A new common fixed point theorem on orthogonal metric spaces and an application Текст научной статьи по специальности «Математика»

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У. ТоиаП, А. Jaid, Е1 Мо^ашакП, Новая общая теорема о неподвижной точке в ортогональных метрических пространствах и ее приложение, Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки, 2023, номер 4, 737-744

001: 10.14498/уБ^и1998

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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki

[J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2023, vol. 27, no. 4, pp. 737-744 ISSN: 2310-7081 (online), 1991-8615 (print) d https://doi

MSC: 47H10, 54H25

A new common fixed point theorem on orthogonal metric spaces and an application

Y. Touail, A. Jaid, D. El Moutawakil

Université Sultan Moulay Slimane, Beni-Mellal, 23000, Morocco.

Abstract

In the present work, a common fixed point result for self-mappings on orthogonal complete metric spaces, which are not necessarily complete, is proved. Furthermore, as an application, we find the existence of solutions to two differential equations.

Keywords: common fixed point, orthogonal metric space.

Received: 28th January, 2023 / Revised: 17th November, 2023 / Accepted: 13th December, 2023 / First online: 25th December, 2023

.org/10.14498/vsgtu1998

1. Introduction and Preliminary. The pioneering mathematician in the area of fixed point theory was Banach, who established and proved the first fixed point theorem is named the Banach contraction theorem [1]. After that, extensions of this theorem have been obtained either by generalizing the distance properties of the underlying metric space or by modifying the contractive condition on the mappings.

In 2017, Eshaghi Gordji et al. [2] defined orthogonal metric spaces as a generalization of metric spaces, as follows:

Differential Equations and Mathematical Physics Short Communication

© Authors, 2023

© Samara State Technical University, 2023 (Compilation, Design, and Layout) 3 ©® The content is published under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/) Please cite this article in press as:

Touail Y., Jaid A., El Moutawakil D. A new common fixed point theorem on orthogonal metric spaces and an application, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2023, vol. 27, no. 4, pp. 737-744. EDN: KPNGNF. DOI: 10.14498/vsgtu1998. Authors' Details:

Youssef Touail https://orcid.org/0000-0003-3593-8253 MATIC, Faculté Polydisciplinaire de Khouribga; e-mail: [email protected]; [email protected] Amine Jaid® https://orcid.org/0000-0001-7322-2008

MATIC, Faculté Polydisciplinaire de Khouribga; e-mail: [email protected]

Driss El Moutawakil © https://orcid.org/0000-0001-7322-2008

MATIC, Faculté Polydisciplinaire de Khouribga; e-mail: [email protected]

Definition 1 [2]. Let X = 0 and let ±c X x X be a binary relation. If ± satisfies the following hypothesis:

: (Vy, y ± xo) or (Vy, xo ± y).

Then (X, ±) is called an orthogonal set (briefly O-set).

The triplet (X, ±, d) is called an orthogonal metric space if (X, d) is a metric space and (X, ±) is an O-set, and x0 is said to be an orthogonal element.

Then, an important extension of the Banach fixed point principal is given, as follows:

Theorem 1 [2]. Let (X, ±,d) be an O-complete metric space and T a self-mapping on X which is ±-preserving and ±-continuous. If there exists k G [0,1) such that for all x,y G X:

x ± y implies d(Tx,Ty) ^ kd(x,y). (1)

Then, T has a unique fixed point.1

Later, many remarkable works in this area can be found in [3-5].

Motivated by [2] and other works concerning the theory of common fixed points [6-9], in this paper we restrict our studies only to orthogonal elements, to prove a result of common fixed points in a new setting and under weak conditions. In other words, we extend condition (1) to two self-mappings f, g : X ^ X, as follows:

x ± y implies d(fx,gy) ^ <fi(d(x,y)),

where 0 G the class of all nondecreasing selfmaps 0 on [0, +rc>) satisfying E+=°1 0n(t) < +œ for all t > 0.

Moreover, an extension of the Banach fixed point theorem is delivered for a large class of mappings, we call it weakly-±-preserving.

