Abstract
The existence of approximate fixed points and approximate endpoints of the multivalued almost -contractions is established. We also develop quantitative estimates of the sets of approximate fixed points and approximate endpoints for multivalued almost
-contractions. The proved results unify and improve recent results of Amini-Harandi (2010), M. Berinde and V. Berinde (2007), Ćirić (2009), M. Păcurar and R. V. Păcurar (2007) and many others.
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1. Introduction and Preliminaries
In fixed point theory, one of the main directions of investigation concerns the study of the fixed point property in topological spaces. Recall that a topological space is said to have the fixed point property if every continuous mapping
has a fixed point. The major contribution to this subject has been provided by Tychonoff. Every compact convex subset of a locally convex space has the fixed point property. Another important branch of fixed point theory is the study of the approximate fixed point property. Recently, the interest in approximate fixed point results arise in the study of some problems in economics and game theory, including, for example, the Nash equilibrium approximation in games; see [1–3] and references therein.
We establish some existence results concerning approximate fixed points, endpoints, and approximate endpoints of multivalued contractions. We also develop quantitative estimates of the sets of approximate fixed points and approximate endpoints for set-valued almost -contractions. The results presented in this paper extend and improve the recent results of [4–10] and many others.
Now, we give some notions and definitions.
Let be a metric space and let
and
denote the families of all nonempty subsets and nonempty closed subsets of
, respectively. Let
and
be two Hausdorff topological spaces and
a multivalued mapping with nonempty values. Then
is said to be
(1)upper semicontinuous (u.s.c.) if, for each closed set ,
is closed in
;
(2)lower semicontinuous (l.s.c.) if, for each open set ,
is open in
;
(3)continuous if it is both u.s.c. and l.s.c.;
(4)closed if its graph is closed;
(5)compact if is a compact subset of
.
For any subsets , of a metric space
, we consider the following notions:
: the distance between the sets
and
;
: the diameter of the sets
and
;
: the diameter of the set
;
: the Hausdorff metric on
induced by the metric
.
Let be a multivalued mapping. An element
such that
is called a fixed point of
. We denote by
the set of all fixed points of
, that is,
A mapping is called
a multivalued contraction (or multivalued-contraction) if there exists a number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ1_HTML.gif)
a multivalued almost contraction [6] or a multivalued-almost contraction if there exist two constants
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ2_HTML.gif)
a generalized multivalued almost contraction [6] if there exists a function satisfying
for every
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ3_HTML.gif)
It is important to note that any mapping satisfying Banach, Kannan, Chatterjea, Zamfirescu, or Ćirić (with the constant in
) type conditions is a single-valued almost contraction; see [5, 6, 8, 11].
2. Approximate Fixed Points of Multivalued Contractions
Definition 2.1.
A multivalued mapping is said to have the approximate fixed point property [2] provided
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ4_HTML.gif)
or, equivalently, for any , there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ5_HTML.gif)
or, equivalently, for any , there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ6_HTML.gif)
where denotes a closed ball of radius
centered at
.
We first prove that every generalized multivalued almost contraction has the approximate fixed point property.
Lemma 2.2.
Every generalized multivalued almost contraction has the approximate fixed point property.
Proof.
Let be an arbitrary metric space and
a generalized multivalued almost contraction. Let
and
be such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ7_HTML.gif)
By passing to the subsequences, if necessary, we may assume that the sequence is convergent. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ8_HTML.gif)
Since , we get
. This completes the proof.
Corollary 2.3 (see [5, Theorem ], [10, Theorem
]).
Let be a metric space and
a single-valued almost contraction. Then
has the approximate fixed point property.
The authors in [5, 10] obtained the following quantitative estimate of the diameter of the set, , of approximate fixed points of single-valued almost contraction
.
Theorem 2.4 (see [5, Theorem ], [10, Theorem
]).
Let be a metric space. If
is a single-valued almost contraction with
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ9_HTML.gif)
The following simple example shows that the conclusion of Theorem 2.4 is not valid for set-valued almost contractions.
Example 2.5.
