Open Access

On some fixed-point theorems for ψ-contraction on metric space involving a graph

Journal of Inequalities and Applications20142014:39

https://doi.org/10.1186/1029-242X-2014-39

Received: 2 October 2013

Accepted: 2 January 2014

Published: 24 January 2014

Abstract

In this paper, we introduce the ( G , ψ ) -contraction and the ( G , ψ ) -graphic contraction in a metric space by using a graph. We explain some conditions for a mapping which is a ( G , ψ ) -contraction to have a unique fixed point and also we give conditions as regards the existence of a fixed point for ( G , ψ ) -graphic contraction by applying the connectivity of the graph in both cases. Moreover, we give examples to show that our results are a substantial improvement of some known results in the literature.

MSC:47H10, 54H25.

Keywords

connected graphfixed pointmetric spaceψ-type contraction

1 Introduction

The metric fixed-point theory has been researched extensively in the past two decades such as in a metric space endowed with a partial ordering, and many results appeared giving sufficient conditions for a mapping to be a Picard operator. For these concepts have been given two main theorems, which are the Banach Contraction Principle and the Knaster-Tarski Theorem [1].

Recently Jachymski [2] and Gwóźdź-Lukawska and Jachymski [3] have given an interesting concept in fixed-point theory with some general structures by using the context of metric spaces endowed with a graph. Jachymski [2] has proved some generalizations of the Banach Contraction Principle to mappings on a metric space endowed with a graph and also has presented its applications to the Kelisky-Rivlin Theorem on iterates of the Bernstein operators on the space C [ 0 , 1 ] . Afterwards different contractions have been studied by various authors. In [4] the contraction principle for set-valued mappings, in [57] Kannan type, Reich type contractions, and φ-contractions have been investigated, respectively. Some new fixed-point results for graphic contractions on a complete metric space with a graph have been presented in [8]; also they gave a particular case of almost contractions.

In this paper, motivated by the work of Jachymski [2] and Petruşel [8], we introduce new contractions for the mappings on complete metric space and prove some fixed-point theorems. Our results generalize and unify some results by the above-mentioned authors.

2 Basic facts and definitions

Let ( X , d ) be a metric space and Δ denote the diagonal of the Cartesian product X × X . Let G be a directed graph such that the set V ( G ) of its vertices coincides with X, and the set E ( G ) of its edges contains all loops; that is, E ( G ) Δ . Assume that G has no parallel edges, so one can identify G with the pair ( V ( G ) , E ( G ) ) .

The conversion of a graph G is denoted by G 1 and this is a graph obtained from G by reversing the direction of the edges. Hence
E ( G 1 ) = { ( x , y ) X × X : ( y , x ) E ( G ) } .
By G ˜ we denote the undirected graph obtained from G by omitting the direction of the edges. Indeed, it is more convenient to treat G ˜ as a directed graph for which the set of its edges is symmetric, and under this convention, we have
E ( G ˜ ) = E ( G ) E ( G 1 ) .
A subgraph of a graph G is a graph H such that V ( H ) V ( G ) and E ( H ) E ( G ) . Let x and y be vertices in a graph G. A path from x to y of length N ( N N { 0 } ) is a sequence ( x i ) i = 0 N of N + 1 distinct vertices such that x 0 = x , x N = y and ( x i 1 , x i ) E ( G ) for i = 1 , , N . The number edges in G forming the path is called the length of the path. A graph G is connected if there is a path between any two vertices. If a graph G is not connected then it is called disconnected and its different paths are called the components of G. Every component of G is a subgraph of it. Furthermore, G is weakly connected if G ˜ is connected. Let G x be the component of G which consists of all edges and vertices contained in some path in G beginning at x. Suppose that G is such that E ( G ) is symmetric; then V ( G ) = [ x ] G where [ x ] G denotes the equivalence class of relations defined on V ( G ) by the rule
y z  if there is a path in  G  from  y  to  z .

Some basic notations related to connectivity of graphs can be found in [9].

If f : X X is an operator, then we denote by
F ( f ) = { x X : x = f x }

the set of all fixed points of f.

