F-cone metric spaces over Fréchet algebra

Abstract

The paper deals with the achievements of introducing the notion of F-cone metric spaces over Fréchet algebra as a generalization of F-cone metric spaces over a Banach algebra, $N_p$-cone metric spaces over a Banach algebra, and $N_b$-cone metric spaces over a Banach algebra. First, we study some of its topological properties. Next, we define a generalized Lipschitz for such spaces. Also, 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 work generalizes some well-known results in the literature.

Keywords:

• F-cone metric spaces over Fréchet algebra
• c-sequence
• generalized Lipschitz mapping
• fixed point

Subjects:

• 46B20
• 46B40
• 46J10
• 54A05
• 47H10

PUBLIC INTEREST STATEMENT

Metric fixed point theory is a powerful tool for solving several problems in various parts of mathematics and its applications. Recently, Fernandez et al. introduced F-cone metric spaces over a Banach algebra, which generalize N_p-cone metric spaces over the Banach algebra and -cone metric spaces over the Banach algebra. Now, in this paper, we introduce the notion of FN_p-cone metric spaces over a Frechet algebra as a generalization of F-cone metric spaces over the Banach algebra, N_p-cone metric spaces over the Banach algebra, and N_b-cone metric space over the Banach algebra, and we study some of its topological properties. Next, we define a generalized Lipschitz for such spaces. Also, we investigate some fixed points for mappings satisfying such conditions in the new framework.

1. Introduction

Malviya and Fisher (2013) introduced the concept of $N$-cone metric spaces, which is a new generalization of the generalized $G$-cone metric (Ismat et al., 2010) and the generalized $D^{\ast }$-metric spaces (Aage & Salunke, 2010). Afterwards, Malviya and Chouhan (2013) defined contractive maps in $N$-cone metric spaces and proved various fixed point theorems for such maps.

Recently, Fernandez et al. (0000) introduced the structure of $N_p$-cone metric spaces over a Banach algebra as a generalization of $N$-cone metric space over the Banach algebra (Fernandez et al., 0002) and partial metric spaces and defined $N_b$-cone metric spaces over a Banach algebra (Fernandez et al., 0001) as a generalization of $N$-cone metric space over the Banach space (Fernandez et al., 0002) and $b$-metric space, respectively.

Following these ideas, very recently, Fernandez et al. (2017) introduced $F$-cone metric spaces over a Banach algebra, which generalize $N_p$-cone metric spaces over the Banach algebra and $N_b$-cone metric spaces over the Banach algebra.

Now, in this paper, we introduce the notion of $F$-cone metric spaces over a Fréchet algebra as a generalization of $F$-cone metric spaces over the Banach algebra, $N_p$-cone metric spaces over the Banach algebra, and $N_b$-cone metric space over the Banach algebra, and we study some of its topological properties. Next, we define a generalized Lipschitz for such spaces. Also, 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 work is a generalization of some well-known results in the literature (see Fernandez et al., 2017, 0000, 0001, 0002).

2. Preliminaries

Throughout this paper, the notations $\mathbb{R},$ ${\mathbb{R}}_{+}$, and $\mathbb{N}$ denote the set of all real numbers, the set of all nonnegative real numbers, and the set of all positive integers, respectively.

Let $\mathbb{A}$ be a real Hausdorff topological vector space ($tvs$ for short) with the zero vector $\hat{I},$. A proper nonempty and closed subset $P$ of $\mathbb{A}$ is called a cone if $P+P\subset P$, $\lambda P\subset P$ for $\lambda \ge 0$ and $P\cap (-P)=\theta$. We always assume that the cone P has a nonempty interior, int $P$ such cones are called solid. Each cone $P$ induces a partial order $\underset{\_}{\prec }$ on $\mathbb{A}$ by $x\underset{\_}{\prec }y\iff y-x\in P$. $x\prec y$ will stand for $x\underset{\_}{\prec }y$ and $x\ne y$, while $x\ll y$ will stand for $y-x\in intP$. The pair $(\mathbb{A},p)$ is an ordered topological vector space (see Kadelburg et al., 2010).

An $OTVS$ $(\mathbb{E},\tau ,\underset{\_}{\prec })$ is said to be order-convex if it has a base of neighborhoods of $\theta$ consisting of order-convex subsets. In this case, the cone $C$ is said to be normal. In the case of a normed space, this condition means that the unit ball is order-convex, which is equivalent to the condition that there is a number $k$ such that $x,y\in \mathbb{A}$ and $0\underset{\_}{\prec }x\underset{\_}{\prec }y$ implies that $\parallel x\parallel \le k\parallel y\parallel .$

The cone $C$ in an $OTVS$ $(\mathbb{E},\tau )$ is called solid if it has a nonempty interior, $intC\ne \mathrm{\varnothing }$ (see Kadelburg et al., 2010).

Now, we first recall some notions in topological algebras.