Let be defined by
. Then
and so
is multivalued almost contraction with
. Further,
and so
. This shows that conclusion of Theorem 2.4 is not true whenever
is multivalued almost contraction.
Theorem 2.6.
Let be a metric space. If
is a generalized multivalued almost contraction, then
has a fixed point provided either
is compact and the function
is lower semicontinuous or
is closed and compact.
Proof.
By Lemma 2.2, we have . The lower semicontinuity of the function
and the compactness of
imply that the infimum is attained. Thus there exists an
such that
and so
.
Suppose that is closed and compact. According to Lemma 2.2,
has the approximate fixed point property. Therefore, for any
, there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ10_HTML.gif)
Now, since is compact, we may assume that
converges to a point
as
. Consequently,
also converges to
as
. Since
is closed, then
This completes the proof.
Let be a single-valued mapping and
a multivalued mapping. Then
is called a multivalued almost
-contraction [6, 8] if there exist constants
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ11_HTML.gif)
We say that is a generalized multivalued almost
-contraction if there exists a function
satisfying
for every
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ12_HTML.gif)
The mappings and
are said to have an approximate coincidence point property provided
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ13_HTML.gif)
or, equivalently, for any , there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ14_HTML.gif)
A point is called a coincidence (common fixed) point of
and
if
(
).
Theorem 2.7.
Every generalized multivalued almost -contraction in a metric space
has the approximate coincidence point property provided each
is
-invariant. Further, if
is compact and the function
is lower semicontinuous, then
and
have a coincidence point.
Proof.
Let be a generalized multivalued almost
-contraction and let
and
be such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ15_HTML.gif)
By passing to the subsequences, if necessary, we may assume that the sequence is convergent. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ16_HTML.gif)
since each is
-invariant, that is, for each
, we have
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ17_HTML.gif)
we get .
Further, the lower semi-continuity of the function and the compactness of
imply that the infimum is attained. Thus there exists
such that
and so
as required. This completes the proof.
Corollary 2.8.
Every multivalued almost -contraction in a metric space
has the approximate coincidence point property provided each
is
-invariant. Further, if
is compact and the function
is lower semicontinuous, then
and
have a coincidence point.
Recently, Ćirić [7] has introduced multivalued contractions and obtained some interesting results which are proper generalizations of the recent results of Klim and Wardowski [9], Feng and Liu [12], and many others. In the results to follow, we obtain approximate fixed point property for these multivalued contractions.
Theorem 2.9.
Let be a metric space and
a multivalued mapping from
into
Suppose that there exist a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ18_HTML.gif)
and and
satisfying the following two conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ19_HTML.gif)
where . Then
has the approximate fixed point property. Further,
has a fixed point provided either
is compact and the function
is lower semicontinuous or
is closed and compact.
Proof.
Let and
be the sequences that satisfy (2.16). By passing to subsequences, if necessary, we may assume that both of the sequences
and
are convergent (note that
is bounded since
). Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ20_HTML.gif)
Since , we get
.
Further, the lower semi-continuity of the function and the compactness of
imply that the infimum is attained. Thus there exists
such that
and so
.
The second assertion follows as in the proof of Theorem 2.6. This completes the proof.
Theorem 2.10.
Let be a metric space and
a multivalued mapping from
into
Suppose that there exist a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ21_HTML.gif)
and and
satisfying the following two conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ22_HTML.gif)
where . Then
has the approximate fixed point property. Further,
has a fixed point provided either
is compact and the function
is lower semicontinuous or
is closed and compact.
Proof.
Let and
satisfy (2.19). By passing to subsequences, if necessary, we may assume that the sequence
is convergent. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ23_HTML.gif)
Since , we get
.
Further, the lower semi-continuity of the function and the compactness of
imply that the infimum is attained. Thus there exists
such that
and so
.
The second assertion follows as in the proof of Theorem 2.6. This completes the proof.
