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Fcone metric spaces over Banach algebra
Fixed Point Theory and Applications volume 2017, Article number: 7 (2016)
Abstract
The paper deals with the achievement of introducing the notion of Fcone metric spaces over Banach algebra as a generalization of \(N_{p}\)cone metric space over Banach algebra and \(N_{b}\)cone metric space over Banach algebra and studying some of its topological properties. Also, here we define generalized Lipschitz and expansive maps for such spaces. Moreover, we investigate some fixed points for mappings satisfying such conditions in the new framework. Subsequently, as an application of our results, we provide an example. Our results generalize some wellknown results in the literature.
Introduction
Partial metric spaces were introduced by Matthews [1] in 1994. He studied a partial metric space as a part of the denotational semantics of dataflow networks and showed that the Banach contraction principle can be generalized to the partial metric context for applications in program verification. Especially, it has the property that differentiates it from other spaces, that is, the selfdistance of any point may not be zero, also a convergent sequence need not have unique limit in these spaces.
On the other hand, in 1989, the concept of bmetric spaces was introduced by Bakhtin [2] as a generalization of metric spaces. He showed the contraction mapping principle in a bmetric space that generalizes the prominent Banach contraction principle in metric spaces.
In the same spirit, recently, Huang and Zhang [3] replaced the set of real numbers by ordering Banach space and defined a cone metric space as a generalization of the metric space. The authors proved some fixed point theorems of contractive mappings on cone metric spaces. They also defined the Cauchy sequence and convergence of a sequence in such spaces in terms of interior points of the underlying cone. After that, in [4], Rezapour and Hamlbarani generalized some results of [5] by omitting the assumption of normality. For fixed point theorems on cone metric spaces, readers may refer to [6–9] and the references therein.
Malviya et al. [10] introduced the concept of Ncone metric spaces, which is a new generalization of the generalized Gcone metric space [11] and the generalized \(D^{*}\)metric space [12]. They proved fixed point theorems for asymptotically regular maps and sequences. Afterwards, in [13], the authors defined contractive maps in Ncone metric spaces and proved various fixed point theorems for such maps.
Despite these features, some authors demonstrated that the fixed point results proved on cone metric spaces are the straightforward outcome of the corresponding results of usual metric spaces where the realvalued metric function \(d^{*}\) is defined by a nonlinear scalarization function \(\xi_{e}\) (see [14]) or by a Minkowski functional \(q_{e}\) (see [5]).
Due to the concrete reasons mentioned above, researchers were losing their interest in a cone metric space. Fortunately, Liu and Xu [15] introduced the approach of cone metric spaces over Banach algebras by replacing Banach spaces E by Banach algebras A as the underlying spaces of cone metric spaces. They verified some fixed point theorems of generalized Lipschitz mappings with weaker conditions on the generalized Lipschitz constant k by means of the spectral radius. Not long ago, Xu and Radenović [16] deleted the normality of cones and greatly generalized the main results of [15]. In particular, some authors have shown recently some fixed point results given in [17–19].
Following these ideas, very recently, Fernandez et al. [6] introduced the notion of partial cone metric spaces over Banach algebra as a generalization of partial metric spaces and cone metric spaces over Banach algebra, which was selected for Young Scientist Award 2016, M.P., India (see [20]).
Recently, proceeding in this direction, Fernandez et al. introduced the structure of \(N_{p}\)cone metric space over Banach algebra [21] as a generalization of Ncone metric space over Banach algebra [22] and partial metric space and \(N_{b}\)cone metric space over Banach algebra [23] as a generalization of Ncone metric space over Banach algebra [22] and bmetric space, respectively.
Inspired and encouraged by the previous works, we present seven sections in this paper. For the reader’s convenience, we recall in Section 2 some definitions that will be used in the sequel. In Section 3, after the preliminaries, we introduce Fcone metric spaces over Banach algebra which generalize \(N_{p}\)cone metric spaces over Banach algebra and \(N_{b}\)cone metric spaces over Banach algebra. In Section 4, we discuss the topological properties. In Section 5, we introduce the notions of generalized Lipschitz and expansive maps. Section 6 is devoted to deriving the existence of fixed point theorems for such spaces by using the mentioned contractive conditions. Finally, in Section 7, we define expansive maps and investigate the existence and uniqueness of the fixed point in the new framework. Our main theorems extend and unify existing results in the recent literature. We also give illustrative examples that verify our results.
Preliminaries
We begin with the following definition as a recall from [15].
Let A always be a real Banach algebra. That is, A is a real Banach space in which an operation of multiplication is defined subject to the following properties (for all \(x, y, z\in A\), \(\alpha\in R\)):

