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Common fixed point theorems for two weakly compatible self-mappings in cone b-metric spaces

Abstract

In this paper, we establish common fixed point theorems for two weakly compatible self-mappings satisfying the contractive condition or the quasi-contractive condition in the case of a quasi-contractive constant λ(0,1/s) in cone b-metric spaces without the normal cone, where the coefficient s satisfies s1. The main results generalize, extend and unify several well-known comparable results in the literature.

1 Introduction and preliminaries

Huang and Zhang [1] introduced the concept of a cone metric space, proved the properties of sequences on cone metric spaces and obtained various fixed point theorems for contractive mappings. The existence of a common fixed point on cone metric spaces was considered recently in [25]. Also, Ilic and Rakocevic [6] introduced a quasi-contraction on a cone metric space when the underlying cone was normal. Later on, Kadelburg et al. obtained a few similar results without the normality of the underlying cone, but only in the case of a quasi-contractive constant λ(0,1/2). However, Gajic [7] proved that result is true for λ(0,1) on a cone metric space by a new way, which answered the open question whether the result is true for λ(0,1). Recently, Hussain and Shah [8] introduced cone b-metric spaces, as a generalization of b-metric spaces and cone metric spaces, and established some important topological properties in such spaces. Following Hussain and Shah, Huang and Xu [9] obtained some interesting fixed point results for contractive mappings in cone b-metric spaces. Although Ion Marian [10] proved some common fixed point theorems in complete b-cone metric spaces, the main ways of the proof depend strongly on the nonlinear scalarization function ξ e :YR. In the present paper, we will show common fixed point theorems for two weakly compatible self-mappings satisfying the contractive condition or quasi-contractive condition in the case of a quasi-contractive constant λ(0,1/s) in cone b-metric spaces without the assumption of normality, where the coefficient s satisfies s1. As consequences, our results generalize, extend and unify several well-known comparable results (see, for example, [27, 913]).

Consistent with Huang and Zhang [1], the following definitions and results will be needed in the sequel.

Let E be a real Banach space and let P be a subset of E. By θ we denote the zero element of E and by intP the interior of P. The subset P is called a cone if and only if:

  1. (i)

    P is closed, nonempty, and P{θ};

  2. (ii)

    a,bR, a,b0, x,yPax+byP;

  3. (iii)

    P(P)={θ}.

On this basis, we define a partial ordering with respect to P by xy if and only if yxP. We write xy to indicate that xy but xy, while xy stands for yxintP. Write as the norm on E. The cone P is called normal if there is a number K>0 such that for all x,yE, θxy implies xKy. The least positive number satisfying the above is called the normal constant of P. It is well known that K1.

In the following, we always suppose that E is a Banach space, P is a cone in E with intP and is a partial ordering with respect to P.

Definition 1.1 [8]

Let X be a nonempty set and let s1 be a given real number. A mapping d:X×XE is said to be cone b-metric if and only if for all x,y,zX the following conditions are satisfied:

  1. (i)

    θd(x,y) with xy and d(x,y)=θ if and only if x=y;

  2. (ii)

    d(x,y)=d(y,x);

  3. (iii)

    d(x,y)s[d(x,z)+d(z,y)].

The pair (X,d) is called a cone b-metric space.

Example 1.2 Consider the space L p (0<p<1) of all real function x(t) (t[0,1]) such that 0 1 | x ( t ) | p dt<. Let X= L p , E= R 2 , P={(x,y)Ex,y0} R 2 and d:X×XE such that

d(x,y)= ( α { 0 1 | x ( t ) y ( t ) | p d t } 1 p , β { 0 1 | x ( t ) y ( t ) | p d t } 1 p ) ,

where α,β0 are constants. Then (X,d) is a cone b-metric space with the coefficient s= 2 1 p 1 .

Remark 1.3 It is obvious that any cone metric space must be a cone b-metric space. Moreover, cone b-metric spaces generalize cone metric spaces, b-metric spaces and metric spaces.

