Open Access

Fixed points for mappings satisfying some multi-valued contractions with w-distance

  • Zeqing Liu1,
  • Xiaochen Wang1,
  • Shin Min Kang2Email author and
  • Sun Young Cho3
Fixed Point Theory and Applications20142014:246

https://doi.org/10.1186/1687-1812-2014-246

Received: 25 August 2014

Accepted: 4 December 2014

Published: 18 December 2014

Abstract

The existence of fixed points and iterative approximations for some nonlinear multi-valued contraction mappings in complete metric spaces with w-distance are proved. Two examples are included. The results presented in this paper extend, improve and unify many known results in recent literature.

MSC:54H25, 47H10.

Keywords

multi-valued contractions w-distancefixed point theorems

1 Introduction and preliminaries

In 1996, Kada et al. [1] introduced the concept of w-distance and got some fixed point theorems for single-valued mappings under w-distance. In 2006, Feng and Liu [[2], Theorem 3.1] proved the following fixed point theorem for a multi-valued contractive mapping, which generalizes the nice fixed point theorem due to Nadler [[3], Theorem 5].

Theorem 1.1 ([2])

Let ( X , d ) be a complete metric space and T be a multi-valued mapping from X into CL ( X ) , where CL ( X ) is the family of all nonempty closed subsets of X. Assume that

(c1) the mapping f : X R + , defined by f ( x ) = d ( x , T ( x ) ) , x X , is lower semi-continuous;

(c2) there exist constants b , c ( 0 , 1 ) with c < b such that for any x X , there is y T ( x ) satisfying
b d ( x , y ) f ( x ) and f ( y ) c d ( x , y ) .

Then T has a fixed point in X.

In 2007, Klim and Wardowski [[4], Theorem 2.1] extended Theorem 1.1 and proved the following result.

Theorem 1.2 ([4])

Let ( X , d ) be a complete metric space and T be a multi-valued mapping from X into CL ( X ) satisfying (c1). Assume that

(c3) there exist b ( 0 , 1 ) and φ : R + [ 0 , b ) satisfying
lim sup r t + φ ( r ) < b , t R + ,
and for any x X , there is y T ( x ) satisfying
b d ( x , y ) d ( x , T ( x ) ) and f ( y ) φ ( d ( x , y ) ) d ( x , y ) .

Then T has a fixed point in X.

In 2009 and 2010, Ćirić [[5], Theorem 2.1] and Liu et al. [[6], Theorems 2.1 and 2.3] established a few fixed point theorems for some multi-valued nonlinear contractions, which include the multi-valued contraction in Theorem 1.1 as a special case.

Theorem 1.3 ([5])

Let ( X , d ) be a complete metric space and T be a multi-valued mapping from X into CL ( X ) satisfying (c1). Assume that

(c4) there exists a function φ : R + [ a , 1 ) , 0 < a < 1 , satisfying
lim sup r t + φ ( r ) < 1 , t R + ,
and for any x X , there is y T ( x ) satisfying
φ ( f ( x ) ) d ( x , y ) f ( x ) and f ( y ) φ ( f ( x ) ) d ( x , y ) .

Then T has a fixed point in X.

Theorem 1.4 ([6])

Let T be a multi-valued mapping from a complete metric space ( X , d ) into CL ( X ) such that
for each  x X , there exists  y T ( x )  satisfying α ( f ( x ) ) d ( x , y ) f ( x ) and f ( y ) β ( f ( x ) ) d ( x , y ) ,
where
B = { [ 0 , sup f ( X ) ] if  sup f ( X ) < , [ 0 , ) if  sup f ( X ) = ,
α : B ( 0 , 1 ] and β : B [ 0 , 1 ) satisfy that
lim inf r 0 + α ( r ) > 0 and lim sup r t + β ( r ) α ( r ) < 1 , t [ 0 , sup f ( X ) ) .

Then

(a1) for each x 0 X , there exist an orbit { x n } n N 0 of T and z X such that lim n x n = z ;

(a2) z is a fixed point of T in X if and only if the function f ( x ) = d ( x , T ( x ) ) , x X , is T-orbitally lower semi-continuous at z.

Theorem 1.5 ([6])

Let T be a multi-valued mapping from a complete metric space ( X , d ) into CL ( X ) such that
for each  x X , there exists  y T ( x )  satisfying α ( d ( x , y ) ) d ( x , y ) f ( x ) and f ( y ) β ( d ( x , y ) ) d ( x , y ) ,
where
A = { [ 0 , diam ( X ) ] if  diam ( X ) < , [ 0 , ) if  diam ( X ) = ,
α : A ( 0 , 1 ] and β : A [ 0 , 1 ) satisfy that
lim inf r t + α ( r ) > 0 and lim sup r t + β ( r ) α ( r ) < 1 , t [ 0 , diam ( X ) ) ,

and one of α and β is nondecreasing. Then

(a1) for each x 0 X , there exist an orbit { x n } n N 0 of T and z X such that lim n x n = z ;

(a2) z is a fixed point of T in X if and only if the function f ( x ) = d ( x , T ( x ) ) , x X , is T-orbitally lower semi-continuous at z.

