Open Access

Fixed points of multivalued mappings in partial metric spaces

Fixed Point Theory and Applications20132013:316

https://doi.org/10.1186/1687-1812-2013-316

Received: 11 April 2013

Accepted: 25 October 2013

Published: 25 November 2013

Abstract

We use the notion of Hausdorff metric on the family of closed bounded subsets of a partial metric space and establish a common fixed point theorem of a pair of multivalued mappings satisfying Mizoguchi and Takahashi’s contractive condition. Our result extends some well-known recent results in the literature.

MSC:46S40, 47H10, 54H25.

Keywords

partial Hausdorff metric common fixed point set-valued mappings partial metric space

1 Introduction and preliminaries

In the last thirty years, the theory of multivalued functions has advanced in a variety of ways. In 1969, the systematic study of Banach-type fixed theorems of multivalued mappings started with the work of Nadler [1], who proved that a multivalued contractive mapping of a complete metric space X into the family of closed bounded subsets of X has a fixed point. His findings were followed by Agarwal et al. [2], Azam et al. [3] and many others (see, e.g., [49]).

In 1994, Matthews [10], introduced the concept of a partial metric space and obtained a Banach-type fixed point theorem on complete partial metric spaces. Later on, several authors (see, e.g., [1117]) proved fixed point theorems of single-valued mappings in partial metric spaces. Recently Aydi et al. [18] proved a fixed point result for multivalued mappings in partial metric spaces. Haghi et al. [19] established that some metric fixed point generalizations to partial metric spaces can be obtained from the corresponding results in metric spaces. In this paper we obtain common fixed points of contractive-type multivalued mappings on partial metric spaces which cannot be deduced from the corresponding results in metric spaces. An example is also established to show that our result is a real generalization of analogous results for metric spaces [1, 9, 10, 18, 20].

We start with recalling some basic definitions and lemmas on a partial metric space.

Definition 1 A partial metric on a nonempty set X is a function p : X × X [ 0 , ) such that for all x , y , z X :

  • (P1) p ( x , x ) = p ( y , y ) = p ( x , y ) if and only if x = y ,

  • (P2) p ( x , x ) p ( x , y ) ,

  • (P3) p ( x , y ) = p ( y , x ) ,

  • (P4) p ( x , z ) p ( x , y ) + p ( y , z ) p ( y , y ) .

The pair ( X , p ) is then called a partial metric space. Also, each partial metric p on X generates a T 0 topology τ p on X with a base of the family of open p-balls { B p ( x , r ) : x X , r > 0 } , where B p ( x , r ) = { y X : p ( x , y ) < p ( x , x ) + r } . If ( X , p ) is a partial metric space, then the function p s : X × X R + given by p s ( x , y ) = 2 p ( x , y ) p ( x , x ) p ( y , y ) , x , y X , is a metric on X. A basic example of a partial metric space is the pair ( R + , p ) , where p ( x , y ) = max { x , y } for all x , y R + .

Lemma 2 [10]

Let ( X , p ) be a partial metric space, then we have the following.
  1. 1.

    A sequence { x n } in a partial metric space ( X , p ) converges to a point x X if and only if lim n p ( x , x n ) = p ( x , x ) .

     
  2. 2.

    A sequence { x n } in a partial metric space ( X , p ) is called a Cauchy sequence if the lim n , m p ( x n , x m ) exists and is finite.

     
  3. 3.

    A partial metric space ( X , p ) is said to be complete if every Cauchy sequence { x n } in X converges to a point x X , that is, p ( x , x ) = lim n , m p ( x n , x m ) .

     
  4. 4.

    A partial metric space ( X , p ) is complete if and only if the metric space ( X , p s ) is complete. Furthermore, lim n p s ( x n , z ) = 0 if and only if p ( z , z ) = lim n p ( x n , z ) = lim n , m p ( x n , x m ) .

     

A subset A of X is called closed in ( X , p ) if it is closed with respect to τ p . A is called bounded in ( X , p ) if there is x 0 X and M > 0 such that a B p ( x 0 , M ) for all a A , i.e., p ( x 0 , a ) < p ( x 0 , x 0 ) + M for all a A .