Definition 2.1 (Azram & Asif, 2013) The algebra $\mathbb{A}$ with Hausdorff topology and continuous algebra operations is called a topological algebra. A topological algebra is called a locally convex algebra if its topology is generated by the seminorms $\{p_{\alpha }{\}}_{\alpha \in I}$ such that for all $\alpha \in I$, it follows that

(i) $p_{\alpha }(x)\ge 0$ and $p_{\alpha }(x)=0$ if $x=0$;

(ii) $p_{\alpha }(x+y)\le p_{\alpha }(x)+p_{\alpha }(y)$ for all $x,y\in \mathbb{A}$;

(iii) $p_{\alpha }(\lambda x)=|\lambda |p_{\alpha }(x)$ for all$x\in \mathbb{A}$ and $\lambda \in \mathbb{C}$.

A locally convex algebra is called locally multiplicatively convex if $p_{\alpha }(xy)\le p_{\alpha }(x)p_{\alpha }(y)$ for all $x,y\in \mathbb{A}.$ A complete metrizable locally multiplicatively convex algebra is called a Fréchet algebra.

The topology of a Fréchet algebra $\mathbb{A}$ can be generated by a sequence $(p_n{)}_n$ of separating submultiplicative seminorm, that is $p_n(xy)\le p_n(x)p_n(y)$ for all $n\in \mathbb{N}$ and every $x,y\in \mathbb{A},$ such that $p_n(x)\le p_{n+1}(x)$ for all $x\in \mathbb{A}$ and $n\in \mathbb{N}.$ If $\mathbb{A}$ is unital, then $p_n$ can be chosen such that $p_n(e)=1.$ The Fréchet algebra $\mathbb{A}$ with the above generating sequence of seminorm is denoted by $(\mathbb{A},(p_n)).$ Note that a sequence $(x_k)$ in the Fréchet $(\mathbb{A},(p_n))$ is convergent to $x\in \mathbb{A}$ if and only if $p_n(x_k-x)\to 0,$ for each $n\in \mathbb{N},$ as $k\to \mathrm{\infty }$ (see Ghasemi Honary, 2011).

Example 2.2 (Lawson & Read, 2008) Let $s\subset {\mathbb{C}}^{\mathbb{N}}$ be the vector space of all complex sequences tending to zero faster than any polynomial, that is, $s=\{(x_k{)}_{k\in \mathbb{N}}:k^n|x_k|\to 0\mathrm{a}\mathrm{s}k\to \mathrm{\infty },\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}n\in \mathbb{N}\}.$ This is known as the space of rapidly decreasing sequence. From now on, $(x_k{)}_{k\in \mathbb{N}}$ is denoted by $x.$ The vector space $\mathbb{A}$ becomes a commutative algebra when equipped with pointwise multiplication and is a Fréchet algebra with respect to the norm $(p_n{)}_{n\in \mathbb{N}},$ given by

$p_n(x)=sup\{k^n|x_k|:k>0,n\in \mathbb{N}\}.$

Example 2.3 (Lawson & Read, 2008) Let $\mathbb{A}\subset {\mathbb{C}}_0(\mathbb{R})$ be the collection of continuous complex-valued functions that tend to zero faster than any polynomial, that is, $\mathbb{A}=\{f\in \mathbb{C}(\mathbb{R}):|x{|}^n|f(x)|\to 0\mathrm{a}\mathrm{s}|x|\to 0\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}n\in \mathbb{N}\}$. Then, with pointwise addition and multiplication, $\mathbb{A}$ is a Fréchet algebra with respect to norms

$p_n(f)=sup\{(1+|x|{)}^n|f(x)|:x\in \mathbb{R}\}.$

Example 2.4 (Lawson & Read, 2008) Let $D\subset \mathbb{C}$ be an open unit disc, and let $\mathbb{A}=\mathbb{O}(D)$ be the algebra of analytic functions $f:D\to \mathbb{C}.$ Then $\mathbb{A}$ is a Fréchet algebra with respect to the system of seminorms $p_n,$ given by

$p_n(f)=\underset{|x|\le 1-\frac{1}{n}}{sup}|f(x)|,n\ge 0.$

Example 2.5 (Goldmann, 1990), pp. 67–77 Let $\mathbb{C}(\mathbb{R})$ be the space of all continuous complex-valued functions. Then $\mathbb{C}(\mathbb{R})$ is a Fr$e^{\prime }$chet algebra with the seminorms $\parallel f{\parallel }_n={sup}_{|t|\le n}\{|f(t)|\}$ for $n\ge 0.$

3. F-cone metric spaces over Fréchet algebra

In this section, we describe the concept of $F$-cone metric spaces on a Fréchet algebra.

Definition 3.1 Let $X$ be a nonempty set. Suppose that a mapping $F:X^3\to \mathbb{E}$ is a function satisfying the following axioms:

(F1) $\theta \underset{\_}{\prec }F(x,x,x)\underset{\_}{\prec }F(x,x,y)\underset{\_}{\prec }F(x,y,z),$ for all $x,y,z,\in X$ with $x\ne y\ne z,$

(F2) $F(x,y,z)\underset{\_}{\prec }s[F(x,x,a)+F(y,y,a)+F(z,z,a)]-F(a,a,a),$

for all $x,y,z,a\in X.$ Then the pair $(X,F)$ is called an $F$-cone metric space over Fréchet algebra $\mathbb{E}.$ The number $s\ge 1$ is called the coefficient of $(X,F).$

Now we give some examples of $F$-cone metric spaces over Fréchet algebras.

