Open Access

A note on approximate fixed point property and Du-Karapinar-Shahzad’s intersection theorems

Journal of Inequalities and Applications20132013:506

https://doi.org/10.1186/1029-242X-2013-506

Received: 30 July 2013

Accepted: 24 September 2013

Published: 8 November 2013

Abstract

In this note, we give new short proofs of Du-Karapinar-Shahzad’s intersection theorems for multivalued non-self-maps in complete metric spaces.

MSC:47H10, 54H25.

Keywords

τ-function MT -function (-function)coincidence pointfixed pointapproximate fixed point property

1 Introduction and preliminaries

Let us begin with some basic definitions and notations that will be needed in this paper. Let ( X , d ) be a metric space. Denote by N ( X ) the family of all nonempty subsets of X and by CB ( X ) the family of all nonempty closed and bounded subsets of X. For each x X and A X , let d ( x , A ) = inf y A d ( x , y ) . A function H : CB ( X ) × CB ( X ) [ 0 , ) defined by
H ( A , B ) = max { sup x B d ( x , A ) , sup x A d ( x , B ) }

is said to be the Hausdorff metric on CB ( X ) induced by the metric d on X. The symbols and are used to denote the sets of positive integers and real numbers, respectively.

Let K be a nonempty subset of X, g : K X be a single-valued non-self-map and T : K N ( X ) be a multivalued non-self-map. A point v in X is a coincidence point (see, for instance, [16]) of g and T if g v T x . If g = id is the identity map, then v = g v T v and call v a fixed point of T. The set of fixed points of T and the set of coincidence points of g and T are denoted by F K ( T ) and COP K ( g , T ) , respectively. In particular, if K X , we use F ( T ) and COP ( g , T ) instead of F K ( T ) and COP K ( g , T ) , respectively. The map T is said to have approximate fixed point property [15] on K provided inf x K d ( x , T x ) = 0 . It is obvious that F K ( T ) implies that T has approximate fixed point property.

A function φ : [ 0 , ) [ 0 , 1 ) is said to be an MT -function (or -function) [311] if lim sup s t + φ ( s ) < 1 for all t [ 0 , ) . Clearly, if φ : [ 0 , ) [ 0 , 1 ) is a nondecreasing function or a nonincreasing function, then φ is an MT -function. So, the set of MT -functions is a rich class and has the questions many of which are worth studying.

The study of fixed points for single-valued non-self-maps or multivalued non-self-maps satisfying certain contractive conditions is an interesting and important direction of research in metric fixed point theory. A great deal of such research has been investigated by several authors, see, e.g., [1119] and the references therein. Very recently, Du, Karapinar and Shahzad [11] established the following intersection existence theorem of coincidence points and fixed points of multivalued non-self-maps of Kannan type and Chatterjea type.

Theorem 1.1 [[11], Theorem 8]

Let ( X , d ) be a complete metric space, K be a nonempty closed subset of X, T : K CB ( X ) be a multivalued map and g : K X be a continuous map. Suppose that

(D1) T x K for all x K ,

(D2) T x K is g-invariant (i.e., g ( T x K ) T x K ) for each x K ,

(D3) there exist a function h : K [ 0 , ) and γ [ 0 , 1 2 ) such that
H ( T x , T y K ) γ [ d ( x , T x K ) + d ( y , T x K ) + d ( y , T y K ) ] + h ( y ) d ( g y , T x K ) for all x , y K .
(1.1)

Then COP K ( g , T ) F K ( T ) .

In [11], they also gave some coincidence and fixed point theorems for multivalued non-self-maps of Mizoguchi-Takahashi type, Berinde-Berinde type and Du type.

Theorem 1.2 [[11], Theorem 19]

Let ( X , d ) be a complete metric space, K be a nonempty closed subset of X, T : K CB ( X ) be a multivalued map and g : K X be a continuous map. Suppose that conditions (D1) and (D2) as in Theorem  1.1 hold. If there exist an MT -function φ : [ 0 , ) [ 0 , 1 ) and a function h : K [ 0 , ) such that
H ( T x , T y K ) φ ( d ( x , y ) ) d ( x , y ) + h ( y ) d ( g y , T x K ) for all x , y K ,
(1.2)

then COP K ( g , T ) F K ( T ) .

In this work, we give new short proofs of Du-Karapinar-Shahzad’s intersection theorems of COP K ( g , T ) and F K ( T ) for multivalued non-self-maps (i.e., Theorems 1.1 and 1.2) by applying an existence theorem for approximate fixed point property.