In 2011, Latif and Abdou [[7], Theorem 2.1] generalized Theorem 1.3 and proved the following fixed point theorem for some multi-valued contractive mapping with w-distance.

Theorem 1.6 ([7])

Let ( X , d ) be a complete metric space with a w-distance w, and let T be a multi-valued mapping from X into CL ( X ) . Assume that

(c5) the mapping f : X R + , defined by f w ( x ) = w ( x , T ( x ) ) , x X , is lower semi-continuous;

(c6) there exists a function φ : R + [ b , 1 ) , 0 < b < 1 , satisfying
lim sup r t + φ ( r ) < 1 , t R +
and for any x X , there is y T ( x ) satisfying
φ ( f w ( x ) ) w ( x , y ) f w ( x ) and f w ( y ) φ ( f w ( x ) ) w ( x , y ) .

Then there exists v 0 X such that f w ( v 0 ) = 0 . Further, if w ( v 0 , v 0 ) = 0 , then v 0 T ( v 0 ) .

The purpose of this paper is to prove the existence of fixed points and iterative approximations for some multi-valued contractive mappings with w-distance. Two examples with uncountably many points are included. The results presented in this paper extend, improve and unify Theorem 3.1 in [2], Theorem 2.1 in [4], Theorems 2.1 and 2.2 in [5], Theorems 2.1 and 2.3 in [6], Theorems 2.1-2.3 and 2.5 in [7], Theorem 6 in [8], Theorems 2.2 and 2.4 in [9] and Theorems 3.1-3.4 in [10].

Throughout this paper, we assume that R + = [ 0 , ) , N 0 = N { 0 } , where denotes the set of all positive integers.

Definition 1.7 ([1])

A function w : X × X R + is called a w-distance in X if it satisfies the following:

(w1) w ( x , z ) w ( x , y ) + w ( y , z ) , x , y , z X ;

(w2) for each x X , a mapping w ( x , ) : X R + is lower semi-continuous, that is, if { y n } n N is a sequence in X with lim n y n = y X , then w ( x , y ) lim inf n w ( x , y n ) ;

(w3) for any ε > 0 , there exists δ > 0 such that w ( z , x ) δ and w ( z , y ) δ imply d ( x , y ) ε .

For any u X , D X , w-distance w and T : X CL ( X ) , put
d ( u , D ) = inf y D d ( u , y ) , w ( u , D ) = inf y D w ( u , y ) , f ( u ) = d ( u , T ( u ) ) , f w ( u ) = w ( u , T ( u ) ) , diam ( X ) = sup { d ( x , y ) : x , y X } , diam ( X w ) = sup { w ( x , y ) : x , y X } , A w = { [ 0 , diam ( X w ) ] if  diam ( X w ) < , [ 0 , ) if  diam ( X w ) =
and
B w = { [ 0 , sup f w ( X ) ] if  sup f w ( X ) < , [ 0 , ) if  sup f w ( X ) = .

A sequence { x n } n N 0 in X is called an orbit of T at x 0 X if x n T ( x n 1 ) for all n N . A function g : X R + is said to be T-orbitally lower semi-continuous at z X if g ( z ) lim inf n g ( x n ) for each orbit { x n } n N 0 X of T with lim n x n = z . A function φ : A w R + is called subadditive in A w if φ ( s + t ) φ ( s ) + φ ( t ) for all s , t A w . A function φ : A w R + is called strictly inverse in A w if φ ( t ) < φ ( s ) implies that t < s .

Lemma 1.8 ([11])

Let ( X , d ) be a metric space with a w-distance w and D CL ( X ) . Suppose that there exists u X such that w ( u , u ) = 0 . Then w ( u , D ) = 0 if and only if u D .

2 Fixed point theorems

In this section we prove the existence of fixed points and iterative approximations for some nonlinear multi-valued contraction mappings in complete metric spaces with w-distance.

Theorem 2.1 Let ( X , d ) be a complete metric space, w be a w-distance in X and T be a multi-valued mapping from X into CL ( X ) such that
for each  x X , there exists  y T ( x )  satisfying α ( f w ( x ) ) φ ( w ( x , y ) ) f w ( x ) and f w ( y ) β ( f w ( x ) ) ψ ( w ( x , y ) ) ,
(2.1)
where
α  and  β  are functions from  B w  into  ( 0 , 1 ]  and  [ 0 , 1 ) , respectively, with  β ( 0 ) < α ( 0 ) , lim inf r 0 + α ( r ) > 0 and lim sup r t + β ( r ) α ( r ) < 1 , t B w ,
(2.2)
φ  and  ψ  are functions from  A w  into  R +  with  ψ ( t ) φ ( t ) , t A w and φ  is subadditive in  A w  and satisfies that either 
(2.3)
φ  is strictly inverse in  A w , φ ( 0 ) = 0 , φ ( t ) > 0 , t A w { 0 }
(2.4)
or
φ  is strictly increasing in  A w  and  lim t 0 + φ 1 ( t ) = 0 , where  φ 1 stands for the inverse function of  φ .
(2.5)

Then

(a1) for each x 0 X , there exists an orbit { x n } n N 0 of T such that lim n x n = u 0 for some u 0 X ;

