We introduced a notion of topological vector space valued cone metric space and obtained some common fixed point results. Our results generalize some recent results in the literature.

1. Introduction

Huang and Zhang  generalized the notion of metric space by replacing the set of real numbers by ordered Banach space, deffined a cone metric space, and established some fixed point theorems for contractive type mappings in a normal cone metric space. Subsequently, several other authors  studied the existence of common fixed point of mappings satisfying a contractive type condition in normal cone metric spaces. Afterwards, Rezapour and Hamlbarani  studied fixed point theorems of contractive type mappings by omitting the assumption of normality in cone metric spaces (see also ). In this paper we obtain common fixed points for a pair of self-mappings satisfying a generalized contractive type condition without the assumption of normality in a class of topological vector space valued cone metric spaces which is bigger than that introduced by Huang and Zhang .

Let (E,τ) be always a topological vector space and P a subset of E. Then, P is called a cone whenever

P is closed, nonempty and P{0},

ax+byP for all x,yP and nonnegative real numbers a,b,

P(-P)={0}.

For a given cone PE, we can define a partial ordering with respect to P by xy if and only if y-xP.x<y will stand for xy and xy, while xy will stand for y-xintP, where intP denotes the interior of P.

Definition 1.1.

Let X be a nonempty set. Suppose that the mapping d:X×XE satisfies

0d(x,y) for all x,yX and d(x,y)=0 if and only if x=y,

d(x,y)=d(y,x) for all x,yX,

d(x,y)d(x,z)+d(z,y) for all x,y,zX.

Then d is called a cone metric on X and (X,d) is called a topological vector space valued cone metric space.

Note that Huang and Zhang  notion of cone metric space is a special case of our notion of topological vector space valued cone metric space.

Example 1.2.

Let X=[0,1],and let  E be the set of all real valued functions on X which also have continuous derivatives on    X,  then E    is a vector space over under the following operations: (f+g)(t)=f(t)+g(t),(αf)(t)=αf(t), for all f,gE,α.    Let τ    be the strongest vector (locally convex) topology on E,  then (X,τ) is a topological vector space which is not normable and is not even metrizable (see ). Define d:X×XE as follows: (d(x,y))(t)=|x-y|et,P={xE:x(t)0  tX}. Then (X,d)    is a topological vector space valued cone metric space.

Example 1.2 shows that this category of cone metric spaces is larger than that considered in  .

Definition 1.3.

Let (X,d) be a topological vector space valued cone metric space, and let xX and {xn}n1 be a sequence in X. Then

(i)  {xn}n1 converges to x whenever for every cE with 0c there is a natural number N such that d(xn,x)c for all nN. We denote this by limnxn=x or xnx.

(ii)  {xn}n1 is a Cauchy sequence whenever for every cE with 0c there is a natural number N such that d(xn,xm)c for all n,mN.

(iii)  (X,d) is a complete topological vector space valued cone metric space if every Cauchy sequence is convergent.

2. Fixed Point

In this section, we shall give some results which generalize [6, Theorems 2.3, 2.6, 2.7, and 2.8] (and so [1, Theorems 1, 3, and 4]).

Theorem 2.1.

Let (X,d) be a complete topological vector space valued cone metric space and let the self-mappings S,T:XX satisfy d(Sx,Ty)kd(x,y)+l(d(x,Ty)+d(y,Sx)), for all x,yX, where k,l[0,1) with k+2l<1. Then S and T have a unique common fixed point.

Proof.