1.
\((xy)z=x(yz)\);

2.
\(x(y + z) = xy + xz\) and \((x + y)z = xz + yz\);

3.
\(\alpha(xy) = (\alpha x)y = x(\alpha y)\);

4.
\(\Vert xy\Vert \leq \Vert x\Vert \Vert y\Vert \).
Throughout this paper, we shall assume that a Banach algebra has a unit (i.e., a multiplicative identity) e such that \(ex = xe = x\) for all \(x\in A \). An element \(x\in A\) is said to be invertible if there is an inverse element \(y\in A\) such that \(xy = yx = e\). The inverse of x is denoted by \(x^{1}\). For more details, we refer the reader to [24].
The following proposition is given in [24].
Proposition 2.1
Let A be a Banach algebra with a unit e, and \(x\in A\). If the spectral radius \(\rho(x)\) of x is less than 1, i.e.,
then \(ex\) is invertible. Actually,
Remark 2.2
From [24] we see that the spectral radius \(\rho(x)\) of x satisfies \(\rho(x)\leq \Vert x\Vert \) for all \(x\in A\), where A is a Banach algebra with a unit e.
Remark 2.3
see [16]
In Proposition 2.1, if the condition \({}^{\prime}\rho(x) < 1^{\prime}\) is replaced by \(\Vert x \Vert \leq1\), then the conclusion remains true.
Remark 2.4
see [16]
If \(\rho(x)< 1\), then \(\Vert x\Vert ^{n}\rightarrow0\) (\(n\rightarrow\infty\)).
Lemma 2.5
see [25]
If E is a real Banach space with a solid cone P and if \(\theta\preccurlyeq u \ll c\) for each \(\theta\ll c\), then \(u = \theta\).
A subset P of A is called a cone of A if

1.
P is nonempty closed and \(\{\theta, e\}\subset P\);

2.
\(\alpha P + \beta P\subset P\) for all nonnegative real numbers α, β;

3.
\(P^{2} = PP\subset P\);

4.
\(P \cap(P) = \{\theta\}\),
where θ denotes the null of the Banach algebra A. For a given cone \(P\subset A\), we can define a partial ordering ≼ with respect to P by \(x\preccurlyeq y\) if and only if \(y  x\in P. x \prec y\) will stand for \(x\preccurlyeq y\) and \(x\neq y\), while \(x\ll y\) will stand for \(yx\in \operatorname{int}P\), where intP denotes the interior of P. If \(\operatorname{int}P\neq\varnothing\), then P is called a solid cone.
The cone P is called normal if there is a number \(M > 0\) such that, for all \(x, y\in A\),
The least positive number satisfying the above is called the normal constant of P [3].
In the following we always assume that A is a Banach algebra with a unit e, P is a solid cone in A and ≼ is the partial ordering with respect to P.
Definition 2.6
Let X be a nonempty set. Suppose that the mapping \(d: X \times X\rightarrow A\) satisfies

1.
\(\theta\prec d(x, y)\) for all \(x, y\in X\) and \(d(x, y) = \theta\) if and only if \(x = y\);

2.
\(d(x, y) = d(y, x)\) for all \(x, y\in X\);

3.
\(d(x, y)\preccurlyeq d(x, z) + d(z, y)\) for all \(x, y, z\in X\).
Then d is called a cone metric on X, and \((X, d)\) is called a cone metric space over Banach algebra A.
For other definitions and related results on cone metric space over Banach algebra, we refer to [15].
Definition 2.7
[2]
Let X be a nonempty set and \(s\geq1\) be a given real number. A function \(d: X\times X\rightarrow R^{+}\) is a bmetric on X if, for all \(x, y, z\in X\), the following conditions hold:

1.
\(d(x,y) = 0\) if and only if \(x=y\);

2.
\(d(x,y) = d(y,x)\);