Definition 1 [2]

A mapping f : X X is a Banach G-contraction or simply G-contraction if f preserves edges of G;
( x , y ) E ( G ) ( f x , f y ) E ( G ) ,
(1)
for all x , y X , and f decreases weights of edges of G: for all x , y X there exists α ( 0 , 1 ) such that
( x , y ) E ( G ) d ( f x , f y ) < α d ( x , y ) .
(2)

Definition 2 [8]

The mapping f : X X is a G-graphic contraction
  1. (i)
    if f preserves edges of G;
    ( x , y ) E ( G ) ( f x , f y ) E ( G ) ,
    (3)
     
for all x , y X ;
  1. (ii)
    there exists α ( 0 , 1 ) such that
    ( x , y ) E ( G ) d ( f x , f 2 x ) α d ( x , f x ) ,
    (4)
     

for all x , y X f .

Definition 3 [2]

A mapping f : X X is called orbitally continuous if for all x , y X and any sequence ( k n ) n N of positive integers,
f k n x y implies f ( f k n x ) f y as  n .

Definition 4 [2]

A mapping f : X X is called orbitally G-continuous if for all x , y X and any sequence ( k n ) n N of positive integers,
f k n x y , ( f k n x , f k n + 1 x ) E ( G ) imply f ( f k n x ) f y as  n .

Now, we give a definition of the class Ψ which is used in several well-known papers to obtain some fixed-point results [1013].

Definition 5 Let us define the class Ψ = { ψ : R + R + ψ  is nondecreasing } which satisfies the following conditions:
  1. (i)

    ψ ( ω ) = 0 if and only if ω = 0 ;

     
  2. (ii)

    for every ( ω n ) R + , ψ ( ω n ) 0 if and only if ω n 0 ;

     
  3. (iii)

    for every ω 1 , ω 2 R + , ψ ( ω 1 + ω 2 ) ψ ( ω 1 ) + ψ ( ω 2 ) .

     

3 ( G , ψ ) -Contraction and related fixed-point theorems

We establish some fixed-point theorems in metric space with a graph by defining the ( G , ψ ) -contraction.

Definition 6 We say that a mapping f : X X is a ( G , ψ ) -contraction if the following hold;
  1. (i)

    f preserves edges of G, i.e. ( ( x , y ) E ( G ) ( f x , f y ) E ( G ) ) , x , y X ;

     
  2. (ii)
    f decreases the weight of edges of G, that is, there exists c ( 0 , 1 ) such that
    ( x , y ) E ( G ) ψ ( d ( f x , f y ) ) c ψ ( d ( x , y ) ) ,
     

for all x , y X .

Lemma 1 If f : X X is a ( G , ψ ) -contraction, then f is both a ( G 1 , ψ ) -contraction and a ( G ˜ , ψ ) -contraction.

Proof The proof can be obtained by the symmetry of d and the definition of the ( G ˜ , ψ ) -contraction. □

Lemma 2 Let f : X X be a ( G , ψ ) -contraction with constant c ( 0 , 1 ) ; for a given x X and y [ x ] G ˜ , there exists r ( x , y ) 0 such that
ψ ( d ( f n x , f n y ) ) c n r ( x , y ) .
(5)
Proof Let x X and y [ x ] G ˜ . Then there is a path ( x i ) i = 0 N in G ˜ from x to y, which means x 0 = x , x N = y , and ( x i 1 , x i ) E ( G ˜ ) for i = 1 , 2 , , N . By Lemma 1, f is a ( G ˜ , ψ ) -contraction. With an easy induction, we have ( f n x i 1 , f n x i ) E ( G ˜ ) and
ψ ( d ( f n x i 1 , f n x i ) ) c ψ ( d ( f n 1 x i 1 , f n 1 x i ) ) c ( c ψ ( d ( f n 2 x i 1 , f n 2 x i ) ) ) c n ψ ( d ( x i 1 , x i ) )

for all n N and i = 1 , 2 , , N .

Hence using the triangle inequality, we get
ψ ( d ( f n x , f n y ) ) i = 1 N ψ ( d ( f n x i 1 , f n x i ) ) c n i = 1 N ψ ( d ( x i 1 , x i ) ) .

