Journal of mathematics and computer science
13 (2014), 336-342
A Best Proximity Point Theorem in Metric Spaces with Generalized Distance Mehdi Omidvari1, S. Mansour Vaezpour2 1
2
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran, Department of Mathematics and Computer Science, Amirkabir University of Technology, Hafez Ave., P. O. Box 15914, Tehran, Iran, 1
[email protected], 2
[email protected]
Abstract In this paper at first, we define the weak P-property with respect to a -distance such as p. Then we state a best proximity point theorem in a complete metric space with generalized distance such that it is an extension of previous research.
Keywords: weak P-property, best proximity point, -distance, weakly contractive mapping, altering distance functions.
1. Introduction The best proximity point is a interesting topic in best proximity theory. Let be two non-empty subsets of a metric space and . A solution , for the equation is called a best proximity point of . If then is called a fixed point of [15]. The existence and convergence of best proximity points has generalized by several authors such as Jleli and Samet [3], Prolla [4], Reich [5], Sadiq Basha [7,8], Sehgal and Singh [10,11], Vertivel, Veermani and Bhattacharyya[13] in many directions. On the other hand Suzuki [12] introduced the concept of distance on a metric space. Many fixed point theorems extended for various contractive mappings with respect to a -distance. In this paper, by using the concept of -distance, we prove a best proximity point theorem. Our results are extension of a best proximity point theorem in metric spaces.
2. Preliminary Let be two non-empty subsets of a metric space throughout this paper: { } { } { } 2
Corresponding author
. The following notations will be used
M. Omidvari, S. M. Vaezpour / J. Math. Computer Sci. 13 (2014), 336-342
{ }. We recall that is a best proximity point of the mapping if . It can be observed that a best proximity reduces to a fixed point if the underlying mapping is a selfmapping. Definition 2.1.[9] Let be a pair of non-empty subsets of a metric space pair is said to have the P-property if and only if
with
. Then the
} where and . It is clear that, for any nonempty subset of , the pair has the P-property. Rhoades [6] introduced a class of contractive mappings called weakly contractive mapping. Harjani and Sadarangani [1] generalized the concept of the weakly contractive mappings in partially ordered metric spaces. Definition 2.2.[2] A function the following conditions: (i) is continuous and non-decreasing. (ii) if and only if . Definition 2.3.[6] Let
is said to be an altering distance function if it satisfies
be a metric space. (
Where
is weakly contractive if )
is a altering distance function.
Suzuki [12] introduced the concept of -distance on a metric space. Definition 2.4.[12] Let be a metric space with metric . A function is called distance on X if there exist a function such that the following are satisfied: ( ( and for all and , and is concave and continuous in it’s second variable. ( and imply for all { ( ) } ( (
{ (
)
} and
Remark 2.5.[12] It can be replaced { } for all Remark 2.6. If
and (
)
imply imply
by the following . and is non-decreasing in it’s second variable.
is a metric space, then the metric d is a -distance on .
In the following examples, we define by , for all . It is easy to see that is a -distance on metric space . Example 2.7. Let be a metric space and be a positive real number. Then for is a -distance on . ‖ ‖ be a normed space. ‖ ‖ ‖ ‖ for Example 2.8. Let by is a -distance on . Example 2.9. Let distance on .
‖ ‖ be a normed space.
by
337
‖ ‖ for
and by
is a -
M. Omidvari, S. M. Vaezpour / J. Math. Computer Sci. 13 (2014), 336-342
Definition 2.10.[12] Let be a metric space and Cauchy if there exists a function such that . { ( ) }
be a -distance on . A sequence { } in is a satisfying ( -( and a sequence { } in
The following lemmas are essential for the next sections. Lemma 2.11.[12] Let be a metric space and be a -distance on sequence, then it is a Cauchy sequence. Moreover if { } is { } , then { } is also -Cauchy sequence and Lemma 2.12.[12] Let for some , then imply
. If { } is a -Cauchy a sequence satisfying
be a metric space and be a -distance on . If { } in satisfies , then { } is a -Cauchy sequence. Moreover if { } in also satisfies In particular, for , and
Lemma 2.13.[12] Let be a metric space and be a -distance on . If { } in { } , then { } is a -Cauchy sequence. Moreover if { } in , then { } is also -Cauchy sequence and
satisfies satisfies
The next result is an immediate consequence of the Lemma 2.11 and Lemma 2.13. Corollary 2.14. Let be a metric space and be a -distance on . If a sequence { } in { } , then { } is a Cauchy sequence.
satisfies
3. Main results Inspire of Sankar Raj[9] and Zhang and others[14], we define the weak -property with respect to a -distance as follows: Definition 3.1. Let be a pair of non-empty subsets of a metric space with . Also let be a -distance on . Then the pair is said to have the weak P-property with respect to if and only if } where and . It is clear that, for any nonempty subset
of , the pair
has the weak P-property with respect to .