3.
\(d(x,z) \leq s [d(x,y)+d(y,z) ]\).
In this case, the pair \((X, d)\) is called a bmetric space.
For more definitions and results on bmetric spaces, we refer the reader to [26].
Definition 2.8
[1]
A partial metric on a nonempty set X is a function \(p:X\times X\rightarrow R^{+}\) such that for all \(x,y,z \in X\), the following conditions hold:

1.
\(x = y\Leftrightarrow p(x, x) = p(x, y) = p(y,y)\);

2.
\(p (x, x)\leq p(x, y)\);

3.
\(p(x, y)= p(y,x)\);

4.
\(p(x, y)\leq p(x, z) + p(z, y)  p (z, z)\).
The pair \((X, p)\) is called a partial metric space. It is clear that if \(p(x, y) = 0\), then from (1) and (2) \(x = y\). But if \(x = y\), \(p(x, y)\) may not be 0.
Definition 2.9
[10]
Let X be a nonempty set. A function \(N:X^{3}\rightarrow A\) is called Ncone metric on X if for any \(x,y,z,a\in X\), the following conditions hold:
 (\(N_{1}\)):

\(0\leq N(x,x,x)\);
 (\(N_{2}\)):

\(N(x,y,z) = \theta\) iff \(x = y = z\);
 (\(N_{3}\)):

\(N(x,y,z)\leq N(x,x,a)+N(y,y,a)+N(z,z,a)\).
Then the pair \((X,N)\) is called an Ncone metric space over Banach algebra A.
Definition 2.10
[23]
An \(N_{b}\)cone metric on a nonempty set X is a function \(N_{b}: X^{3}\rightarrow A\) such that for all \(x, y, z, a\in X\):
 (\(N_{b_{1}}\)):

\(\theta\preccurlyeq N_{b}(x,y,z)\);
 (\(N_{b_{2}}\)):

\(N_{b}(x,y,z) = \theta\) iff \(x = y = z\);
 (\(N_{b_{3}}\)):

\(N_{b}(x,y,z)\preccurlyeq s[N_{b}(x,x,a)+N_{b}(y,y,a)+ N_{b}(z,z,a)]\).
The pair \((X,N_{b})\) is called an \(N_{b}\)cone metric space over Banach algebra A. The number \(s\geq1\) is called the coefficient of \((X,N_{b})\).
Definition 2.11
[21]
An \(N_{p}\)cone metric on a nonempty set X is a function \(N_{p}: X^{3}\rightarrow A\) such that for all \(x, y, z, a\in X\):
 (\(N_{p}1\)):

\(x = y = z\Leftrightarrow N_{p}(x,x,x)=N_{p}(y,y,y)=N _{p}(z,z,z)=N_{p}(x,y,z)\);
 (\(N_{p}2\)):

\(\theta\preccurlyeq N_{p}(x,x,x)\preccurlyeq N_{p}(x,x,y) \preccurlyeq N_{p}(x,y,z)\), for all \(x,y,z \in X\) with \(x\neq y\neq z\);
 (\(N_{p}3\)):

\(N_{p}(x,y,z)\preccurlyeq N_{p}(x,x,a)+N_{p}(y,y,a)+N_{p}(z,z,a) N_{p}(a,a,a)\).
The pair \((X,N_{p})\) is called an \(N_{p}\)cone metric space over Banach algebra A.
Fcone metric space over Banach algebra
In this section, we define a new structure, i.e., Fcone metric spaces over Banach algebra.
Definition 3.1
Let X be a nonempty set. A function \(F:X^{3}\rightarrow A\) is called Fcone metric on X if for any \(x,y,z,a\in X\), the following conditions hold:
 (\(F_{1}\)):

\(x=y=z\) iff \(F(x,x,x)=F(y,y,y)=F(z,z,z)=F(x,y,z)\);
 (\(F_{2}\)):

\(\theta\preccurlyeq F(x,x,x)\preccurlyeq F(x,x,y) \preccurlyeq F(x,y,z)\), for all \(x,y,z\in X\) with \(x\neq y\neq z\);
 (\(F_{3}\)):