So it qualifies to set r ( x , y ) : = i = 1 N ψ ( d ( x i 1 , x i ) ) . □

Lemma 3 Let ( X , d ) be a complete metric space endowed with a graph G and f : X X be a ( G , ψ ) -contraction for which there exists x 0 X such that f x 0 [ x 0 ] G ˜ . Let G ˜ x 0 be the component of G ˜ containing x 0 . Then [ x 0 ] G ˜ is f-invariant and f | [ x ] G ˜ is a ( G ˜ x 0 , ψ ) -contraction. Furthermore, x , y [ x 0 ] G ˜ , and the sequences ( f n x ) n N and ( f n y ) n N are Cauchy equivalent.

Proof The proof of this lemma can obtained by using similar arguments as given in [7]. So we omit the proof. □

The following result shows that there is a close relation between convergence of an iteration sequence which can be obtained by using a ( G , ψ ) -contraction mapping and connectivity of the graph.

Theorem 1 Let ( X , d ) be a metric space endowed with a graph G and f : X X be a ( G , ψ ) -contraction, then the following statements are equivalent:
  1. (i)

    G is weakly connected;

     
  2. (ii)

    for given x , y X , the sequences ( f n x ) n N and ( f n y ) n N are Cauchy equivalent;

     
  3. (iii)

    card F ( f ) 1 .

     
Proof (i) (ii) Let f be a ( G , ψ ) -contraction and x , y X . By hypothesis, [ x ] G ˜ = X , so f x [ x ] G ˜ . By Lemma 2, we get
ψ ( d ( f n x , f n + 1 x ) ) c n r ( x , f x )
for all n N . Hence
n = 0 ψ ( d ( f n x , f n + 1 x ) ) <
and if we use a standard argument, then ( f n x ) n N is obtained as a Cauchy sequence. Since also y [ x ] G ˜ , Lemma 2 leads to ψ ( d ( f n x , f n y ) ) c n r ( x , y ) . Therefore, ( f n x ) n N and ( f n y ) n N are equivalent. Clearly, because ( f n x ) n N is a Cauchy sequence, so is ( f n y ) n N .
  1. (ii)

    (iii) Let f be a ( G , ψ ) -contraction and x , y F ( f ) . By (ii), ( f n x ) n N and ( f n y ) n N are equivalent, which yields x = y .

     
  2. (iii)
    (ii) Suppose, to the contrary, G is not weakly connected, that is, G ˜ is disconnected. Let x 0 X . Then the sets [ x 0 ] G ˜ and X [ x 0 ] G ˜ both are nonempty. Let y 0 X [ x 0 ] G ˜ and define
    f x = { x 0 , if  x [ x 0 ] G ˜ , y 0 , if  x X [ x 0 ] G ˜ .
     

Obviously, F ( f ) = { x 0 , y 0 } . We show f is a ( G , ψ ) -contraction. Let ( x , y ) E ( G ) . Then [ x ] G ˜ = [ y ] G ˜ , so either x , y [ x 0 ] G ˜ or x , y X [ x 0 ] G ˜ . Hence in both cases f x = f y , so ( f x , f y ) E ( G ) as E ( G ) Δ , and ψ ( d ( f x , f y ) ) = 0 . Thereby, f is a ( G , ψ ) -contraction having two fixed points which violates the assumption. □

The following result is an easy consequence of Theorem 1.

Corollary 1 Let ( X , d ) be a complete metric space endowed with a graph G and f : X X be a ( G , ψ ) -contraction, then the following statements are equivalent:
  1. (i)

    G is weakly connected;

     
  2. (ii)

    there is x X such that lim n f n x = x , for all x X .

     

Now, we give an example of f being a ( G , ψ ) -contraction and this example shows that we could not add that x is a fixed point of f in Corollary 1.

Example 1 Let X = [ 0 , 1 ] be endowed with the usual metric. Take
E ( G ) = { ( 0 , 0 ) } { ( 0 , 1 ) } { ( x , y ) ( 0 , 1 ] × ( 0 , 1 ] : x y } ,
and f : X X as follows:
f x = { x 3 , if  x ( 0 , 1 ] , 1 2 , if  x = 0 .

Then f is a ( G , ψ ) -contraction where ψ ( ω ) = ω ω + 1 .