Example 3.2 By using Example 2.5, $\mathbb{A}=\mathbb{C}(\mathbb{R})$ is a Fréchet algebra with respect to the seminorm $(p_n{)}_{n\in \mathbb{N}},$ given by

$p_n(f)=\underset{|x|\le n}{sup}|f(x)|$

for $n\ge 0$. Also, the constant function $1$ acts as an identity. Set $\{f\in \mathbb{A}:f(t)\ge 0,t\in \mathbb{R}\}$ as a cone in $\mathbb{A}$. Suppose that $X=\mathbb{R}$. Define the mapping $F:X^3\to \mathbb{A}$ by $F(x,y,z)(t)=(|x^2-y^2|+|y^2-z^2|+|x^2-z^2|)e^t$ for all $x,y,z\in X$. Thus $(X,F)$ with $s=1$ is an $F$-cone metric space over Fréchet algebra $\mathbb{A}$.

Proposition 3.3. Let $(X,F)$ be an $F$-cone metric space over a Fr$e^{\prime }$chet algebra $\mathbb{A}.$ Then, for all $x,y,z\in X,$ we have $F(x,x,y)=F(y,y,x).$

Proof. From Definition 3.1 $(F_2)$, we have $F(x,x,y)\underset{\_}{\prec }2sF(x,x,x)+sF(y,y,x)-F(x,x,x)$, and similarly, we can write $F(y,y,x)\underset{\_}{\prec }2sF(y,y,y)+sF(x,x,y)-F(y,y,y)$ for all $s\ge 1$. Therefore, we get $(2se-e)F(x,x,x)+sF(y,y,x)-F(x,x,y)=(2s-1)F(x,x,x)+sF(y,y,x)-F(x,x,y)\in P$ and $(2se-e)F(y,y,y)+sF(x,x,y)-F(y,y,x)=(2s-1)F(y,y,y)+sF(x,x,y)-F(y,y,x)\in P$. On the other hand, from condition $(F_1)$, we have $(2s-1)F(x,x,x)\in P$ and $(2s-1)F(y,y,y)\in P$. Thus, $sF(x,x,y)\underset{\_}{\prec }F(y,y,x)$ and $sF(y,y,x)\underset{\_}{\prec }F(x,x,y)$ for $s\ge 1$. These imply that $F(x,x,y)=F(y,y,x)$.

Proposition 3.4 Let $(X,F)$ be an $F$-cone metric space over Fr$e^{\prime }$chet algebra $\mathbb{A}.$ Then, for all $x,y,x\in X$, we have $F(x,y,y)\prec F(x,x,y)$.

Proof. By using Definition 3.1 $(F_2)$, we can write $F(x,y,y)\prec sF(x,x,x)+2sF(y,y,x)$. Thus, $sF(x,x,x)-F(x,y,y)+2sF(y,y,x)\in P$. On the other hand, from condition $(F_1)$, we have $sF(x,x,x)\in P$. Therefore, we deduce that $2sF(y,y,x)-F(x,y,y)\in P$. Using Proposition 3.3 and using again condition $(F_1)$, we obtain $sF(x,x,y)-F(x,y,y)\in P$ for $s\ge 1$. This implies that $F(x,y,y)\prec F(x,x,y)$.

4. Topology on $F$-cone metric over Fréchet algebra

Now, we define the topology of $F$-cone metric spaces over a Fréchet algebra and study its topological properties.

Definition 4.1 Assume that $(X,F)$ is an $F$-cone metric space over a Fréchet algebra $\mathbb{A}.$ Then, for $x\in X$ and $c$, the $F$-ball with center $x$ and radius $c>\theta$ are

$B_F(x,c)=\{y\in X:F(x,x,y)\ll F(x,x,x)+c\}.$

Definition 4.2 Assume that $(X,F)$ is an $F$-cone metric space over a Fr$e^{\prime }$chet algebra $\mathbb{A}.$ For each $x\in X$ and $\theta \ll c,$ put $B_F(x,c)=\{y\in X:F(x,x,y)\ll F(x,x,x)+c\}.$ Then, $B=\{B_F(x,c):x\in X,\theta \ll c\}$ is a subbase for topology $\tau$ on $X$ and $U$ denotes the base generated by the subbase $B.$

Theorem 4.3 Let $(X,F)$ be an $F$-cone metric space over a Fréchet algebra $\mathbb{A}$ and let $P$ be a solid cone in $\mathbb{A}.$ Then $(X,F)$ is a Hausdorff space.

Proof. Assume that $x,y\in X$ with $x\ne y.$ Consider two cases:

Case1 $\theta \prec F(x,x,x)$: Choose $F(x,x,x)=\frac{c}{6s}$ and $F(x,x,y)=2c$. Let $U=B(x,\frac{c}{2s})$ and let $V=B(y,\frac{c}{2s})$, where $s\ge 1$. Then $x\in B$ and $y\in V.$ We intend to show that $U\cap V=\mathrm{\varnothing }.$ We assume on the contrary that there exists $z\in U\cap V.$ Then $F(x,x,z)\prec \frac{2c}{3s},$ $F(y,y,z)\prec \frac{2c}{3s}.$ Therefore, we have

$\begin{array}{rl}& 2c=F(x,x,y)s[2F(x,x,z)+F(y,y,z)]-F(z,z,z)\\ & \prec s[2F(x,x,z)+F(y,y,z)]\\ & \prec s[\frac{4c}{3s}+\frac{2c}{3s}]\\ & \prec 2c,\\ & \end{array}$

which means $2c\prec 2c,$ a contradiction. Thus, $B\cap V=\mathrm{\varnothing }$ and $X$ is a Hausdorff space.