\(F(x,y,z)\preccurlyeq s [F(x,x,a)+F(y,y,a)+F(z,z,a) ]F(a,a,a)\).
Then the pair \((X,F)\) is called an Fcone metric space over Banach algebra A. The number \(s\geq1\) is called the coefficient of \((X,F)\).
Remark 3.2
In an Fcone metric space over Banach algebra \((X,F)\), if \(x,y,z\in X\) and \(F(x,y,z)=\theta\), then \(x=y=z\), but the converse may not be true.
Example 3.3
Let \(A=C_{1}^{R}[0,1]\) and define a norm on A by \(\Vert x\Vert = \Vert x\Vert _{\infty} + \Vert x^{\prime} \Vert _{\infty}\) for \(x\in A\). Define multiplication in A as just pointwise multiplication. Then A is a real unit Banach algebra with unit \(e=1\). Set \(P=\{x\in A: x\geq0 \}\) is a cone in A. Moreover, P is not normal (see [4]). Let \(X=[0,\infty)\). Define a mapping \(F:X^{3}\rightarrow A\) by \(F(x,y,z)(t)= ((\max \{x,z\})^{2} + (\max \{y,z\})^{2}, (\max \{x,z\})^{2} + (\max \{y,z\})^{2} )e^{t}\) for all \(x,y,z\in X\), and let \(\alpha> 0\) be any constant. Then \((X,F)\) is an Fcone metric space over Banach algebra A with the coefficient \(s=2\). But it is not an \(N_{p}\)cone metric space over Banach algebra since the triangle inequality is not satisfied; neither it is an \(N_{b}\)cone metric space over Banach algebra A since for any \(x > 0\), we have \(N_{b}(x,x,x)(t) = 2x^{2}.e^{t}\neq\theta\).
Lemma 3.4
Let \((X,F)\) be an Fcone metric space over Banach algebra A. Then

(a)
if \(F(x,y,z)=\theta\), then \(x=y=z\).

