Fixed point theorem for orthogonal contraction of Hardy-Rogers-type mapping on $O$-complete metric spaces

1.   Introduction

In 1922, Banach ^[1] introduced and established a fixed point result which is well known as "Banach fixed point theorem", it is one of the most famous and important results in metric fixed point theory. Since the simplicity and usefulness of the Banach's fixed point theorem, many authors have extended, improved and generalized Banach's fixed point theorem from different perspectives. For more details, we cite the readers to ^[2,3] and the references therein.

In ^[5], Gordji et al. introduced the concept of an orthogonal set and extended Banach's fixed point theorem to the setting of orthogonal sets as follows :

Theorem 1.1. Let $(X, \bot, d)$ be an orthogonal complete metric space. Assume that $T : X \to X$ is a $\bot$-continuous mapping such that the following conditions hold :

(i) there exists $0 < k < 1$ such that

(ii) $T$ is $\bot$-preserving.

Then, $T$ has a unique fixed point in $X$. Moreover, for each $x \in X$, the Picard sequence $\{T^n x\}$ converges to the fixed point of $T$.

Theorem 1.1 is a real generalization of Banach's fixed point theorem, it can be applied to show the existence of a solution for a differential equation which can not be applied by the Banach's fixed point theorem, the results of Nieto and Rodr${\rm \acute{i}}$guez ^[6] and the results of Ran and Reurings ^[7].

We know that by using some control functions can generalize the Banach contractive condition (see ^{[8,9,12,14,15]}). Recently, Sawangsup et al. ^[10] modified the concept of $F$-contraction mappings to orthogonal sets and given some new fixed point theorems for $F$-contraction mappings in orthogonal complete metric spaces.

Definition 1.1. (^[13]) Let $\mathcal{F}$ be the set of functions $F : (0, +\infty) \to \mathbb{R}$, such that

(ⅰ) $F$ is strictly increasing;

(ⅱ) For every sequence $\{t_n\} \subset (0, +\infty)$, $\lim_{n \to +\infty} F(t_n) = - \infty \ \Leftrightarrow \ \lim_{n \to +\infty} t_n = 0$;

(ⅲ) There exists $k \in (0, 1)$ such that $\lim_{t \to 0^+} t^k F(t) = 0$.

Recently, Sawangsup et al. proved the following fixed point theorem in orthogonal complete orthogonal metric space.

Theorem 1.2. (^[10]) Let $(X, \bot, d)$ be an orthogonal complete orthogonal metric space with an orthogonal element $x_0$ and $T$ be a self-mapping on $X$ satisfying the following conditions:

(i) $T$ is $\bot$-preserving;

(ii) $T$ is an $F_{\bot}$-contraction mapping;

(iii) $T$ is $\bot$-continuous.

Then, $T$ has a unique fixed point in $X$. Moreover, the Picard sequence $\{T^n x_0\}$ converges to the fixed point of $T$.

In ^[4], Cosentino and Vetro introduced the notion of $F$-contraction of Hardy-Rogers-type and given a new fixed point theorem which is generalized the result of Wardowski ^[13].

Definition 1.2. (^[4]) Let $(X, d)$ be a complete metric space. A mapping $T : X \to X$ is called an $F$-contraction of Hardy-Rogers-type if there exist $\tau > 0$ and $F \in \mathcal{F}$ such that

holds for any $x, y \in X$ with $d(x, y) > 0$, where $\alpha, \beta, \gamma, \delta, L$ are non-negative numbers, $\gamma \not = 1$ and $\alpha + \beta + \gamma + 2 \delta = 1$.

By the above definition, Cosentino and Vetro given the following new fixed point theorem in complete metric space.

Theorem 1.3. (^[4]) Let $(X, d)$ be a complete metric space and let $T$ be a self-mapping on $X$. Assume that $T$ is an $F$-contraction of Hardy-Rogers-type. Then $T$ has a fixed point. Moreover, if $\alpha + \delta + L \le 1$, then the fixed point of $T$ is unique.

Inspired by the work of ^[4,10,11], we initiate the concept of an orthogonal generalized contraction of Hardy-Rogers-type mapping and establish some fixed point results for such contraction mapping in orthogonal metric spaces. Our main result is a generalization of Theorem 3.3 in ^[10].

2.   Preliminaries

Firstly, we introduce the following control functions.

Let $\Psi$ be the set of functions $\psi : [0, +\infty) \to (-\infty, 0)$ such that $\psi$ is upper semi-continuous from the right. And let $\mathcal{F}$ be the set of functions as in Definition 1.1.

Now, we give the following concepts, which are used in this paper.

Definition 2.1. Let $X \not = \emptyset$ and $\bot \subset X \times X$ be a binary relation. If $\bot$ satisfies the following condition :

then it is called an orthogonal set (briefly $O$-set) and $x_0$ is called an orthogonal element. We denote this $O$-set by $(X, \bot)$.