Case2 $F(x,x,x)=\theta$: Let $F(x,x,y)=c$. Then, $F(x,x,z)\prec \frac{c}{3s}$ and $F(y,y,z)\prec \frac{c}{3s}$. Thus, we get

$\begin{array}{rl}& c=F(x,x,y)s[2F(x,x,z)+F(y,y,z)]-F(z,z,z)\\ & \prec s[2F(x,x,z)+F(y,y,z)]\\ & \prec s[\frac{2c}{3s}+\frac{c}{3s}]\\ & \prec c,\end{array}$

which is a contradiction. Thus $B\cap V=\mathrm{\varnothing }$.

Now, we define a $\theta$-Cauchy sequence and a convergent sequence in an $F$-cone metric space over a Fréchet algebra $\mathbb{A}.$

Definition 4.4 Let $(X,F)$ be an $F$-cone metric space over a Fréchet algebra $\mathbb{A}.$ A sequence $\{x_q\}$ in $(X,F)$ converges to a point $x\in X$ whenever for every $c\gg \theta$ there is a natural number $N$ such that $F(x_q,x,x)\ll c$ for all $q\ge N.$ We denote this by ${lim}_{q\to \mathrm{\infty }}{x}_q=x$ or $x_q\to x$ as $q\to \mathrm{\infty }.$

Definition 4.5 The sequence $\{x_q\}$ is a $\theta$-Cauchy sequence in $(X,F)$ if $\{F(x_q,x_p,x_p)\}$ is a $c$-sequence in $\mathbb{A},$ that is, if for every $c\gg \theta$ there exists $q_0\in \mathbb{N}$ such that $F(x_q,x_p,x_p)\ll c$ for all $q,p\ge q_0.$

Definition 4.6 The space $(X,F)$ is $\theta$-complete if every $\theta$-Cauchy sequence converges to $x\in X$ such that $F(x,x,x)=\theta .$

5. Generalized Lipschitz maps

In this section, we define generalized Lipschitz maps in an $F$-cone metric space over a Fr$e^{\prime }$chet algebra with unit $e.$

Definition 5.1. Let $(X,F)$ be an $F$-cone metric space with the coefficient $s$ over a Fr$e^{\prime }$chet algebra $\mathbb{A}$ and let $P$ be a cone in $\mathbb{A}.$ A map $T:X\to X$ is said to be a generalized Lipschitz mapping if there exists a vector $k\in P$ with $\rho (k)<1$ (the spectral radius) such that

$F(Tx,Tx,Ty)\underset{\_}{\prec }kF(x,x,y)$

for all $x,y\in X.$

Example 5.2 Let the Fréchet algebra $\mathbb{A}$, the cone $P,$ and the mapping $F:X^3\to \mathbb{A}$ be the same ones as those in Example 3.2. Then $(X,F)$ is an $F$-cone metric space over the Fréchet algebra $\mathbb{A}.$ Now, we define the mapping $T:X\to X$ by $T(x)=\frac{x}{3}.$ We have $F(Tx,Tx,Ty)=\frac{2\left|x^2-y^2\right|}{9}{e}^t\underset{\_}{\prec }\frac{2}{9}\left|x^2-y^2\right|e^t=\frac{1}{9}F(x,x,y)(t)$ for $k=\frac{1}{9}.$ Then $T$ is a generalized Lipschitz map in $X.$

Proposition 5.3 Let $\mathbb{A}$ be a Fréchet algebra with a cone $P$ and $k\in P$ such that $\rho (k)<1.$ Then $(p_n(k){)}^q\to 0$ as $q\to \mathrm{\infty }$.

Proof. Since $\sigma (k)$ is nonempty (see Goldmann, 1990, p. 72). Then there exists a $\lambda \in \sigma (k)$ such that $\lambda e-k$ is not ivertible. Also, since a Fr$e^{\prime }$chet algebra, which is a field, is isomorphic to $\mathbb{C}$ (see Goldmann, 1990, p. 80), then $k=\lambda e$. On the other hand, $p_n(k)=|\lambda |p_n(e)=|\lambda |<1$. Hence, we get $(p_n(k){)}^q\to 0$ as $q\to \mathrm{\infty }$.

Definition 5.4 (Simon, 2017) Let $E$ be a separated seminormed space, let $\{\parallel .{\parallel }_{E;\text{^}I1/2}:\text{^}I1/2\in N_E\}$ be its family of seminorms, and let $(u_n{)}_{n\in N}\subset E$. The sequence $(x_n{)}_{n\in N}$ converges to a limit $u\in E$, which is denoted by $u_n\to u$ as $n\to \mathrm{\infty }$, if, for all $\nu \in N_E$,

$\parallel u_n-u{\parallel }_{E;\nu }\to 0,$

when $n\to \mathrm{\infty }$. That is if, for all $\nu \in N_E$ and $\epsilon >0$, there exists $m\in N$ such that $n\ge m$ yields $\parallel u_n-u{\parallel }_{E;\nu }\le \epsilon$.

