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Common fixed point theorems for two weakly compatible selfmappings in cone bmetric spaces
Fixed Point Theory and Applications volume 2013, Article number: 120 (2013)
Abstract
In this paper, we establish common fixed point theorems for two weakly compatible selfmappings satisfying the contractive condition or the quasicontractive condition in the case of a quasicontractive constant \lambda \in (0,1/s) in cone bmetric spaces without the normal cone, where the coefficient s satisfies s\ge 1. The main results generalize, extend and unify several wellknown 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 [2–5]. Also, Ilic and Rakocevic [6] introduced a quasicontraction 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 quasicontractive constant \lambda \in (0,1/2). However, Gajic [7] proved that result is true for \lambda \in (0,1) on a cone metric space by a new way, which answered the open question whether the result is true for \lambda \in (0,1). Recently, Hussain and Shah [8] introduced cone bmetric spaces, as a generalization of bmetric 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 bmetric spaces. Although Ion Marian [10] proved some common fixed point theorems in complete bcone metric spaces, the main ways of the proof depend strongly on the nonlinear scalarization function {\xi}_{e}:Y\to \mathbb{R}. In the present paper, we will show common fixed point theorems for two weakly compatible selfmappings satisfying the contractive condition or quasicontractive condition in the case of a quasicontractive constant \lambda \in (0,1/s) in cone bmetric spaces without the assumption of normality, where the coefficient s satisfies s\ge 1. As consequences, our results generalize, extend and unify several wellknown comparable results (see, for example, [2–7, 9–13]).
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:

(i)
P is closed, nonempty, and P\ne \{\theta \};

(ii)
a,b\in \mathbb{R}, a,b\ge 0, x,y\in P\Rightarrow ax+by\in P;

(iii)
P\cap (P)=\{\theta \}.
On this basis, we define a partial ordering ⪯ with respect to P by x\u2aafy if and only if yx\in P. We write x\prec y to indicate that x\u2aafy but x\ne y, while x\ll y stands for yx\in intP. Write \parallel \cdot \parallel as the norm on E. The cone P is called normal if there is a number K>0 such that for all x,y\in E, \theta \u2aafx\u2aafy implies \parallel x\parallel \le K\parallel y\parallel. The least positive number satisfying the above is called the normal constant of P. It is well known that K\ge 1.
In the following, we always suppose that E is a Banach space, P is a cone in E with intP\ne \mathrm{\varnothing} and ⪯ is a partial ordering with respect to P.
Definition 1.1 [8]
Let X be a nonempty set and let s\ge 1 be a given real number. A mapping d:X\times X\to E is said to be cone bmetric if and only if for all x,y,z\in X the following conditions are satisfied:

(i)
\theta \prec d(x,y) with x\ne y and d(x,y)=\theta if and only if x=y;

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

(iii)
d(x,y)\u2aafs[d(x,z)+d(z,y)].
The pair (X,d) is called a cone bmetric space.
Example 1.2 Consider the space {L}_{p} (0<p<1) of all real function x(t) (t\in [0,1]) such that {\int}_{0}^{1}{x(t)}^{p}\phantom{\rule{0.2em}{0ex}}dt<\mathrm{\infty}. Let X={L}_{p}, E={\mathbb{R}}^{2}, P=\{(x,y)\in E\mid x,y\ge 0\}\subset {\mathbb{R}}^{2} and d:X\times X\to E such that
where \alpha ,\beta \ge 0 are constants. Then (X,d) is a cone bmetric space with the coefficient s={2}^{\frac{1}{p}1}.
Remark 1.3 It is obvious that any cone metric space must be a cone bmetric space. Moreover, cone bmetric spaces generalize cone metric spaces, bmetric spaces and metric spaces.
Definition 1.4 [8]
Let (X,d) be a cone bmetric space, x\in X and \{{x}_{n}\} be a sequence in X. Then

(i)
\{{x}_{n}\} converges to x whenever, for every c\in E with \theta \ll c, there is a natural number N such that d({x}_{n},x)\ll c for all n\ge N. We denote this by {lim}_{n\to \mathrm{\infty}}{x}_{n}=x or {x}_{n}\to x (n\to \mathrm{\infty}).