For x0X and n0, define x2n+1=Sx2n and x2n+2=Tx2n+1. Then, d(x2n+1,x2n+2)=d(Sx2n,Tx2n+1)kd(x2n,x2n+1)+l[d(x2n,Tx2n+1)+d(x2n+1,Sx2n)]=kd(x2n,x2n+1)+l[d(x2n,Tx2n+1)]kd(x2n,x2n+1)+l[d(x2n,x2n+1)+d(x2n+1,x2n+2)]=[k+l]d(x2n,x2n+1)+ld(x2n+1,x2n+2). It implies that d(x2n+1,x2n+2)[(k+l)/(1-l)]d(x2n,x2n+1). Similarly, d(x2n+2,x2n+3)=d(Sx2n+2,Tx2n+1)kd(x2n+2,x2n+1)+l[d(x2n+2,Tx2n+1)+d(x2n+1,Sx2n+2)]kd(x2n+2,x2n+1)+l[d(x2n+2,x2n+3)+d(x2n+1,x2n+2)]=[k+l]d(x2n+1,x2n+2)+ld(x2n+2,x2n+3). Hence, d(x2n+2,x2n+3)[(k+l)/(1-l)]d(x2n+1,x2n+2). Thus, d(xn,xn+1)λnd(x0,x1), for all n0, where λ=((k+l)/(1-l))<1. Now, for    n>m we have d(xn,xm)d(xn,xn-1)+d(xn-1,xn-2)++d(xm+1,xm)(λn-1+λn-2++λm)d(x0,x1)λm1-λd(x0,x1). Let 0c. Take a symmetric neighborhood V of 0 such that c+VintP. Also, choose a natural number N1 such that (λm/(1-λ))d(x1,x0)V, for all mN1. Then, (λm/(1-λ))d(x1,x0)c, for all mN1. Thus, d(xn,xm)λm1-λd(x1,x0)c, for all n>m. Therefore, {xn}n1 is a Cauchy sequence in (X,d). Since X is complete, there exists uX such that xnu. Choose a natural number N2 such that d(xn,u)[c(1-l)/2(1+l)] for all nN2. Thus, d(u,Tu)d(u,x2n+1)+d(x2n+1,Tu)=d(u,x2n+1)+d(Sx2n,Tu)d(u,x2n+1)+kd(u,x2n)+l[d(u,Sx2n)+d(x2n,Tu)]d(u,x2n+1)+kd(u,x2n)+l[d(u,x2n+1)+d(x2n,u)+d(u,Tu)]=(1+l)d(u,x2n+1)+(k+l)d(u,x2n)+ld(u,Tu). So, d(u,Tu)[1+l1-l]d(u,x2n+1)+[k+l1-l]d(u,x2n)[1+l1-l]d(u,x2n+1)+[1+l1-l]d(u,x2n)=c2+c2=c, for all nN2. Therefore, d(u,Tu)c/i for all i1. Hence, (c/i)-d(u,Tu)P for all    i1. Since P is closed, -d(u,Tu)P and so d(u,Tu)=0. Hence, u is a fixed point of T. Similarly, we can show that u=Su. Now, we show that S and T have a unique fixed point. For this, assume that there exists another point u* in X such that u*=Tu*=Su*. Then, d(u,u*)=d(Su,Tu*)kd(u,u*)+l[d(u,Tu*)+d(u*,Su)]kd(u,u*)+l[d(u,u*)+d(u*,u)](k+2l)d(u,u*). Since k+2l<1,d(u,u*)=0 and so u=u*.

The following corollary generalizes [6, Theorems 2.3, 2.7, and 2.8] (and so [1, Theorems 1 and 4]).

Corollary 2.2.

Let (X,d) be a complete topological vector space valued cone metric space and let the self-mapping T:XX satisfy d(Tx,Ty)ad(x,y)+bd(x,Ty)+cd(y,Tx) for all x,yX, where a,b,c[0,1) with a+b+c<1. Then T has a unique fixed point.

Proof.

The symmetric property of d and the above inequality imply that d(Tx,Ty)ad(x,y)+b+c2[d(x,Ty)+d(y,Tx)]. By substituting S=Ta=k and (b+c)/2=l in Theorem 2.1, we obtain the required result.

Theorem 2.3.

Let (X,d) be a complete topological vector space valued cone metric space and let the self-mappings S,T:XX satisfy d(Sx,Ty)kd(x,y)+l(d(x,Sx)+d(y,Ty)), for all x,yX, where k,l[0,1) with k+2l<1. Then S and T have a unique common fixed point.

Proof.