Remark 3.2. If then is said to have the weak -property where . (See [14]) It is easy to see that if has the -property then has the weak -property. Example 3.3. Let with the usual metric and be two -distances that defined in Example 2.8 and Example 2.9, respectively. Consider, { }, { } { } Then has the weak -property with respect to and has not the weak -property with respect to . By the definition of we obtain, ( ) ( ) √ where and We have, and ( ) ( ) √ √ and ( ) ( ) √ √ . Therefore has the weak -property with respect to . On the other hand, we have and ( ) ( ) √ . 338
M. Omidvari, S. M. Vaezpour / J. Math. Computer Sci. 13 (2014), 336-342
This implies that
has not the weak -property with respect to
.
Sankar Raj[9] stated a best proximity point theorem for weakly contractive non-self mappings in metric spaces. The following Theorem is an extension of his results in a metric spaces with generalized distance. Theorem 3.4. Let and be non-empty closed subsets of the metric space such that . Let be a -distance on and satisfies the following conditions:
(a) (b)
and has the has the weak -property with respect to . is a continuous function on such that ( ) ( ) where is an altering distance function and is non-decreasing function also if and only if .
Then has a best proximity point in . Moreover, if , then . Proof. Choose . Since , there exists Again, , there exists such that we can find a sequence { } in such that
for some such that . Continuing this process,
{ } satisfies the weak -property with respect to , therefore from (1) we obtain,
We will prove that the sequence { that
} is convergent in (
(1)
(2) is non-decreasing function we receive
. Since
)
(3)
Also by the definition of , we have (4) From (3) and (4), we receive that
for all
( ) ( ) is non-decreasing function, we have
Since
Therefore, the sequence { such that We claim that
(
)
} is monotone non-increasing and bounded. Hence there exists
. Suppose to the contrary, that ( )
. From the inequality ( ) (
)
we obtain (
and
Since
)
is non-decreasing function, (
)
So, ( which
is
a
contradiction.
Now we show that subsequence { }{ means that
Hence
} such that
) Similarly
we
for . In contrary case, there exists is smallest index for which , (
339
receive
that
and two This )
M. Omidvari, S. M. Vaezpour / J. Math. Computer Sci. 13 (2014), 336-342
(
)
(5)
So, by the triangle inequality and (5), we have ( (
) )
( )
( Letting
)
, we receive that (
)
By triangle inequality, we have ( ) ( ) ( ( ) ( ) ( Letting in above two inequality and using (6), we get ( )
(6) ) )
( (
) )
So, ( (
))
( (
))
( ( From continuity of
( ( in the above inequality, we obtain that ( (
(
From
)
, we can find
( (
)) )) ))
such that for any
(
))
(7) ,
)
This implies that, ( )
( (
))
and this contradicts to (7). Thus
for {
and this implies that, }
Therefore by Corollary 2.14, { } is a Cauchy sequence in . Since is a complete metric space and is a closed subset of , there exists such that . is continuous, therefore with letting in (1), we obtain Now let
such that
We claim that . Suppose to the contrary, that . Hence ( therefore by the definition of , we obtain that, ( ) ( ) ( ) ( ) ( ) which is a contradiction. Hence and this completes the proof of the theorem. The next result is an immediate consequence of the Theorem 3.4 by taking Corollary 3.5. Let and be non-empty closed subsets of the metric space be a -distance on and satisfies the following conditions:
(a) (b)
)
and
for all such that
. . Let
and has the has the weak -property with respect to . is a continuous function on such that
340
M. Omidvari, S. M. Vaezpour / J. Math. Computer Sci. 13 (2014), 336-342
where
is non-decreasing function also
Then has a best proximity point in . Moreover, if , then .
if and only if
.
for some
The following result is the special case of the Corollary 3.5, obtained by setting . Corollary 3.6.[9] Let be a pair of two nonempty, closed subsets of a complete metric space such that is non-empty. Let be a weakly contractive mapping such that . Assume that the pair has the -property. Then there exists a unique in A such that .
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