2 Some auxiliary key results

Let ( X , d ) be a metric space. Recall that a function p : X × X [ 0 , ) is said to be a τ-function [35, 7, 8, 2022], first introduced and studied by Lin and Du, if the following conditions hold:

(τ 1) p ( x , z ) p ( x , y ) + p ( y , z ) for all x , y , z X ;

(τ 2) if x X and { y n } in X with lim n y n = y such that p ( x , y n ) M for some M = M ( x ) > 0 , then p ( x , y ) M ;

(τ 3) for any sequence { x n } in X with lim n sup { p ( x n , x m ) : m > n } = 0 , if there exists a sequence { y n } in X such that lim n p ( x n , y n ) = 0 , then lim n d ( x n , y n ) = 0 ;

(τ 4) for x , y , z X , p ( x , y ) = 0 and p ( x , z ) = 0 imply y = z .

Note that with the additional condition

(τ 5) p ( x , x ) = 0 for all x X ,

a τ-function becomes a τ 0 -function [35, 7, 8] introduced by Du.

Clearly, any metric d is a τ 0 -function. Observe further that if p is a τ 0 -function, then, from (τ 4) and (τ 5), p ( x , y ) = 0 if and only if x = y .

Example A [7]

Let X = R with the metric d ( x , y ) = | x y | and 0 < a < b . Define the function p : X × X [ 0 , ) by
p ( x , y ) = max { a ( y x ) , b ( x y ) } .

Then p is nonsymmetric and hence p is not a metric. It is easy to see that p is a τ 0 -function.

Lemma 2.1 [[22], Lemma 2.1]

Let ( X , d ) be a metric space and p : X × X [ 0 , ) be a function. Assume that p satisfies the condition (τ 3). If a sequence { x n } in X with lim n sup { p ( x n , x m ) : m > n } = 0 , then { x n } is a Cauchy sequence in X.

Let ( X , d ) be a metric space and p be a τ-function. A multivalued map T : X N ( X ) is said to have p-approximate fixed point property on X provided
inf x X p ( x , T x ) = 0 .

The following characterizations of MT -functions proved first by Du [6] are quite useful for proving our main results.

Theorem 2.1 [[6], Theorem 2.1]

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

    φ is an MT -function.

     
  2. (b)

    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. (c)

    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. (d)

    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. (e)

    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. (f)

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

     
  7. (g)

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

     

The following result was essentially proved by Du et al. in [4], but we give the proof for the sake of completeness and the readers convenience.

Lemma 2.2 [[4], Lemma 3.1]

Let ( X , d ) be a metric space, p be a τ 0 -function and T : X N ( X ) be a multivalued map. Then the following statements are equivalent.

(Q1) There exist a function ξ : [ 0 , ) [ 0 , ) and an MT -function φ : [ 0 , ) [ 0 , 1 ) such that for each x X , if y T x with y x , then there exists z T y such that
p ( y , z ) φ ( ξ ( p ( x , y ) ) ) p ( x , y ) .
(Q2) There exist a function τ : [ 0 , ) [ 0 , ) and an MT -function κ : [ 0 , ) [ 0 , 1 ) such that for each x X ,
p ( y , T y ) κ ( τ ( p ( x , y ) ) ) p ( x , y ) for all y T x .
Proof If (Q1) holds, then it is easy to verify that (Q2) also holds with κ φ and τ ξ . So it suffices to prove that ‘(Q2) (Q1)’. Suppose that (Q2) holds. Define φ : [ 0 , ) [ 0 , 1 ) by φ ( t ) = 1 + κ ( t ) 2 . Then φ is also an MT -function. Indeed, it is obvious that 0 κ ( t ) < φ ( t ) < 1 for all t [ 0 , ) . Let { x n } n N be a strictly decreasing sequence in [ 0 , ) . Since κ is an MT -function, by (g) of Theorem 2.1, we get
0 sup n N κ ( x n ) < 1
and hence
0 < sup n N φ ( x n ) = 1 2 [ 1 + sup n N κ ( x n ) ] < 1 .

So, by Theorem 2.1 again, we prove that φ is an MT -function.