For x0X and n0, define x2n+1=Sx2n and x2n+2=Tx2n+1. Then, d(x2n+1,x2n+2)=d(Sx2n,Tx2n+1)kd(x2n,x2n+1)+l[d(x2n,Sx2n)+d(x2n+1,Tx2n+1)]=kd(x2n,x2n+1)+l[d(x2n,x2n+1)+d(x2n+1,x2n+2)]=[k+l]d(x2n,x2n+1)+ld(x2n+1,x2n+2). It implies that d(x2n+1,x2n+2)[(k+l)/(1-l)]d(x2n,x2n+1). Similarly, d(x2n+2,x2n+3)=d(Sx2n+2,Tx2n+1)kd(x2n+2,x2n+1)+l[d(x2n+2,Sx2n+2)+d(x2n+1,Tx2n+1)]=kd(x2n+2,x2n+1)+l[d(x2n+2,x2n+3)+d(x2n+1,x2n+2)]=[k+l]d(x2n+1,x2n+2)+ld(x2n+2,x2n+3). Hence, d(x2n+2,x2n+3)[(k+l)/(1-l)]d(x2n+1,x2n+2). Thus, d(xn,xn+1)λnd(x0,x1), for all n0, where λ=((k+l)/(1-l))<1. Now, for    n>m we have d(xn,xm)d(xn,xn-1)+d(xn-1,xn-2)++d(xm+1,xm)(λn-1+λn-2++λm)d(x0,x1)λm1-λd(x0,x1). Let 0c. Take a symmetric neighborhood V of 0 such that c+VintP. Also, choose a natural number N1 such that (λm/(1-λ))d(x1,x0)V, for all mN1. Then, (λm/(1-λ))d(x1,x0)c, for all mN1. Thus, d(xn,xm)λm1-λd(x1,x0)c, for all n>m. Therefore, {xn}n1 is a Cauchy sequence in (X,d). Since X is complete, there exists uX such that xnu. Choose a natural number N2 such that d(xn,u)[c(1-l)/2(1+l)] for all nN2. Thus, d(u,Tu)d(u,x2n+1)+d(x2n+1,Tu)=d(u,x2n+1)+d(Sx2n,Tu)d(u,x2n+1)+kd(u,x2n)+l[d(u,Tu)+d(x2n,Sx2n)]d(u,x2n+1)+kd(u,x2n)+l[d(u,x2n+1)+d(x2n,u)+d(u,Tu)]=(1+l)d(u,x2n+1)+(k+l)d(u,x2n)+ld(u,Tu). So, d(u,Tu)[1+l1-l]d(u,x2n+1)+[k+l1-l]d(u,x2n)[1+l1-l]d(u,x2n+1)+[1+l1-l]d(u,x2n)c2+c2=c, for all nN2. Therefore, d(u,Tu)c/i for all i1. Hence, (c/i)-d(u,Tu)P for all i1. Since P is closed, -d(u,Tu)P and so d(u,Tu)=0. Hence, u is a fixed point of T. Similarly, we can show that u=Su. Now, we show that S and T have a unique fixed point. For this, assume that there exists another point u* in X such that u*=Tu*=Su*. Then, d(u,u*)=d(Su,Tu*)kd(u,u*)+l[d(u,u*)+d(u*,u)]=kd(u,u*). Since k<1,  d(u,u*)=0 and so u=u*.

The following corollary generalizes [6, Theorem 2.6] (and so [1, Theorem 3]).

Corollary 2.4.

Let (X,d) be a complete topological vector space valued cone metric space and let the self-mapping T:XX satisfy d(Tx,Ty)ad(x,y)+bd(x,Tx)+cd(y,Ty) for all x,yX, where a,b,c[0,1) with a+b+c<1. Then T has a unique fixed point.

Proof is similar to the proof of Corollary 2.2.

Example 2.5.

Let (X,d)    be a topological vector space valued cone metric space of Example 1.2. Define S,T:XX    as follows: S(t)=T(t)={t3ifx1,16ifx=1. Then, |Sx-Ty|etk|x-y|et+l[|x-Sx|et+|y-Ty|et], if k=1/6,l=5/18.    Hence all conditions of Theorem 2.3 are satisfied.

Acknowledgments

The present version of the paper owes much to the precise and kind remarks of the learned referees.