Definition 1.4 [8]

Let (X,d) be a cone b-metric space, xX and { x n } be a sequence in X. Then

  1. (i)

    { x n } converges to x whenever, for every cE with θc, there is a natural number N such that d( x n ,x)c for all nN. We denote this by lim n x n =x or x n x (n).

  2. (ii)

    { x n } is a Cauchy sequence whenever, for every cE with θc, there is a natural number N such that d( x n , x m )c for all n,mN.

  3. (iii)

    (X,d) is a complete cone b-metric space if every Cauchy sequence is convergent.

Lemma 1.5 [8]

Let (X,d) be a cone b-metric space. The following properties are often used while dealing with cone b-metric spaces in which the cone is not necessarily normal.

  1. (1)

    If uv and vw, then uw;

  2. (2)

    If θuc for each cintP, then u=θ;

  3. (3)

    If ab+c for each cintP, then ab;

  4. (4)

    If θd( x n ,x) b n and b n θ, then x n x;

  5. (5)

    If aλa, where aP and 0<λ<1, then a=θ;

  6. (6)

    If cintP, θ a n and a n θ, then there exists n 0 N such that a n c for all n> n 0 .

Lemma 1.6 [8]

The limit of a convergent sequence in a cone b-metric space is unique.

Definition 1.7 [2]

The mappings f,g:XX are weakly compatible if for every xX, fgx=gfx holds whenever fx=gx.

Definition 1.8 [3]

Let f and g be self-maps of a set X. If w=fx=gx for some x in X, then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g.

Lemma 1.9 [3]

Let f and g be weakly compatible self-maps of a set X. If f and g have a unique point of coincidence w=fx=gx, then w is the unique common fixed point of f and g.

Definition 1.10 [13]

Let (X,d) be a cone metric space. A mapping f:XX is such that, for some constant λ(0,1) and for every x,yX, there exists an element

uC(g,x,y)= { d ( g x , g y ) , d ( g x , f x ) , d ( g y , f y ) , d ( g x , f y ) , d ( g y , f x ) }

for which d(fx,fy)λu is called a g-quasi-contraction.

2 Main results

In this section, we give some common fixed point results for two weakly compatible self-mappings satisfying the contractive condition and quasi-contractive condition in the case of a contractive constant λ(0,1/s) in cone b-metric spaces without the assumption of normality.

Theorem 2.1 Let (X,d) be a cone b-metric space with the coefficient s1 and let a i 0 (i=1,2,3,4,5) be constants with 2s a 1 +(s+1)( a 2 + a 3 )+( s 2 +s)( a 4 + a 5 )<2. Suppose that the mappings f,g:XX satisfy the condition, for all x,yX,

d(fx,fy) a 1 d(gx,gy)+ a 2 d(gx,fx)+ a 3 d(gy,fy)+ a 4 d(gx,fy)+ a 5 d(gy,fx).
(2.1)

If the range of g contains the range of f and g(X) or f(X) is a complete subspace of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point in X.

Proof For an arbitrary x 0 X, since f(X)g(X), there exists an x 1 X such that f x 0 =g x 1 . By induction, a sequence { x n } can be chosen such that f x n =g x n + 1 (n1). If g x n 0 1 =g x n 0 =f x n 0 1 for some natural number n 0 , then x n 0 1 is a coincidence point of f and g in X. Suppose that g x n 1 g x n for all n1.

Thus, by (2.1) for any nN, we have

d ( g x n + 1 , g x n ) = d ( f x n , f x n 1 ) a 1 d ( g x n , g x n 1 ) + a 2 d ( g x n , f x n ) + a 3 d ( g x n 1 , f x n 1 ) + a 4 d ( g x n , f x n 1 ) + a 5 d ( g x n 1 , f x n )

and

d ( g x n , g x n + 1 ) = d ( f x n 1 , f x n ) a 1 d ( g x n 1 , g x n ) + a 2 d ( g x n 1 , f x n 1 ) + a 3 d ( g x n , f x n ) + a 4 d ( g x n 1 , f x n ) + a 5 d ( g x n , f x n 1 ) .