Proof It can be easily seen that G is a weakly connected graph and f is a ( G , ψ ) -contraction where ψ ( ω ) = ω ω + 1 . It is a fact that ( f n x ) 0 , for all x X but f has no fixed point. □

For any mapping which satisfies the condition of Corollary 1 to have a fixed point we need to add condition (6), which is given in the following theorem.

Theorem 2 Let ( X , d ) be a complete metric space and the triple ( X , d , G ) have the following condition:
for any  ( x n ) n N  in  X ,  if  x n x  and  ( x n , x n + 1 ) E ( G )  for  n N , then there is a subsequence  ( x k n ) n N  with  ( x k n , x ) E ( G )  for  n N .
(6)
Let f : X X be a ( G , ψ ) -contraction, and X f = { x X : ( x , f x ) E ( G ) } . Then the following statements hold.
  1. (i)

    card F ( f ) = card { [ x ] G ˜ : x X f } .

     
  2. (ii)

    F ( f ) iff X f .

     
  3. (iii)

    f has a unique fixed point iff there exists x 0 X f such that X f [ x 0 ] G ˜ .

     
  4. (iv)

    For any x X f , f | [ x ] G ˜ is a Picard operator.

     
  5. (v)

    If X f and G is weakly connected, then f is a Picard operator.

     
  6. (vi)

    If X : = { [ x ] G ˜ : x X f } , then f | X is a weakly Picard operator.

     
  7. (vii)

    If f E ( G ) , then f is a weakly Picard operator.

     
Proof Initially, we prove the items (iv) and (v). Take x X f and then f x [ x ] G ˜ , so by Lemma 3, if y [ x ] G ˜ , then ( f n x ) n N and ( f n y ) n N are Cauchy equivalent. Since X is complete, ( f n x ) n N converges to some x X . It is obvious that lim n f n y = x . Then by using induction we get
( f n x , f n + 1 x ) E ( G )
(7)
for all n N , since ( x , f x ) E ( G ) . By (6), there is a subsequence ( f k n x ) n N such that ( f k n x , x ) E ( G ) for all n N . If we use (7), we conclude that ( x , f x , f 2 x , , f k 1 , x ) is a path in G and also in G ˜ from x to x , and this means that x [ x ] G ˜ . Since f is a ( G , ψ ) -contraction we have
ψ ( d ( f k n + 1 x , f x ) ) c ψ ( d ( f k n x , x ) ) ,

for all n N . By taking the limit as n , we deduce f x = x . Thereby, f | [ x ] G ˜ is a Picard operator. Also, we conclude that f is a Picard operator, when [ x ] G ˜ = X , since there is weakly connectedness of G.

(vi) is obvious from (iv). For proof of (vii), if f E ( G ) then X f = X and so X = X holds. Thus f is a weakly Picard operator because of (vi).

Let us define a mapping to prove (i): ρ ( x ) = [ x ] G ˜ for all x F ( f ) . It is sufficient to show that ρ : F ( f ) C = { [ x ] G ˜ : x X f } is a bijection. Because E ( G ) Δ , we deduce F ( f ) X f and then ρ ( F ( f ) ) C . Beside, if x X f , then by (iv), lim n f n x [ x ] G ˜ F ( f ) , which implies ρ ( lim n f n x ) = [ x ] G ˜ and so ρ is a surjective mapping. We show that f is injective. Take x 1 , x 2 F ( f ) which are such that ρ ( x 1 ) = ρ ( x 2 ) [ x 1 ] G ˜ = [ x 2 ] G ˜ , then x 2 [ x 1 ] G ˜ and so, by (i),
lim n f n x 2 [ x 1 ] G ˜ F ( f ) = { x 1 } ,

which gives x 1 = x 2 . Thus, f is injective and this is the desired result. Finally, one can see that (ii) and (iii) are easy consequences of (i). □

Corollary 2 Let ( X , d ) be complete metric space and ( X , d , G ) obey condition (6). The following are equivalent:
  1. (i)

    G is weakly connected;

     
  2. (ii)

    every ( G , ψ ) -contraction f : X X such that ( x 0 , f x 0 ) E ( G ) , for some x 0 X , is a Picard operator;

     
  3. (iii)

    for any ( G , ψ ) -contraction, card F ( f ) 1 .