3. Endpoints of Multivalued Nonlinear Contractions
Let be a multivalued mapping. An element
is said to be a endpoint (or stationary point) [13] of
if
. We say that a multivalued mapping
has the approximate endpoint property [4] if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ24_HTML.gif)
Let be a single-valued mapping and
a multivalued contraction. We say that the mappings
and
have an approximate endpoint property provided
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ25_HTML.gif)
A point is called an endpoint of
and
if
.
For each , set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ26_HTML.gif)
Lemma 3.1.
Let be a metric space. Let
be a single-valued mapping such that
for all
, where
is a constant. If
is a multivalued almost
-contraction with
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ27_HTML.gif)
Proof.
For any , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ28_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ29_HTML.gif)
Since , from (3.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ30_HTML.gif)
The following simple example shows that under the assumptions of Lemma 3.1, may be empty.
Example 3.2.
Let be a multivalued mapping defined by
for each
and
the identity mapping. Then
and so
is a multivalued almost
-contraction with
. However,
for each
.
Lemma 3.3.
Let be a metric space. Let
be a continuous single-valued mapping and
a lower semicontinuous multivalued mapping. Then, for each
,
is closed.
Proof.
Let be such that with
as
. Let
. Since
is lower semicontinuous, then there exists
such that
. Since
, then
and so
. Since
is continuous,
. Therefore,
, that is,
. This completes the proof.
Theorem 3.4.
Let be a complete metric space. Let
be a continuous single-valued mapping such that
, where
is a constant. Let
be a lower semicontinuous multivalued almost
-contraction. Then
and
have a unique endpoint if and only if
and
have the approximate endpoint property.
Proof.
It is clear that, if and
have an endpoint, then
and
have the approximate endpoint property. Conversely, suppose that
and
have the approximate endpoint property. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ31_HTML.gif)
Also it is clear that, for each ,
. By Lemma 3.3,
is closed for each
. Since
and
have the approximate endpoint property, then
for each
. Now, we show that
. To show this, let
. Then, from Lemma 3.1,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ32_HTML.gif)
and so . It follows from the Cantor intersection theorem that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ33_HTML.gif)
Thus is the unique endpoint of
and
.
If is the identity mapping on
, then the above result reduces to the following.
Corollary 3.5.
Let be a metric space. If
is a multivalued almost contraction with
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ34_HTML.gif)
where .
Corollary 3.6.
Let be a complete metric space. Let
be a lower semicontinuous multivalued almost contraction with
. Then
has a unique endpoint if and only if
has the approximate endpoint property.
Corollary 3.7 (see [4, Corollary ]).
Let be a complete metric space. Let
be a multivalued
-contraction. Then
has a unique endpoint if and only if
has the approximate endpoint property.
Theorem 3.8.
Let be a complete metric space and
a multivalued mapping from
into
. Suppose that there exist a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ35_HTML.gif)
and and
satisfying the two following conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ36_HTML.gif)
where . Then
has the approximate endpoint property. Further,
has an endpoint provided
is compact and the function
is lower semicontinuous.
Proof.
We first prove that has the approximate endpoint property. Let
and
that satisfy (3.13). By passing to subsequences, if necessary, we may assume that the sequence
is convergent. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ37_HTML.gif)
Since , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ38_HTML.gif)
Thus has the approximate endpoint property. The lower semi-continuity of the function
and the compactness of
imply that the infimum is attained. Thus there exists
such that
. Therefore,
. This completes the proof.
The following theorem extends and improves Theorem in [4].
Theorem 3.9.
Let be a complete metric space. Let
be a continuous single-valued mapping such that
, where
is a constant. Let
be a multivalued mapping satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ39_HTML.gif)
where is a function such that
and
for each
. Then
and
have a unique endpoint if and only if
and
have the approximate endpoint property.
Proof.