(b)
if \(x\neq y\), then \(F(x,x,y)>\theta\).
Proof
The proof is obvious. □
Proposition 3.5
If \((X,F)\) is an Fcone metric space over Banach algebra, then for all \(x, y, z \in X\), we have \(F(x,x,y)=F(y,y,x)\).
Definition 3.6
Let \((X,F)\) be an Fcone metric space over Banach algebra A. Then, for \(x\in X\) and \(c>\theta\), the Fballs with center x and radius \(c>\theta\) are
Topology on Fcone metric space over Banach algebra
In this section, we define the topology of Fcone metric space over Banach algebra and study its topological properties.
Definition 4.1
Let \((X,F)\) be an Fcone metric space over Banach algebra A with coefficient \(s\geq1\). For each \(x\in X\) and each \(\theta\ll c\), put \(B_{F}(x,c) = \{y\in X: F(x,x,y)\ll F(x,x,x)+c \}\) and put \(B=\{B_{F}(x,c): x\in X\mbox{ and }\theta\ll c\}\). Then B is a subbase for some topology τ on X.
Remark 4.2
Let \((X,F)\) be an Fcone metric space over Banach algebra A. In this paper, τ denotes the topology on X, B denotes a subbase for the topology on τ and \(B_{F}(x,c)\) denotes the Fball in \((X,F)\), which are described in Definition 4.1. In addition, U denotes the base generated by the subbase B.
Theorem 4.3
Let \((X,F)\) be an Fcone metric space over Banach algebra A, and let P be a solid cone in A. Let \(k\in P\) be an arbitrarily given vector, then \((X,F)\) is a Hausdorff space.
Proof
Let \((X,F)\) be an Fcone metric space over Banach algebra, and let \(x,y\in X\) with \(x\neq y\).
Let \(F(x,x,y)=c\).
Suppose \(U=B (x,\frac{c}{3} )\) and \(V=B (y,\frac {c}{3} )\).
Then \(x\in U\) and \(y\in V\).
We claim that \(U\cap V=\phi\).
If not, there exists \(z\in U\cap V\).
But then
So, we get
i.e., \(c < c\), which is a contradiction.
Hence \(U\cap V=\phi\) and X is a Hausdorff space.
Now, we define θCauchy sequence and convergent sequence in an Fcone metric space over Banach algebra A. □
Definition 4.4
Let \((X, F)\) be an Fcone metric space over Banach algebra A. A sequence \(\{x_{n}\}\) in \((X,F)\) converges to a point \(x\in X\) whenever for every \(c\gg\theta\) there is a natural number N such that \(F(x_{n}, x, x)\ll c\) for all \(n\geq N\). We denote this by
Definition 4.5
Let \((X, F)\) be an Fcone metric space over Banach algebra A. A sequence \(\{x_{n}\}\) in X is said to be a θCauchy sequence in \((X,F)\) if \(\{F(x_{n}, x_{m}, x_{m})\}\) is a csequence in A, i.e., if for every \(c\gg\theta\) there exists \(n_{0}\in N\) such that \(F(x_{n}, x_{m}, x_{m})\ll c\) for all \(n,m \geq n_{0}\).
Definition 4.6
Let \((X, F)\) be an Fcone metric space over Banach algebra A. Then X is said to be θcomplete if every θCauchy sequence \(\{x_{n}\}\) in \((X,F)\) converges to \(x\in X\) such that \(F(x, x, x)=\theta\).
Definition 4.7
Let \((X, F)\) and \((X^{\prime}, F^{\prime})\) be an Fcone metric space over Banach algebra A. Then a function \(f: X\rightarrow X^{\prime}\) is said to be continuous at a point \(x\in X\) if and only if it is sequentially continuous at x, that is, whenever \(\{x_{n}\}\) is convergent to x, we have \(\{fx_{n}\}\) is convergent to \(f(x)\).
Generalized Lipschitz maps
In this section, we define generalized Lipschitz maps in Fcone metric spaces over Banach algebra.
Definition 5.1
Let \((X,F)\) be an Fcone metric space over Banach algebra A and P be a cone in A. A map \(T:X\rightarrow X\) is said to be a generalized Lipschitz mapping if there exists a vector \(k\in P \) with \(\rho(k)< 1\) for all \(x,y\in X\) such that
Example 5.2
Let the Banach algebra A and the cone P be the same ones as those in Example 3.3, and let \(X=R^{+}\). Define a mapping \(F: X^{3}\rightarrow A\) as in Example 3.3. Then \((X,F)\) is an Fcone metric space over Banach algebra A. Now define the mapping \(T:X\rightarrow X\) by \(T(x)=\frac{x}{3} \cos\frac{x}{3}\). Since \(u \cos u\leq u\) for each \(u\in[0,\infty)\), for all \(x,y\in X\), we have
where \(k=\frac{1}{9}\). Clearly, T is a generalized Lipschitz map in X.
Now we review some facts on csequence theory.
Definition 5.3
[27]
Let P be a solid cone in a Banach space E. A sequence \(\{u_{n}\}\subset P\) is said to be a csequence if for each \(c\gg\theta\) there exists a natural number N such that \(u_{n}\ll c\) for all \(n > N\).
Lemma 5.4
[28]
If E is a real Banach space with a solid cone P and \(\{u_{n}\}\subset P\) is a sequence with \(\Vert u_{n}\Vert \rightarrow0\) (\(n\rightarrow\infty\)), then \(u_{n}\) is a csequence.
Lemma 5.5
[24]
Let A be a Banach algebra with a unit \(e, k\in A\), then \(\lim_{n\rightarrow\infty} \Vert k^{n}\Vert ^{\frac{1}{n}}\) exists and the spectral radius \(\rho(k)\) satisfies
If \(\rho(k) < \vert \lambda \vert \), then \((\lambda e  k)\) is invertible in A; moreover,
where λ is a complex constant.
Lemma 5.6
[24]
Let A be a Banach algebra with a unit \(e, a, b\in A\). If a commutes with b, then
Lemma 5.7
[28]
If E is a real Banach space with a solid cone P

(1)
If \(a, b, c\in E\) and \(a\leq b\ll c\), then \(a\ll c\).

