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Generalization of Mizoguchi-Takahashi type contraction and related fixed point theorems

Journal of Inequalities and Applications20142014:458

https://doi.org/10.1186/1029-242X-2014-458

  • Received: 29 March 2014
  • Accepted: 1 September 2014
  • Published:

Abstract

In this paper, we introduce a new notion to generalize a Mizoguchi-Takahashi type contraction. Then, using this notion, we obtain a fixed point theorem for multivalued maps. Our results generalize some results by Minak and Altun, Kamran and those contained therein.

MSC:47H10, 54H25.

Keywords

  • Mizoguchi-Takahashi contraction
  • α-admissible maps
  • α -admissible maps

1 Introduction and preliminaries

The notions of α-ψ-contractive and α-admissible mappings were introduced by Samet et al. [1]. They proved some fixed point results for such mappings in complete metric spaces. These notions were generalized by Karapınar and Samet [2]. Asl et al. [3] extended these notions to multifunctions and introduced the notions of α -ψ-contractive and α -admissible mappings. Afterwards Ali and Kamran [4] further generalized the notion of α -ψ-contractive mappings and obtained some fixed point theorems for multivalued mappings. Some interesting extensions of results by Samet et al. [1] are available in [513]. Nadler initiated a fixed point theorem for multivalued mappings. Some extensions of Nadler’s result can also be found in [1431]. Mizoguchi and Takahashi [32] extended the Nadler fixed point theorem. Recently, Minak and Altun generalized Mizoguchi and Takahashi’s theorem by introducing a function α : X × X [ 0 , ) . In this paper, we introduce the notion of α -Mizoguchi-Takahashi type contraction. By using this notion, we generalize some fixed point theorems presented by Minak and Altun [7], Kamran [26] and those contained therein.

We denote by CL ( X ) the class of all nonempty closed subsets of X and by CB ( X ) the class of all nonempty closed and bounded subsets of X. For A CL ( X ) or CB ( X ) and x X , d ( x , A ) = inf { d ( x , a ) : a A } , and H is a generalized Hausdorff metric induced by d. Now we recollect some basic definitions and results for the sake of completeness.

If, for x 0 X , there exists a sequence { x n } in X such that x n T x n 1 , then O ( T , x 0 ) = { x 0 , x 1 , x 2 , } is said to be an orbit of T : X CL ( X ) at x 0 . A mapping h : X R is said to be T-orbitally lower semicontinuous at ξ X , if { x n } is a sequence in O ( T , x 0 ) and x n ξ implies h ( ξ ) lim inf h ( x n ) . The following definition is due to Asl et al. [3].

Definition 1.1 [3]

Let ( X , d ) be a metric space, α : X × X [ 0 , ) and T : X CL ( X ) . Then T is α -admissible if for each x , y X with α ( x , y ) 1 α ( T x , T y ) 1 , where α ( T x , T y ) = inf { α ( a , b ) : a T x , b T y } .

Minak and Altun [7] generalized Mizoguchi and Takahashi’s theorem in the following way.

Theorem 1.2 [7]

Let ( X , d ) be a complete metric space, T : X CB ( X ) be a mapping satisfying
α ( T x , T y ) H ( T x , T y ) ϕ ( d ( x , y ) ) d ( x , y ) for each  x , y X ,
where ϕ : [ 0 , ) [ 0 , 1 ) such that lim sup r t + ϕ ( r ) < 1 for each t [ 0 , ) . Also assume that
  1. (i)

    T is α -admissible;

     
  2. (ii)

    there exists x 0 X with α ( x 0 , x 1 ) 1 for some x 1 T x 0 ;

     
  3. (iii)
    1. (a)

      T is continuous,

       
    or
    1. (b)

      if { x n } is a sequence in X with x n x as n and α ( x n , x n + 1 ) 1 for each n N { 0 } , then we have α ( x n , x ) 1 for each n N { 0 } .

       
     

Then T has a fixed point.

Kamran in [26] generalized Mizoguchi and Takahashi’s theorem in the following way.

Theorem 1.3 [26]

Let ( X , d ) be a complete metric space and T : X CL ( X ) be a mapping satisfying
d ( y , T y ) ϕ ( d ( x , y ) ) d ( x , y ) for each  x X  and  y T x ,
where ϕ : [ 0 , ) [ 0 , 1 ) such that lim sup r t + ϕ ( r ) < 1 for each t [ 0 , ) . Then,
  1. (i)

    for each x 0 X , there exists an orbit { x n } of T and ξ X such that lim n x n = ξ ;

     
  2. (ii)

    ξ is a fixed point of T if and only if the function h ( x ) : = d ( x , T x ) is T-orbitally lower semicontinuous at ξ.

