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Fixed Point Theorems in Cone Banach Spaces
Fixed Point Theory and Applications volume 2009, Article number: 609281 (2009)
Abstract
In this manuscript, a class of self-mappings on cone Banach spaces which have at least one fixed point is considered. More precisely, for a closed and convex subset of a cone Banach space with the norm
, if there exist
,
,
and
satisfies the conditions
and
for all
, then
has at least one Fixed point.
1. Introduction and Preliminaries
In 1980, Rzepecki [1] introduced a generalized metric on a set
in a way that
, where
is Banach space and
is a normal cone in
with partial order
. In that paper, the author generalized the fixed point theorems of Maia type [2].
Let
be a nonempty set endowed in two metrics
,
and
a mapping of
into itself. Suppose that
for all
, and
is complete space with respect to
, and
is continuous with respect to
, and
is contraction with respect to
, that is,
for all
, where
. Then
has a unique fixed point in
.
Seven years later, Lin [3] considered the notion of -metric spaces by replacing real numbers with cone
in the metric function, that is,
. In that manuscript, some results of Khan and Imdad [4] on fixed point theorems were considered for
-metric spaces. Without mentioning the papers of Lin and Rzepecki, in 2007, Huang and Zhang [5] announced the notion of cone metric spaces (CMS) by replacing real numbers with an ordering Banach space. In that paper, they also discussed some properties of convergence of sequences and proved the fixed point theorems of contractive mapping for cone metric spaces: any mapping
of a complete cone metric space
into itself that satisfies, for some
, the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ1_HTML.gif)
for all , has a unique fixed point.
Recently, many results on fixed point theorems have been extended to cone metric spaces (see, e.g., [5–9]). Notice also that in ordered abstract spaces, existence of some fixed point theorems is presented and applied the resolution of matrix equations (see, e.g., [10–12]).
In this manuscript, some of known results (see, e.g., [13, 14]) are extended to cone Banach spaces which were defined and used in [15, 16] where the existence of fixed points for self-mappings on cone Banach spaces is investigated.
Throughout this paper stands for real Banach space. Let
always be a closed nonempty subset of
.
is called cone if
for all
and nonnegative real numbers
where
and
.
For a given cone , one can define a partial ordering (denoted by
or
) with respect to
by
if and only if
. The notation
indicates that
and
, while
will show
, where
denotes the interior of
. From now on, it is assumed that
The cone is called
normal if there is a number such that for all
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ2_HTML.gif)
regular if every increasing sequence which is bounded from above is convergent. That is, if is a sequence such that
for some
, then there is
such that
.
In , the least positive integer
, satisfying (1.2), is called the normal constant of
.
-
(i)
Every regular cone is normal.
(ii)For each , there is a normal cone with normal constant
.
-
(iii)
The cone
is regular if every decreasing sequence which is bounded from below is convergent.
Proofs of (i) and (ii) are given in [6] and the last one follows from definition.
Definition 1.2 (see [5]).
Let be a nonempty set. Suppose the mapping
satisfies
for all
,
if and only if
,
for all
,
for all
,
then is called cone metric on
, and the pair
is called a cone metric space (CMS).
Example 1.3.
Let ,
, and
. Define
by
, where
are positive constants. Then
is a CMS. Note that the cone
is normal with the normal constant
It is quite natural to consider Cone Normed Spaces (CNS).
Definition 1.4 (see [15, 16]).
Let be a vector space over
. Suppose the mapping
satisfies
for all
,
if and only if
,
for all
,
for all
,
then is called cone norm on
, and the pair
is called a cone normed space (CNS).
Note that each CNS is CMS. Indeed, .
Definition 1.5.
Let be a CNS,
and
a sequence in
. Then
(i)converges to
whenever for every
with
there is a natural number
, such that
for all
. It is denoted by
or
;
(ii) is a Cauchy sequence whenever for every
with
there is a natural number
, such that
for all
;
(iii) is a complete cone normed space if every Cauchy sequence is convergent.
Complete cone normed spaces will be called cone Banach spaces.
Lemma 1.6.
Let be a CNS,
a normal cone with normal constant
, and
a sequence in
. Then,
(i)the sequence converges to
if and only if
, as
;
(ii)the sequence is Cauchy if and only if
as
;
(iii)the sequence converges to
and the sequence
converges to
then
.
The proof is direct by applying [5, Lemmas ,
, and
] to the cone metric space
, where
, for all
.
Let be a CNS over a cone
in
. Then
and
,
.
If
then there exists
such that
implies
.
For any given
and
, there exists
such that
.
If
are sequences in
such that
,
, and
, for all
then
.
The proofs of the first two parts followed from the definition of . The third part is obtained by the second part. Namely, if
is given then find
such that
implies
. Then find
such that
and hence
. Since
is closed, the proof of fourth part is achieved.
Definition 1.8 (see [17]).
is called minihedral cone if
exists for all
, and strongly minihedral if every subset of
which is bounded from above has a supremum.
Lemma 1.9 (see [18]).
Every strongly minihedral normal cone is regular.
Example 1.10.