(a2) f w ( u 0 ) = 0 if and only if the function f w is T-orbitally lower semi-continuous at u 0 ;

(a3) u 0 T ( u 0 ) provided that w ( u 0 , u 0 ) = 0 = f w ( u 0 ) ;

(a4) T has a fixed point in X if for each orbit { z n } n N 0 of T in X and v X with v T ( v ) , one of the following conditions is satisfied:
inf { w ( z n , v ) + φ ( w ( z n , z n + 1 ) ) : n N 0 } > 0 ;
(2.6)
inf { w ( z n , v ) + w ( z n , T ( z n ) ) : n N 0 } > 0 .
(2.7)
Proof Firstly, we prove (a1). Let
γ ( t ) = β ( t ) α ( t ) , t B w .
(2.8)
It follows from (2.1) that for each x 0 X , there exists x 1 T ( x 0 ) satisfying
α ( f w ( x 0 ) ) φ ( w ( x 0 , x 1 ) ) f w ( x 0 ) and f w ( x 1 ) β ( f w ( x 0 ) ) ψ ( w ( x 0 , x 1 ) ) ,
which together with (2.3) and (2.8) yields that
f w ( x 1 ) β ( f w ( x 0 ) ) ψ ( w ( x 0 , x 1 ) ) β ( f w ( x 0 ) ) φ ( w ( x 0 , x 1 ) ) β ( f w ( x 0 ) ) f w ( x 0 ) α ( f w ( x 0 ) ) = γ ( f w ( x 0 ) ) f w ( x 0 ) .
Continuing this process, we choose easily an orbit { x n } n N 0 of T satisfying
x n + 1 T ( x n ) , α ( f w ( x n ) ) φ ( w ( x n , x n + 1 ) ) f w ( x n ) and f w ( x n + 1 ) β ( f w ( x n ) ) ψ ( w ( x n , x n + 1 ) ) , n N 0 .
(2.9)
It follows from (2.3), (2.8) and (2.9) that
f w ( x n + 1 ) β ( f w ( x n ) ) ψ ( w ( x n , x n + 1 ) ) β ( f w ( x n ) ) φ ( w ( x n , x n + 1 ) ) β ( f w ( x n ) ) f w ( x n ) α ( f w ( x n ) ) = γ ( f w ( x n ) ) f w ( x n ) , n N 0 .
(2.10)
Now we claim that
lim n f w ( x n ) = 0 .
(2.11)
Notice that the ranges of α and β, (2.2) and (2.8) ensure that
0 γ ( t ) < 1 , t B w .
(2.12)
Using (2.10) and (2.12), we conclude that { f w ( x n ) } n N 0 is a nonnegative and nonincreasing sequence, which means that there is a constant a 0 satisfying
lim n f w ( x n ) = a .
(2.13)
Suppose that a > 0 . Using (2.2), (2.8), (2.10), (2.12) and (2.13), we obtain that
a = lim sup n f w ( x n + 1 ) lim sup n [ γ ( f w ( x n ) ) f w ( x n ) ] lim sup n γ ( f w ( x n ) ) lim sup n f w ( x n ) a lim sup r a + γ ( r ) < a ,

which is a contradiction. Thus a = 0 , that is, (2.11) holds.

Next we claim that { x n } n N 0 is a Cauchy sequence. Put
b = lim sup n γ ( f w ( x n ) ) and c = lim inf n α ( f w ( x n ) ) .
(2.14)
It follows from (2.2), (2.8), (2.12) and (2.14) that
0 b < 1 and c > 0 .
(2.15)
Let p ( 0 , c ) and q ( b , 1 ) . Because of (2.14) and (2.15), we deduce that there exists some n 0 N such that
γ ( f w ( x n ) ) < q and α ( f w ( x n ) ) > p , n n 0 ,
which together with (2.9) and (2.10) yields that
f w ( x n + 1 ) q f w ( x n ) and φ ( w ( x n , x n + 1 ) ) f w ( x n ) p , n n 0 ,
which implies that
f w ( x n + 1 ) q n + 1 n 0 f w ( x n 0 ) and φ ( w ( x n , x n + 1 ) ) f w ( x n 0 ) p q n n 0 , n n 0 .
(2.16)
By means of (w1), (2.3) and (2.16), we deduce that
φ ( w ( x n , x m ) ) k = n m 1 φ ( w ( x k , x k + 1 ) ) k = n m 1 f w ( x n 0 ) p q k n 0 f w ( x n 0 ) p ( 1 q ) q n n 0 , m > n n 0 .
(2.17)
Given ε > 0 , denote by δ the constant in (w3) corresponding to ε. Assume that (2.4) holds. It follows from φ ( δ ) > 0 and q ( b , 1 ) that there exists a positive integer N n 0 satisfying
f w ( x n 0 ) p ( 1 q ) q n n 0 < φ ( δ ) , n N .
(2.18)
Combining (2.17) and (2.18), we infer that
max { φ ( w ( x N , x m ) ) , φ ( w ( x N , x n ) ) } f w ( x n 0 ) p ( 1 q ) q n n 0 < φ ( δ ) , m > n N ,
which together with (2.4) guarantees that
max { w ( x N , x m ) , w ( x N , x n ) } < δ , m > n > N .
(2.19)
It follows from (w3) and (2.19) that
d ( x m , x n ) ε , m > n > N .
(2.20)

It is clear that (2.20) yields that { x n } n N 0 is a Cauchy sequence.