Let CB p ( X ) be the collection of all nonempty, closed and bounded subsets of X with respect to the partial metric p. For A CB p ( X ) , we define
p ( x , A ) = inf y A p ( x , y ) .
For A , B CB p ( X ) ,
δ p ( A , B ) = sup a A p ( a , B ) , δ p ( B , A ) = sup b B p ( b , A ) , H p ( A , B ) = max { δ p ( A , B ) , δ p ( B , A ) } .

Note that [18] p ( x , A ) = 0 p s ( x , A ) = 0 , where p s ( x , A ) = inf y A p s ( x , y ) .

Proposition 3 [18]

Let ( X , p ) be a partial metric space. For any A , B , C CB p ( X ) , we have
  1. (i):

    δ p ( A , A ) = sup { p ( a , a ) : a A } ;

     
  2. (ii):

    δ p ( A , A ) δ p ( A , B ) ;

     
  3. (iii):

    δ p ( A , B ) = 0 implies that A B ;

     
  4. (iv):

    δ p ( A , B ) δ p ( A , C ) + δ p ( C , B ) inf c C p ( c , c ) .

     

Proposition 4 [18]

Let ( X , p ) be a partial metric space. For any A , B , C CB p ( X ) , we have

  • (h1): H p ( A , A ) H p ( A , B ) ;

  • (h2): H p ( A , B ) = H p ( B , A ) ;

  • (h3): H p ( A , B ) H p ( A , C ) + H p ( C , B ) inf c C p ( c , c ) .

It is immediate [18] to check that H p ( A , B ) = 0 A = B . But the converse does not hold always.

Remark 5 [18]

Let ( X , p ) be a partial metric space and A be a nonempty set in ( X , p ) , then a A ¯ if and only if
p ( a , A ) = p ( a , a ) ,

where A ¯ denotes the closure of A with respect to the partial metric p. Note that A is closed in ( X , p ) if and only if A ¯ = A .

Lemma 6 [21]

Let A and B be nonempty, closed and bounded subsets of a partial metric space ( X , p ) and 0 < h R . Then, for every a A , there exists b B such that p ( a , b ) H p ( A , B ) + h .

Definition 7 [22]

A function φ : [ 0 , + ) [ 0 , 1 ) is said to be an MT-function if it satisfies Mizoguchi and Takahashi’s condition (i.e., lim sup r t + φ ( r ) < 1 for all t [ 0 , + ) ). Clearly, if φ : [ 0 , + ) [ 0 , 1 ) is a nondecreasing function or a nonincreasing function, then it is an MT-function. So, the set of MT-functions is a rich class.

Proposition 8 [22]

Let φ : [ 0 , + ) [ 0 , 1 ) be a function. Then the following statements are equivalent.
  1. 1.

    φ is an MT-function.

     
  2. 2.

    For each t [ 0 , ) , there exist r t ( 1 ) [ 0 , 1 ) and ε t ( 1 ) > 0 such that φ ( s ) r t ( 1 ) for all s ( t , t + ε t ( 1 ) ) .

     
  3. 3.

    For each t [ 0 , ) , there exist r t ( 2 ) [ 0 , 1 ) and ε t ( 2 ) > 0 such that φ ( s ) r t ( 2 ) for all s [ t , t + ε t ( 2 ) ] .

     
  4. 4.

    For each t [ 0 , ) , there exist r t ( 3 ) [ 0 , 1 ) and ε t ( 3 ) > 0 such that φ ( s ) r t ( 3 ) for all s ( t , t + ε t ( 3 ) ] .

     
  5. 5.

    For each t [ 0 , ) , there exist r t ( 4 ) [ 0 , 1 ) and ε t ( 4 ) > 0 such that φ ( s ) r t ( 4 ) for all s [ t , t + ε t ( 4 ) ) .

     
  6. 6.

    For any nonincreasing sequence { x n } n N in [ 0 , ) , we have 0 sup n N φ ( x n ) < 1 .

     
  7. 7.

    φ is a function of contractive factor [23], that is, for any strictly decreasing sequence { x n } n N in [ 0 , ) , we have 0 sup n N φ ( x n ) < 1 .

