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Best Proximity Points of Cyclic
-Contractions on Reflexive Banach Spaces
Fixed Point Theory and Applications volume 2010, Article number: 946178 (2010)
Abstract
We provide a positive answer to a question raised by Al-Thagafi and Shahzad (Nonlinear Analysis, 70 (2009), 3665-3671) about best proximity points of cyclic -contractions on reflexive Banach spaces.
1. Introduction
As a generalization of Banach contraction principle, Kirk et al. proved, in 2003, the following fixed point result; see [1].
Theorem 1.1.
Let and
be nonempty closed subsets of a complete metric space
. Suppose that
is a map satisfying
,
and there exists
such that
for all
and
. Then,
has a unique fixed point in
.
Let and
be nonempty closed subsets of a metric space
and
a strictly increasing map. We say that
is a cyclic
-contraction map [2] whenever
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ1_HTML.gif)
for all and
, where
. Also,
is called a best proximity point if
. As a special case, when
, in which
is a constant,
is called cyclic contraction.
In 2005, Petruel proved some periodic point results for cyclic contraction maps [3]. Then, Eldered and Veeramani proved some results on best proximity points of cyclic contraction maps in 2006 [4]. They raised a question about the existence of a best proximity point for a cyclic contraction map in a reflexive Banach space. In 2009, Al-Thagafi and Shahzad gave a positive answer to the question [2]. More precisely, they proved some results on the existence and convergence of best proximity points of cyclic contraction maps defined on reflexive (and strictly convex) Banach spaces [2, Theorems 9, 10, 11, and 12]. They also introduced cyclic
-contraction maps and raised the following question in [2].
Question 1.
It is interesting to ask whether Theorems 9 and 10 (resp., Theorems 11 and 12) hold for cyclic -contraction maps where the space is only reflexive (resp., reflexive and strictly convex) Banach space.
In this paper, we provide a positive answer to the above question. For obtaining the answer, we use some results of [2].
2. Main Results
First, we give the following extension of [4, Proposition 3.3] for cyclic -contraction maps, where
is unbounded.
Theorem 2.1.
Let be a strictly increasing unbounded map. Also, let
and
be nonempty subsets of a metric space
,
a cyclic
-contraction map,
and
for all
. Then, the sequences
and
are bounded.
Proof.
Suppose that (the proof when
is similar). By [2, Theorem 3],
. Hence, it is sufficient to prove that
is bounded. Since
is unbounded, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ2_HTML.gif)
If is not bounded, then there exists a natural number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ3_HTML.gif)
Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ4_HTML.gif)
Since for all
and
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ5_HTML.gif)
Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ6_HTML.gif)
Thus, we obtain . Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ7_HTML.gif)
. Hence,
. This contradiction completes the proof.
Since the proof of last result was classic, we presented it separately. Here, we provide our key result via a special proof which is a general case of Theorem 2.1.
Theorem 2.2.
Let be a strictly increasing map. Also, let
and
be nonempty subsets of a metric space
,
a cyclic
-contraction map,
, and
for all
. Then, the sequences
and
are bounded.
Proof.
Suppose that (the proof when
is similar). By [2, Theorem 3],
. Hence, either
and
are bounded or both are unbounded. Suppose that both sequences are unbounded. Fix
and define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ8_HTML.gif)
for all . Since
is unbounded,
for all
. Thus, we can choose a strictly increasing subsequence
of the sequence
. Since
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ9_HTML.gif)
Again, we can choose a strictly increasing subsequence of the sequence
such that
. By continuing this process, for each natural number
, we can choose a strictly increasing subsequence
of the sequence
such that
. By the construction, if we consider the sequence
, then
,
is a strictly increasing subsequence of
and
for all
. Now, define
. Also, by induction define the sequence
by
. Note that, the sequence
is strictly increasing and
. Since
is a cyclic
-contraction map,
is a decreasing sequence. Hence by the construction of the sequence
,
is a decreasing sequence. Let
be given. Since
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ10_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ11_HTML.gif)
for all . Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ12_HTML.gif)
for all . Since
for all
and
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ13_HTML.gif)
for all . Consequently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ14_HTML.gif)
for all . This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ15_HTML.gif)
for all , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ16_HTML.gif)
is a constant. But, for all
. This contradiction completes the proof.