(2)
If \(a\in P\) and \(a\ll c\) for each \(c\gg\theta\), then \(a = \theta\).
Lemma 5.8
[16]
Let P be a solid cone in a Banach algebra A. Suppose that \(k\in P\) and \(\{u_{n}\}\) is a csequence in P. Then \(\{ku_{n}\}\) is a csequence.
Lemma 5.9
[28]
Let A be a Banach algebra with a unit e and \(k\in A\). If λ is a complex constant and \(\rho(k) < \vert \lambda \vert \), then
Lemma 5.10
[28]
Let A be a Banach algebra with a unit e and P be a solid cone in A. Let \(a,k, l\in P\) hold \(l\preccurlyeq k\) and \(a\preccurlyeq la\). If \(\rho(k) < 1\), then \(a = \theta\).
Applications to fixed point theory
In this section, we prove some famous fixed point theorems satisfying generalized Lipschitz maps in the framework of Fcone metric space over Banach algebra A.
Theorem 6.1
Let \((X,F)\) be a θcomplete Fcone metric space over Banach algebra A and suppose that \(T:X\rightarrow X\) is a mapping satisfying the following condition:
where \(\rho(k)< 1\). Then T has a unique fixed point. For each \(x\in X\), the sequence of iterates \(\{T^{n}x\}_{n\geq1}\) converges to the fixed point.
Proof
For each \(x_{0}\in X\) and \(n\geq1\), set \(x_{1}=Tx _{0}\) and \(x_{n+1}=T^{n+1}x_{0}\). Then
So, for \(m>n\),
By Remark 2.4, \(\Vert (sk)^{n}\cdot F(x_{0},x_{0},x_{1}) \Vert \leq \Vert (sk)^{n}\Vert \Vert F(x_{0},x_{0},x_{1})\Vert \rightarrow0\). By Lemma 5.4, we have \(\{(sk)^{n}F(x_{0},x_{0},x_{1}) \}\) is a csequence. Next, by using Lemmas 5.7 and 5.8, we conclude that \(\{x_{n}\}\) is a θCauchy sequence in X.
By the θcompleteness of X, there exists \(u\in X\) such that
Furthermore, one has
Now that \(\{F(u,u,x_{n+1})\}\) and \(\{F(u,u,x_{n})\}\) are csequences, by using Lemmas 5.7 and 5.8, we conclude that \(Tu=u\). Thus u is a fixed point of T.
Finally, we prove the uniqueness of the fixed point. In fact, if v is another fixed point,
That is,
Multiplying both sides above by
we get \(F(u,u,v)\preccurlyeq\theta\). Thus, \(F(u,u,v)=\theta\), which implies that \(u=v\), a contradiction.
Hence, the fixed point is unique. □
Corollary 6.2
Let \((X,F)\) be a θcomplete Fcone metric space over Banach algebra A. Suppose that a mapping \(T:X\rightarrow X\) satisfies, for some positive integer n,
for all \(x,y\in X\), where k is a vector with \(\rho(k)< \frac{1}{s}\). Then T has a unique fixed point in X.
Proof
From Theorem 6.1, \(T^{n}\) has a unique fixed point \(x^{*}\). But \(T^{n}(Tx^{*}) = T(T^{n}x^{*}) = Tx^{*}\). So, \(Tx^{*}\) is also a fixed point of \(T^{n}\). Hence \(Tx^{*} = x^{*}\), \(x^{*}\) is a fixed point of T. Since the fixed point of T is also a fixed point of \(T^{n}\), then the fixed point of T is unique. □
We now prove Chatterjee’s fixed point theorem in the new space.
Theorem 6.3
Let \((X,F)\) be a θcomplete Fcone metric space over a Banach algebra A, and let P be the underlying cone with \(k\in P\) with \(\rho(k)< \frac{1}{s+1}\). Suppose that a mapping \(T:X\rightarrow X\) satisfies the generalized Lipschitz condition
for all \(x,y\in X\). Then T has a unique fixed point in X. And for any \(x\in X\), the iterative sequence \(\{T^{n}x\}\) converges to the fixed point.
Proof
Let \(x_{0}\in X\) be arbitrarily given and set \(x_{n}=T^{n}x\), \(n\geq1\). We have
which implies
Note that \(\rho(k)< (s+1)\rho(k)<1\).
Then by Lemma 5.5 it follows that \((ek)\) is invertible.
Multiplying both sides of (6.3.1) by \((ek)^{1}\), we get
As is shown in the proof of Theorem 6.1, \(\{x_{n}\}\) is a θCauchy sequence, and by the θcompleteness of X, there exists \(z\in X\) such that