     

2 Main results

Mizoguchi and Takahashi proved the following theorem in [20].

Theorem 9 Let ( X , d ) be a complete metric space, S : X CB ( X ) be a multivalued map and φ : [ 0 , + ) [ 0 , 1 ) be an MT-function. Assume that
H ( S x , S y ) φ ( d ( x , y ) ) d ( x , y )
(2.1)

for all x , y X , then S has a fixed point in X.

In the following we show that in partial metric spaces Mizoguchi and Takahashi’s contractive condition (2.1) is useful to achieve common fixed points of two distinct mappings. Whereas this condition is not feasible to obtain a common fixed point of two distinct mappings on a metric space.

Theorem 10 Let ( X , p ) be a complete partial metric space, S , T : X CB p ( X ) be multivalued mappings and φ : [ 0 , + ) [ 0 , 1 ) be an MT-function. Assume that
H p ( S x , T y ) φ ( p ( x , y ) ) p ( x , y )
(2.2)

for all x , y X , then there exists z X such that z S z and z T z .

Proof Let x 0 X and x 1 S x 0 . If p ( x 0 , x 1 ) = 0 , then x 0 = x 1 and
H p ( S x 0 , T x 1 ) φ ( p ( x 0 , x 1 ) ) p ( x 0 , x 1 ) = 0 .
Thus S x 0 = T x 1 , which implies that
x 1 = x 0 S x 0 = T x 1 = T x 0
and we finished. Assume that p ( x 0 , x 1 ) > 0 . By Lemma 6, we can take x 2 T x 1 such that
p ( x 1 , x 2 ) H p ( S x 0 , T x 1 ) + p ( x 0 , x 1 ) 2 .
(2.3)
If p ( x 1 , x 2 ) = 0 , then x 1 = x 2 and
H p ( T x 1 , S x 2 ) φ ( p ( x 1 , x 2 ) ) p ( x 1 , x 2 ) = 0 .
Then, T x 1 = S x 2 . That is,
x 2 = x 1 T x 1 = S x 2 = S x 2
and we finished. Assume that p ( x 1 , x 2 ) > 0 . Now we choose x 3 S x 2 such that
p ( x 2 , x 3 ) H p ( T x 1 , S x 2 ) + p ( x 1 , x 2 ) 2 .
(2.4)
By repeating this process, we can construct a sequence x n of points in X and a sequence A n of elements in CB p ( X ) such that
x j + 1 A j = { S x j , j = 2 k , k 0 , T x j , j = 2 k + 1 , k 0
(2.5)
and
p ( x j , x j + 1 ) H p ( A j 1 , A j ) + p ( x j 1 , x j ) 2 with  j 0 ,
(2.6)
along with the assumption that p ( x j , x j + 1 ) > 0 for each j 0 . Now, for j = 2 k + 1 , we have
p ( x j , x j + 1 ) H p ( A j 1 , A j ) + p ( x j 1 , x j ) 2 H p ( S x 2 k , T x 2 k + 1 ) + p ( x 2 k , x 2 k + 1 ) 2 φ ( p ( x 2 k , x 2 k + 1 ) ) p ( x 2 k , x 2 k + 1 ) + p ( x 2 k , x 2 k + 1 ) 2 ( φ ( p ( x j 1 , x j ) ) + 1 2 ) p ( x j 1 , x j ) p ( x j 1 , x j ) .
Similarly, for j = 2 k + 2 , we obtain
p ( x j , x j + 1 ) H p ( T x 2 k + 1 , S x 2 k + 2 ) + p ( x j 1 , x j ) 2 ( φ ( p ( x j 1 , x j ) ) + 1 2 ) p ( x j 1 , x j ) p ( x j 1 , x j ) .
It follows that the sequence { p ( x n , x n + 1 ) } is decreasing and converges to a nonnegative real number t 0 . Define a function ψ : [ 0 , ) [ 0 , 1 ) as follows:
ψ ( ξ ) = φ ( ξ ) + 1 2 .
Then
lim sup ξ t + ψ ( ξ ) < 1 .