(ii)
\{{x}_{n}\} is a Cauchy sequence whenever, for every c\in E with \theta \ll c, there is a natural number N such that d({x}_{n},{x}_{m})\ll c for all n,m\ge N.

(iii)
(X,d) is a complete cone bmetric space if every Cauchy sequence is convergent.
Lemma 1.5 [8]
Let (X,d) be a cone bmetric space. The following properties are often used while dealing with cone bmetric spaces in which the cone is not necessarily normal.

(1)
If u\ll v and v\u2aafw, then u\ll w;

(2)
If \theta \u2aafu\ll c for each c\in intP, then u=\theta;

(3)
If a\u2aafb+c for each c\in intP, then a\u2aafb;

(4)
If \theta \u2aafd({x}_{n},x)\u2aaf{b}_{n} and {b}_{n}\to \theta, then {x}_{n}\to x;

(5)
If a\u2aaf\lambda a, where a\in P and 0<\lambda <1, then a=\theta;

(6)
If c\in intP, \theta \u2aaf{a}_{n} and {a}_{n}\to \theta, then there exists {n}_{0}\in \mathbb{N} such that {a}_{n}\ll c for all n>{n}_{0}.
Lemma 1.6 [8]
The limit of a convergent sequence in a cone bmetric space is unique.
Definition 1.7 [2]
The mappings f,g:X\to X are weakly compatible if for every x\in X, fgx=gfx holds whenever fx=gx.
Definition 1.8 [3]
Let f and g be selfmaps 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 selfmaps 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:X\to X is such that, for some constant \lambda \in (0,1) and for every x,y\in X, there exists an element
for which d(fx,fy)\u2aaf\lambda u is called a gquasicontraction.
2 Main results
In this section, we give some common fixed point results for two weakly compatible selfmappings satisfying the contractive condition and quasicontractive condition in the case of a contractive constant \lambda \in (0,1/s) in cone bmetric spaces without the assumption of normality.
Theorem 2.1 Let (X,d) be a cone bmetric space with the coefficient s\ge 1 and let {a}_{i}\ge 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:X\to X satisfy the condition, for all x,y\in X,
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}\in X, since f(X)\subset g(X), there exists an {x}_{1}\in 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} (n\ge 1). 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}_{n1}\ne g{x}_{n} for all n\ge 1.
Thus, by (2.1) for any n\in \mathbb{N}, we have
and
Hence
Since 2s{a}_{1}+(s+1)({a}_{2}+{a}_{3})+({s}^{2}+s)({a}_{4}+{a}_{5})<2, we have
where k=\frac{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\in [0,\frac{1}{s}).
Thus, setting any positive integers m and n, we have
Since k\in [0,1/s), we notice that \frac{s{k}^{n}}{1sk}d(g{x}_{1},g{x}_{0})\to \theta as n\to \mathrm{\infty} for any m\in {\mathbb{N}}_{+}. By Lemma 1.5, for any c\in intP, we can choose {n}_{0}\in \mathbb{N} such that \frac{s{k}^{n}}{1sk}d(g{x}_{1},g{x}_{0})\ll c for all n>{n}_{0}. Thus, for each c\in intP, d(g{x}_{n+m},g{x}_{n})\ll c for all n>{n}_{0}, m\ge 1. Therefore \{g{x}_{n}\} is a Cauchy sequence in g(X).
If g(X)\subset X is complete, there exist q\in g(X) and p\in X such that g{x}_{n}\to q as n\to \mathrm{\infty} and gp=q. (If f(X) is complete, there exists q\in f(X) such that f{x}_{n}\to q as n\to \mathrm{\infty}. Since f(X)\subset g(X), we can find p\in X such that gp=q.)
Now, from (2.1) we show that fp=q,
Similarly,
thus, we have
Since 0\le {a}_{2}+{a}_{3}+{a}_{4}+{a}_{5}<2/s, by the triangular inequality, it follows that
Since \{g{x}_{n}\} is a Cauchy sequence and g{x}_{n}\to q (n\to \mathrm{\infty}), for any c\in intP, we can choose {n}_{1}\in \mathbb{N} such that for all n\ge {n}_{1},
and
Thus, for any c\in intP, d(g{x}_{n+2},fp)\ll c for all n\ge {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.
Since 0\le {a}_{1}+{a}_{4}+{a}_{5}<1, by Lemma 1.5, we can obtain that d(gu,gp)=\theta, 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}_{\mathbb{R}}^{1}([0,1]), P=\{\phi \in E:\phi \ge 0\}\subset E, X=[1,\mathrm{\infty}) and d(x,y)={xy}^{2}{e}^{t}. Then (X,d) is a cone bmetric space with the coefficient s=2, but it is not a cone metric space. We consider the functions f,g:X\to X defined by fx=\frac{1}{6}lnx+1, gx=lnx+1. Hence
