Skip to main content

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

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.

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 (NN{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

yz 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:XX is an operator, then we denote by

F(f)={xX:x=fx}

the set of all fixed points of f.

Definition 1 [2]

A mapping f:XX is a Banach G-contraction or simply G-contraction if f preserves edges of G;

(x,y)E(G)(fx,fy)E(G),
(1)

for all x,yX, and f decreases weights of edges of G: for all x,yX there exists α(0,1) such that

(x,y)E(G)d(fx,fy)<αd(x,y).
(2)

Definition 2 [8]

The mapping f:XX is a G-graphic contraction

  1. (i)

    if f preserves edges of G;

    (x,y)E(G)(fx,fy)E(G),
    (3)

for all x,yX;

  1. (ii)

    there exists α(0,1) such that

    (x,y)E(G)d ( f x , f 2 x ) αd(x,fx),
    (4)

for all x,y X f .

Definition 3 [2]

A mapping f:XX is called orbitally continuous if for all x,yX and any sequence ( k n ) n N of positive integers,

f k n xyimpliesf ( f k n x ) fyas n.

Definition 4 [2]

A mapping f:XX is called orbitally G-continuous if for all x,yX and any sequence ( k n ) n N of positive integers,

f k n xy, ( f k n x , f k n + 1 x ) E(G)implyf ( f k n x ) fyas 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:XX is a (G,ψ)-contraction if the following hold;

  1. (i)

    f preserves edges of G, i.e. ((x,y)E(G)(fx,fy)E(G)), x,yX;

  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 ) ) cψ ( d ( x , y ) ) ,

for all x,yX.

Lemma 1 If f:XX 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:XX be a (G,ψ)-contraction with constant c(0,1); for a given xX 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 xX 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 nN 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:XX 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:XX be a (G,ψ)-contraction, then the following statements are equivalent:

  1. (i)

    G is weakly connected;

  2. (ii)

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

  3. (iii)

    cardF(f)1.

Proof (i) (ii) Let f be a (G,ψ)-contraction and x,yX. By hypothesis, [ x ] G ˜ =X, so fx [ x ] G ˜ . By Lemma 2, we get

ψ ( d ( f n x , f n + 1 x ) ) c n r(x,fx)

for all nN. 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,yF(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

    fx= { 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,yX [ x 0 ] G ˜ . Hence in both cases fx=fy, so (fx,fy)E(G) as E(G)Δ, and ψ(d(fx,fy))=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:XX 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 xX.

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:XX as follows:

fx= { 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 xX 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:XX be a (G,ψ)-contraction, and X f ={xX:(x,fx)E(G)}. Then the following statements hold.

  1. (i)

    cardF(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 fE(G), then f is a weakly Picard operator.

Proof Initially, we prove the items (iv) and (v). Take x X f and then fx [ 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 nN, since (x,fx)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 nN. If we use (7), we conclude that (x,fx, 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 ) ) cψ ( d ( f k n x , x ) ) ,

for all nN. 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 fE(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 xF(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:XX 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, cardF(f)1.

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

  1. (ii)

    (iii): Let f:XX 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, cardF(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:XX is called a (G,ψ)-graphic contraction if the following conditions hold:

  1. (i)

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

  2. (ii)

    there exists a ψΨ with constants c[0,1) such that

    ψ ( d ( f x , f 2 x ) ) cψ ( d ( x , f x ) )

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

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

Lemma 4 If f:XX 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:XX 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 nN, where r(x):=ψ(d(x,fx)).

Lemma 6 Suppose that f:XX 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 nN. 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 fx [ 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 =fx,f x N =fy)

is a path in G ˜ from x 0 to fy such that (fx,fy)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:XX 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 nN. Now assume that (x,fx)E(G). (A similar deduction can be made if (fx,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 nN. A path in G can be formed by using the points x,fx,, 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 ΔE(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:XX,

fx= { 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:XX as follows:

fx= { 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 ) ) cψ ( 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].

References

  1. 1.

    Granas A, Dugundji J: Fixed Point Theory. Springer, New York; 2003.

    Google Scholar 

  2. 2.

    Jachymski J: The contraction principle for mappings on a metric space endowed with a graph. Proc. Am. Math. Soc. 2008, 136: 1359-1373.

    MATH  MathSciNet  Article  Google Scholar 

  3. 3.

    Gwóźdź LG, Jachymski J: IFS on a metric space with a graph structure and extensions of the Kelisky-Rivlin theorem. J. Math. Anal. Appl. 2009, 356: 453-463. 10.1016/j.jmaa.2009.03.023

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    Beg I, Butt RA, Radojević S: The contraction principle for set valued mappings on a metric space with a graph. Comput. Math. Appl. 2010, 60: 1214-1219. 10.1016/j.camwa.2010.06.003

    MATH  MathSciNet  Article  Google Scholar 

  5. 5.

    Bojor F: Fixed points of Kannan mappings in metric spaces endowed with a graph. An. Univ. “Ovidius” Constanţa, Ser. Mat. 2012,20(1):31-40.

    MATH  MathSciNet  Google Scholar 

  6. 6.

    Bojor F: Fixed point theorems for Reich type contractions on metric spaces with a graph. Nonlinear Anal. 2012, 75: 3895-3901. 10.1016/j.na.2012.02.009

    MATH  MathSciNet  Article  Google Scholar 

  7. 7.

    Bojor F: Fixed point of φ -contraction in metric spaces endowed with a graph. An. Univ. Craiova, Ser. Mat. Inform. 2010,37(4):85-92.

    MATH  MathSciNet  Google Scholar 

  8. 8.

    Petruşel GR, Chifu CI: Generalized contractions in metric spaces endowed with a graph. Fixed Point Theory Appl. 2012., 2012: Article ID 161

    Google Scholar 

  9. 9.

    Johnsonbaugh R: Discrete Mathematics. Prentice Hall, New Jersey; 1997.

    Google Scholar 

  10. 10.

    Shatanawi W, Postolache M: Common fixed point theorems for dominating and weak annihilator mappings in ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 271 10.1186/1687-1812-2013-271

    Google Scholar 

  11. 11.

    Shatanawi W, Pitea A: Some coupled fixed point theorems in quasi-partial metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 153 10.1186/1687-1812-2013-153

    Google Scholar 

  12. 12.

    Shatanawi W: Some fixed point results for a generalized ψ -weak contraction mappings in orbitally metric spaces. Chaos Solitons Fractals 2012,2012(45):520-526.

    MathSciNet  Article  Google Scholar 

  13. 13.

    Shatanawi W, Samet B:On (ψ,φ)-weakly contractive condition in partially ordered metric spaces. Comput. Math. Appl. 2011, 62: 3204-3214. 10.1016/j.camwa.2011.08.033

    MATH  MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

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

Author information

Affiliations

Authors

Corresponding author

Correspondence to Mahpeyker Öztürk.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally in writing this article. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Öztürk, M., Girgin, E. On some fixed-point theorems for ψ-contraction on metric space involving a graph. J Inequal Appl 2014, 39 (2014). https://doi.org/10.1186/1029-242X-2014-39

Download citation

Keywords

  • connected graph
  • fixed point
  • metric space
  • ψ-type contraction