Using Proposition 8, for t 0 , we can find δ ( t ) > 0 , λ t < 1 , such that t r δ ( t ) + t implies ψ ( r ) < λ t and there exists a natural number N such that t p ( x n , x n + 1 ) δ ( t ) + t , whenever n > N . Hence
ψ ( p ( x n , x n + 1 ) ) < λ t , whenever  n > N .
Then, for n = 1 , 2 , 3 ,  ,
p ( x n , x n + 1 ) ( φ ( p ( x n 1 , x n ) ) + 1 2 ) p ( x n 1 , x n ) ψ ( p ( x n 1 , x n ) ) p ( x n 1 , x n ) max { max n = 1 N ψ ( p ( x n 1 , x n ) ) , λ t } p ( x n 1 , x n ) [ max { max n = 1 N ψ ( p ( x n 1 , x n ) ) , λ t } ] 2 p ( x n 2 , x n 1 ) [ max { max n = 1 N ψ ( p ( x n 1 , x n ) ) , λ t } ] n p ( x 0 , x 1 ) .
Put max { max n = 1 N ψ ( p ( x n 1 , x n ) ) , λ t } = Ω , then Ω < 1 ,
p ( x n , x n + 1 ) Ω n p ( x 0 , x 1 )
(2.7)
and
p ( x n , x n + m ) i = 1 m p ( x n + i 1 , x n + i ) i = 1 m p ( x n + i , x n + i ) p ( x n , x n + 1 ) + p ( x n + 1 , x n + 2 ) + + p ( x n + m 1 , x n + m ) ( Ω n + Ω n + 1 + + Ω n + m 1 ) p ( x 0 , x 1 ) ( Ω n 1 Ω ) p ( x 0 , x 1 ) 0 as  n ( since  0 < Ω < 1 ) .
By the definition of p s , we get, for any m N ,
p s ( x n , x n + m ) 2 p ( x n , x n + m ) 0 as  n + .
Which implies that { x n } is a Cauchy sequence in ( X , p s ) . Since ( X , p ) is complete, so the corresponding metric space ( X , p s ) is also complete. Therefore, the sequence { x n } converges to some z X with respect to the metric p s , that is, lim n + p s ( x n , z ) = 0 . Since,
p ( x n , x n ) p ( x n , x n + 1 ) Ω n p ( x 0 , x 1 ) 0 as  n .
Therefore
p ( z , z ) = lim n + p ( x n , z ) = lim n p ( x n , x n ) = 0 .
(2.8)
Now from (P4) and (2.2), we get
p ( S z , z ) p ( S z , x 2 n + 2 ) + p ( x 2 n + 2 , z ) p ( x 2 n + 2 , x 2 n + 2 ) p ( x 2 n + 2 , S z ) + p ( x 2 n + 2 , z ) sup u T x 2 n + 1 p ( u , S z ) + p ( x 2 n + 2 , z ) δ p ( T x 2 n + 1 , S z ) + p ( x 2 n + 2 , z ) H p ( T x 2 n + 1 , S z ) + p ( x 2 n + 2 , z ) φ ( p ( x 2 n + 1 , z ) ) p ( x 2 n + 1 , z ) + p ( x 2 n + 2 , z ) p ( x 2 n + 1 , z ) + p ( x 2 n + 2 , z ) .
Taking limit as n , we get
p ( S z , z ) = 0 .
(2.9)
Thus from (2.8) and (2.9), we get
p ( z , z ) = p ( S z , z ) .

Thus by Remark 5, we get that z S z . It follows similarly that z T z . This completes the proof of the theorem. □

Remark 11 The above theorem cannot be deduced from an analogous result of metric spaces. Indeed the contractive condition (2.2) for a pair S , T : X X of mappings on a metric space ( X , d ) , that is,
H d ( S x , T y ) k d ( x , y ) for all  x , y X ,
is not feasible. Because S T implies that S u T u , for some u X , then
H d ( S u , T u ) > 0 = k d ( u , u )

and condition (2.2) is not satisfied for x = y = u . However, the same condition in a partial metric space is practicable to find a common fixed point result for a pair of mappings. This fact can been seen again in the following example.