Assume that (2.5) holds. Since φ is strictly increasing, so does φ 1 . It follows from (2.5) and q ( b , 1 ) that there exists a positive integer N n 0 satisfying
φ 1 ( f w ( x n 0 ) p ( 1 q ) q n n 0 ) < δ , n N ,
which together with (2.5) and (2.17) means that
w ( x n , x m ) = φ 1 ( φ ( w ( x n , x m ) ) ) φ 1 ( f w ( x n 0 ) p ( 1 q ) q n n 0 ) < δ , m > n N ,

which ensures that (2.19) and (2.20) hold. Consequently, { x n } n N 0 is a Cauchy sequence.

It follows from completeness of ( X , d ) that there is some u 0 X such that lim n x n = u 0 .

Secondly, we prove (a2). Suppose that f w is T-orbitally lower semi-continuous at u 0 . Let { x n } n N 0 be the orbit of T defined by (2.9) and satisfy (2.11). It follows from (2.11) that
0 w ( u 0 , T ( u 0 ) ) = f w ( u 0 ) lim inf n f w ( x n ) = 0 ,
which means that f w ( u 0 ) = 0 . Conversely, suppose that f w ( u 0 ) = 0 for some u 0 X . Let { y n } n N 0 be an arbitrary orbit of T in X with lim n y n = u 0 . It follows that
f w ( u 0 ) = 0 lim inf n f w ( y n ) ,

that is, f w is T-orbitally lower semi-continuous at u 0 .

Thirdly, we prove (a3). Note that T ( u 0 ) is closed and
w ( u 0 , u 0 ) = 0 = f w ( u 0 ) = w ( u 0 , T ( u 0 ) ) .

It follows from Lemma 1.8 that u 0 T ( u 0 ) .

Finally, we prove (a4). Assume that { x n } n N 0 is the orbit of T defined by (2.9) and that it satisfies (2.11), (2.16), (2.17) and lim n x n = u 0 X . Clearly, (2.16) and q ( b , 1 ) mean that
lim n φ ( w ( x n , x n + 1 ) ) = 0 .
(2.21)
Now we claim that
lim n w ( x n , u 0 ) = 0 .
(2.22)

In order to prove (2.22), we consider two possible cases as follows.

Case 1. Assume that (2.4) holds. Let ε > 0 be given. Notice that φ ( ε ) > 0 and q ( b , 1 ) . It follows that there exists a positive integer N > n 0 satisfying
f w ( x n 0 ) p ( 1 q ) q n n 0 < φ ( ε ) , n N ,
which together with (2.17) yields that
φ ( w ( x n , x m ) ) f w ( x n 0 ) p ( 1 q ) q n n 0 < φ ( ε ) , m > n N .
Since φ is strictly inverse, it follows that
w ( x n , x m ) < ε , m > n N .
Letting m in the above inequality and using (w2), we get that
w ( x n , u 0 ) lim inf m w ( x n , x m ) ε , n N ,

that is, (2.22) holds.

Case 2. Assume that (2.5) holds. It follows from (2.5) and (2.17) that
w ( x n , x m ) = φ 1 ( φ ( w ( x n , x m ) ) ) φ 1 ( f w ( x n 0 ) p ( 1 q ) q n n 0 ) , m > n n 0 ,
which together with (w2) and (2.5) ensures that
w ( x n , u 0 ) lim inf m w ( x n , x m ) φ 1 ( f w ( x n 0 ) p ( 1 q ) q n n 0 ) 0 as  n ,

that is, (2.22) holds.

Suppose that u 0 T ( u 0 ) . Let v = u 0 and z n = x n for each n N 0 . Assume that (2.6) holds. Making use of (2.6), (2.21) and (2.22), we conclude that
0 < inf { w ( x n , u 0 ) + φ ( w ( x n , x n + 1 ) ) : n N 0 } = 0 ,
which is a contradiction. Assume that (2.7) holds. By virtue of (2.7), (2.11) and (2.22), we infer that
0 < inf { w ( x n , u 0 ) + w ( x n , x n + 1 ) : n N 0 } = 0 ,

which is also a contradiction. Consequently, u 0 T ( u 0 ) . This completes the proof. □

Theorem 2.2 Let ( X , d ) be a complete metric space, w be a w-distance in X and T be a multi-valued mapping from X into CL ( X ) such that (2.3) and one of (2.4) and (2.5) hold and
for each  x X , there exists  y T ( x )  satisfying α ( w ( x , y ) ) φ ( w ( x , y ) ) f w ( x ) and f w ( y ) β ( w ( x , y ) ) ψ ( w ( x , y ) ) ,
(2.23)
where
α  and  β  are functions from  A w  into  ( 0 , 1 ]  and  [ 0 , 1 ) , respectively, with β ( 0 ) < α ( 0 ) , lim inf r 0 + α ( r ) > 0 and lim sup r t + β ( r ) α ( r ) < 1 , t A w
(2.24)
and
one of  α  and  β  is nondecreasing in  A w .
(2.25)

Then (a1)-(a4) hold.