     
Proof (i) (ii): This can be obtained directly from Theorem 2(v).
  1. (ii)

    (iii): Let f : X X be a ( G , ψ ) -contraction. If X f is empty, so is F ( f ) , because F ( f ) is a subset of X f . If X f is nonempty, then by (ii), F ( f ) is singleton. In these two cases, card F ( f ) 1 .

     
  2. (iii)

    (i): This implication follows from Theorem 1. □

     

Remark 1 In the above results by taking ψ ( ω ) = ω , we obtain Corollary 3.2, which is given in [2].

4 ( G , ψ ) -Graphic contraction and fixed-point theorems

Now, we define ( G , ψ ) -graphic contraction and give some results and examples.

Definition 7 Let ( X , d ) be a metric space and G be a graph. The mapping f : X X is called a ( G , ψ ) -graphic contraction if the following conditions hold:
  1. (i)

    ( x , y ) E ( G ) implies ( f x , f y ) E ( G ) (f is edge preserving);

     
  2. (ii)
    there exists a ψ Ψ with constants c [ 0 , 1 ) such that
    ψ ( d ( f x , f 2 x ) ) c ψ ( d ( x , f x ) )
     

for all x X f , where X f : = { x X : ( x , f x ) E ( G )  or  ( f x , x ) E ( G ) } .

Firstly, we give the following lemmas which can be proved as in the above section.

Lemma 4 If f : X X is a ( G , ψ ) -graphic contraction, then f is both a ( G 1 , ψ ) -graphic contraction and a ( G ˜ , ψ ) -graphic contraction.

Lemma 5 Let f : X X be a ( G , ψ ) -graphic contraction with constant c [ 0 , 1 ) . Then, given x X f , there exists r ( x ) 0 such that
ψ ( d ( f n x , f n + 1 x ) ) c n r ( x ) ,
(8)

for all n N , where r ( x ) : = ψ ( d ( x , f x ) ) .

Lemma 6 Suppose that f : X X is a ( G , ψ ) -graphic contraction. Then for each x X f , there exists x X such that the sequence ( f n x ) n N converges to x as n .

Proof Take an arbitrary element x in X f . By Lemma 5, we obtain
ψ ( d ( f n x , f n + 1 x ) ) c n r ( x ) ,

for all n N . Therefore, n = 0 ψ ( d ( f n x , f n + 1 x ) ) < and so ψ ( d ( f n x , f n + 1 x ) ) 0 ; consequently using the property of ψ we have d ( f n x , f n + 1 x ) 0 . Then we say that ( f n x ) n N is a Cauchy sequence. By the completeness of X, there exists x X such that ( f n x ) n N converges as n . □

Lemma 7 The self-mapping f is a ( G , ψ ) -graphic contraction for which there exists x 0 X such that f x 0 [ x 0 ] G ˜ . Then the set [ x 0 ] G ˜ invariant with respect to f and f | [ x 0 ] G ˜ is a ( G ˜ x 0 , ψ ) -graphic contraction, where G ˜ x 0 is the component of G ˜ containing x 0 .

Proof Let x be an element in [ x 0 ] G ˜ . Then there exist ( x i ) i = 0 N in G ˜ from x 0 to x, i.e., x N = x and ( x i 1 , x i ) E ( G ˜ ) for i = 1 , 2 , , N . Since f is a ( G , ψ ) -graphic contraction we get ( f x i 1 , f x i ) E ( G ˜ ) for i = 1 , 2 , , N . So we have a path from f x 0 to fx. Therefore f x [ f x 0 ] G ˜ = [ x 0 ] G ˜ since f x 0 [ x 0 ] G ˜ . Consequently [ x 0 ] G ˜ is invariant with respect to f.

Take ( x , y ) E ( G ˜ x 0 ) ; then there is a path ( x i ) i = 0 N in G ˜ from x 0 to y such that x N 1 = x . Also let ( y i ) i = 0 M be a path in G ˜ from x 0 to f x 0 . Then we realize
( y 0 , y 1 , , y M , f x 1 , f x 2 , , f x N 1 = f x , f x N = f y )

is a path in G ˜ from x 0 to fy such that ( f x , f y ) E ( G ˜ x 0 ) . Furthermore, f is a ( G ˜ x 0 , ψ ) -graphic contraction because E ( G ˜ x 0 ) E ( G ˜ ) and f is a ( G ˜ , ψ ) -graphic contraction. □