Hence

2 d ( g x n , g x n + 1 ) = d ( g x n + 1 , g x n ) + d ( g x n , g x n + 1 ) ( 2 a 1 + a 2 + a 3 + s a 4 + s a 5 ) d ( g x n , g x n 1 ) + ( a 2 + a 3 + s a 4 + s a 5 ) d ( g x n + 1 , g x n ) .

Since 2s a 1 +(s+1)( a 2 + a 3 )+( s 2 +s)( a 4 + a 5 )<2, we have

d ( g x n , g x n + 1 ) 2 a 1 + a 2 + a 3 + s a 4 + s a 5 2 a 2 a 3 s a 4 s a 5 d ( g x n , g x n 1 ) = k d ( g x n , g x n 1 ) k 2 d ( g x n 1 , g x n 2 ) k 3 d ( g x n 2 , g x n 3 ) k n d ( g x 1 , g x 0 ) ,

where k= 2 a 1 + a 2 + a 3 + s a 4 + s a 5 2 a 2 a 3 s a 4 s a 5 . Obviously, k[0, 1 s ).

Thus, setting any positive integers m and n, we have

d ( g x n , g x n + m ) s d ( g x n , g x n + 1 ) + s d ( g x n + 1 , g x n + m ) s d ( g x n , g x n + 1 ) + s 2 d ( g x n + 1 , g x n + 2 ) + s 2 d ( g x n + 2 , g x n + m ) s d ( g x n , g x n + 1 ) + s 2 d ( g x n + 1 , g x n + 2 ) + s 3 d ( g x n + 2 , g x n + 3 ) + + s m 1 d ( g x n + m 2 , g x n + m 1 ) + s m 1 d ( g x n + m 1 , g x n + m ) s d ( g x n , g x n + 1 ) + s 2 d ( g x n + 1 , g x n + 2 ) + s 3 d ( g x n + 2 , g x n + 3 ) + + s m 1 d ( g x n + m 2 , g x n + m 1 ) + s m d ( g x n + m 1 , g x n + m ) ( s k n + s 2 k n + 1 + + s m k n + m 1 ) d ( g x 1 , g x 0 ) = s k n [ 1 ( s k ) m ] 1 s k d ( g x 1 , g x 0 ) s k n 1 s k d ( g x 1 , g x 0 ) .

Since k[0,1/s), we notice that s k n 1 s k d(g x 1 ,g x 0 )θ as n for any m N + . By Lemma 1.5, for any cintP, we can choose n 0 N such that s k n 1 s k d(g x 1 ,g x 0 )c for all n> n 0 . Thus, for each cintP, d(g x n + m ,g x n )c for all n> n 0 , m1. Therefore {g x n } is a Cauchy sequence in g(X).

If g(X)X is complete, there exist qg(X) and pX such that g x n q as n and gp=q. (If f(X) is complete, there exists qf(X) such that f x n q as n. Since f(X)g(X), we can find pX such that gp=q.)

Now, from (2.1) we show that fp=q,

d ( g x n + 2 , f p ) = d ( f x n + 1 , f p ) a 1 d ( g x n + 1 , q ) + a 2 d ( g x n + 1 , g x n + 2 ) + a 3 d ( q , f p ) + a 4 d ( g x n + 1 , f p ) + a 5 d ( q , g x n + 2 ) .

Similarly,

d ( f p , g x n + 2 ) = d ( f p , f x n + 1 ) a 1 d ( q , g x n + 1 ) + a 2 d ( q , f p ) + a 3 d ( g x n + 1 , g x n + 2 ) + a 4 d ( q , g x n + 2 ) + a 5 d ( g x n + 1 , f p ) ,

thus, we have

2 d ( g x n + 2 , f p ) 2 a 1 d ( g x n + 1 , q ) + ( a 2 + a 3 ) d ( g x n + 1 , g x n + 2 ) + ( a 2 + a 3 ) d ( q , f p ) + ( a 4 + a 5 ) d ( g x n + 1 , f p ) + ( a 4 + a 5 ) d ( q , g x n + 2 ) ( 2 s a 1 + s a 2 + s a 3 + a 4 + a 5 ) d ( g x n + 2 , q ) + ( s a 2 + s a 3 + s a 4 + s a 5 ) d ( g x n + 2 , f p ) + ( 2 s a 1 + a 2 + a 3 + s a 4 + s a 5 ) d ( g x n + 1 , g x n + 2 ) .