Proof Firstly, we prove (a1). Let
γ ( t ) = β ( t ) α ( t ) , t A w .
(2.26)
Notice that the ranges of α and β, (2.24) and (2.26) ensure that
0 γ ( t ) < 1 , t A w .
(2.27)
It follows from (2.23) that for each x 0 X , there exists x 1 T ( x 0 ) satisfying
α ( w ( x 0 , x 1 ) ) φ ( w ( x 0 , x 1 ) ) f w ( x 0 ) and f w ( x 1 ) β ( w ( x 0 , x 1 ) ) ψ ( w ( x 0 , x 1 ) ) ,
which together with (2.3) and (2.26) means that
f w ( x 1 ) β ( w ( x 0 , x 1 ) ) ψ ( w ( x 0 , x 1 ) ) β ( w ( x 0 , x 1 ) ) φ ( w ( x 0 , x 1 ) ) β ( w ( x 0 , x 1 ) ) f w ( x 0 ) α ( w ( x 0 , x 1 ) ) = γ ( w ( x 0 , x 1 ) ) f w ( x 0 ) .
Continuing this process, we choose easily an orbit { x n } n N 0 of T satisfying
x n + 1 T ( x n ) , α ( w ( x n , x n + 1 ) ) φ ( w ( x n , x n + 1 ) ) f w ( x n ) and f w ( x n + 1 ) β ( w ( x n , x n + 1 ) ) ψ ( w ( x n , x n + 1 ) ) , n N 0 ,
(2.28)
which together with (2.3) and (2.26) gives that
f w ( x n + 1 ) β ( w ( x n , x n + 1 ) ) ψ ( w ( x n , x n + 1 ) ) β ( w ( x n , x n + 1 ) ) φ ( w ( x n , x n + 1 ) ) β ( w ( x n , x n + 1 ) ) f w ( x n ) α ( w ( x n , x n + 1 ) ) = γ ( w ( x n , x n + 1 ) ) f w ( x n ) , n N 0
(2.29)
and
φ ( w ( x n + 1 , x n + 2 ) ) f w ( x n + 1 ) α ( w ( x n + 1 , x n + 2 ) ) β ( w ( x n , x n + 1 ) ) α ( w ( x n + 1 , x n + 2 ) ) ψ ( w ( x n , x n + 1 ) ) , n N 0 .
(2.30)
Now we claim that
w ( x n + 1 , x n + 2 ) w ( x n , x n + 1 ) , n N 0 .
(2.31)
Suppose that there exists n 0 N 0 satisfying
w ( x n 0 + 1 , x n 0 + 2 ) > w ( x n 0 , x n 0 + 1 ) .
(2.32)
Let (2.4) hold. It follows from (2.3), (2.25), (2.26), (2.30) and (2.32) that
φ ( w ( x n 0 + 1 , x n 0 + 2 ) ) β ( w ( x n 0 , x n 0 + 1 ) ) α ( w ( x n 0 + 1 , x n 0 + 2 ) ) ψ ( w ( x n 0 , x n 0 + 1 ) ) max { γ ( w ( x n 0 , x n 0 + 1 ) ) , γ ( w ( x n 0 + 1 , x n 0 + 2 ) ) } φ ( w ( x n 0 , x n 0 + 1 ) ) .
(2.33)
If φ ( w ( x n 0 , x n 0 + 1 ) ) = 0 , it follows from (2.33) that φ ( w ( x n 0 + 1 , x n 0 + 2 ) ) = 0 . Thus (2.4) and (2.32) guarantee that
0 w ( x n 0 , x n 0 + 1 ) < w ( x n 0 + 1 , x n 0 + 2 ) = 0 ,
which is a contradiction; if φ ( w ( x n 0 , x n 0 + 1 ) ) > 0 , (2.4), (2.26), (2.27) and (2.33) yield that
φ ( w ( x n 0 + 1 , x n 0 + 2 ) ) max { γ ( w ( x n 0 , x n 0 + 1 ) ) , γ ( w ( x n 0 + 1 , x n 0 + 2 ) ) } φ ( w ( x n 0 , x n 0 + 1 ) ) < φ ( w ( x n 0 , x n 0 + 1 ) ) .
(2.34)
Since φ is strictly inverse, it follows from (2.32) and (2.34) that
w ( x n 0 + 1 , x n 0 + 2 ) < w ( x n 0 , x n 0 + 1 ) < w ( x n 0 + 1 , x n 0 + 2 ) ,

which is impossible.