     

2 Main results

We begin this section with the following definition.

Definition 2.1 Let ( X , d ) be a metric space, T : X CL ( X ) is said to be an α -Mizoguchi-Takahashi type contraction if there exist two functions α : X × X [ 0 , ) and ϕ : [ 0 , ) [ 0 , 1 ) satisfying lim sup r t + ϕ ( r ) < 1 for every t [ 0 , ) such that
α ( T x , T y ) d ( y , T y ) ϕ ( d ( x , y ) ) d ( x , y ) for each  x X  and  y T x .
(2.1)

Before moving toward our main results, we prove some lemmas.

Lemma 2.2 Let ( X , d ) be a metric space, { A k } be a sequence in CL ( X ) , { x k } be a sequence in X such that x k A k 1 . Let ϕ : [ 0 , ) [ 0 , 1 ) be a function satisfying lim sup r t + ϕ ( r ) < 1 for every t [ 0 , ) . Suppose that { d ( x k 1 , x k ) } is a nonincreasing sequence such that
d ( x k , A k ) ϕ ( d ( x k 1 , x k ) ) d ( x k 1 , x k ) ,
(2.2)
d ( x k , x k + 1 ) d ( x k , A k ) + ϕ n k ( d ( x k 1 , x k ) ) ,
(2.3)

where n 1 < n 2 <  , k , n k N . Then { x k } is a Cauchy sequence in X.

Proof The proof runs on the same lines as the proof of [[18], Lemma 3.2]. We include its details for completeness. Let d k : = d ( x k 1 , x k ) . Since d k is a nonincreasing sequence of nonnegative real numbers, therefore lim k d k = c 0 . By hypothesis, for t = c , we get lim sup t c + ϕ ( t ) < 1 . Therefore, there exists k 0 such that k k 0 implies that ϕ ( d k ) < h , where lim sup t c + ϕ ( t ) h < 1 . From (2.2) and (2.3), we have
d k + 1 ϕ ( d k ) d k + ϕ n k ( d k ) ϕ ( d k ) ϕ ( d k 1 ) d k 1 + ϕ ( d k ) ϕ n k 1 ( d k 1 ) + ϕ n k ( d k ) i = 1 k ϕ ( d i ) d 1 + m = 1 k 1 i = m + 1 k ϕ ( d i ) ϕ n m ( d m ) + ϕ n k ( d k ) i = 1 k ϕ ( d i ) d 1 + m = 1 k 1 i = max { k 0 , m + 1 } k ϕ ( d i ) ϕ n m ( d m ) + ϕ n k ( d k ) .
(2.4)
We have deleted some factors of ϕ from the product in (2.4) using the fact that ϕ < 1 . Let S denote the second term on the right-hand side of (2.4),
S ( k 0 1 ) h k k 0 + 1 m = 1 k 0 1 ϕ n m ( d m ) + m = k 0 k 1 h k m ϕ n m ( d m ) ( k 0 1 ) h k k 0 + 1 m = 1 k 0 1 ϕ n m ( d m ) + m = k 0 k 1 h k m + n m C h k + m = k 0 k 1 h k m + n m C h k + h k + n k 0 k 0 + h k + n k 0 1 ( k 0 1 ) + + h k + n k 1 ( k 1 ) C h k + m = k + n k 0 k 0 k + n k 1 ( k 1 ) h m = C h k + h k + n k 0 k 0 + 1 h k + n k 1 k + 2 1 h < C h k + h k h n k 0 k 0 + 1 1 h = C h k ,
where C is a generic positive constant. Now, it follows from (2.4) that
d k + 1 i = 1 k ϕ ( d i ) d 1 + C h k + ϕ n k ( d k ) < h k k 0 + 1 i = 1 k 0 1 ϕ ( d i ) d 1 + C h k + h n k < C h k + C h k + k = C h k ,
C again being a generic constant. Now, for k k 0 , m N ,
d ( x k , x k + m ) i = k + 1 k + m d i < i = k + 1 k + m C h i 1 = C h k + 1 h k + m 1 h h k ,

which shows that { x k } is a Cauchy sequence in X. □

Lemma 2.3 Let ( X , d ) be a metric space, T : X CL ( X ) be an α -Mizoguchi-Takahashi type contraction. Let { x k } be an orbit of T at x 0 such that α ( T x k 1 , T x k ) 1 and
d ( x k , x k + 1 ) d ( x k , T x k ) + ϕ n k ( d ( x k 1 , x k ) ) ,
(2.5)

where x k T x k 1 , n 1 < n 2 < and k , n k N and { d ( x k 1 , x k ) } is a nonincreasing sequence. Then { x k } is a Cauchy sequence in X.

