,
(s4) .
It is known (see [12, 13]) that a semi-inner-product space is a normed linear space with the norm and that every Banach space can be endowed with a semi-inner-product (and in general in infinitely many different ways, but a Hilbert space in a unique way).
In [6] the authors considered several fixed point results for nonexpansive mappings with unbounded domains satisfying additional asymptotic contractive-type conditions in terms of a function under the following assumptions:
(G1) for any
and
,
(G2) for any
,
(G3) for any
,
(G4) there exists an such that
for every
.
3. Zeroes for Accretive Operators
We begin with the definition of a certain kind of functions on which we will be concerned. This class is more general than the corresponding one considered in [6]. Let be a real Banach space and
a mapping which satisfies the following conditions:
(g1) for any
and
,
(g2) there exists such that
for any
with
,
(g3) for any
,
(g4) for each , there exists
(depending on
), such that if
, then
.
Notice that if we consider either or
, then
satisfies (g1)–(g4).
Let be a Banach space with the (FPP). If
is an
-accretive operator such that its domain
is a bounded set, then it is well known that
(see, e.g., [7, 9]). If
is not bounded, then we give the following result.
Theorem 3.1.
Let be a Banach space with the (FPP). Let
be a mapping satisfying (g1) and (g2). If
is an
-accretive operator such that there exists
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ3_HTML.gif)
whenever , then,
.
Proof.
Since is
-accretive and
has the (FPP), by Theorem 2.2 we know that
if and only if the set
is bounded.
In order to get a contradiction we assume that is an unbounded set. This fact means that for each
there exists
such that
.
Since , then there exist
and
such that
. This means that
.
Consequently, for every , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ4_HTML.gif)
which is a contradiction.
In the following theorem we are going to give a characterization in terms of a particular function , (in the framework of the Banach spaces with the (FPP)), of those
-accretive operators which have zeroes.
Theorem 3.2.
Let be a Banach space with the (FPP). Let
be the mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ5_HTML.gif)
If is an
-accretive operator, then the following conditions are equivalent:
(1)there exists such that
whenever
and
;
(2).
Proof.
()
(
) It is clear that
satisfies conditions (g1) and (g2), thus by Theorem 3.1 we obtain that
.
()
(
) In order to get a contradiction, assume that for each
there exists
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ6_HTML.gif)
The above inequality implies that for each , there exist
and
such that
.
By definition of , we have that
, and thus
.
From the above fact, we derive that for each ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ7_HTML.gif)
that is, is unbounded. By Theorem 2.2, it follows that if
is unbounded, then
; therefore, we have a contradiction.
As a consequence of the above characterization it is easy to capture the following result which is related to [7, Theorems 2 and 3].
Corollary 3.3.
Let be a real Banach space with the (FPP). Suppose that
is an
-accretive operator for which there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ8_HTML.gif)
for all with
. Then
.
Proof.
Without loss of generality we may assume that . Otherwise, we work with the operator
defined by
.
If we take as in Theorem 3.2, to obtain the conclusion it is enough to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ9_HTML.gif)
whenever .
In order to get a contradiction, assume that for each there exists
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ10_HTML.gif)
The above inequality implies that for each , there exist
and
such that
.
By definition of , we have that
, and thus
.
By hypothesis, we know that the inequality holds for every
.
This means that there exists such that
. Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ11_HTML.gif)
On the other hand, since is an accretive operator, it is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ12_HTML.gif)
which is a contradiction.
The above corollary allows us to recapture the following well-known result.
Corollary 3.4.
Let be a real Banach space with the (FPP). Suppose that
is an
-accretive operator; if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ13_HTML.gif)
then .
Corollary 3.5.
Let be a Banach space with the (FPP). Let
be a mapping satisfying (g1) and (g2). If
is an
-accretive operator such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ14_HTML.gif)
then .
Proof.
It is clear that condition (3.12) implies assumption (3.1).
Corollary 3.6.
Let be a real Hilbert space. Let
be a convex proper lower semicontinuous mapping with effective domain
. Suppose that for some
there exists
such that
for all
with
. Then
has an absolute minimum on
.