Example 12 Let X = [ 0 , 1 ] and p ( x , y ) = max { x , y } , and let S , T : X CB p ( X ) be defined by
S x = B ( 0 , 1 7 x ) ¯ , T x = B ( 0 , 2 7 x ) ¯ .
Then
H p ( B ( 0 , 1 7 x ) ¯ , B ( 0 , 2 7 x ) ¯ ) = max { 1 7 x , 2 7 x } and H p ( S x , T y ) = 1 7 max { x , 2 y } H p ( S x , T y ) 3 10 max { x , y } k p ( x , y ) .
Therefore, for φ ( t ) = 3 10 , all the conditions of Theorem 10 are satisfied to find a common fixed point of S and T. However, note that for any metric d on X,
H d ( S 1 , T 1 ) = H d ( B ( 0 , 1 7 ) ¯ , B ( 0 , 2 7 ) ¯ ) > k d ( 1 , 1 ) = 0 for any  k [ 0 , 1 ) .

Therefore common fixed points of S and T cannot be obtained from an analogous metric fixed point theorem.

In the following we present a partial metric extension of the results in [1, 9, 10, 18, 20].

Theorem 13 (see [9, 10])

Let ( X , p ) be a complete partial metric space, S : X CB p ( X ) be a multivalued mapping and φ : [ 0 , + ) [ 0 , 1 ) be an MT-function. Assume that
H p ( S x , S y ) φ ( p ( x , y ) ) p ( x , y )

for all x , y X , then S has a fixed point.

For φ ( t ) = k t , we have the following result as a special case of the above theorem.

Corollary 14 Let ( X , p ) be a complete partial metric space, and let S , T : X CB p ( X ) be a multivalued mapping satisfying the following condition:
H p ( S x , T y ) k p ( x , y )

for all x , y X and k [ 0 , 1 ) , then S and T have a common fixed point.

Corollary 15 [18] (see also [1])

Let ( X , p ) be a complete partial metric space, and let S : X CB p ( X ) be a multivalued mapping satisfying the following condition:
H p ( S x , S y ) k p ( x , y )

for all x , y X and k [ 0 , 1 ) , then S has a fixed point.

Now we deduce the results for single-valued self-mappings from Theorem 10.

Theorem 16 Let ( X , p ) be a complete partial metric space, S, T be two self-mappings on X and φ : [ 0 , + ) [ 0 , 1 ) be an MT-function. Assume that
p ( S x , T y ) φ ( p ( x , y ) ) p ( x , y )

for all x , y X , then S and T have a common fixed point.

Corollary 17 [10]

Let ( X , p ) be a complete partial metric space, and let S : X X be a mapping satisfying the following condition:
p ( S x , S y ) k p ( x , y )

for all x , y X and k [ 0 , 1 ) , then S has a fixed point.

Declarations

Acknowledgements

The authors thank the editor and the referees for their valuable comments and suggestions which improved greatly the quality of this paper.

Authors’ Affiliations

(1)
Department of Mathematics, COMSATS Institute of Information Technology
(2)
Department of Mathematics, International Islamic University