Proof Given that { x k } is an orbit of T at x 0 , i.e., x k T x k 1 for each k N , with α ( T x k 1 , T x k ) 1 for each k N , as T is an α -Mizoguchi-Takahashi type contraction. From (2.1), we have
d ( x k , T x k ) α ( T x k 1 , T x k ) d ( x k , T x k ) ϕ ( d ( x k 1 , x k ) ) d ( x k 1 , x k ) .
From (2.5), we have
d ( x k , x k + 1 ) d ( x k , T x k ) + ϕ n k ( d ( x k 1 , x k ) ) .

Since all the conditions of Lemma 2.2 are satisfied, { x k } is a Cauchy sequence in X. □

Theorem 2.4 Let ( X , d ) be a complete metric space, T : X CL ( X ) be an α -Mizoguchi-Takahashi type contraction and α -admissible. Suppose that there exist x 0 X and x 1 T x 0 such that α ( x 0 , x 1 ) 1 . Then,
  1. (i)

    there exists an orbit { x n } of T and x X such that lim x n = x ;

     
  2. (ii)

    x is a fixed point of T if and only if h ( x ) = d ( x , T x ) is T-orbitally lower semicontinuous at x .

     
Proof By hypothesis, we have x 0 X and x 1 T x 0 with α ( x 0 , x 1 ) 1 . Thus, for x 1 T x 0 , we can choose a positive integer n 1 such that
ϕ n 1 ( d ( x 0 , x 1 ) ) [ 1 ϕ ( d ( x 0 , x 1 ) ) ] d ( x 0 , x 1 ) .
(2.6)
There exists x 2 T x 1 such that
d ( x 1 , x 2 ) d ( x 1 , T x 1 ) + ϕ n 1 ( d ( x 0 , x 1 ) ) .
(2.7)
As T is α -admissible, we have α ( T x 0 , T x 1 ) 1 . From (2.6) and (2.7) it follows that
d ( x 1 , x 2 ) d ( x 1 , T x 1 ) + ϕ n 1 ( d ( x 0 , x 1 ) ) α ( T x 0 , T x 1 ) d ( x 1 , T x 1 ) + ϕ n 1 ( d ( x 0 , x 1 ) ) ϕ ( d ( x 0 , x 1 ) ) d ( x 0 , x 1 ) + [ 1 ϕ ( d ( x 0 , x 1 ) ) ] d ( x 0 , x 1 ) = d ( x 0 , x 1 ) .
Now we can choose a positive integer n 2 > n 1 such that
ϕ n 2 ( d ( x 1 , x 2 ) ) [ 1 ϕ ( d ( x 1 , x 2 ) ) ] d ( x 1 , x 2 ) .
(2.8)
There exists x 3 T x 2 such that
d ( x 2 , x 3 ) d ( x 2 , T x 2 ) + ϕ n 2 ( d ( x 1 , x 2 ) ) .
(2.9)
As T is α -admissible, then α ( x 1 , x 2 ) α ( T x 0 , T x 1 ) 1 implies α ( T x 1 , T x 2 ) 1 . Using (2.8) and (2.9) we have that
d ( x 2 , x 3 ) d ( x 2 , T x 2 ) + ϕ n 2 ( d ( x 1 , x 2 ) ) α ( T x 1 , T x 2 ) d ( x 2 , T x 2 ) + ϕ n 2 ( d ( x 1 , x 2 ) ) ϕ ( d ( x 1 , x 2 ) ) d ( x 1 , x 2 ) + [ 1 ϕ ( d ( x 1 , x 2 ) ) ] d ( x 1 , x 2 ) = d ( x 1 , x 2 ) .
By repeating this process for all k N , we can choose a positive integer n k such that
ϕ n k ( d ( x k 1 , x k ) ) [ 1 ϕ ( d ( x k 1 , x k ) ) ] d ( x k 1 , x k ) .
(2.10)
There exists x k T x k 1 such that
d ( x k , x k + 1 ) d ( x k , T x k ) + ϕ n k ( d ( x k 1 , x k ) ) .
(2.11)
Also, by α -admissibility of T, we have α ( T x k 1 , T x k ) 1 for each k N . From (2.10) and (2.11) it follows that
d ( x k , x k + 1 ) d ( x k , T x k ) + ϕ n k ( d ( x k 1 , x k ) ) α ( T x k 1 , T x k ) d ( x k , T x k ) + ϕ n k ( d ( x k 1 , x k ) ) ϕ ( d ( x k 1 , x k ) ) d ( x k 1 , x k ) + [ 1 ϕ ( d ( x k 1 , x k ) ) ] d ( x k 1 , x k ) = d ( x k 1 , x k ) ,
which implies that { d ( x k , x k + 1 ) } is a nonincreasing sequence of nonnegative real numbers. Thus, by Lemma 2.3, { x k } is a Cauchy sequence in X. Since X is complete, there exists x X such that x k x as k . Since x k T x k 1 , it follows from (2.1) that
d ( x k , T x k ) α ( T x k 1 , T x k ) d ( x k , T x k ) ϕ ( d ( x k 1 , x k ) ) d ( x k 1 , x k ) < d ( x k 1 , x k ) .
Letting k , in the above inequality, we have
lim k d ( x k , T x k ) = 0 .
(2.12)
Suppose that h ( x ) = d ( x , T x ) is T-orbitally lower semicontinuous at x , then
d ( x , T x ) = h ( x ) lim inf k h ( x k ) = lim inf k d ( x k , T x k ) = 0 .