Let (2.5) hold. Notice that φ is strictly increasing. It follows from (2.3), (2.26), (2.27), (2.30) and (2.32) that
φ ( w ( x n 0 + 1 , x n 0 + 2 ) ) β ( w ( x n 0 , x n 0 + 1 ) ) α ( w ( x n 0 + 1 , x n 0 + 2 ) ) ψ ( w ( x n 0 , x n 0 + 1 ) ) max { γ ( w ( x n 0 , x n 0 + 1 ) ) , γ ( w ( x n 0 + 1 , x n 0 + 2 ) ) } φ ( w ( x n 0 , x n 0 + 1 ) ) φ ( w ( x n 0 , x n 0 + 1 ) ) < φ ( w ( x n 0 + 1 , x n 0 + 2 ) ) ,

which is absurd. Hence (2.31) holds. That is, { w ( x n , x n + 1 ) } n N 0 is a nonincreasing and nonnegative sequence. It follows that lim n w ( x n , x n + 1 ) = d for some d 0 .

Now we claim that (2.11) holds. Using (2.27) and (2.29), we conclude that { f w ( x n ) } n N 0 is a nonnegative and nonincreasing sequence. Consequently, (2.13) is satisfied for some a 0 . Suppose that a > 0 . Using (2.13), (2.24), (2.27) and (2.29), we obtain that
a = lim sup n f w ( x n + 1 ) lim sup n [ γ ( w ( x n , x n + 1 ) ) f w ( x n ) ] lim sup n γ ( w ( x n , x n + 1 ) ) lim sup n f w ( x n ) a lim sup t d + γ ( t ) < a ,

which is a contradiction. Thus a = 0 , that is, (2.11) holds.

Next we claim that { x n } n N 0 is a Cauchy sequence. Put
b = lim sup n γ ( w ( x n , x n + 1 ) ) and c = lim inf n α ( w ( x n , x n + 1 ) ) .
(2.35)
It follows from (2.24), (2.27), (2.29) and (2.35) that (2.15) holds. Let p ( 0 , c ) and q ( b , 1 ) . Because of (2.15) and (2.35), we deduce that there exists some n 0 N such that
γ ( w ( x n , x n + 1 ) ) < q and α ( w ( x n , x n + 1 ) ) > p , n n 0 ,
which together with (2.28) and (2.29) yields that
f w ( x n + 1 ) q f w ( x n ) and φ ( w ( x n , x n + 1 ) ) f w ( x n ) p , n n 0 .

The rest of the proof is similar to that of Theorem 2.1 and is omitted. This completes the proof. □

3 Remarks and illustrative examples

In this section we construct two nontrivial examples to illustrate the results in Section 2.

Remark 3.1 Theorem 2.1 extends Theorem 3.1 in [2], Theorem 2.1 in [5], Theorem 2.1 in [6], Theorems 2.1 and 2.2 in [7], Theorems 2.2 and 2.4 in [9], and Theorems 3.1 and 3.2 in [10]. Example 3.2 below shows that Theorem 2.1 extends substantially Theorem 3.1 in [2] and Theorem 2.1 in [5] and differs from Theorems 5 and 6 in [8] and Theorem 2.1 in [4].

Example 3.2 Let X = [ 0 , 1 ] { 6 5 } be endowed with the Euclidean metric d = | | and u 0 = 0 . Define w : X × X R + , T : X CL ( X ) , α : [ 0 , 1 4 ] ( 0 , 1 ] , β : [ 0 , 1 4 ] [ 0 , 1 ) and φ , ψ : [ 0 , 6 5 ] R + by
w ( x , y ) = y , x , y X , T ( x ) = { { x 4 } , x [ 0 , 2 5 ) ( 2 5 , 1 ] , { 1 10 , 1 3 } , x { 2 5 , 6 5 } , α ( t ) = 8 + t 9 , β ( t ) = 2 + t 3 , t [ 0 , 1 4 ]
and
φ ( t ) = t , ψ ( t ) = min { t , | 1 t | } , t [ 0 , 6 5 ] .
It is easy to see that A w = [ 0 , 6 5 ] , B w = [ 0 , 1 4 ] , (2.3), (2.4) and (2.5) hold and
f w ( x ) = w ( x , T ( x ) ) = { x 4 , x [ 0 , 2 5 ) ( 2 5 , 1 ] , 1 10 , x { 2 5 , 6 5 } ,
is T-orbitally lower semi-continuous at u 0 ,
β ( 0 ) = 2 3 < 8 9 = α ( 0 ) , lim inf r 0 + α ( r ) = 8 9 > 0 , lim sup r t + β ( r ) α ( r ) = lim sup r t + ( 2 + r 3 9 8 + r ) = 6 + 3 t 8 + t < 1 , t [ 0 , 1 4 ] .
For x [ 0 , 2 5 ) ( 2 5 , 1 ] , there exists y = x 4 T ( x ) = { x 4 } satisfying
α ( f w ( x ) ) φ ( w ( x , y ) ) = 8 + x 4 9 x 4 x 4 = f w ( x )
and
f w ( y ) = x 16 2 + x 4 3 min { x 4 , 1 x 4 } = β ( f w ( x ) ) ψ ( w ( x , y ) ) .
For x { 2 5 , 6 5 } , there exists y = 1 10 T ( x ) = { 1 10 , 1 3 } satisfying
α ( f w ( x ) ) φ ( w ( x , y ) ) = 8 + 1 10 9 1 10 1 10 = f w ( x )
and
f w ( y ) = 1 40 2 + 1 10 3 min { 1 10 , 1 1 10 } = β ( f w ( x ) ) ψ ( w ( x , y ) ) .
Put v X { 0 } and { z n } n N 0 is an orbit of T in X. It is easy to verify that lim n z n = u 0 = 0 and
inf { w ( z n , v ) + φ ( w ( z n , z n + 1 ) ) : n N 0 } = inf { v + z n + 1 : n N 0 } = v + u 0 = v > 0 .