Put \(h=(ek)^{1}\cdot k\), \(\therefore F(x_{n+1},x_{n+1},x_{n})\preccurlyeq hF(x_{n1},x_{n1},x _{n})\)
We shall show that z is a fixed point of T.
We have
which implies that
Since \(\rho(sk)< (s+1)\rho(k)<1\), it concludes by Lemma 5.5 that \((esk)\) is invertible. So,
Now that \(\{F(z,z,x_{n+1})\}\) and \(\{F(x_{n+1},x_{n+1},x_{n})\}\) are csequences, then by Lemmas 5.7 and 5.8, it concludes that \(Tz=z\). Then z is a fixed point of T.
Finally, we prove the uniqueness of the fixed point. In fact, if v is another fixed point, then
So, \(F(v,v,z)\preccurlyeq\theta\), which is a contradiction. Hence \(F(v,v,z)=\theta\).
So, \(v=z\).
Hence the fixed point is unique. □
Example 6.4
Let the Banach algebra A and the cone P be the same ones as those in Example 3.3, and let \(X=R^{+}\). Define a mapping \(F: X^{3}\rightarrow A\) as in Example 3.3. We make a conclusion that \((X,F)\) is a θcomplete Fcone metric space over Banach algebra A. Now define the mapping \(T:X\rightarrow X\) by \(T(x)= \cos\frac{x}{2}1\).
Then
where \(k=\frac{1}{4}\), then all the conditions of Theorem 6.1 hold trivially good and 0 is the unique fixed point of T. Clearly, T is a generalized Lipschitz map in X.
Expansive mapping on Fcone metric space over Banach algebra
In this section, we define expansive maps in Fcone metric spaces over Banach algebra.
Definition 7.1
Let \((X,F)\) be an Fcone metric space over Banach algebra A and P be a cone in A. A map \(T:X\rightarrow X\) is said to be an expansive mapping where \(k,k^{1}\in P\) are called the generalized Lipschitz constants with \(\rho(k^{1})<1\) for all \(x,y\in X\) such that
Example 7.2
Let the Banach algebra A and the cone P be the same ones as those in Example 3.3, and let \(X=R^{+}\). Define a mapping \(F: X^{3}\rightarrow A\) as in Example 3.3. Then \((X,F)\) is an Fcone metric space over Banach algebra A. Now define the mapping \(T:X\rightarrow X\) by \(T(x)=2x\). Then, for all \(x,y\in X\), we have
where \(k=4\). Clearly, T is an expansive map in X.
Now we present a fixed point theorem for such maps.
Theorem 7.3
Let \((X,F)\) be a θcomplete Fcone metric space over Banach algebra, and let P be an underlying solid cone with \(k\in P\) with \(\rho((e+k+ska)(b+c2sk)^{1})<\frac{1}{s}\). Let f and g be two surjective selfmaps of X satisfying
for all \(x,y\in X\), and then f and g have a unique common fixed point in X.
Proof
We define a sequence \(x_{n}\) as follows for \(n=0,1,2,3,\ldots \) :
If \(x_{2n} = x_{2n+1} = x_{2n+2}\) for some n, then we say that \(x_{2n}\) is a fixed point of f and g. Therefore, we suppose that no two consecutive terms of the sequence \(\{x_{n}\}\) are equal.
Now, putting \(x=x_{2n+1}\) and \(y=x_{2n+2}\) in (7.3.1), we get
Put \(b+c2sk=r\), then
Since r is invertible, to multiply \(r^{1}\) on both sides of (7.3.2),
where \(h=(e+k+ska)(b+c2sk)^{1}\).
Note that \(\rho(h)<\frac{1}{s}\).
Hence by the proof of Theorem 6.1, we can easily see that the sequence \(\{x_{n}\}\) is a θCauchy sequence. Moreover, by the θcompleteness of X, there exists \(x^{*}\in X\) such that
Since f and g are surjective maps and hence there exist two points y and \(y^{\prime}\) in X such that \(x^{*}=fy\) and \(x^{*}=gy ^{\prime}\).
Consider
Then
Since
so
which implies
Since \(2sb+csk = r\) is invertible, we have
Now that \(\{F(x_{2n+1},x_{2n+1},x^{*})\}\) and \(\{F(x_{2n+1},x_{2n+1},x _{2n})\}\) are csequences, then by using Lemmas 5.7 and 5.8, we conclude that \(gx_{n+2} = y^{\prime}\).
Finally, we prove the uniqueness of the fixed point. In fact, if \(y^{*}\) is another common fixed point of f and g, that is, \(fy^{*}=y^{*}\) and \(gy^{*}=y^{*}\),
