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Research Article

Open Access Peer-reviewed

Qiancheng Wang, Hongyan Guan^{ }

Received August 01, 2023; Revised September 01, 2023; Accepted September 08, 2023

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.

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.

**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

**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 ,

**H****ypothesis**** ****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

**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.

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.

[1] | Banach, S. Sur les operations dans les ensembles abstraits et leurs applications aux equations integrals [J]. Fundam. Math., 1992, 3, 133–181. | ||

In article | |||

[2] | Istratescu, V. Some ﬁxed point theorems for convex contraction mappings and mappings with convex diminishing diameters (I) [J]. Ann. Mat. Pure Appl, 1982, 130, 89–104. | ||

In article | View Article | ||

[3] | Istratescu, V. Some ﬁxed point theorems for convex contraction mappings and mappings with convex diminishing diameters (II) [J]. Ann. Mat. Pura Appl, 1983, 134, 327–362. | ||

In article | View Article | ||

[4] | Berinde, V. Approximating ﬁxed points of weak contractions using the Picard iteration [J]. Nonlinear Anal Forum, 2004, 9, 43–53. | ||

In article | View Article | ||

[5] | Czerwik, S. Contraction mappings in b- metric spaces [J]. Acta Math Inform Univ Ostrav,1993,1:5-11. | ||

In article | View Article | ||

[6] | Abbas, J. Common fixed point of four maps in b- metric spaces.[J]. Hacet. J Math. Stat., 2014, 43(4): 613-624. | ||

In article | |||

[7] | Aydi, H., Bota, M.F., Karapinar, E., et al. A commonfixed point for weak の- contractions on b- metric spaces [J]. Fixed Point Theory, 2012, 13(2): 337-346. | ||

In article | |||

[8] | Suzuki, T. A new type of fixed point theorem in metric spaces [J]. Nonlinear Anal., 2009, 71(11): 5313-5317. | ||

In article | View Article | ||

[9] | Wardowski, D. Fixed points of a new type of contractive mappings in complete metric spaces [J]. Fixed Point Theory Appl., 2012, 2012(1): 1-6. | ||

In article | View Article | ||

[10] | Gordji, M.E., Habibi, H. Fixed point theory in generalized orthogonal metric space. [J]. Linear Topol. Algebra, 2017, 6, 251–260. | ||

In article | View Article | ||

[11] | Aiman, M., Arul Joseph, G., Absar, U.H., Senthil Kumar, P., Gunaseelan, M., Imran, A.B. Solving an integral equation via orthogonal Brianciari metric spaces. [J]. J Funct. Spaces, 2022, 2022, 7251823. | ||

In article | View Article | ||

[12] | Dhanraj, M., Gnanaprakasam, A.J., Mani, G., Ege, O., De la Sen, M. Solution to integral equation in an o- complete Branciari b- metric spaces, Axioms, 2022,11(12): 728, 1-14. | ||

In article | |||

[13] | Al-Mazrooei, A.E., Ahmad, J. Fixed point approach to solve nonlinear fractional differential equations in orthogonal F- metric spaces, Aims Math., 2023,8(3): 5080-5098. | ||

In article | View Article | ||

[14] | Gardasevic-Filipovic, M., Kukic, K., Gardasevic, D., Mitrovic, Z.D. Some best proximity point results in the orthogonal o- complete b- metric - like spaces, J Contemp. Math. Anal. , 2023,58, 105-115. | ||

In article | View Article | ||

[15] | Gnanaprakasam, A.J., Mani, G., Ege, O., Aloqaily, A., Mlaiki, N. New fixed point results in orthogonal b- metric spaces with related applications, Mathematics, 2023,11(3): 677, 1-18. | ||

In article | View Article | ||

[16] | Prakasam, S.K., Gnanaprakasam, A.J., Mani, G., Jarad, F. Solving an integral equation via orthogonal generalized a-Ψ- Geraghty contractions, Aims Mathematics, 2023, 8(3): 5899-5917. | ||

In article | View Article | ||

[17] | Gnanaprakasam, A.J., Nallaselli, G., Haq, A.U., Mani, G., Baloch, I.A., Nonlaopon, K. Common fixed-points technique for the existence of a solution to fractional integro differential equations via orthogonal branciari metric spaces [J]. Symmetry, 2022, 14, 1859. | ||

In article | View Article | ||

[18] | Prakasam, S.K., Gnanaprakasam, A.J., Kausar, N., Mani, G., Munir, M. Solution of integral equation via orthogonally modiﬁed F- contraction mappings on o- complete metric-like space [J]. Int. J. Fuzzy Log. Intell. Syst, 2022, 22, 287–295. | ||

In article | View Article | ||

[19] | Senthil Kumar, P., Arul Joseph, G., Ege, O., Gunaseelan, M., Haque, S., Mlaiki, N. Fixed point for an OgF-c in o- complete b- metric-like spaces [J]. Aims Math., 2022, 8, 1022–1039. | ||

In article | View Article | ||

Published with license by Science and Education Publishing, Copyright © 2023 Qiancheng Wang and Hongyan Guan

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Qiancheng Wang, Hongyan Guan. Common Fixed Point of *F-* type Contractive Mappings in Generalized Orthogonal Metric Spaces. *American Journal of Applied Mathematics and Statistics*. Vol. 11, No. 2, 2023, pp 77-82. http://pubs.sciepub.com/ajams/11/2/6

Wang, Qiancheng, and Hongyan Guan. "Common Fixed Point of *F-* type Contractive Mappings in Generalized Orthogonal Metric Spaces." *American Journal of Applied Mathematics and Statistics* 11.2 (2023): 77-82.