Lemma 5.5. Let $\mathbb{A}$ be a Fréchet algebra with a solid cone $P.$ Suppose that $\{x_q\}$ is a sequence in $\mathbb{A}$ such that $p_n(x_q)\to 0$ as $q\to \mathrm{\infty };$ then $x_q$ is a $c$-sequence.

Proof. By the assumption, we have $p_n(x_q)\to 0$. Therefore from Definition 5.4, for every $\epsilon >0$, there exists $q_0\in \mathbb{N}$ such that $p_n(x_q)<\epsilon$ for $q\ge q_0$. On the other hand, for each $c\gg \theta$, there is $\delta >0$ such that $B(c,\delta )=\{y\in \mathbb{A}:p_n(y-c)<\delta \}\subset P$. We know that $p_n(x_q)=p_n(c-x_q-c)<\epsilon$. Since $\epsilon$ is arbitrary, choose $\epsilon =\delta$. Then we have $p_n(x_q)=p_n(c-x_q-c)<\delta$, this leads to $c-x_q\in B(c,\delta )\subset P$. It means $c-x_q\in intP$. Hence $x_q\ll c$.

Lemma 5.6 (Xun & Shou, 2014) Let $\mathbb{E}$ be a topological vector space with a $tvs$-cone $p$. Then the following properties hold:

(1) If $a\gg \theta$, then $ra\gg \theta$ for each $r\in {\mathbb{R}}_{+}$.

(2) If $a_1\gg {\beta }_1$ and $a_2\ge {\beta }_2$, then $a_1+a_2\gg {\beta }_1+{\beta }_2$ and $a_2\ge {\beta }_2\iff a_2-{\beta }_2\ge \theta \iff a_2-{\beta }_2\in p$.

Lemma 5.7 (Arandelović & Kćkć, 2012) Let $(\mathbb{E},P)$ be an ordered TVS. Then if $x\in P$ and $y\in intP$, then $x+y\in intP$. Consequently, if $x\le y$ and $y\ll z$, then $x\ll z$ ($x\le y$, which we say “x is less then y”, if $y-x\in p$).

6. Applications to fixed point theory

In this section, we prove fixed point theorems for generalized Lipschitz maps on an $F$-cone metric space over a Fr$e^{\prime }$chet algebra.

Theorem 6.1. Let $(X,F)$ be a $\theta$-complete $F$-cone metric space over a Fréchet algebra $\mathbb{A}$ and let $P$ be a solid cone in $\mathbb{A}.$ Let $k\in P$ be a generalized Lipschitz constant with $\rho (k)<1$ and let the mapping $T:X\to X$ satisfy the following condition

$F(Tx,Tx,Ty)\underset{\_}{\prec }kF(x,x,y)$

for all $x,y\in X$. Moreover, $(e-2s^{2}k)\succ \theta$. Then, $T$ has a unique fixed point in $X.$ For each $x\in X,$ the sequence of iterates $\{T^{q}x\}$ converges to the fixed point.

Proof. Let $x_0\in X$ and let $q\ge 1.$ Consider $x_1=T{x}_0$ and $x_{q+1}=T^{q+1}{x}_0.$ Therefore, from the assumptions of the theorem, we get

$\begin{array}{rl}& F(x_q,x_q,x_{q+1})=F(T{x}_{q-1},T{x}_{q-1},T{x}_q)\\ & kF(x_{q-1},x_{q-1},x_q)\\ & k^{2}F(x_{q-2},x_{q-2},x_{q-1})\\ & ⋮\\ & k^{q}F(x_0,x_0,x_1).\end{array}$

Now, we assume $p>q$. By Definition 3.1 $(F_2)$, and the assumptions of the theorem, we get

$\begin{array}{rl}& F(x_q,x_q,x_p)2sF(x_q,x_q,x_{p+1})+sF(x_p,x_p,x_{p+1})\\ & -F(x_{p+1},x_{p+1},x_{p+1})\\ & \prec 4s^{2}F(x_q,x_q,x_{q+1})+2s^{2}F(x_{q+1},x_{q+1},x_{p+1})\\ & +2s^{2}F(x_p,x_p,x_{p+1})-sF(x_{q+1},x_{q+1},x_{q+1})\\ & \prec 4s^{2}F(x_q,x_q,x_{q+1})+2s^{2}F(x_{q+1},x_{q+1},x_{p+1})\\ & +sF(x_p,x_p,x_{p+1})\\ & 4s^2{k}^{q}F(x_0,x_0,x_1)+s{k}^{p}F(x_0,x_0,x_1)\\ & +2s^{2}kF(x_q,x_q,x_p).\end{array}$

Therefore,

$(e-2s^{2}k)F(x_q,x_q,x_p)\underset{\_}{\prec }4s^2{k}^{q}F(x_0,x_0,x_1)+s{k}^{p}F(x_0,x_0,x_1).$

Similarly to the proof of Proposition 5.3, there exists a $\lambda \in \sigma (k)$ such that $k=\lambda e$. Also, from the assumption of the theorem, we deduce that $1-2s^2\lambda >0$. Then, we have