Proof.
Consider the subdifferential associated to
, that is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ15_HTML.gif)
It is well known that is an
-accretive operator on
(see [14]). Now, we consider
defined as in Theorem 3.2.
In order to get a contradiction, suppose that given there is
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ16_HTML.gif)
By definition of we have that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ17_HTML.gif)
This means that , hence
. Consequently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ18_HTML.gif)
By hypothesis, when , we obtain the following contradiction:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ19_HTML.gif)
This contradiction allows us to conclude that there exists such that if
with
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ20_HTML.gif)
Since has the (FPP), from Theorem 3.1 we conclude that
; that is, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ21_HTML.gif)
and therefore is an absolute minimum of
.
If has the (FPP),
is an accretive operator with the range condition, and
is convex and bounded, then,
; see [8]. For the case that
is not bounded we have the following result.
Theorem 3.7.
Let be a Banach space. Suppose that
is a mapping satisfying conditions (g1)–(g4).
If has the (FPP),
is an accretive operator with the range condition,
is convex, and
satisfies condition (3.1), then
.
Proof.
Since is accretive with the range condition, then the following two conditions hold:
(i)
(ii) is a nonexpansive mapping.
Fix . For each positive integer
, from (i) there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ22_HTML.gif)
Hence, . It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ23_HTML.gif)
We claim that is a bounded sequence. Indeed, otherwise we can assume that there exists a subsequence
of
such that
. Without loss of generality we may assume that
,
, where the constants
and
are given in the definitions of conditions (g2) and (g4), respectively.
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ24_HTML.gif)
Consequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ25_HTML.gif)
This is a contradiction which proves our claim.
Since is a bounded sequence, it is clear that
goes to
as
goes to infinity.
Now we claim that has a bounded a.f.p. sequence. Indeed, consider for each positive integer
,
. It is not difficult to see that
because
. In this case, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ26_HTML.gif)
Finally, if we call , we obtain that the following set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ27_HTML.gif)
is bounded closed convex and -invariant. Thus, since
enjoys the (FPP), there exists
such that
and then
.
Remark 3.8.
If we check the proof of Theorem 3.7, we may notice that such theorem still holds if we omit conditions (g3) and (g4) but we add .
Corollary 3.9.
Let be a Banach space. Suppose that
is a mapping satisfying conditions (g1)–(g4).
If has the (FPP),
is an accretive operator with the range condition,
is convex, and
satisfies condition (3.12), then
.
4. Fixed Point Results
Theorem 4.1.
Let be a Banach space with the (FPP). Suppose that
is a mapping satisfying conditions (g1) and (g2). Let
be a closed convex and unbounded subset of
with
. Let
be a continuous pseudocontractive mapping. Assume that the following conditions are satisfied.
(a) is weakly inward on
.
(b)There exists such that for every
with
the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ28_HTML.gif)
holds.
Then has a fixed point in
.
Proof.
Since is a continuous, pseudocontractive mappings weakly inward on
, then
is an accretive operator with the range condition (see [11, 15]).
Let us see that condition (3.1) is satisfied. Indeed, if with
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ29_HTML.gif)
The above equality along with (4.1) allows us conclude that condition (3.1) holds.
On the other hand, since , by Remark 3.8 and following the same argument developed in the proof of Theorem 3.7, it is not difficult to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ30_HTML.gif)
has a bounded a.f.p. sequence , and thus, if we call
, we obtain that the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ31_HTML.gif)
is bounded closed convex and -invariant. Thus, since
enjoys the (FPP), there exists
such that
and then
.
Corollary 4.2.
Let be a Banach space with the (FPP). Suppose that
is a mapping satisfying conditions (g1) and (g2). Let
be a closed convex and unbounded subset of
with
. If
is a continuous pseudocontractive mapping weakly inward on
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ32_HTML.gif)
then has a fixed point in
.
Proof.
Clearly inequality (4.5) implies condition (4.1).
Corollary 4.3.
Let be a Banach space with the (FPP). Suppose that
is a mapping satisfying conditions (g1)–(g4). Let
be a closed convex and unbounded subset of
. If
is a continuous pseudocontractive mapping weakly inward on
and satisfies condition (4.1), then
has a fixed point in
.