For each x X , let y T x with y x . Then p ( x , y ) > 0 . By (Q2), we have
p ( y , T y ) < φ ( τ ( p ( x , y ) ) ) p ( x , y ) .
Since φ ( t ) > 0 for all t [ 0 , ) , there exists z T y such that
p ( y , z ) < φ ( τ ( p ( x , y ) ) ) p ( x , y ) ,

which shows that (Q1) holds with ξ τ . So, by above, we prove ‘(Q1) (Q2)’. □

Now, we present an existence theorem for p-approximate fixed point property and approximate fixed point property, which is indeed a somewhat generalized form of [[4], Theorem 3.3] and is one of the key technical devices in the new short proofs of Theorems 1.1 and 1.2.

Theorem 2.2 Let ( X , d ) be a metric space, p be a τ 0 -function and T : X N ( X ) be a multivalued map. Assume that one of (L1) and (L2) is satisfied, where

(L1) there exist a nondecreasing function ξ : [ 0 , ) [ 0 , ) and an MT -function φ : [ 0 , ) [ 0 , 1 ) such that for each x X , if y T x with y x , then there exists z T y such that
p ( y , z ) φ ( ξ ( p ( x , y ) ) ) p ( x , y ) ;
(L2) there exist a nondecreasing function τ : [ 0 , ) [ 0 , ) and an MT -function κ : [ 0 , ) [ 0 , 1 ) such that for each x X ,
p ( y , T y ) κ ( τ ( p ( x , y ) ) ) p ( x , y ) for all y T x .
Then the following statements hold.
  1. (a)
    There exists a Cauchy sequence { x n } n N in X such that
    1. (i)

      x n + 1 T x n for all n N ,

       
    2. (ii)

      inf n X p ( x n , x n + 1 ) = lim n p ( x n , x n + 1 ) = lim n d ( x n , x n + 1 ) = inf n N d ( x n , x n + 1 ) = 0 .

       
     
  2. (b)

    inf x X p ( x , T x ) = inf x X d ( x , T x ) = 0 ; that is, T has p-approximate fixed point property and approximate fixed point property on X.

     
Proof By Lemma 2.2, it suffices to prove that the conclusions hold under assumption (L1). Let u X be given. If u T u , then
inf x X p ( x , T x ) p ( u , T u ) p ( u , u ) = 0 ,
and
inf x X d ( x , T x ) d ( u , u ) = 0 ,
which implies that inf x X p ( x , T x ) = inf x X d ( x , T x ) = 0 . Let w n = u for all n N . Thus we have
w n + 1 = u T u = T w n for all  n N , lim n p ( w n , w n + 1 ) = inf n N p ( w n , w n + 1 ) = p ( u , u ) = 0 ,
and
lim n d ( w n , w n + 1 ) = inf n N d ( w n , w n + 1 ) = d ( u , u ) = 0 .
Clearly,
p ( w n + 1 , w n + 2 ) = 0 = φ ( ξ ( p ( w n , w n + 1 ) ) ) p ( w n , w n + 1 ) for all  n N .
So, conclusions (a) and (b) hold in this case u T u , no matter what condition one begins with. Suppose that u T u . Put x 1 = u and x 2 T x 1 . Then x 2 x 1 and hence p ( x 1 , x 2 ) > 0 . Assume that condition (L1) is satisfied. Then there exists x 3 T x 2 such that
p ( x 2 , x 3 ) φ ( ξ ( p ( x 1 , x 2 ) ) ) p ( x 1 , x 2 ) .
If x 2 = x 3 T x 2 , then, following a similar argument as above, the conclusions are also proved. If x 3 x 2 , then there exists x 4 T x 3 such that
p ( x 3 , x 4 ) φ ( ξ ( p ( x 2 , x 3 ) ) ) p ( x 2 , x 3 ) .
By induction, we can obtain a sequence { x n } in X satisfying x n + 1 T x n and
p ( x n + 1 , x n + 2 ) φ ( ξ ( p ( x n , x n + 1 ) ) ) p ( x n , x n + 1 ) for all  n N .
(2.1)
Since φ ( t ) < 1 for all t [ 0 , ) , inequality (2.1) implies that the sequence { p ( x n , x n + 1 ) } n N is strictly decreasing in [ 0 , ) . Hence
lim n p ( x n , x n + 1 ) = inf n N p ( x n , x n + 1 ) 0  exists .
(2.2)
Since ξ is nondecreasing, { ξ ( p ( x n , x n + 1 ) ) } n N is a nonincreasing sequence in [ 0 , ) . Since φ is an MT -function, by (g) of Theorem 2.1, we have
0 sup n N φ ( ξ ( p ( x n , x n + 1 ) ) ) < 1 .
Let λ : = sup n N φ ( ξ ( p ( x n , x n + 1 ) ) ) . So λ [ 0 , 1 ) and we get from (2.1) that
p ( x n + 1 , x n + 2 ) λ p ( x n , x n + 1 ) λ n p ( x 1 , x 2 ) for each  n N .
(2.3)
Since λ [ 0 , 1 ) , lim n λ n = 0 and hence the last inequality implies
lim n p ( x n , x n + 1 ) = 0 .
(2.4)
By (2.2) and (2.4), we obtain
inf n N p ( x n , x n + 1 ) = lim n p ( x n , x n + 1 ) = 0 .
(2.5)
Now, we claim that { x n } is a Cauchy sequence in X. Let α n = λ n 1 1 λ p ( x 1 , x 2 ) , n N . For m , n N with m > n , by (2.3), we have
p ( x n , x m ) j = n m 1 p ( x j , x j + 1 ) < α n .
Since λ [ 0 , 1 ) , lim n α n = 0 and hence
lim n sup { p ( x n , x m ) : m > n } = 0 .
Applying Lemma 2.1, we show that { x n } is a Cauchy sequence in X. Hence lim n d ( x n , x n + 1 ) = 0 . Since inf n N d ( x n , x n + 1 ) d ( x m , x m + 1 ) for all m N and lim m d ( x m , x m + 1 ) = 0 , one also obtains
lim n d ( x n , x n + 1 ) = inf n N d ( x n , x n + 1 ) = 0 .
(2.6)
So conclusion (a) is proved. To see (b), since x n + 1 T x n for each n N , we have
inf x X p ( x , T x ) p ( x n , T x n ) p ( x n , f x n + 1 )
(2.7)
and
inf x X d ( x , T x ) d ( x n , T x n ) d ( x n , f x n + 1 )
(2.8)
for all n N . Combining (2.6), (2.7) and (2.8), we get
inf x X p ( x , T x ) = inf x X d ( x , T x ) = 0 .