or
which means \(F(x,x,y^{*})=\theta\), which implies that \(x=y^{*}\), a contradiction. Hence the fixed point is unique. □
Corollary 7.4
Let \((X, F)\) be a θcomplete Fcone metric space over Banach algebra, and let P be an underlying solid cone, where \(c\in P\) is a generalized Lipschitz constant with \(\rho(c)^{1}< \frac{1}{s}\). Let f and g be two surjective selfmaps of X satisfying
Then f and g have a unique common fixed point in X.
Proof
If we put \(k,a,b=\theta\) in Theorem 7.3, then we get the above Corollary 7.4. □
Corollary 7.5
Let \((X, F)\) be a θcomplete Fcone metric space over Banach algebra, and let P be an underlying solid cone, where \(c\in P\) is a generalized Lipschitz constant with \(\rho(c)^{1}< \frac{1}{s}\). Let f be a surjective selfmap of X satisfying
Then f has a unique fixed point in X.
Proof
If we put \(f=g\) in Corollary 7.4, then we get the above Corollary 7.5 which is an extension of Theorem 1 of Wang et al. [29] in an Fcone metric space over Banach algebra. □
Corollary 7.6
Let \((X, F)\) be a θcomplete Fcone metric space over Banach algebra, and let P be an underlying solid cone, where \(c\in P\) is a generalized Lipschitz constant with \(\rho(c)^{1}< \frac{1}{s}\). Let f be a surjective selfmap of X, and suppose that there exists a positive integer n satisfying
Then f has a unique fixed point in X.
Proof
From Corollary 7.5 \(f^{n}\) has a unique fixed point z. But \(f^{n} (fz)=f(f^{n} z)=fz\), so fz is also a fixed point of \(f^{n}\). Hence \(fz=z\), z is a fixed point of f. Since the fixed point of f is also a fixed point of \(f^{n}\), the fixed point of f is unique. □
Corollary 7.7
Let \((X, F)\) be a θcomplete Fcone metric space over Banach algebra, and let P be an underlying solid cone, where \(a, b, c, a\in P\) are generalized Lipschitz constants with \(\rho [(e  a) (b +c)^{1} ] < \frac{1}{s}\). Let f and g be two surjective selfmaps of X satisfying
Then f and g have a unique common fixed point in X.
Proof
If we put \(k=\theta\) in Theorem 7.3, then we get the above Corollary 7.7. □
Corollary 7.8
Let \((X, F)\) be a θcomplete Fcone metric space over Banach algebra, and let P be an underlying solid cone, where \(a, b, c, a\in P\) are generalized Lipschitz constants with \(\rho [(e  a) (b +c)^{1} ] < \frac{1}{s}\). Let f be a surjective selfmap of X satisfying
Then f has a unique fixed point in X.
Proof
If we put \(f=g\) in Corollary 7.7, then we get the above Corollary 7.8 which is an extension of Theorem 2 of Wang et al. [29] in an Fcone metric space over Banach algebra. □
Conclusion
In this paper, we introduce an Fcone metric space over Banach algebra which generalizes an \(N_{p}\)cone metric space over Banach algebra and an \(N_{b}\)cone metric space over Banach algebra. We introduce the concept of generalized Lipschitz and expansive mapping in the new structure. Also we derive the existence and uniqueness of some fixed point theorems for such spaces. Our main theorems extend and unify the existing results in the recent literature. Example is constructed to support our result.
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Fernandez, J., Malviya, N., Radenovič, S. et al. Fcone metric spaces over Banach algebra. Fixed Point Theory Appl 2017, 7 (2016). https://doi.org/10.1186/s1366301706005
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MSC
 46B20
 46B40
 46J10
 54A05
 47H10
Keywords
 Fcone metric space over Banach algebra
 csequence
 generalized Lipschitz mapping
 expansive mapping
 fixed point