By the closedness of T it follows that x T x . Conversely, suppose that x is a fixed point of T, then h ( x ) = 0 lim inf k h ( x k ) . □

Example 2.5 Let X = { 1 n : n N } { 0 } ( 1 , ) be endowed with the usual metric d. Define T : X CL ( X ) by
T x = { { 0 } if  x = 0 , { 1 n + 2 , 1 n + 3 } if  x = 1 n : 1 n 6 , { 1 n , 0 } if  x = 1 n : n > 6 , [ 2 x , ) if  x > 1 ,
and α : X × X [ 0 , ) by
α ( x , y ) = { 1 if  x , y { 1 n : n N } { 0 } , 0 otherwise .
Define ϕ : [ 0 , ) [ 0 , 1 ) by
ϕ ( t ) = { 4 5 if  0 t 1 6 , 1 2 if  t > 1 6 .
One can check that for each x X and y T x , we have
α ( T x , T y ) d ( y , T y ) ϕ ( d ( x , y ) ) d ( x , y ) .

Also, T is α -admissible and for x 0 = 1 we have x 1 = 1 3 T x 0 with α ( x 0 , x 1 ) = 1 . Moreover, all the other conditions of Theorem 2.4 are satisfied. Therefore T has a fixed point. Note that Theorem 5 of Minak and Altun [7] is not applicable here; see, for example, x = 1 7 and y = 1 8 . Further Theorem 2.1 of Kamran [26] is also not applicable; see, for example, x = 2 and y = 4 T x .

The proofs of the following theorems run on the same lines as the proof of Theorem 2.4.

Theorem 2.6 Let ( X , d ) be a complete metric space, T : X CL ( X ) be an α -admissible mapping such that
α ( y , T y ) d ( y , T y ) ϕ ( d ( x , y ) ) d ( x , y ) for each  x X  and  y T x ,
(2.13)
where ϕ : [ 0 , ) [ 0 , 1 ) satisfying lim sup r t + ϕ ( r ) < 1 for every t [ 0 , ) . Suppose that there exist x 0 X and x 1 T x 0 such that α ( x 0 , x 1 ) 1 . Then,
  1. (i)

    there exists an orbit { x n } of T and x X such that lim x n = x ;

     
  2. (ii)

    x is a fixed point of T if and only if h ( x ) = d ( x , T x ) is T-orbitally lower semicontinuous at x .