It is clear that, if and
have an endpoint, then
and
have the approximate endpoint property. Conversely, suppose that
and
have the approximate endpoint property. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ40_HTML.gif)
Also it is clear that, for each ,
. Since the mapping
is continuous (note that
and
are continuous), we have that
is closed. Now we show that
. On the contrary, assume that
. Since
, then
(note that the sequences
and
are nonincreasing and bounded below and then they have the limits). Let
be such that
. Given
, from (3.16) and triangle inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ41_HTML.gif)
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ42_HTML.gif)
From (3.19), we have for each
and so we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ43_HTML.gif)
Hence we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ44_HTML.gif)
From (3.21), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ45_HTML.gif)
Since , from (3.22), we get
. Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ46_HTML.gif)
which is a contradiction and so . It follows from the Cantor intersection theorem that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ47_HTML.gif)
Thus and hence
. To prove the uniqueness of the endpoints of
and
, let
be an arbitrary endpoint of
and
. Then
=0 and so
. Thus
. This completes the proof.
From Theorem 3.9, we obtain the following improved version of the main result of [4].
Corollary 3.10.
Let be a complete metric space. Let
be a multivalued mapping satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ48_HTML.gif)
where is a function such that
and
for each
. Then
has a unique endpoint if and only if
has the approximate endpoint property.
Example 3.11.
Let with the usual metric
. Let
be a multivalued mapping defined by
and
be a function defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ49_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F614867/MediaObjects/13663_2010_Article_1313_Equ50_HTML.gif)
Then and
satisfy the conditions of Corollary 3.10, but the conditions of Theorem
in [4] are not satisfied (note that
).
References
Barroso CS: The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete and Continuous Dynamical Systems 2009,25(2):467–479.
Khamsi MA: On asymptotically nonexpansive mappings in hyperconvex metric spaces. Proceedings of the American Mathematical Society 2004,132(2):365–373. 10.1090/S0002-9939-03-07172-7
Lin PK, Sternfeld Y: Convex sets with the Lipschitz fixed point property are compact. Proceedings of the American Mathematical Society 1985,93(4):633–639. 10.1090/S0002-9939-1985-0776193-5
Amini-Harandi A: Endpoints of set-valued contractions in metric spaces. Nonlinear Analysis: Theory, Methods & Applications 2010,72(1):132–134. 10.1016/j.na.2009.06.074
Berinde M: Approximate fixed point theorems. Studia. Universitatis Babeş-Bolyai. Mathematica 2006,51(1):11–25.
Berinde M, Berinde V: On a general class of multi-valued weakly Picard mappings. Journal of Mathematical Analysis and Applications 2007,326(2):772–782. 10.1016/j.jmaa.2006.03.016
Ćirić L: Multi-valued nonlinear contraction mappings. Nonlinear Analysis: Theory, Methods & Applications 2009,71(7–8):2716–2723. 10.1016/j.na.2009.01.116
Hussain N, Cho YJ: Weak contractions, common fixed points, and invariant approximations. Journal of Inequalities and Applications 2009, 2009:-10.
Klim D, Wardowski D: Fixed point theorems for set-valued contractions in complete metric spaces. Journal of Mathematical Analysis and Applications 2007,334(1):132–139. 10.1016/j.jmaa.2006.12.012
Păcurar M, Păcurar RV: Approximate fixed point theorems for weak contractions on metric spaces. Carpathian Journal of Mathematics 2007,23(1–2):149–155.
Hussain N: Common fixed points in best approximation for Banach operator pairs with Ćirić type -contractions. Journal of Mathematical Analysis and Applications 2008,338(2):1351–1363. 10.1016/j.jmaa.2007.06.008
Feng Y, Liu S: Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings. Journal of Mathematical Analysis and Applications 2006,317(1):103–112. 10.1016/j.jmaa.2005.12.004
Aubin J-P, Siegel J: Fixed points and stationary points of dissipative multivalued maps. Proceedings of the American Mathematical Society 1980,78(3):391–398. 10.1090/S0002-9939-1980-0553382-1
Acknowledgments
The authors would like to thank the referees for their valuable suggestions to improve the paper. This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).
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Hussain, N., Amini-Harandi, A. & Cho, Y. Approximate Endpoints for Set-Valued Contractions in Metric Spaces. Fixed Point Theory Appl 2010, 614867 (2010). https://doi.org/10.1155/2010/614867
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DOI: https://doi.org/10.1155/2010/614867