Open Access

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: 19 November 2014

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 contractionα-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, Islamabad, Pakistan
(2)
Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, Islamabad, Pakistan
(3)
Department of Mathematics, Quaid-i-azam University, Islamabad, Pakistan

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