Proof.
From the above theorem, we know that is an accretive operator with the range condition and with condition (3.1). Therefore by Theorem 3.7 we obtain the result.
Remark 4.4.
In order to give an alternative proof of Corollary 4.3, it is enough to see that condition (4.5) implies that has an a.f.p. sequence
and thus, using [16, Theorem 4.3], we obtain the same conclusion. In this case, if we assume that
is a reflexive Banach space and
is demiclosed at zero, then we can remove the assumption on the (FPP) for the space
. Nevertheless, it is well known that there exist nonreflexive Banach spaces with the FPP (see [13]). On the other hand, if
is a reflexive Banach space such that for every nonexpansive mapping, say
, the mapping
is demiclosed at
, then the Banach space has the FPP.
Remark 4.5 (Theorem 3.2 in [6] reads).
Let be a reflexive Banach space. Suppose that
satisfies conditions (G1), (G2), (G3), and (G4). Let
be a nonempty unbounded closed convex set. If
is a nonexpansive mapping such that
,
is demiclosed and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ33_HTML.gif)
for some , then
has a fixed point in
.
Notice that Corollary 4.3 generalizes this theorem in several senses.
(i)Our assumptions (g1)–(g4) on mapping are weaker than the corresponding in that theorem.
(ii)Every nonexpansive mapping is in fact continuous and pseudocontractive.
(iii)The inwardness condition is more general than the assumption .
(iv)Condition (4.6) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ34_HTML.gif)
Interchanging the roles of and
we can conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ35_HTML.gif)
for every . Therefore, there exists
such that if
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ36_HTML.gif)
which is just condition (4.1) of Theorem 4.1.
In the same sense, Theorem 4.1 is a generalization of Theorem 3.1 of [6].
Corollary 4.6.
Let be a Banach space with the (FPP). Let
be a closed convex and unbounded subset of
such that
. Let
be a continuous pseudocontractive mapping. Assume that the following conditions are satisfied.
(a) is weakly inward on
.
(b)There exists such that for every
and for every
,
.
Then has a fixed point in
.
Proof.
It is enough to apply Theorem 4.1, where is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ37_HTML.gif)
and if , then
.
Remark 4.7.
Notice that the above condition (b) is similar to the well-known Leray-Schauder boundary condition. Some results of this type can be found in [17–19].
Corollary 4.8.
Let be a Banach space with the (FPP). Let
be a closed convex and unbounded subset of
. If
is a continuous pseudocontractive mapping weakly inward on
and for every
and
large enough
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ38_HTML.gif)
for some , then
has a fixed point in
.
Proof.
Let be the function defined by
. It is clear that
satisfies conditions (g1) and (g3) . Moreover,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ39_HTML.gif)
for some . Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ40_HTML.gif)
Since is a fix element of
, clearly there exists
such that
whenever
. This means that
satisfies (g2).
To see that satisfies condition (g4) we argue as follows.
Given a fix , we know that
.
Since as
, we can find
such that
for every
.
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ41_HTML.gif)
Now, we will see that satisfies inequality (4.1) in Corollary 4.3. Indeed, if
, we have, for some
, that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ42_HTML.gif)
Thus the conclusion follows from Corollary 4.3.
Remark 4.9.
In the case that for all ,
, then the mapping
is said to have
as a center; see [20], where some fixed point theorems are given for this class of mappings.
On the other hand, in [21, Corollary 1.6, page 54] one can read a similar condition, where the domain of the mapping is required to be bounded.
If is asymptotically contractive in the sense due to Penot, then it is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ43_HTML.gif)
which implies condition (4.11) of Corollary 4.8 for , and therefore Penot's fixed point theorem is a consequence of Corollary 4.8.
Example 4.10.
Next, we are concerned with the solvability of the following Hammerstein's integral equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ44_HTML.gif)
in . Here
,
is a bounded domain of
such that its Lebesgue's measure
, and
. Suppose that
and
satisfy the following conditions:
(1) is a Carathéodory function,
(2), where
and
,
(3),
(4)the function is strongly measurable and
whenever
,
(5)there exists a function , belonging to
such that
for all
,
(6) and
.