Definition 2.2. The triplet $(X, \bot, d)$ is called an orthogonal metric space if $(X, \bot)$ is an $O$-set and $(X, d)$ is a metric space.

Definition 2.3. Let $(X, \bot)$ be an $O$-set. A sequence $\{x_n\}$ is called an orthogonal sequence (briefly, $O$-sequence) if

Definition 2.4. Let $(X, \bot, d)$ be an orthogonal metric space. Then, a mapping $f : X \to X$ is said to be orthogonally continuous (or $\bot$-continuous) in $x \in X$ if, for each $O$-sequence $\{x_n\}$ in $X$ with $x_n \to x$ as $n \to \infty$, we have $f(x_n) \to f(x)$ as $n \to \infty$. Also, $f$ is said to be $\bot$-continuous on $X$ if $f$ is $\bot$-continuous in each $x \in X$.

Remark 2.1. By ^[5], we know that every continuous mapping is $\bot$-continuous, but the converse is not true.

Definition 2.5. Let $(X, \bot, d)$ be an orthogonal metric space. Then, $X$ is said to be orthogonally complete (briefly $O$-complete) if every Cauchy $O$-sequence is convergent.

Remark 2.2. By ^[5], we know that every complete metric space is $O$-complete and the converse is not true.

Definition 2.6. Let $(X, \bot)$ be an $O$-set. A mapping $f : X \to X$ is said to be $\bot$-preserving if $f(x) \bot f(y)$ whenever $x \bot y$. Also, $f : X \to X$ is said to be weakly $\bot$-preserving if $f(x) \bot f(y)$ or $f(y) \bot f(x)$ whenever $x \bot y$.

Let $(X, d)$ be a metric space. For a given mapping $T : X \to X$, set

where $\alpha, \beta, \gamma, \delta, L$ are non-negative numbers, $\gamma \not = 1$ and $\alpha + \beta + \gamma + 2 \delta = 1$.

Now we give the concept of orthogonal $(F, \psi)$-contraction of Hardy-Rogers-type as follows:

Definition 2.7. Let $(X, \bot, d)$ be an orthogonal metric space. A mapping $T : X \to X$ is said to be orthogonal $(F, \psi)$-contraction of Hardy-Rogers-type, if there exists $(F, \psi) \in \mathcal{F} \times \Psi$ such that

for all $(x, y) \in X \times X$ with $M_T(x, y) > 0$.

Remark 2.3. If $\psi \equiv - \tau$, $\alpha = 1$ and $\beta = \gamma = \delta = L = 0$, then Definition 2.7 reduced to the Definition 3.1 in ^[10].

3.   Main results

In this section, we will prove some fixed point theorems for an orthogonal $(F, \psi)$-contraction of Hardy-Rogers-type mapping in an orthogonal metric space.

Theorem 3.1. Let $(X, \bot, d)$ be an $O$-complete orthogonal metric space with an orthogonal element $x_0$ and $T : X \to X$ satisfying the following conditions:

(i) $T$ is $\bot$-preserving;

(ii) $T$ is an orthogonal $(F, \psi)$-contraction of Hardy-Rogers-type mapping;

(iii) $T$ is $\bot$-continuous.

Then $T$ has a fixed point. Moreover, if $\beta + L \le \gamma + \delta < 1$, then the fixed point of $T$ is unique.

Proof. By the definition of the orthogonality, we have that $x_0 \bot T(x_0)$ or $T(x_0) \bot x_0$. Let $x_n : = T x_{n-1} = \cdots = T^n x_0$, for all $n \in \mathbb{N}$. If $x_{n_*} = x_{n_*+1}$ for some $n^* \in \mathbb{N} \cup \{0\}$, then $x_{n_*}$ is a fixed point of $T$ and so the proof is completed. Thus, we may assume that $x_n \not = x_{n+1}$ for all $n \in \mathbb{N} \cup \{0\}$. So, we get $d(Tx_n, Tx_{n+1}) > 0$ for all $n \in \mathbb{N} \cup \{0\}$. Since $T$ is $\bot$-preserving, we have

which implies that $\{x_n\}$ is $O$-sequence. Since $T$ is an orthogonal $(F, \psi)$-contraction mapping, we have

For convenience, set $d_n = d(x_n, x_{n+1})$. Then, by (3.1) one has

which implies that

by $\psi \in \Psi$. It from the monotony of $F$, we obtain

By the conditions $\gamma + \delta < 1$ and $\alpha + \beta + \gamma + 2 \delta = 1$, we get $1-\gamma-\delta > 0$ and

by (3.3). So, the sequence $\{d_n\}$ is strictly decreasing, and $\lim_{n\to\infty} d_n = d$. Assume that $d > 0$. Since $\psi$ is upper semi-continuous from the right, there exists some $q \in \mathbb{N}$ such that