Hence (2.1), (2.2) and (2.6) hold, that is, the conditions of Theorem 2.1 are fulfilled. Thus Theorem 2.1 guarantees that (a1)-(a4) hold. Moreover, T has a fixed point u 0 = 0 X .

Now we show that Theorem 2.1 in [5] is unapplicable in proving the existence of fixed points for the multi-valued mapping T. Otherwise there exists a function φ : R + [ a , 1 ) , 0 < a < 1 , such that
lim sup r t + φ ( r ) < 1 , t R + ,
(3.1)
and for any x X there is y T ( x ) satisfying
φ ( f ( x ) ) d ( x , y ) f ( x )
(3.2)
and
f ( y ) φ ( f ( x ) ) d ( x , y ) .
(3.3)
Note that
f ( x ) = d ( x , T ( x ) ) = { 3 4 x , x [ 0 , 2 5 ) ( 2 5 , 1 ] , 1 15 , x = 2 5 , 13 15 , x = 6 5 .

Put x = 2 5 . For y T ( x ) = { 1 10 , 1 3 } , we discuss two cases as follows.

Case 1. y = 1 10 . It follows from (3.2) and (3.3) that
3 10 φ ( 1 15 ) = φ ( f ( 2 5 ) ) d ( 2 5 , 1 10 ) = φ ( f ( x ) ) d ( x , y ) f ( x ) = f ( 2 5 ) = 1 15
and
3 40 = f ( 1 10 ) = f ( y ) φ ( f ( x ) ) d ( x , y ) = φ ( f ( 2 5 ) ) d ( 2 5 , 1 10 ) = 3 10 φ ( 1 15 ) ,
which imply that
0.25 = 1 4 φ ( 1 15 ) 4 81 = 0.049 ,

which is impossible.

Case 2. y = 1 3 . It follows from (3.3) that
1 4 = f ( 1 3 ) = f ( y ) φ ( f ( x ) ) d ( x , y ) = φ ( f ( 2 5 ) ) d ( 2 5 , 1 3 ) = 1 15 φ ( 1 15 ) ,
which together with φ ( R + ) [ a , 1 ) yields that
15 4 φ ( 1 15 ) < 1 ,

which is absurd.

Next we show that Theorem 5 in [8] is useless in proving the existence of fixed points for the multi-valued mapping T. Otherwise there exists a function φ : R + [ 0 , 1 ) such that (3.1) holds, and for any x X there is y T ( x ) satisfying
d ( x , y ) ( 2 φ ( d ( x , y ) ) ) f ( x )
(3.4)
and
f ( y ) φ ( d ( x , y ) ) d ( x , y ) .
(3.5)

Put x = 2 5 . For y T ( x ) = { 1 10 , 1 3 } , we discuss two cases as follows.

Case 1. y = 1 10 . It follows from (3.4) that
3 10 = d ( 2 5 , 1 10 ) = d ( x , y ) ( 2 φ ( d ( x , y ) ) ) f ( x ) = ( 2 φ ( 3 10 ) ) 1 15 ,
which together with φ ( R + ) [ 0 , 1 ) yields that
0 φ ( 3 10 ) 5 2 < 0 ,

which is a contradiction.

Case 2. y = 1 3 . It follows from (3.4) that
1 4 = f ( 1 3 ) = f ( y ) φ ( d ( x , y ) ) d ( x , y ) = φ ( d ( 2 5 , 1 3 ) ) d ( 2 5 , 1 3 ) = 1 15 φ ( 1 15 ) ,
which together with φ ( R + ) [ 0 , 1 ) gives that
15 4 φ ( 1 15 ) < 1 ,

which is impossible.

Finally we show that Theorem 6 in [8] is futile in proving the existence of fixed points for the multi-valued mapping T. Otherwise there exist functions φ : R + ( 0 , 1 ) , b : R + [ b , 1 ) , b > 0 such that
φ ( t ) < b ( t ) , lim sup r t + φ ( r ) < lim sup r t + b ( r ) , t R + ,
(3.6)
and for any x X , there is y T ( x ) satisfying (3.5) and
b ( d ( x , y ) ) d ( x , y ) f ( x ) .
(3.7)

Put x = 2 5 . For y T ( x ) = { 1 10 , 1 3 } , we discuss two cases as follows.

Case 1. y = 1 10 . It follows from (3.7) and (3.5) that
3 10 b ( 3 10 ) = b ( d ( 2 5 , 1 10 ) ) d ( 2 5 , 1 10 ) = b ( d ( x , y ) ) d ( x , y ) f ( x ) = f ( 2 5 ) = 1 15
and
3 40 = f ( 1 10 ) = f ( y ) φ ( d ( x , y ) ) d ( x , y ) = 3 10 φ ( 3 10 ) ,
which together with (3.6) means that
b ( 3 10 ) 2 9 < 1 4 φ ( 3 10 ) < b ( 3 10 ) ,

which is absurd.