Let with the supremum norm and
Then
is a cone with normal constant
which is not regular. This is clear, since the sequence
is monotonically decreasing, but not uniformly convergent to
. This cone, by Lemma 1.9, is not strongly minihedral. However, it is easy to see that the cone mentioned in Example 1.3 is strongly minihedral.
Definition 1.11.
Let be a closed and convex subset of a cone Banach space with the norm
and
a mapping which satisfies the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ3_HTML.gif)
for all . Then,
is said to satisfy the condition
.
For , the set of fixed points of
is denoted by
.
Definition 1.12 (see [14]).
Let be a closed and convex subset of a cone Banach space with the norm
and
a mapping. Consider the conditions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ4_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ5_HTML.gif)
Then is called nonexpansive (resp., quasi-nonexpansive) if it satisfies the condition (1.4) (resp., (1.5)).
2. Main Results
From now on, will be a cone Banach space,
a normal cone with normal constant
and
a self-mapping operator defined on a subset
of
.
Theorem 2.1.
Let with
and let
be a complete cone metric space
an onto mapping which satisfies the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ6_HTML.gif)
Then, has a unique fixed point.
Proof.
Let and
, then by (2.1), one can observe
which is a contradiction. Thus,
is one-to-one and it has an inverse, say
. Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ7_HTML.gif)
By [5, Theorem ],
has a unique fixed point which is equivalent to saying that
has a unique fixed point.
The following statement is consequence of Definition 1.11.
Proposition 2.2.
Every nonexpansive mapping satisfies the condition .
Proposition 2.3.
Let satisfy the condition
and
, then
is a quasi-nonexpansive.
Proof.
Let and
. Since
and satisfies the condition
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ8_HTML.gif)
Theorem 2.4.
Let be a closed and convex subset of a cone Banach space
with the norm
and
a mapping which satisfies the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ9_HTML.gif)
for all , where
. Then,
has at least one fixed point.
Proof.
Let be arbitrary. Define a sequence
in the following way:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ10_HTML.gif)
Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ11_HTML.gif)
which yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ12_HTML.gif)
for Combining this observation with the condition (2.4), one can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ13_HTML.gif)
Thus, , where
. Hence,
is a Cauchy sequence in
and thus converges to some
Regarding the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ14_HTML.gif)
and by the help of Lemma 1.6(iii), one can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ15_HTML.gif)
Taking into account (2.6) and (2.4), substituting and
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ16_HTML.gif)
Thus, when , one can get
, that is,
.
Notice that identity map, , satisfies the condition (2.4). Thus, maps that satisfy the condition (2.4) may have fixed points.
From the triangle inequality,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ17_HTML.gif)
By (2.4),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ18_HTML.gif)
Thus, letting implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ19_HTML.gif)
Hence we have the following conclusion.
Theorem 2.5.
Let be a closed and convex subset of a cone Banach space with the norm
and
a mapping which satisfies the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ20_HTML.gif)
for all , where
. Then
has a fixed point.
Theorem 2.6.
Let be a closed and convex subset of a cone Banach space with the norm
and
a mapping which satisfies the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ21_HTML.gif)
for all , where
. Then
has at least one fixed point.
Proof.
Construct a sequence as in the proof of Theorem 2.4, that is, (2.5), (2.6) and also
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ22_HTML.gif)
hold. Thus the triangle inequality implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ23_HTML.gif)
Then, by (2.17) and (2.7) we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ24_HTML.gif)
Replacing and
in (2.16) and regarding (2.7) and (2.19), one can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ25_HTML.gif)
and thus, . Since
, the sequence
is a Cauchy sequence that converges to some
. Since
also converges to
as in the proof of Theorem 2.4, the inequality (2.16) (under the assumption
and
) by the help of Lemma 1.6(iii) yields that
which is equivalent to saying that
Theorem 2.7.
Let be a closed and convex subset of a cone Banach space with the norm
. If there exist
and
satisfies the conditions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ26_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ27_HTML.gif)
for all . Then,
has at least one fixed point.
Proof.
Construct a sequence as in the proof of Theorem 2.4. We claim that the inequality (2.22) for
and
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ28_HTML.gif)
for all that satisfy (2.21). For the proof of the claim, first recall from (2.7) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ29_HTML.gif)
The case is trivially true. Indeed, taking into account (2.22) with
and
together with (2.24) and (2.19), one can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ30_HTML.gif)
which is equivalent to (2.23) since . For the case
, consider the inequality
which is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ31_HTML.gif)
By substituting and
in (2.22) together with (2.24), (2.26) and (2.17), one can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ32_HTML.gif)
which is equivalent to (2.23) since . Hence, the claim is proved.
By (2.23), one can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ33_HTML.gif)
Due to (2.21), we have . Thus, the sequence
is a Cauchy sequence that converges to some
. By substituting
with
and
with
in (2.22), one can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F609281/MediaObjects/13663_2009_Article_1162_Equ34_HTML.gif)
as . This last condition is equivalent to saying that
as
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Karapınar, E. Fixed Point Theorems in Cone Banach Spaces. Fixed Point Theory Appl 2009, 609281 (2009). https://doi.org/10.1155/2009/609281
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DOI: https://doi.org/10.1155/2009/609281