By (3.2), (3.4) and decreasing of $\{d_n\}$, for $n > q$, we have

Letting $n \to \infty$ in the inequality (3.5), we get

which implies from Definition 1.1 (ⅱ) that

In the following, we will prove that $\{d_n\}$ is a Cauchy sequence. By (3.6) and Definition (ⅲ), there exists some $k \in (0, 1)$ such that

From (3.5), we get

Let $n \to + \infty$, and using (3.6) and (3.7), we obtain that

which implies that there exists $n_1 \in \mathbb{N}$ such that

Using (3.9) and the triangle inequality, for $n \ge n_1$ and $m \in \mathbb{N^*}$, we have

Since $0 < k < 1$, the series $\sum \limits_{n = 1}^{\infty} \frac{1}{n^{1/k}}$ is converge, which yields $\{x_n\}$ is a Cauchy $O$–sequence in $X$. Since $X$ is $O$-complete, there exists $x_* \in X$ such that $x_n \to x_*$ as $n \to \infty$. Thus, we have

by $T$ is $\bot$-continuous. So, $x_*$ is a fixed point of $T$.

Moreover, we will show that $x_*$ is a unique fixed point of $T$. Suppose that $x^*$ is an another fixed point of $T$. If $x_n \to x^*$ as $n \to \infty$, one has $x_* = x^*$. If $x_n$ does not converge to $x^*$ as $n \to \infty$, there is a subsequence $\{x_{n_k}\}$ such that $T x_{n_k} \not = x^*$ for all $k \in \mathbb{N}$. By the choice of $x_0$ in the first of the proof, we have

Since $T$ is $\bot$-preserving and $T^n x^* = x^*$ for all $n \in \mathbb{N}$, we get

Since $T$ is an orthogonal $(F, \psi)$-contraction of Hardy-Rogers-type mapping, we obtain

By condition $\beta + L \le \delta + \gamma$, one has $1-\beta-L > 0$, and by $\alpha + \beta + \gamma + 2 \delta = 1$ we have $\alpha + 2 \beta + \delta + L \le 1$. Thus, by (3.10) and the monotony of $F$, we obtain

Let $\mu_{n_k} : = d(x_{n_k}, x^*)$, (3.11) implies that the sequence $\{\mu_{n_k}\}$ is strictly decreasing. Thus, $\lim_{k \to \infty} \mu_{n_k} = \mu \ge 0$. Similar to the proof of (3.4), if $\mu > 0$, then there exists some $p \in \mathbb{N}$ such that

By (3.10), (3.12) and the decreasing of $\mu_{n_k}$, for $k > p$, we get

which yields that $F(\mu_{n_k}) \to - \infty$ as $k \to \infty$ and so it follows from the condition of Definition 1.1 (ⅱ) that $d(x_{n_k}, x^*) = \mu_{n_k} \to 0$ as $k \to \infty$. This implies that $x_n \to x^*$ as $n \to \infty$, which is a contradiction. Thus, $T$ has a unique fixed point.

Remark 3.1. If $\psi \equiv - \tau$, $\alpha = 1$ and $\beta = \gamma = \delta = L = 0$, then Theorem 3.1 reduces to the Theorem 3.3 in ^[10].

We now illustrate Theorem 3.1 by the following example.

Example 3.1. Let $X = [0, \infty)$ be endowed with the metric $d(x, y) = |x - y|$ for all $x, y \in X$.

Define the sequence $\{S_n\}$ by $S_n = n 2^n$ for all $n \in \mathbb{N} \cup \{0\}$. Define the orthogonality relation $\bot$ on $X$ by

Clearly, $(X, \bot, d)$ is an $O$-complete metric space.

Consider the mapping $T : X \to X$ given as

Similar to Example in ^[10], it is easy to check that $T$ is $\bot$-continuous and $T$ is $\bot$-preserving. Now we will show that $T$ is an orthogonal $(F, \psi)$-contraction of Hardy-Rogers-type mapping. Let $x, y \in X$ with $x \bot y$ and $d(Tx, Ty) > 0$. Without loss of generality, we may assume that $x < y$. This implies that $x \in \{S_0, S_1\}$ and $y = S_k$ for some $k \ge 2$. Thus, we get

Taking $\alpha = \frac{1}{3}$, $\beta = \gamma = L = 0$ and $\delta = \frac{1}{3}$. Then

Hence we have

which implies that

with $F(t) = t + \ln t$ and $\psi \equiv - 2$. This means that $T$ is an orthogonal $(F, \psi)$-contraction of Hardy-Rogers-type mapping. Hence, all the conditions of Theorem 3.1 are satisfied and so $T$ has a unique fixed point $x = S_0$.

Remark 3.2. Theorem 3.1 is a powerful generalization of some fixed point theorems in orthogonal complete orthogonal metric space, the result of Example 3.1 can not be obtained by those classical fixed point theorems.

Acknowledgments

The authors thanks editor and anonymous referees for their remarkable comments, suggestion that help to improve this paper. The authors thank the financial support from Natural Science Foundation of China (No. 11571136).

Conflict of interest

The authors declare that there are no conflicts of interest.