The proof is completed. □

The following existence theorem is obviously an immediate result from Theorem 2.2.

Theorem 2.3 Let ( X , d ) be a metric space, p be a τ 0 -function and T : X N ( X ) be a multivalued map. Assume that one of (H1) and (H2) is satisfied, where

(H1) there exists an MT -function α : [ 0 , ) [ 0 , 1 ) such that for each x X , if y T x with y x , then there exists z T y such that
p ( y , z ) α ( p ( x , y ) ) p ( x , y ) ;
(H2) there exists an MT -function β : [ 0 , ) [ 0 , 1 ) such that for each x X ,
p ( y , T y ) β ( p ( x , y ) ) p ( x , y ) for all y T x .
Then the following statements hold.
  1. (a)
    There exists a Cauchy sequence { x n } n N in X such that
    1. (i)

      x n + 1 T x n for all n N ,

       
    2. (ii)

      inf n X p ( x n , x n + 1 ) = lim n p ( x n , x n + 1 ) = lim n d ( x n , x n + 1 ) = inf n N d ( x n , x n + 1 ) = 0 .

       
     
  2. (b)

    inf x X p ( x , T x ) = inf x X d ( x , T x ) = 0 ; that is, T has p-approximate fixed point property and approximate fixed point property on X.

     

Lemma 2.3 Let τ : [ 0 , ) [ 0 , ) be a nondecreasing function and κ : [ 0 , ) [ 0 , 1 ) be an MT -function. Then κ τ is an MT -function.

Proof Let { x n } n N be a strictly decreasing sequence in [ 0 , ) . Since τ is a nondecreasing function, { τ ( x n ) } n N is a nonincreasing sequence in [ 0 , ) . Since κ is an MT -function, by (f) of Theorem 2.1, we get
0 sup n N κ ( τ ( x n ) ) < 1 ,
or, equivalently,
0 sup n N ( κ τ ) ( x n ) < 1 .

So, by Theorem 2.1 again, we prove that κ τ is an MT -function. □

Applying Lemma 2.3, we conclude that Theorem 2.2 is also a special case of Theorem 2.3. Therefore we obtain the following important fact.

Theorem 2.4 Theorem  2.2 and Theorem  2.3 are equivalent.

3 Short proofs of Theorems 1.1 and 1.2

Let us see how we can utilize Theorem 2.3 to prove Theorem 1.1.