$(1-2s^2\lambda )eF(x_q,x_q,x_p)\underset{\_}{\prec }4s^2{k}^{q}F(x_0,x_0,x_1)+s{k}^{p}F(x_0,x_0,x_1).$

Thus,

$\begin{array}{rl}& F(x_q,x_q,x_p)4s^2(1-2s^2\lambda {)}^{-1}{k}^{q}F(x_0,x_0,x_1)\\ & +s(1-2s^2\lambda {)}^{-1}{k}^{p}F(x_0,x_0,x_1)\end{array}$

Since $p_n(k^{q}F(x_0,x_0,x_1))\le p_n(k^q)p_n(F(x_0,x_0,x_1)$ and

$p_n(k^{p}F(x_0,x_0,x_1))$ $\le p_n(k^p)p_n(F(x_0,x_0,x_1))$, and by applying Proposition 5.3, we have $p_n(k^q)\to 0$ as $q\to \mathrm{\infty }$ and $p_n(k^p)\to 0$ as $p\to \mathrm{\infty }$. Now from Lemma 5.5, we have $\{k^{q}F(x_0,x_0,x_1)\}$ and $\{k_{p}F(x_0,x_0,x_1)\}$ are $c$-sequences. By Lemma 5.6, we deduce that $\{4s^2(1-2s^2\lambda {)}^{-1}{k}^{q}F(x_0,x_0,x_1)+s(1-2s^2\lambda {)}^{-1}{k}^{p}F(x_0,x_0,x_1)\}$ is a $c$-sequence. Finally, from Proposition 3.4 and Lemma 5.7, we get $F(x_q,x_p,x_p)\ll c$ and $\{x_q\}$ is a $\theta$-Cauchy sequence in $X.$ Therefore, from $\theta$-complete of $X$, there exists $u\in X$ such that

$\underset{q\to \mathrm{\infty }}{lim}F(x_q,x_q,u)=F(u,u,u)=\theta .$

We are going to show that $u$ is a fixed point of $T.$ Now, by Definition 3.1 $(F_2),$ Proposition 3.3, and the assumptions of the theorem, we have

$\begin{array}{rl}& F(x_q,x_q,Tu)\underset{\_}{\prec }2sF(x_q,x_q,x_{q+1})+sF(Tu,Tu,x_{q+1})\\ & -F(x_{q+1},x_{q+1},x_{q+1})\\ & \prec 2sF(x_q,x_q,x_{q+1})+sF(T{x}_q,T{x}_q,Tu)\\ & \underset{\_}{\prec }2sF(x_q,x_q,x_{q+1})+skF(x_q,x_q,u).\end{array}$

By using a similar method to the proof of the Proposition 5.3, there exists a $\lambda \in \sigma (k)$ such that $k=\lambda e$. Then, we have

$F(x_q,x_q,Tu)\underset{\_}{\prec }2sF(x_q,x_q,x_{q+1})+s\lambda eF(x_q,x_q,u).$

We know that $\{F(x_q,x_q,x_{q+1})\}$ and $\{F(x_q,x_q,u)\}$ are $c$-sequences. Now, from Lemma 5.6, we deduce that

$\{2sF(x_q,x_q,x_{q+1})+s\lambda F(x_q,x_q,u)\}$ is a $c$-sequence. Next, by Lemma 5.7 and Proposition 3.4, we have $F(x_q,Tu,Tu)\ll c$ and ${lim}_{q\to \mathrm{\infty }}{x}_q=Tu.$ From Theorem 4.3, $u=Tu$ and $u$ is a fixed point for $T.$ Now, we study the uniqueness of the fixed point of $T.$ Assume that $T$ has another fixed point $v,$ such that $v\ne u.$ For this aim, we consider the following two cases:

Case1: $F(x_q,x_q,v)\prec F(x_q,x_q,u)$: Since $F(x_q,x_q,u)$ is a $c$-sequence, we yield

$F(x_q,x_q,v)\prec F(x_q,x_q,u)\ll c$. Hence, from Lemma 5.7 and Proposition 3.4, we have $F(x_q,v,v)\ll c$, and by using Theorem 4.3, it is a contradiction. Therefore $u=v.$

Case2: $F(x_q,x_q,u)\prec F(x_q,x_q,v)$: By applying Definition 3.1 $(F_2)$, Proposition 3.3, and from the assumptions of the theorem, we get

$F(x_q,x_q,v)\underset{\_}{\prec }2sF(x_q,x_q,u)+sF(v,v,u)-F(u,u,u)$

$\prec 2sF(x_q,x_q,u)+sF(v,v,u)$

$\underset{\_}{\prec }2sF(x_q,x_q,u)+skF(v,v,u)$

$\underset{\_}{\prec }2sF(x_q,x_q,u)+2s^{2}kF(v,v,x_q)+s^{2}kF(u,u,x_q)$

$-skF(x_q,x_q,x_q)$

$\prec 2sF(x_q,x_q,u)+2s^{2}kF(v,v,x_q)+s^{2}kF(u,u,x_q)$

$\underset{\_}{\prec }2sF(x_q,x_q,u)+s^{2}kF(x_q,x_q,u)+2s^{2}kF(x_q,x_q,v).$

Therefore,

$(e-2s^{2}k)F(x_q,x_q,v)\underset{\_}{\prec }(2se+s^{2}k)F(x_q,x_q,u).$

Using again the process of Proposition 5.3. Then, there exists a $\lambda \in \sigma (k)$ such that $\lambda e=k$. From the assumption of the theorem, we observe that $1-2s^2\lambda >0$. Hence, we get