     
Theorem 2.7 Let ( X , d ) be a complete metric space, T : X CL ( X ) be an α -admissible mapping such that
α ( x , y ) d ( y , T y ) ϕ ( d ( x , y ) ) d ( x , y ) for each  x X  and  y T x ,
(2.14)
where ϕ : [ 0 , ) [ 0 , 1 ) satisfying lim sup r t + ϕ ( r ) < 1 for every t [ 0 , ) . Suppose that there exist x 0 X and x 1 T x 0 such that α ( x 0 , x 1 ) 1 . Then,
  1. (i)

    there exists an orbit { x n } of T and x X such that lim x n = x ;

     
  2. (ii)

    x is a fixed point of T if and only if h ( x ) = d ( x , T x ) is T-orbitally lower semicontinuous at x .

     

Corollary 2.8 [26]

Let ( X , d ) be a complete metric space and T : X CL ( X ) be a mapping satisfying
d ( y , T y ) ϕ ( d ( x , y ) ) d ( x , y ) for each  x X  and  y T x ,
where ϕ : [ 0 , ) [ 0 , 1 ) such that lim sup r t + ϕ ( r ) < 1 for each t [ 0 , ) . Then,
  1. (i)

    for each x 0 X , there exists an orbit { x n } of T and ξ X such that lim n x n = ξ ;

     
  2. (ii)

    ξ is a fixed point of T if and only if the function h ( x ) : = d ( x , T x ) is T-orbitally lower semicontinuous at ξ.

     

Proof Define α : X × X [ 0 , ) by α ( x , y ) = 1 for each x , y X . Then the proof follows from Theorem 2.4 as well as from Theorem 2.6, and from Theorem 2.7. □

3 Application

From Definition 2.1, we get the following definition by considering only those x X and y T x for which we have α ( T x , T y ) 1 .

Definition 3.1 Let ( X , d ) be a metric space, T : X CL ( X ) is said to be a modified α -Mizoguchi-Takahashi type contraction if there exist two functions α : X × X [ 0 , ) and ϕ : [ 0 , ) [ 0 , 1 ) satisfying lim sup r t + ϕ ( r ) < 1 for every t [ 0 , ) such that for each x X and y T x ,
α ( T x , T y ) 1 d ( y , T y ) ϕ ( d ( x , y ) ) d ( x , y ) .
(3.1)
Lemma 3.2 Let ( X , d ) be a metric space, T : X CL ( X ) be a modified α -Mizoguchi-Takahashi contraction. Let { x k } be an orbit of T at x 0 such that α ( T x k 1 , T x k ) 1 and
d ( x k , x k + 1 ) d ( x k , T x k ) + ϕ n k ( d ( x k 1 , x k ) ) ,
(3.2)

where x k T x k 1 , n 1 < n 2 < and k , n k N and { d ( x k 1 , x k ) } is a nonincreasing sequence. Then { x k } is a Cauchy sequence in X.

Proof Given that { x k } is an orbit of T at x 0 , i.e., x k T x k 1 for each k N , with α ( T x k 1 , T x k ) 1 for each k N , as T is a modified α -Mizoguchi-Takahashi contraction. From (3.1), we have
d ( x k , T x k ) ϕ ( d ( x k 1 , x k ) ) d ( x k 1 , x k ) .
From (3.2), we have
d ( x k , x k + 1 ) d ( x k , T x k ) + ϕ n k ( d ( x k 1 , x k ) ) .

Since all the conditions of Lemma 2.2 are satisfied, { x k } is a Cauchy sequence in X. □

Working on the same lines as the proof of Theorem 2.4 is done, one may obtain the proof of the following result.

Theorem 3.3 Let ( X , d ) be a complete metric space, T : X CL ( X ) be a modified α -Mizoguchi-Takahashi contraction and α -admissible. Suppose that there exist x 0 X and x 1 T x 0 such that α ( x 0 , x 1 ) 1 . Then,
  1. (i)

    there exists an orbit { x n } of T and x X such that lim x n = x ;

     
  2. (ii)

    x is a fixed point of T if and only if h ( x ) = d ( x , T x ) is T-orbitally lower semicontinuous at x .

     

Declarations

Acknowledgements

Authors are thankful to referees for their valuable suggestions.

Authors’ Affiliations

(1)
School of Electrical Engineering and Computer Sciences, National University of Sciences and Technology, H-12 Islamabad, Pakistan
(2)
Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, H-12 Islamabad, Pakistan
(3)
Department of Mathematics, Quaid-i-azam University, Islamabad, Pakistan

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© Kiran et al.; licensee Springer. 2014

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/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

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