Proposition 4.11.
Assume that conditions (1)–(6) are satisfied, then problem (4.17) has at least one solution in .
Proof.
First notice that (4.17) may be written in the form where
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ45_HTML.gif)
Let us see that satisfies the conditions of Corollary 4.8. In this sense, we are going to prove that
is a nonexpansive mapping. Indeed,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ46_HTML.gif)
Since , by Holder's inequality, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ47_HTML.gif)
Finally, we are going to show that there exists such that if
, then
. Indeed, we know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ48_HTML.gif)
hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ49_HTML.gif)
Applying again Holder's inequality, we derive that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ50_HTML.gif)
Moreover, it is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ51_HTML.gif)
therefore there exists such that if
, then
as we claimed.
Notice that if then, Corollary
in [4] does not apply because under this condition we cannot guarantee that
is asymptotically contractive on
.
Let be a closed convex subset of a Banach space
. A family of mappings
is called a one-parametric strongly continuous semigroup of nonexpansive mappings (nonexpansive semigroup, for short) on
if the following assumptions are satisfied:
(1) for all
,
(2)for each , the mapping
from
into
is continuous,
(3)for each ,
is a nonexpansive mapping.
In the next result we study when a nonexpansive semigroup has a common fixed point.
Theorem 4.12.
Let be a Banach space with the (FPP). Suppose that
is a mapping satisfying conditions (g1)–(g4). Let
be a closed convex and unbounded subset of
. If
is a nonexpansive semigroup such that there exist
with
satisfying that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ52_HTML.gif)
whenever large enough, then the semigroup has a least one common fixed point.
Proof.
By Theorem of [22] in order to get the conclusion it is enough to show that, given
, the mapping
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ53_HTML.gif)
has a fixed point.
By hypotheses we know that there exists such that for every
with
the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ54_HTML.gif)
holds. Since satisfies conditions (g1)–(g4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ55_HTML.gif)
The above inequality means that satisfies the conditions of Corollary 4.3 and therefore
has a fixed point, which implies by Theorem
of [22] that the semigroup has a common fixed point.
Corollary 4.13.
Let be a Banach space with the (FPP). Let
be a closed convex and unbounded subset of
. If
is a nonexpansive semigroup such that there exist
,
with
satisfying that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ56_HTML.gif)
whenever large enough, then the semigroup has a least one common fixed point.
Proof.
It is enough to apply the above theorem with (see the proof of Corollary 4.8).
We conclude this section by presenting a corollary of Theorem 4.1 which guarantees the existence of positive eigenvalues.
Corollary 4.14.
Let be a Banach space with the (FPP). Suppose that
is a mapping satisfying conditions (g1) and (g2). Let
be a closed convex and unbounded subset of
with
. Let
be a continuous pseudocontractive mapping. Assume that the following conditions are satisfied.
(a).
(b)There exists such that for every
with
the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ57_HTML.gif)
holds for some .
Then any is an eigenvalue of
associated to an eigenvector in
.
Proof.
Consider a fixed . Let us see that
is a continuous pseudocontractive mapping such that
. Indeed, since
,
is convex,
, and
, then
.
To see that is a pseudocontractive mapping, it is enough to prove that
is an accretive mapping:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ58_HTML.gif)
The above inequality holds since is a pseudocontractive mapping and therefore
.
Finally, if with
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F769858/MediaObjects/13663_2009_Article_1340_Equ59_HTML.gif)
The above facts show that is under the assumption of Theorem 4.1 and hence there exists
such that
.
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Acknowledgments
Both authors were partially supported by MTM 2009-10696-C02-02. This work is dedicated to Professor W. A. Kirk.
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García-Falset, J., Llorens-Fuster, E. Fixed Points for Pseudocontractive Mappings on Unbounded Domains. Fixed Point Theory Appl 2010, 769858 (2009). https://doi.org/10.1155/2010/769858
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DOI: https://doi.org/10.1155/2010/769858