Since 0 a 2 + a 3 + a 4 + a 5 <2/s, by the triangular inequality, it follows that

d ( g x n + 2 , f p ) 2 s a 1 + s a 2 + s a 3 + a 4 + a 5 2 s a 2 s a 3 s a 4 s a 5 d ( g x n + 2 , q ) + 2 s a 1 + a 2 + a 3 + s a 4 + s a 5 2 s a 2 s a 3 s a 4 s a 5 d ( g x n + 1 , g x n + 2 ) .

Since {g x n } is a Cauchy sequence and g x n q (n), for any cintP, we can choose n 1 N such that for all n n 1 ,

d(g x n + 1 ,g x n + 2 ) ( 2 s a 2 s a 3 s a 4 s a 5 ) c 2 ( 2 s a 1 + a 2 + a 3 + s a 4 + s a 5 )

and

d(g x n + 2 ,q) ( 2 s a 2 s a 3 s a 4 s a 5 ) c 2 ( 2 s a 1 + s a 2 + s a 3 + a 4 + a 5 ) .

Thus, for any cintP, d(g x n + 2 ,fp)c for all n n 1 . Therefore, by Lemma 1.5, we have fp=q=gp.

Assume that there exist u, w in X such that fu=gu=w.

d ( g u , g p ) = d ( f u , f p ) a 1 d ( g u , g p ) + a 2 d ( f u , g u ) + a 3 d ( f p , g p ) + a 4 d ( f p , g u ) + a 5 d ( f u , g p ) = ( a 1 + a 4 + a 5 ) d ( g u , g p ) .

Since 0 a 1 + a 4 + a 5 <1, by Lemma 1.5, we can obtain that d(gu,gp)=θ, i.e., w=gu=gp=q. Moreover, the mappings f and g are weakly compatible, by Lemma 1.9, we know that q is the unique common fixed point of f and g. □

Example 2.2 Let E= C R 1 ([0,1]), P={φE:φ0}E, X=[1,) and d(x,y)= | x y | 2 e t . Then (X,d) is a cone b-metric space with the coefficient s=2, but it is not a cone metric space. We consider the functions f,g:XX defined by fx= 1 6 lnx+1, gx=lnx+1. Hence

d ( f x , f y ) = | 1 6 ln x + 1 1 6 ln y 1 | 2 e t | 1 6 ln x + 1 6 ln y | 2 e t = | 1 5 ( ln x 1 6 ln y ) + 1 5 ( ln y 1 6 ln x ) | 2 e t 2 25 | ln x 1 6 ln y | 2 e t + 2 25 | ln y 1 6 ln x | 2 e t = 2 25 d ( g x , f y ) + 2 25 d ( g y , f x ) .

Here 1X is the unique common fixed point of f and g.

Example 2.3 Let X be the set of Lebesgue measurable functions on [0,1] such that 0 1 | u ( x ) | 2 dx<, E= C R ([0,1]), P={φE:φ0}E. We define d:X×XE as

d ( u ( t ) , v ( t ) ) = e t 0 1 | u ( s ) v ( s ) | 2 ds,

for all x,yX. Then (X,d) is a cone b-metric space with the coefficient s=2, but it is not a cone metric space. Considering the functions fu= 1 4 u(t) and gu= 1 2 u(t) (t[0,1]), we have

d ( f u , f v ) = e t 0 1 | 1 4 u ( s ) 1 4 v ( s ) | 2 d s = e t 4 0 1 | 1 2 u ( s ) 1 2 v ( s ) | 2 d s = 1 4 d ( g u , g v ) .

Clearly, 0X is the unique common fixed point of f and g.