Now by using this key result, we provide our main results which give positive answer to the question. Their proofs are basically due to Al-Thagafi and Shahzad [2]. However, the crucial role is played by our key result. Weak convergence of to
is denoted by
.
Theorem 2.3.
Let be a strictly increasing map. Also, let
and
be nonempty weakly closed subsets of a reflexive Banach space and
a cyclic
-contraction map. Then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ17_HTML.gif)
Proof.
If , the result follows from [2, Theorem 1]. So, we assume that
. For
, define
for all
. By Theorem 2.2, the sequences
and
are bounded. Since
is reflexive and
is weakly closed, the sequence
has a subsequence
such that
. As
is bounded and
is weakly closed, we can say, without loss of generality, that
as
. Since
as
, there exists a bounded linear functional
such that
and
. For each
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ18_HTML.gif)
Since , by using [2, Theorem 3] we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ19_HTML.gif)
Hence, .
Definition 2.4.
(see [2]) Let and
be nonempty subsets of a normed space
,
,
, and
. We say that
satisfies the proximal property if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ20_HTML.gif)
Theorem 2.5.
Let be a strictly increasing map. Also, let
and
be nonempty subsets of a reflexive Banach space
such that
is weakly closed and
a cyclic
-contraction map. Then, there exists
such that
provided that one of the following conditions is satisfied
(a) is weakly continuous on
.
(b) satisfies the proximal property.
Proof.
If , the result follows from [2, Theorem 1]. So, we assume that
. For
, define
for all
. By Theorem 2.2, the sequence
is bounded. Since
is reflexive and
is weakly closed, the sequence
has a subsequence
such that
as
.
(a)Since is weakly continuous on
and
, we have
as
. So
as
. The rest of the proof is similar to that of Theorem 2.3.
(b)By [2, Theorem 3], we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ21_HTML.gif)
as . Since
satisfies the proximal property, we have
.
Theorem 2.6.
Let be a strictly increasing map. Also, let
and
be nonempty closed and convex subsets of a reflexive and strictly convex Banach space and
a cyclic
-contraction map. If
, then there exists a unique
such that
and
.
Proof.
If , the result follows from [2, Theorem 1]. So, we assume that
. Since
is closed and convex, it is weakly closed. By Theorem 2.3, there exists
with
. To show the uniqueness of
, suppose that there exists another
with
. Since
, we conclude that
. As both
and
are convex, by the strict convexity of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ22_HTML.gif)
which is a contradiction. Since , we obtain, from the uniqueness of
, that
. Hence
,
and
.
Theorem 2.7.
Let be a strictly increasing map. Also, let
and
be nonempty subsets of a reflexive and strictly convex Banach space
such that
is closed and convex and
a cyclic
-contraction map. Then, there exists a unique
such that
and
provided that one of the following conditions is satisfied
(a) is weakly continuous on
.
(b) satisfies the proximal property.
Proof.
If , the result follows from [2, Theorem 1]. So, we assume that
. Since
is closed and convex, it is weakly closed. By Theorem 2.5 that there exists
with
. Thus,
. Indeed, if we assume that
. Then from the convexity of
and the strict convexity of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F946178/MediaObjects/13663_2009_Article_1369_Equ23_HTML.gif)
which is a contradiction. The uniqueness of follows as in the proof of [2, Theorem  8].
References
Kirk WA, Srinivasan PS, Veeramani P: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 2003,4(1):79–89.
Al-Thagafi MA, Shahzad N: Convergence and existence results for best proximity points. Nonlinear Analysis. Theory, Methods & Applications 2009,70(10):3665–3671. 10.1016/j.na.2008.07.022
Petruşel G: Cyclic representations and periodic points. Universitatis Babeş-Bolyai. Studia. Mathematica 2005,50(3):107–112.
Eldred AA, Veeramani P: Existence and convergence of best proximity points. Journal of Mathematical Analysis and Applications 2006,323(2):1001–1006. 10.1016/j.jmaa.2005.10.081
Acknowledgment
The authors express their gratitude to the referees for their helpful suggestions concerning the final version of this paper.
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Rezapour, S., Derafshpour, M. & Shahzad, N. Best Proximity Points of Cyclic -Contractions on Reflexive Banach Spaces.
Fixed Point Theory Appl 2010, 946178 (2010). https://doi.org/10.1155/2010/946178
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DOI: https://doi.org/10.1155/2010/946178