Here 1\in X 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 {\int}_{0}^{1}{u(x)}^{2}\phantom{\rule{0.2em}{0ex}}dx<\mathrm{\infty}, E={C}_{\mathbb{R}}([0,1]), P=\{\phi \in E:\phi \ge 0\}\subset E. We define d:X\times X\to E as
for all x,y\in X. Then (X,d) is a cone bmetric space with the coefficient s=2, but it is not a cone metric space. Considering the functions fu=\frac{1}{4}u(t) and gu=\frac{1}{2}u(t) (t\in [0,1]), we have
Clearly, 0\in X 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 bcone metric spaces in [10] and the fixed point results in cone bmetric 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 bmetric space with the coefficient s\ge 1. A mapping f:X\to X is such that, for some constant \lambda \in (0,1/s) and for every x,y\in X, there exists an element
for which d(fx,fy)\u2aaf\lambda u is called a gquasicontraction.
Theorem 2.6 Let (X,d) be a cone bmetric space with the coefficient s\ge 1 and let the mapping f:X\to X be a gquasicontraction. 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}\in X, set g{x}_{1}=f{x}_{0} and g{x}_{n+1}=f{x}_{n} (n\in \mathbb{N}). 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}_{n1}\ne g{x}_{n} for all n\ge 1. Now we prove that \{g{x}_{n}\} is a Cauchy sequence. First, we show that
Clearly, we note (2.3) holds when n=1. We assume that (2.3) holds for some n\le N1 (N\in {\mathbb{N}}_{+}), then we prove that (2.3) holds for all n=N. Because f is a gquasicontractive mapping, there exists a real number k\le N such that
In order to prove that (2.4) holds, we show that for all 1\le i,j\le N, there exists 1\le k\le N such that
Clearly, (2.5) is true for N=1. Suppose that (2.5) is true for each N=P\in \mathbb{N}, that is, for all 1\le i,j\le P, there exists 1\le k\le P such that
Let us prove (2.5) holds for N=P+1.
By (2.6), we only show that for any 1\le {i}_{0}\le P+1, there exists 1\le k\le P+1 such that
Since f is a gquasicontractive mapping, there exists
such that d(g{x}_{P+1},g{x}_{{i}_{0}})\u2aaf\lambda {\nu}_{{i}_{0}}.
By (2.6), we discuss that there exists an element
such that d(g{x}_{P+1},g{x}_{{i}_{0}})\u2aaf\lambda d(g{x}_{P+1},g{x}_{{i}_{1}}) (1\le {i}_{1}\le P+1).
If the above inequality does not hold for 1\le {i}_{1}\le 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\le {i}_{j}\le P+1 (0\le j\le P+1) such that
Notice that there exist 0\le r<s\le P+1 such that {i}_{r}={i}_{s}. That is,
As \lambda \in (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 N\in \mathbb{N}, which implies that (2.4) holds for N\in \mathbb{N}.
Next, let us prove that for all n\in {\mathbb{N}}_{+},
Using the triangular inequality, from (2.3) we obtain
Now, we show that \{g{x}_{n}\} is a Cauchy sequence. For all n>m, there exists
such that d(g{x}_{m},g{x}_{n})=d(f{x}_{m1},f{x}_{n1})\u2aaf\lambda {\nu}_{1}.
By the contractive condition, there exist but not all
such that
In fact, from (2.8) we have
Let {\nu}_{1}=d(g{x}_{i},g{x}_{j})=d(f{x}_{i1},f{x}_{j1})\u2aaf\lambda {\nu}_{2}, where
In general, if there exists
then we have
such that {\nu}_{k}=d(g{x}_{i},g{x}_{j})=d(f{x}_{i1},f{x}_{j1})\u2aaf\lambda {\nu}_{k+1} (1\le k\le m1).
As
we can obtain (2.9).
Using the triangular inequality, we get
so we obtain
Since \frac{2{s}^{2}{\lambda}^{m}}{1s\lambda}d(g{x}_{1},g{x}_{0})\to \theta as m\to \mathrm{\infty}, by Lemma 1.5, it is easy to see that for any c\in intP, there exists {n}_{0}\in \mathbb{N} such that for all n>m>{n}_{0},
So, \{g{x}_{n}\} is a Cauchy sequence in g(X). If g(X)\subset X is complete, there exist q\in g(X) and p\in X such that g{x}_{n}\to q as n\to \mathrm{\infty} and g(p)=q.
Now, from (2.2) we get
such that d(f{x}_{n},fp)\u2aaf\lambda \nu.
We have the following five cases:

(1)
d(f{x}_{n},fp)\u2aaf\lambda d(g{x}_{n},gp)\u2aafs\lambda d(g{x}_{n+1},gp)+s\lambda d(g{x}_{n+1},g{x}_{n});

(2)
d(f{x}_{n},fp)\u2aaf\lambda d(g{x}_{n},f{x}_{n})=\lambda d(g{x}_{n},g{x}_{n+1});

(3)
d(f{x}_{n},fp)\u2aaf\lambda d(gp,fp)\u2aafs\lambda d(g{x}_{n+1},gp)+s\lambda d(g{x}_{n+1},fp), that is, d(f{x}_{n},fp)\u2aaf\frac{s\lambda}{1s\lambda}d(g{x}_{n+1},gp);

(4)
d(f{x}_{n},fp)\u2aaf\lambda d(g{x}_{n},fp)\u2aafs\lambda d(g{x}_{n+1},fp)+s\lambda d(g{x}_{n+1},g{x}_{n}), that is, d(f{x}_{n},fp)\u2aaf\frac{s\lambda}{1s\lambda}d(g{x}_{n+1},g{x}_{n});

(5)
d(f{x}_{n},fp)\u2aaf\lambda d(f{x}_{n},gp)=\lambda d(g{x}_{n+1},gp).
As \frac{s\lambda}{1s\lambda}>s\lambda, then we obtain that
Since g{x}_{n}\to q as n\to \mathrm{\infty}, for any c\in intP, there exists {n}_{1}\in \mathbb{N} such that for all n>{n}_{1},
By Lemmas 1.5 and 1.6, we have g{x}_{n}\to fp as n\to \mathrm{\infty} and q=fp.
Now, if w is another point such that gu=fu=w, then
where \lambda \in (0,\frac{1}{s}) and
It is obvious that d(w,q)=\theta, 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=\mathbb{R}, E={C}_{\mathbb{R}}^{1}[0,1] and P=\{f\in E:f\ge 0\}. Define d:X\times X\to E by d(x,y)={xy}^{\frac{3}{2}}\phi where \phi :[0,1]\to \mathbb{R} such that \phi (t)={e}^{t}. It is easy to see that (X,d) is a cone bmetric space with the coefficient s={2}^{\frac{1}{2}}, but it is not a cone metric space. The mappings f,g:X\to X are defined by fx=\alpha x and gx=\sqrt{\alpha}x (\alpha \in [\frac{1}{\sqrt[3]{8}},\frac{1}{\sqrt[3]{4}})). The mapping f is a gquasicontraction with the constant \lambda ={\alpha}^{\frac{3}{4}}\in [\frac{1}{2},\frac{\sqrt{2}}{2}). Moreover, 0\in X 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 quasicontractive constant \lambda \in (0,1/2) (see [[11], Theorem 2.2]). However, Ljiljana [7] proved the result is true for \lambda \in (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 \lambda \in (0,1/s) in cone bmetric 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 selfmappings in cone bmetric spaces. Fixed Point Theory Appl 2013, 120 (2013). https://doi.org/10.1186/168718122013120
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DOI: https://doi.org/10.1186/168718122013120
Keywords
 common fixed point
 weakly compatible selfmappings
 (quasi)contractive condition
 cone bmetric space