Remark 2.4 Compared with the common fixed point results on cone metric spaces in [2, 3, 5], the common fixed point theorems in complete b-cone metric spaces in [10] and the fixed point results in cone b-metric spaces in [9], Theorem 2.1 is shown to be a proper generalization by Examples 2.2 and 2.3. Furthermore, Theorem 2.1 generalizes and unifies [[9], Theorem 2.1 and 2.3].

Definition 2.5 Let (X,d) be a cone b-metric space with the coefficient s1. A mapping f:XX is such that, for some constant λ(0,1/s) and for every x,yX, there exists an element

νC(g,x,y)= { d ( g x , g y ) , d ( g x , f x ) , d ( g y , f y ) , d ( g x , f y ) , d ( g y , f x ) }
(2.2)

for which d(fx,fy)λu is called a g-quasi-contraction.

Theorem 2.6 Let (X,d) be a cone b-metric space with the coefficient s1 and let the mapping f:XX be a g-quasi-contraction. If the range of g contains the range of f and g(X) or f(X) is a complete subspace of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point in X.

Proof For each x 0 X, set g x 1 =f x 0 and g x n + 1 =f x n (nN). If g x n 0 1 =g x n 0 =f x n 0 1 for some natural number n 0 , then x n 0 1 is a coincidence point of f and g in X.

Suppose that g x n 1 g x n for all n1. Now we prove that {g x n } is a Cauchy sequence. First, we show that

d(g x n ,g x 1 )=d(f x n 1 ,f x 0 ) s λ 1 s λ d(g x 1 ,g x 0 )for all n N + .
(2.3)

Clearly, we note (2.3) holds when n=1. We assume that (2.3) holds for some nN1 (N N + ), then we prove that (2.3) holds for all n=N. Because f is a g-quasi-contractive mapping, there exists a real number kN such that

d(g x N ,g x 1 )λd(g x k ,g x 0 ).
(2.4)

In order to prove that (2.4) holds, we show that for all 1i,jN, there exists 1kN such that

d(g x i ,g x j )λd(g x k ,g x 0 ).
(2.5)

Clearly, (2.5) is true for N=1. Suppose that (2.5) is true for each N=PN, that is, for all 1i,jP, there exists 1kP such that

d(g x i ,g x j )λd(g x k ,g x 0 ).
(2.6)

Let us prove (2.5) holds for N=P+1.

By (2.6), we only show that for any 1 i 0 P+1, there exists 1kP+1 such that

d(g x P + 1 ,g x i 0 )λd(g x k ,g x 0 ).

Since f is a g-quasi-contractive mapping, there exists

ν i 0 C ( g , x P , x i 0 1 ) = { d ( g x P , g x i 0 1 ) , d ( g x P , g x P + 1 ) , d ( g x i 0 1 , g x i 0 ) , d ( g x P , g x i 0 ) , d ( g x i 0 1 , g x P + 1 ) }

such that d(g x P + 1 ,g x i 0 )λ ν i 0 .

By (2.6), we discuss that there exists an element

d ( g x P + 1 , g x i 1 ) { d ( g x P , g x i 0 1 ) , d ( g x P , g x P + 1 ) , d ( g x i 0 1 , g x i 0 ) , d ( g x P , g x i 0 ) , d ( g x i 0 1 , g x P + 1 ) }

such that d(g x P + 1 ,g x i 0 )λd(g x P + 1 ,g x i 1 ) (1 i 1 P+1).

If the above inequality does not hold for 1 i 1 P+1, then (2.5) is true for N=P+1 by (2.6).

We continue in the same way, and after P+1 steps, we get 1 i j P+1 (0jP+1) such that

d(g x P + 1 ,g x i j )λd(g x P + 1 ,g x i j + 1 )(0jP).

Notice that there exist 0r<sP+1 such that i r = i s . That is,

d(g x P + 1 ,g x i r ) λ s r d(g x P + 1 ,g x i s )= λ s r d(g x P + 1 ,g x i r )(0r<sP+1).

As λ(0,1), by Lemma 1.5(5), we get a contradiction. From (2.6), (2.5) is true for N=P+1.