Wang, Q. , & Guan, H. (2023). Common Fixed Point of *F-* type Contractive Mappings in Generalized Orthogonal Metric Spaces. *American Journal of Applied Mathematics and Statistics*, *11*(2), 77-82.

Wang, Qiancheng, and Hongyan Guan. "Common Fixed Point of *F-* type Contractive Mappings in Generalized Orthogonal Metric Spaces." *American Journal of Applied Mathematics and Statistics* 11, no. 2 (2023): 77-82.

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[1] | Banach, S. Sur les operations dans les ensembles abstraits et leurs applications aux equations integrals [J]. Fundam. Math., 1992, 3, 133–181. | ||

In article | |||

[2] | Istratescu, V. Some ﬁxed point theorems for convex contraction mappings and mappings with convex diminishing diameters (I) [J]. Ann. Mat. Pure Appl, 1982, 130, 89–104. | ||

In article | View Article | ||

[3] | Istratescu, V. Some ﬁxed point theorems for convex contraction mappings and mappings with convex diminishing diameters (II) [J]. Ann. Mat. Pura Appl, 1983, 134, 327–362. | ||

In article | View Article | ||

[4] | Berinde, V. Approximating ﬁxed points of weak contractions using the Picard iteration [J]. Nonlinear Anal Forum, 2004, 9, 43–53. | ||

In article | View Article | ||

[5] | Czerwik, S. Contraction mappings in b- metric spaces [J]. Acta Math Inform Univ Ostrav,1993,1:5-11. | ||

In article | View Article | ||

[6] | Abbas, J. Common fixed point of four maps in b- metric spaces.[J]. Hacet. J Math. Stat., 2014, 43(4): 613-624. | ||

In article | |||

[7] | Aydi, H., Bota, M.F., Karapinar, E., et al. A commonfixed point for weak の- contractions on b- metric spaces [J]. Fixed Point Theory, 2012, 13(2): 337-346. | ||

In article | |||

[8] | Suzuki, T. A new type of fixed point theorem in metric spaces [J]. Nonlinear Anal., 2009, 71(11): 5313-5317. | ||

In article | View Article | ||

[9] | Wardowski, D. Fixed points of a new type of contractive mappings in complete metric spaces [J]. Fixed Point Theory Appl., 2012, 2012(1): 1-6. | ||

In article | View Article | ||

[10] | Gordji, M.E., Habibi, H. Fixed point theory in generalized orthogonal metric space. [J]. Linear Topol. Algebra, 2017, 6, 251–260. | ||

In article | View Article | ||

[11] | Aiman, M., Arul Joseph, G., Absar, U.H., Senthil Kumar, P., Gunaseelan, M., Imran, A.B. Solving an integral equation via orthogonal Brianciari metric spaces. [J]. J Funct. Spaces, 2022, 2022, 7251823. | ||

In article | View Article | ||

[12] | Dhanraj, M., Gnanaprakasam, A.J., Mani, G., Ege, O., De la Sen, M. Solution to integral equation in an o- complete Branciari b- metric spaces, Axioms, 2022,11(12): 728, 1-14. | ||

In article | |||

[13] | Al-Mazrooei, A.E., Ahmad, J. Fixed point approach to solve nonlinear fractional differential equations in orthogonal F- metric spaces, Aims Math., 2023,8(3): 5080-5098. | ||

In article | View Article | ||

[14] | Gardasevic-Filipovic, M., Kukic, K., Gardasevic, D., Mitrovic, Z.D. Some best proximity point results in the orthogonal o- complete b- metric - like spaces, J Contemp. Math. Anal. , 2023,58, 105-115. | ||

In article | View Article | ||

[15] | Gnanaprakasam, A.J., Mani, G., Ege, O., Aloqaily, A., Mlaiki, N. New fixed point results in orthogonal b- metric spaces with related applications, Mathematics, 2023,11(3): 677, 1-18. | ||

In article | View Article | ||

[16] | Prakasam, S.K., Gnanaprakasam, A.J., Mani, G., Jarad, F. Solving an integral equation via orthogonal generalized a-Ψ- Geraghty contractions, Aims Mathematics, 2023, 8(3): 5899-5917. | ||

In article | View Article | ||

[17] | Gnanaprakasam, A.J., Nallaselli, G., Haq, A.U., Mani, G., Baloch, I.A., Nonlaopon, K. Common fixed-points technique for the existence of a solution to fractional integro differential equations via orthogonal branciari metric spaces [J]. Symmetry, 2022, 14, 1859. | ||

In article | View Article | ||

[18] | Prakasam, S.K., Gnanaprakasam, A.J., Kausar, N., Mani, G., Munir, M. Solution of integral equation via orthogonally modiﬁed F- contraction mappings on o- complete metric-like space [J]. Int. J. Fuzzy Log. Intell. Syst, 2022, 22, 287–295. | ||

In article | View Article | ||

[19] | Senthil Kumar, P., Arul Joseph, G., Ege, O., Gunaseelan, M., Haque, S., Mlaiki, N. Fixed point for an OgF-c in o- complete b- metric-like spaces [J]. Aims Math., 2022, 8, 1022–1039. | ||

In article | View Article | ||