Short proof of Theorem 1.1 Since K is a nonempty closed subset of X and X is complete, ( K , d ) is also a complete metric space. Let x K . Put k = γ 1 γ and λ = 1 + k 2 . So, 0 k < λ < 1 . Let y T x K be arbitrary. So, d ( y , T x K ) = 0 . By (D2), we have d ( g y , T x K ) = 0 . Hence inequality (1.1) implies
H ( T x , T y K ) γ [ d ( x , T x K ) + H ( T x , T y K ) ] for all  y T x K .
(3.1)
Inequality (3.1) shows that
d ( y , T y K ) H ( T x , T y K ) k d ( x , T x K ) < λ d ( x , y ) for all  y T x K .
(3.2)
Define G : K CB ( K ) by
G x = T x K for all  x K ,
and let μ : [ 0 , ) [ 0 , 1 ) be defined by
η ( t ) = λ for all  t [ 0 , ) .
Then μ is an MT -function. By (3.2), we obtain
d ( y , G y ) μ ( d ( x , y ) ) d ( x , y ) for all  y G x .
Applying Theorem 2.3 with p d , there exists a Cauchy sequence { x n } n N in K such that
x n + 1 G x n = T x n K for all  n N
(3.3)
and
lim n d ( x n , x n + 1 ) = inf n N d ( x n , x n + 1 ) = 0 .
(3.4)
By the completeness of K, there exists v K such that x n v as n . By (3.3) and (D2), we have
g x n + 1 T x n K for each  n N .
(3.5)
Since g is continuous and lim n x n = v , we have
lim n g x n = g v .
(3.6)
Since the function x d ( x , T v ) is continuous, by (1.1), (3.3), (3.4), (3.5) and (3.6), we get
d ( v , T v K ) = lim n d ( x n + 1 , T v K ) lim n H ( T x n , T v K ) lim n { γ [ d ( x n , T x n K ) + d ( v , T x n K ) + d ( v , T v K ) ] + h ( v ) d ( g v , T x n K ) } lim n { γ [ d ( x n , x n + 1 ) + d ( v , x n + 1 ) + d ( v , T v K ) ] + h ( v ) d ( g v , g x n + 1 ) } = γ d ( v , T v K ) ,

which implies d ( v , T v K ) = 0 . By the closedness of Tv, we have v T v K . From (D2), g v T v K T v . Hence we verify v COP K ( g , T ) F K ( T ) . The proof is complete. □

In order to finish off our work, let us prove Theorem 1.2 by applying Theorem 2.3.

Short proof of Theorem 1.2 Since K is a nonempty closed subset of X and X is complete, ( K , d ) is also a complete metric space. Note first that for each x K , by (D2), we have d ( g y , T x K ) = 0 for all y T x K . So, for each x K , by (1.2), we obtain
d ( y , T y K ) φ ( d ( x , y ) ) d ( x , y ) for all  y T x K .
(3.7)
Define G : K CB ( K ) by
G x = T x K for all  x K .
From (3.7), we obtain
d ( y , G y ) φ ( d ( x , y ) ) d ( x , y ) for all  y G x .
By using Theorem 2.3, there exists a Cauchy sequence { x n } n N in K such that
x n + 1 G x n = T x n K for all  n N
(3.8)
and
lim n d ( x n , x n + 1 ) = inf n N d ( x n , x n + 1 ) = 0 .
(3.9)
By the completeness of K, there exists v K such that x n v as n . Thanks to (3.8) and (D2), we have
g x n + 1 T x n K for each  n N .
(3.10)
Since g is continuous and lim n x n = v , we have
lim n g x n = g v .
(3.11)
Since the function x d ( x , T v ) is continuous, by (1.2), (3.8), (3.10) and (3.11), we get
d ( v , T v K ) = lim n d ( x n + 1 , T v K ) lim n H ( T x n , T v K ) lim n { φ ( d ( x n , v ) ) d ( x n , v ) + h ( v ) d ( g v , T x n K ) } lim n { φ ( d ( x n , v ) ) d ( x n , v ) + h ( v ) d ( g v , g x n + 1 ) } = 0 ,

which implies d ( v , T v K ) = 0 . By the closedness of Tv, we have v T v K . By (D2), g v T v K T v and hence v COP K ( g , T ) F K ( T ) . The proof is complete. □

Declarations

Acknowledgements

In this research, the author was supported by grant No. NSC 102-2115-M-017-001 of the National Science Council of the Republic of China.

Authors’ Affiliations

(1)
Department of Mathematics, National Kaohsiung Normal University

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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.