$(1-2s^2\lambda )eF(x_q,x_q,v)\underset{\_}{\prec }(2se+s^2\lambda e)F(x_q,x_q,u)$

Thus,

$F(x_q,x_q,v)\underset{\_}{\prec }(2s+s^2\lambda )(1-2s^2\lambda {)}^{-1}F(x_q,x_q,u)$

And since, $\{F(x_q,x_q,u)\}$ is a $c$-sequence. Thus, from Lemma 5.6 $(1)$, we deduce that $\{(s+2s^2\lambda )(1-2s^2\lambda {)}^{-1}F(x_q,x_q,u)\}$ is a $c$-sequence. Therefore, by using Lemma 5.7 and Proposition 3.4, we obtain $F(x_q,v,v)\ll c$ and $x_q\to v$ as $q\to \mathrm{\infty }.$ Finally, by Theorem 4.3, it is a contradiction. Thus, $v=u.$

Corollary 6.2. Let $(X,F)$ be an $F$-cone metric space over a Fréhet algebra. Assume that a mapping $T:X\to X$ satisfies, for some positive integer $q$,

$F(T^{q}x,T^{q}x,T^{q}y)\underset{\_}{\prec }kF(x,x,y)$

for all $x,y\in X$, where $k$ is a vector in $P.$ Then $T$ has a unique fixed point in $X.$

Proof. By using Theorem 6.1, $T^n$ has a unique fixed point $x^{\ast }.$ So, we have $T^q(T{x}^{\ast })=T(T^q{x}^{\ast })=T{x}^{\ast };$ therefore $T{x}^{\ast }=x^{\ast }$ is a fixed point of $T.$ Also, the fixed point of $T$ is a fixed point of $T^n,$ thus $T$ has a unique fixed point.

Now, we prove Chatterjee’s fixed point theorem in $F$-cone metric spaces over a Fréchet algebra.

Theorem 6.3 Assume that $(X,F)$ is a $\theta$-complete $F$-cone metric space over a Fréchet algebra and let $P$ be the underlying solid cone with $k\in P$, where $\rho (k)<1$. Suppose the mapping $T:X\to X$ satisfies the generalized Lipschitz condition:

$F(Tx,Tx,Ty)\underset{\_}{\prec }k[F(Tx,Tx,x)+F(Ty,Ty,y)],$

for all $x,y\in X$ such that $(e-3sk)\succ \theta$. Then $T$ has a unique fixed point in $X$ for any $x\in X$ and the iterative sequence $\{T^{q}x\}$ converges to the fixed point.

Proof. Let $x_0\in X$ be arbitrarily given, and set $x_q=T^{q}x,$ $q\ge 1$. Then by using the assumptions of the theorem and Proposition 3.3, we have

$F(x_q,x_q,x_{q+1})=F(T{x}_{q-1},T{x}_{q-1},T{x}_q)$

$\underset{\_}{\prec }kF(T{x}_{q-1},T{x}_{q-1},x_{q-1})+kF(T{x}_q,T{x}_q,x_q)$

$=kF(x_{q-1},x_{q-1},x_q)+kF(x_q,x_q,x_{q+1}).$

Hence, $(e-k)F(x_q,x_q,x_{q+1})\underset{\_}{\prec }kF(x_{q-1},x_{q-1},x_q)$. Applying a similar metod the proof of the Proposition 5.3 thus there exists a $\lambda \in \sigma (k)$ such that $\lambda e=k$. Also, by the assumption of the theorem we get $1-3s\lambda >0$. Therefore, we have $F(x_q,x_q,x_{q+1})\underset{\_}{\prec }\lambda (1-\lambda {)}^{-1}eF(x_{q-1},x_{q-1},x_q)$. On the other hand, since $\lambda <\frac{1}{3s}$ then $0<\lambda (1-\lambda {)}^{-1}<1$. Therefore $\rho (k(e-k{)}^{-1})<1$. Now, we use a procedure similar to the proof of Theorem 6.1, so $\{x_q\}$ is a $\theta$-Cauchy sequence. From $\theta$-completeness of $X,$ there exists $u\in X$ such that

$\underset{q\to \mathrm{\infty }}{lim}F(x_q,x_q,u)=F(u,u,u)=\theta .$

Next, we intend to show that $u$ is a fixed point of $T.$ Employing Definition 3.1 $(F_2),$ Proposition 3.3, we obtain

$F(T{x}_q,T{x}_q,Tu)\underset{\_}{\prec }kF(T{x}_q,T{x}_q,x_q)+kF(Tu,,Tu,u)$

$\underset{\_}{\prec }kF(T{x}_q,T{x}_q,x_q)+kF(u,u,Tu)$

$\underset{\_}{\prec }kF(x_{q+1},x_{q+1},x_q)+2skF(u,u,x_{q+1})$

$+skF(Tu,Tu,T{x}_q)-kF(x_{q+1},x_{q+1},x_{q+1})$

$\prec kF(x_q,x_q,x_{q+1})+2skF(x_{q+1},x_{q+1},u)$

$+skF(T{x}_q,T{x}_q,Tu).$

Therefore,

$(e-sk)F(T{x}_q,T{x}_q,Tu)\prec kF(x_q,x_q,T{x}_q)+2skF(x_{q+1},x_{q+1},u)$

Similar to the proof of the Proposition 5.3 there exists a $\lambda \in \sigma (k)$ such that $\lambda e=k$ also, from the assumption of the theorem we observe that $1-s\lambda >0$. Hence, we get