Hence, (2.5) is true for all NN, which implies that (2.4) holds for NN.

Next, let us prove that for all n N + ,

d(g x n ,g x 0 ) s 1 s λ d(g x 0 ,g x 1 ).
(2.7)

Using the triangular inequality, from (2.3) we obtain

d ( g x n , g x 0 ) s [ d ( g x n , g x 1 ) + d ( g x 1 , g x 0 ) ] s 2 λ 1 s λ d ( g x 0 , g x 1 ) + s d ( g x 1 , g x 0 ) = s 1 s λ d ( g x 1 , g x 0 ) .

Now, we show that {g x n } is a Cauchy sequence. For all n>m, there exists

ν 1 C ( g , x m 1 , x n 1 ) = { d ( g x m 1 , g x n 1 ) , d ( g x m 1 , g x m ) , d ( g x n 1 , g x n ) , d ( g x m 1 , g x n ) , d ( g x m , g x n 1 ) }
(2.8)

such that d(g x m ,g x n )=d(f x m 1 ,f x n 1 )λ ν 1 .

By the contractive condition, there exist but not all

ν k { d ( g x i , g x j ) | 0 i < j n } (k=1,2,3,,m)

such that

ν k λ ν k + 1 (k=1,2,3,,m1).
(2.9)

In fact, from (2.8) we have

ν 1 C ( g , x m 1 , x n 1 ) = { d ( g x m 1 , g x n 1 ) , d ( g x m 1 , g x m ) , d ( g x n 1 , g x n ) , d ( g x m 1 , g x n ) , d ( g x m , g x n 1 ) } A m 1 , n 1 = { d ( g x i , g x j ) | j = m , n , n 1 , i = m , m 1 , n 1 ; i < j } .

Let ν 1 =d(g x i ,g x j )=d(f x i 1 ,f x j 1 )λ ν 2 , where

ν 2 C ( g , x i 1 , x j 1 ) A i 1 , j 1 = { d ( g x r , g x s ) | s = i , j , j 1 , r = i , i 1 , j 1 ; r < s } = { d ( g x r , g x s ) | s = m , m 1 , n , n 1 , n 2 , r = m , m 1 , m 2 , n 1 , n 2 ; r < s } .

In general, if there exists

ν k { d ( g x i , g x j ) | j = m , m 1 , m 2 , , m k + 1 , n , n 1 , n 2 , , n k , i = m , m 1 , m 2 , , m k , n 1 , n 2 , , n k ; i < j } (1km),

then we have

ν k + 1 C(g, x i 1 , x j 1 ) A i 1 , j 1 = { d ( g x r , g x s ) | s = i , j , j 1 , r = i , i 1 , j 1 ; r < s } (1km1)

such that ν k =d(g x i ,g x j )=d(f x i 1 ,f x j 1 )λ ν k + 1 (1km1).

As

we can obtain (2.9).

Using the triangular inequality, we get

d(g x i ,g x j )sd(g x i ,g x 0 )+sd(g x 0 ,g x j )(0i,jn),

so we obtain

d ( g x n , g x m ) = d ( f x n 1 , f x m 1 ) λ ν 1 λ 2 ν 2 λ m ν m λ m s d ( g x i , g x 0 ) + λ m s d ( g x 0 , g x j ) 2 s 2 λ m 1 s λ d ( g x 1 , g x 0 ) .

Since 2 s 2 λ m 1 s λ d(g x 1 ,g x 0 )θ as m, by Lemma 1.5, it is easy to see that for any cintP, there exists n 0 N such that for all n>m> n 0 ,

d(g x n ,g x m ) 2 s 2 λ m 1 s λ d(g x 1 ,g x 0 )c.

So, {g x n } is a Cauchy sequence in g(X). If g(X)X is complete, there exist qg(X) and pX such that g x n q as n and g(p)=q.

Now, from (2.2) we get

νC(g, x n ,p)= { d ( g x n , g p ) , d ( g x n , f x n ) , d ( g p , f p ) , d ( g x n , f p ) , d ( f x n , g p ) }

such that d(f x n ,fp)λν.