Theorem 3 Let ( X , d ) be a complete metric space and let the triple ( X , d , G ) have the following condition:
for any  ( x n ) n N  in  X ,  if  x n x  and  ( x n , x n + 1 ) E ( G ) ( or , respectively , ( x n + 1 , x n ) E ( G ) )  for all  n N ,  then there is a subsequence  ( x k n ) n N  with  ( x k n , x ) E ( G ) ( or , respectively , ( x , x k n ) E ( G ) )  for all  n N .
(9)
Let f : X X be a ( G , ψ ) -graphic contraction and f is orbitally G-continuous. Then the following statements hold:
  1. (i)

    F ( f ) if and only if X f .

     
  2. (ii)

    If X f and G is weakly connected, then f is a weakly Picard operator.

     
  3. (iii)

    For any x X f , we see that f | [ x ] G ˜ is a weakly Picard operator.

     

Proof We begin with the statement (iii). Let x X f ; by Lemma 6, there exists x X such that lim n f n x = x . Since x X f , then f n x X f for every n N . Now assume that ( x , f x ) E ( G ) . (A similar deduction can be made if ( f x , x ) E ( G ) .) By condition (9), there is a subsequence ( f k n x ) n N of ( f n x ) n N such that ( f k n x , x ) E ( G ) for each n N . A path in G can be formed by using the points x , f x , , f k 1 x , x and hence x [ x ] G ˜ . Since f is orbitally G-continuous, we see that x is a fixed point for f | [ x ] G ˜ .

To prove (i), using (iii) we have F ( f ) if X f . Suppose that F ( f ) . By using the assumption that Δ E ( G ) , we immediately obtain X f . Hence (i) holds.

For proving (ii) let x X f . If we use weak connectivity of G, we have X = [ x ] G ˜ and by applying (iii) we obtain the desired result. □

The next example illustrates that f must be orbitally G-continuous in order to obtain statements which are given in Theorem 3.

Example 2 Let X = [ 0 , 1 ] be endowed with the usual metric. Consider
E ( G ) = { ( 0 , 0 ) } { ( 0 , x ) : x 1 / 2 } { ( x , y ) : x , y ( 0 , 1 ] } ,
and f : X X ,
f x = { x 2 , if  x ( 0 , 1 ] ; 1 2 , if  x = 0 .

Then G is weakly connected, X f is nonempty and f is a ( G , ψ ) -graphic contraction where ψ ( ω ) = ω 3 , but it is not orbitally G-continuous. Thus, f does not have a fixed point.

Remark 2 In Theorem 3, by replacing the condition that the triple ( X , d , G ) satisfies (9) and f is orbitally G-continuous with the mapping f is orbitally continuous, we have the above result, too.

The following example demonstrates that the ( G , ψ ) -graphic contraction is more general than the ( G , ψ ) -contraction.

Example 3 Let X = [ 0 , 1 ] be endowed with the usual metric. Take
E ( G ) = { ( 0 , 0 ) } { ( 0 , 1 ) } { ( x , y ) ( 0 , 1 ] × ( 0 , 1 ] : x y } ,
and f : X X as follows:
f x = { x 2 , if  x ( 0 , 1 ] , 3 4 , if  x = 0 .

Then G is weakly connected and X f is nonempty and f is a ( G , ψ ) -graphic contraction with ψ ( ω ) = ω 2 which is not a ( G , ψ ) -contraction.

Proof It is clear that G is weakly connected, X f , and with simple calculations it can be easily seen that f is a ( G , ψ ) -graphic contraction. Take
ψ ( d ( f 0 , f 1 2 ) ) c ψ ( d ( 0 , 1 2 ) ) 1 4 c 1 4 ,

which is a contradiction since c [ 0 , 1 ) . Thus, f is not ( G , ψ ) -contraction. □

Remark 3 In Theorem 3, if we take ψ ( ω ) = ω , then we get Theorem 2.1, which is given in [8].

Declarations

Acknowledgements

The authors are grateful to the reviewers for their careful reviews and useful comments.

Authors’ Affiliations

(1)
Department of Mathematics, Sakarya University

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Copyright

© Öztürk and Girgin; 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/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.