$(1-s\lambda )eF(T{x}_q,T{x}_q,Tu)\prec \lambda eF(x_q,x_q,T{x}_q)+2s\lambda eF(x_{q+1},x_{q+1},u).$

Thus

$F(T{x}_q,T{x}_q,Tu)\prec \lambda (1-s\lambda {)}^{-1}F(x_q,x_q,T{x}_q)+2s\lambda (1-s\lambda {)}^{-1}F(x_{q+1},x_{q+1},u).$

Since $\{F(x_q,x_q,x_{q+1})\}$ and $\{F(x_{q+1},x_{q+1},u)\}$ are $c$-sequences. Then by applying Lemma 5.6, we have $\{\lambda (1-s\lambda {)}^{-1}F(x_q,x_q,x_{q+1})+2s\lambda (1-s\lambda {)}^{-1}F(u,u,x_q)\}$ is a $c$-sequence. Therefore from Lemma 5.7 and Proposition 3.4, we get $F(x_q,Tu,Tu)\ll c$ and ${lim}_{q\to \mathrm{\infty }}{x}_q=Tu.$ Finally, applying Theorem 4.3, we get $u=Tu.$ Now, we show that $T$ has at most one fixed point. Let $u,v\in X$ be two fixed points of $T$ such that $u\ne v.$ Next, we consider two following cases:

Case1 $F(T{x}_q,T{x}_q,Tv)\prec F(T{x}_q,T{x}_q,Tu)$: We know that $F(T{x}_q,T{x}_q,u)$ is a $c$-sequence. Then by Lemma 5.7 and Proposition 3.4, we deduce that

$F(x_{q+1},v,v)\ll c.$ By applying Theorem 4.3, we obtain a contradiction, so $v=u.$

Case2 $F(x_{q+1},x_{q+1},Tu)\prec F(x_{q+1},x_{q+1},Tv)$: From Definition 3.1 $(F_2),$ Proposition 3.3 and the assumptions of the theorem, we deduce that

$\begin{array}{rl}& F(T{x}_q,T{x}_q,Tv)kF(T{x}_q,T{x}_q,x_q)+kF(Tv,Tv,v)\\ & kF(T{x}_q,T{x}_q,x_q)+3skF(Tv,Tv,x_{q+1})\\ & -kF(x_{q+1},x_{q+1},x_{q+1})\\ & \prec kF(x_q,x_q,x_{q+1})+3skF(x_{q+1},x_{q+1},Tv).\end{array}$

Hence,

$(e-3sk)F(T{x}_q,T{x}_q,Tv)\prec kF(x_q,x_q,x_{q+1}).$

Again similarly the proof of the Proposition 5.3 there exists a $\lambda \in \sigma (k)$ such that $\lambda e=k$. On the other hand, by using the assumption of the theorem we deduce that $1-3s\lambda >0$ then, we have

$(1-3s\lambda )eF(T{x}_q,T{x}_q,Tv)\prec \lambda eF(x_q,x_q,T{x}_q)$

Thus,

$F(T{x}_q,T{x}_q,Tv)\prec \lambda (1-3s\lambda {)}^{-1}F(x_q,x_q,T{x}_q)$

Since $\{F(x_q,x_q,x_{q+1})\}$ is a $c$-sequence. Therefore by Lemma 5.6 $(1)$, we get $\{\lambda (1-3s\lambda {)}^{-1}F(x_q,x_q,x_{q+1})\}$ is a $c$-sequence. From Lemma 5.7 and Proposition 3.4, we get $F(x_{q+1},v,v)\ll c.$ By using Theorem 4.3, we obtain a contradiction. Then $u=v.$

Example 6.4 Choose Example 5.2. Therefore, $(X,F)$ is an $F$-cone metric space over the Fr$e^{\prime }$chet algebra $\mathbb{A}$ and the mapping $T:X\to X$ by $T(x)=\frac{x}{3}$ is a generalized Lipschitz with $k=\frac{1}{9}$. Also, we get $k=\lambda e=\frac{1}{9}$. Therefore $\lambda =\frac{1}{9}<\frac{1}{8}=\frac{1}{2s^2}$. Hence, the conditions of Theorem 6.1 hold. Thus, $T$ has a unique fixed point $0.$

Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

Reza Allahyari

The authors have introduced the notion of F-cone metric spaces over Frechet algebra as a generalization of F-cone metric spaces over a Banach algebra, N_p-cone metric spaces over a Banach algebra, and N_b-cone metric spaces over a Banach algebra. They studied some of its topological properties. Next, they defined a generalized Lipschitz for such spaces. Also, they investigated some fixed points for mappings satisfying such conditions in the new framework. The results of their article generalized some well-known results in the literature.