We have the following five cases:

  1. (1)

    d(f x n ,fp)λd(g x n ,gp)sλd(g x n + 1 ,gp)+sλd(g x n + 1 ,g x n );

  2. (2)

    d(f x n ,fp)λd(g x n ,f x n )=λd(g x n ,g x n + 1 );

  3. (3)

    d(f x n ,fp)λd(gp,fp)sλd(g x n + 1 ,gp)+sλd(g x n + 1 ,fp), that is, d(f x n ,fp) s λ 1 s λ d(g x n + 1 ,gp);

  4. (4)

    d(f x n ,fp)λd(g x n ,fp)sλd(g x n + 1 ,fp)+sλd(g x n + 1 ,g x n ), that is, d(f x n ,fp) s λ 1 s λ d(g x n + 1 ,g x n );

  5. (5)

    d(f x n ,fp)λd(f x n ,gp)=λd(g x n + 1 ,gp).

As s λ 1 s λ >sλ, then we obtain that

d(g x n + 1 ,fp) s λ 1 s λ [ d ( g x n + 1 , g x n ) + d ( g x n + 1 , q ) ] .

Since g x n q as n, for any cintP, there exists n 1 N such that for all n> n 1 ,

d(g x n + 1 ,g x n ) ( 1 s λ ) c 2 s λ andd(g x n + 1 ,q) ( 1 s λ ) c 2 s λ .

By Lemmas 1.5 and 1.6, we have g x n fp as n and q=fp.

Now, if w is another point such that gu=fu=w, then

d(w,q)=d(fu,fp)λν,

where λ(0, 1 s ) and

νC(f;u,p)= { d ( g u , g p ) , d ( g u , f u ) , d ( g p , f p ) , d ( g u , f p ) , d ( f u , g p ) } .

It is obvious that d(w,q)=θ, i.e., w=q. Therefore, q is the unique point of coincidence of f, g in X. Moreover, the mappings f and g are weakly compatible, by Lemma 1.9 we know that q is the unique common fixed point of f and g.

Similarly, if f(X) is complete, the above conclusion is also established. □

Example 2.7 Let X=R, E= C R 1 [0,1] and P={fE:f0}. Define d:X×XE by d(x,y)= | x y | 3 2 φ where φ:[0,1]R such that φ(t)= e t . It is easy to see that (X,d) is a cone b-metric space with the coefficient s= 2 1 2 , but it is not a cone metric space. The mappings f,g:XX are defined by fx=αx and gx= α x (α[ 1 8 3 , 1 4 3 )). The mapping f is a g-quasi-contraction with the constant λ= α 3 4 [ 1 2 , 2 2 ). Moreover, 0X is the unique common fixed point of f and g.

Remark 2.8 Kadelburg and Radenovi [11] obtained a fixed point result without the normality of the underlying cone, but only in the case of a quasi-contractive constant λ(0,1/2) (see [[11], Theorem 2.2]). However, Ljiljana [7] proved the result is true for λ(0,1) on a cone metric space by a new way. Referring to this way, Theorem 2.6 presents a similar common fixed point result in the case of the contractive constant λ(0,1/s) in cone b-metric spaces without the assumption of normality. Moreover, it is obvious that Example 2.7 given above shows that Theorem 2.6 not only improves and generalizes [[11], Theorem 2.2], but also generalizes and unifies [[7], Theorem 3].

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Acknowledgements

The first author thanks Doctor Hao Liu for his help and encouragement. Besides, the authors are extremely grateful to the referees for their useful comments and suggestions which helped to improve this paper. The research is partially supported by the Foundation of Education Ministry, Hubei Province, China (No: D20102502).

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Shi, L., Xu, S. Common fixed point theorems for two weakly compatible self-mappings in cone b-metric spaces. Fixed Point Theory Appl 2013, 120 (2013). https://doi.org/10.1186/1687-1812-2013-120

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Keywords

  • common fixed point
  • weakly compatible self-mappings
  • (quasi-)contractive condition
  • cone b-metric space