Case 2. y = 1 3 . It follows from (3.5) that
1 4 = f ( 1 3 ) = f ( y ) φ ( d ( x , y ) ) d ( x , y ) = φ ( d ( 2 5 , 1 3 ) ) d ( 2 5 , 1 3 ) = 1 15 φ ( 1 15 ) ,
which together with φ ( R + ) [ 0 , 1 ) gives that
15 4 φ ( 1 15 ) < 1 ,

which is impossible.

Observe that Theorem 6 in [8] extends Theorem 3.1 in [2], Theorem 2.1 in [4] and Theorem 2.2 in [5]. It follows that Theorem 3.1 in [2], Theorem 2.1 in [4] and Theorem 2.2 in [5] are not applicable in proving the existence of fixed points for the multi-valued mapping T.

Remark 3.3 Theorem 2.2 extends, improves and unifies Theorem 3.1 in [2], Theorem 2.1 in [4], Theorem 2.2 in [5], Theorem 2.3 in [6], Theorems 2.3 and 2.5 in [7], Theorem 6 in [8], and Theorems 3.3 and 3.4 in [10]. The following example reveals that Theorem 2.2 generalizes indeed the corresponding results in [2, 4, 5, 8].

Example 3.4 Let X = [ 0 , ) be endowed with the Euclidean metric d = | | and p 1 be a constant. Put u 0 = 0 . Define w : X × X R + , T : X CL ( X ) , α : [ 0 , ) ( 0 , 1 ] and φ , ψ : [ 0 , ) R + by β : [ 0 , ) [ 0 , 1 ) by
w ( x , y ) = y p , x , y X , T ( x ) = { [ x 2 2 , x 2 ] , x [ 0 , 1 ) , [ 1 9 , 1 4 ] , x [ 1 , ) , α ( t ) = 5 + t 1 p 10 , β ( t ) = 3 + t 1 p 10 , t [ 0 , )
and
φ ( t ) = t , t [ 0 , ) , ψ ( t ) = { t , t [ 0 , 1 ) , 1 2 , t [ 1 , ) .
It is easy to see that A w = [ 0 , ) , (2.3), (2.4) and (2.5) hold, w is a w-distance in X and
f w ( x ) = w ( x , T ( x ) ) = { ( x 2 2 ) p , x [ 0 , 1 ) , 1 9 p , x [ 1 , )
is T-orbitally lower semi-continuous in X, α and β are nondecreasing,
β ( 0 ) = 3 10 < 1 2 = α ( 0 ) , lim inf r 0 + α ( r ) = 1 2 > 0
and
lim sup r t + β ( r ) α ( r ) = 3 + t 1 p 5 + t 1 p < 1 , t A w .
Put x [ 0 , 1 ) and y = x 2 2 T ( x ) . Note that
5 + y 10 and ( y 2 ) p 1 4 p 3 + y 10
imply that
α ( w ( x , y ) ) φ ( w ( x , y ) ) = 5 + y 10 y p y p = f w ( x )
and
f w ( y ) = ( y 2 2 ) p 3 + y 10 y p = β ( w ( x , y ) ) ψ ( w ( x , y ) ) .
Put x [ 1 , ) and y = 1 9 T ( x ) = [ 1 9 , 1 4 ] . It follows that
α ( w ( x , y ) ) φ ( w ( x , y ) ) = 5 + 1 9 10 1 9 p 1 9 p = f w ( x )
and
f w ( y ) = 1 182 p 3 + 1 9 10 1 9 p = β ( w ( x , y ) ) ψ ( w ( x , y ) ) .
Let v X { 0 } and { z n } n N 0 be an orbit of T. It is easy to verify that lim n z n = 0 and
inf { w ( z n , v ) + φ ( w ( z n , z n + 1 ) ) : n N 0 } = inf { v p + z n + 1 p : n N 0 } = v p > 0 .

That is, (2.6) and (2.23)-(2.25) hold. Thus the conditions of Theorem 2.2 are satisfied. Consequently, Theorem 2.2 ensures that (a1)-(a4) hold and u 0 = 0 is a fixed point of the multi-valued mapping T in X.

Notice that
f ( x ) = d ( x , T ( x ) ) = { x 2 , x [ 0 , 1 ) , x 1 4 , x [ 1 , )
and
lim inf x 1 f ( x ) = 1 2 < 3 4 = f ( 1 ) ,

which implies that f is not lower semi-continuous at 1. Thus Theorem 3.1 in [2], Theorem 2.1 in [4], Theorem 2.2 in [5] and Theorem 6 in [8] could not be used to judge the existence of fixed points of the multi-valued mapping T in X.

Declarations

Acknowledgements

The authors would like to thank the referees for useful comments and suggestions. This research was supported by the Science Research Foundation of Educational Department of Liaoning Province (L2012380).

Authors’ Affiliations

(1)
Department of Mathematics, Liaoning Normal University
(2)
Department of Mathematics and RINS, Gyeongsang National University
(3)
Department of Mathematics, Gyeongsang National University

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© Liu et al.; licensee Springer. 2014

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