Abstract

In this paper, we propose a new class of orthogonal F- type contractive mappings, and prove one common fixed point theorem in complete orthogonal b- metric spaces. We also provide an example that supports our result.

1. Introduction

Fixed point theory is an important part of modern analysis. In particular, Banach contraction mapping principle ¹ is an effective method to solve the problem of the existence and uniqueness of fixed points in complete metric space, and plays an important role in nonlinear analysis. About a century ago, Banach started as an abstract successive approximation method for solving differential equations, and later defined it as the concept of contraction mapping. Thus, the first fixed point theorem was produced. Later, many scholars gave some important generalizations of this result by changing the space type or contractive conditions. Istratescu ^{2, 3} provided one of the most important ideas of convex contraction and proved some fixed point results. Another interesting extension of fixed point theory, known as "almost contraction map", was introduced by Berinde ⁴. In contrast, there are multiple ways in which this concept of measurement has developed. In 1993, Czerwik ⁵ gave a generalized concept of metric spaces, called metric spaces, by changing the form of trigonometric inequality defined by metric spaces, and the author also proved some new fixed point theorems in this kind of spaces. Afterwards, many scholars carried out researches and got a lot of excellent results in this kind of space (see ^{6, 7, 8, 9}), and the literatures cited therein. In 2012, Wardowski ⁹ gave a new type of compression mapping in complete metric space. That is, type contraction, and some sufficient conditions for the existence and uniqueness of fixed point of this type of mapping are obtained. Recently, Gordji et al. ¹⁰ introduced the concept of orthogonality, and proved the fixed point theorem in orthogonal complete metric space. In 2022, Eiman et al. ¹¹ introduced the concept of orthogonal contraction mappings and proved the fixed point theorem. Also in 2022, Dhanraj et al. ¹² adopted the orthogonal Geraghty type for admissible contraction mapping, fixed-point theorem are proved on orthogonal complete Branciari metric spaces. In 2023, many researchers have deeply studied different types of contraction mapping based on complete orthogonal spaces, and have given applications (see ^{13, 14, 15, 16}). In addition, many researchers have improved and generalized the concept of orthogonal metric spaces (see ^{17, 18, 19}).

In this paper, we propose a new class of contraction for double mappings of square and quadratic forms, and prove some fixed point theorems in an orthogonal complete metric space. Meanwhile, we provide a specific example to demonstrate the effectiveness of the result.

2. Preliminaries

Definition 2.1. Supposeis a constant andis a nonempty set. A function is said to be a metric if for any ,

Generally, is called a metric space.

Definition 2.2. Supposeis a metric space,andis a sequence in.

is convergent inand converges to, if for each, there existssuch that for all We denote this as or as

is a Cauchy sequence in , if for eachthere existssuch that

for all

Definition 2.3. Let be a nonempty set and be a binary relation. Ifholds with the constraintthen is said to be an orthogonal set (briefly set ).

Definition 2.4. Letbe an orthogonal metric space. Then, is said to be complete if every orthogonal Cauchy sequence is convergent.

Definition 2.5. A tripledis called an if is an orthogonal set and is a metric space.

Definition 2.6. Let be an orthogonal set. A sequence is called an orthogonal sequence (sequence) if

Definition 2.7. Suppose is an Then, is said to be orthogonally continuous at if, for each sequence inwith,we have Also, is said to be orthogonal continuous onif is orthogonal continuous at each

Definition 2.8. Letbe a nonempty set, andbe two self mappings on. andare called a pair of weakly compatible mappings, if they are commutative at each coincidence point, that is,

Definition 2.9. Letbe an orthogonal set. A functionis called an orthogonal-preserving mapping ifwhenever

Definition 2.10. Let be a complete metric space with parameter and . Then, is said to beadmissible, if with ,

Hypothesis 2.11. Let be a complete metric space with parameterlet be a function.

If is a sequence insuch that as then there exists a subsequence of with for all

For allwe have

For allwe have

Definition 2.12. Letdenote the family of all functionssatisfying the following properties:

is strictly increasing；

for each sequence of positive numbers, we have

there existssuch that

If we have

Lemma 2.13. Let be a metric space with parameter Assume thatandare convergent to and respectively. Then, we have

In particular, if then we have Moreover, for eachwe have

3. Main Results

Theorem 3.1. Letbe an orthogonal complete metric space, with parameter Suppose satisfy the following conditions:

(1) is orthogonal continuous, are weakly compatible；

(2) , and is closed；

(3) is a admissible mapping；

(4) are orthogonal preserving；

(5) there is an orthogonal element satisfying ；

(6) If ,we have：

, (1)

where is a function such that, is a constant, satisfying that

, and properties and. Thenandpossess a common fixed point in . Moreover, possess a unique common fixed point in

.

Proof：By the definition of orthogonality, we find that with or , for all . Since, there exists , such that In turn define sequences and in by for

Since andis orthogonal-preserving, without loss of generality, then we obtain and It follows from and are orthogonal-preserving that Thus, we have , which imply that are orthogonal sequences.

For orthogonal element, in light of condition, we obtain ,

Hence, for all we deduce Replacingby and by in , we have

That is,

Since ,

then

whereIf , then we have

Since is strictly increasing, and , this is a contradiction. Thus, and the inequality becomes

.

According to , we get

By calculation, we get

Obtained through organization

In, letting ,we have

.

Thus, .

According to,there exists such that.

In , multiplying , we have

Taking in (4), we have

Hence there exists such thatand as .

Next, we are going to prove is Cauchy. For ease of use, set

So

and

Since ,and,, then

and Therefore, there exists such that

Since is closed, there is a satisfying Next, we will prove thatIn view of the property, one can get a subsequenceof with for all

Since , andis orthogonal continuous, we have In view of the conditionandis orthogonal preserving, one can deduce that

Because is orthogonal preserving and , we have . Sincethus

Replacingbyandbyin, we have

Sinceis strictly increasing,we get

Letting , from Lemma 2.13, we obtain

This is a contradiction. So and.

Since are weakly compatible, one can get By the continuity of , we haveTherefore,, that is, and possess a common fixed point in .

Next, we will prove that and possess a unique common fixed point in

.

First, is nonempty set, because

If there exists and is a common fixed point of , then Replacing by and by in ,

We have

Sinceandis strictly increasing, then a contradiction. It follows that That is, possess a unique common fixed point in .

Example 3.2 Let and be a mapping defined by , for all Define the binary relation on by if where

Then is an complete metric space. Define the mappings by

Clearly, are orthogonal preserving, is orthogonal continuous, are weakly compatible,,and is closed. Now, let us consider the mapping defined by

.

Let . If, we have

If , we have So is orthogonal element in . It is easy to show that

which imply that is anadmissible mapping. Next we show that are orthogonal preserving.

Case 1: . We have

Case 2: . We obtain

Case 3: . Clearly,

Case 4: . It is obvious that

Hence, are orthogonal preserving.

Consider

Case 1: . Obviously,

It is clear that is satisfied.

Case 2: . It is easy to show

That is condition holds.

Case 3: or . Then .

Hence, (1) fulfills. Therefore, all the conditions of Theorem 3.1 are satisfied. Therefore, one can conclude that and possess a common fixed point in. Obviously, a common fixed point.

4. Conclusions

In this paper, we proved a fixed point theorem of a new class of orthogonal type contractive mappings, in orthogonal metric space. In addition, we also provided an example to explain in detail the practicality of the obtained results.

References