References

  1. Nadler S: Multi-valued contraction mappings. Pac. J. Math. 1969, 20: 475–488.MathSciNetView ArticleGoogle Scholar
  2. Agarwal RP, Cho YJ, O’Regan D: Fixed point and homotopy invariant results for multi-valued maps on complete gauge spaces. Bull. Aust. Math. Soc. 2003, 67: 241–248. 10.1017/S0004972700033700MathSciNetView ArticleGoogle Scholar
  3. Azam A, Arshad M: Fixed points of sequence of locally contractive multivalued maps. Comput. Math. Appl. 2009, 57: 96–100. 10.1016/j.camwa.2008.09.039MathSciNetView ArticleGoogle Scholar
  4. Beg I, Azam A: Fixed points of asymptotically regular multivalued mappings. J. Aust. Math. Soc. A 1992, 53: 313–326. 10.1017/S1446788700036491MathSciNetView ArticleGoogle Scholar
  5. Cho YJ, Hirunworakit S, Petrot N: Set-valued fixed points theorems for generalized contractive mappings without the Hausdorff metric. Appl. Math. Lett. 2011, 24: 1957–1967.MathSciNetGoogle Scholar
  6. Dhompongsa S, Yingtaweesittikul H: Fixed points for multivalued mappings and the metric completeness. Fixed Point Theory Appl. 2009. 10.1155/2009/972395Google Scholar
  7. Hussain N, Amin-Harandi A, Cho YJ: Approximate endpoints for set-valued contractions in metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 614867Google Scholar
  8. Kikkawa M, Suzuki T: Three fixed point theorems for generalized contractions with constants in complete metric spaces. Nonlinear Anal., Theory Methods Appl. 2008, 69: 2942–2949. 10.1016/j.na.2007.08.064MathSciNetView ArticleGoogle Scholar
  9. Suzuki T: Mizoguchi-Takahashi’s fixed point theorem is a real generalization of Nadler’s. J. Math. Anal. Appl. 2008, 340: 752–755. 10.1016/j.jmaa.2007.08.022MathSciNetView ArticleGoogle Scholar
  10. Matthews SG: Partial metric topology. Ann. New York Acad. Sci. 728. Proc. 8th Summer Conference on General Topology and Applications 1994, 183–197.Google Scholar
  11. Abbas M, Ali B, Vetro C: A Suzuki type fixed point theorem for a generalized multivalued mapping on partial Hausdorff metric spaces. Topol. Appl. 2013, 160: 553–563. 10.1016/j.topol.2013.01.006MathSciNetView ArticleGoogle Scholar
  12. Abdeljawad T: Fixed points for generalized weakly contractive mappings in partial metric spaces. Math. Comput. Model. 2011, 54: 2923–2927. 10.1016/j.mcm.2011.07.013MathSciNetView ArticleGoogle Scholar
  13. Ahmad J, Di Bari C, Cho YJ, Arshad M: Some fixed point results for multi-valued mappings in partial metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 175Google Scholar
  14. Di Bari C, Vetro P: Fixed points for weak ϕ -contractions on partial metric spaces. Int. J. Eng. Contemp. Math. Sci. 2011, 1: 4–9.Google Scholar
  15. Erduran A: Common fixed point of g -approximative multivalued mapping in ordered partial metric space. Fixed Point Theory Appl. 2013. 10.1186/1687-1812-2013-36Google Scholar
  16. Karapinar E, Erhan IM: Fixed point theorems for operators on partial metric spaces. Appl. Math. Lett. 2011, 24: 1894–1899. 10.1016/j.aml.2011.05.013MathSciNetView ArticleGoogle Scholar
  17. Kutbi MA, Ahmad J, Hussain N, Arshad M: Common fixed point results for mappings with rational expressions. Abstr. Appl. Anal. 2013., 2013: Article ID 549518Google Scholar
  18. Aydi H, Abbas M, Vetro C: Partial Hausdorff metric and Nadler’s fixed point theorem on partial metric spaces. Topol. Appl. 2012, 159: 3234–3242. 10.1016/j.topol.2012.06.012MathSciNetView ArticleGoogle Scholar
  19. Haghi RH, Rezapour S, Shahzad N: Be careful on partial metric fixed point results. Topol. Appl. 2012, 160: 450–454.MathSciNetView ArticleGoogle Scholar
  20. Mizoguchi N, Takahashi W: Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl. 1989, 141: 177–188. 10.1016/0022-247X(89)90214-XMathSciNetView ArticleGoogle Scholar
  21. Aydi H, Abbas M, Vetro C: Common fixed points for multi-valued generalized contractions on partial metric spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 2013. 10.1007/s13398-013-0120-zGoogle Scholar
  22. Du WS: On coincidence point and fixed point theorems for nonlinear multivalued maps. Topol. Appl. 2012, 159: 49–56. 10.1016/j.topol.2011.07.021View ArticleGoogle Scholar
  23. Du WS: Coupled fixed point theorems for nonlinear contractions satisfied Mizoguchi-Takahashi’s condition in quasi-ordered metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 876372 10.1155/2010/876372Google Scholar

Copyright

© Ahmad et al.; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.