In addition, we give an example to support the proven theorem and to show the usability of this new direction of research.

At the end of the results, an application to the study of the existence of common solutions for a class of differential equations is presented.

Finally, we assert some definitions that will be needed in the topic:

Definition 2 [2]. Let (X, ±) be an O-set. A mapping T : X ^ X is said to be ^-preserving if Tx ± Ty whenever x ± y.

Definition 3 [2]. Let (X, ±) be an O-set. A sequence {xn} is called an orthogonal sequence (briefly, O-sequence) if

(Vn, xn ± xn+i) or (Vn, Xn+1 ± xn).

Definition 4 [2]. Let (X, ±,d) be an orthogonal metric space. Then, a mapping T : X ^ X is said to be orthogonally continuous (briefly ^-continuous) in x G X, if for each O-sequence {xn} c X such that xn ^ x as n ^ œ, we obtain as n ^ œ. In addition, T is said to be ^-continuous on X if T is ^-continuous in each x G X.

xIn the sequel, we will recall the related basic notions of orthogonality.

Definition 5 [2]. Let (X, ±,d) be an orthogonal metric space. Then, X is said to be orthogonally complete (or ^-complete) if every Cauchy O-sequence is convergent.

Remark 1 [2]. Every complete metric space (continuous mapping) is O-complete metric space (^-continuous mapping) and the converse is not true.

Example 1 [2]. Let X = Z. Define the binary relation ± in X by m ± n if there exists k e Z such that m = kn. It is easy to see that 0 ± n for all n e Z. Hence, (X, is an O-set.

2. Main results. The main result of this article is the following:

Theorem 2. Let (X, ±,d) be an O-complete metric space and f, g : X ^ X be ±-continuous mappings such that:

1) x ± y (fx ± gy or gy ± fx) and (gx ± fy or fy ± gx);

2) x ± y d(gx, fy) ^ <fi(d(x, y)), for all x,y e X, where $ e Then, f, g have a common fixed point.

Proof. Since X is an O-set, there exists at least x0 e X such that

Vy e X, x0 ^ y or Vy e X, y ± x0.

So, in particular we have x0 ^ fx0 or fx0 ^ x0. We can choose a sequence {xn} defined by x2n+i = fx2n and x2n+2 = gx2n+i for all n e N*. The condition 1) implies

Vn e N*, xn ± xn+\ or Vn e N*, xn+\ ± xn.

Then, {xn} is an O-sequence. Hence, we have

d(x2n+i,x2n+2) = d(fx2n,gx2n+i) ^ ^(d(x2n,x2n+i)).

Similarly, we have

d(X2n+2,X2n+3) = d(fX2n+2,gX2n+\) ^ $(d(X2n+l, X2n+2)).

Therefore,

d(xn,xn+i) ^ <ft(d(xn-i,xn)) ^ 4>2(d(xn-2,xn-i)) ^ ••• ^ 4>a(d(xo,xi)), for all n e N. Let n, m e N*, we have

k=n+m-i k=n+m-i

d(xn,xn+m) ^ ^ d(xk,xk+i) ^ ^ 4>k(d(xk,xk+i)). (2)

k=n k=n

Letting n, m ^ œ in (2), we deduce that {xn} is an Cauchy O-sequence. Since X is an O-complete space there exists u G X such that lim On the other

side, the orthogonal continuity of f, g implies lim fxn — fu and lim gxn — gu,

which leads to u — fu — gu. □

Example 2. Let X — Q and d(x,y) — \x — y\ for all x, y G X is the usual metric on X.

Define a binary relation on X by

x ^y ^^ x = 0 or y = 0.

Note that (X, ±, d) is not a complete metric space, but is an O-complete metric space.

Consider the mappings f, g : X ^ X defined by

, ) = \ 1, if x = 1,

I(X)\x/3, if x = 1,

and

( ) = I 1, if x = 1,

9(x) = \x/2, if x = 1.

Without loss of generality, let xn ± xn+\ for each n e N. Then we have xn = 0, which leads to fxn = xn/3 = 0 = f0 and gun = xn/2 = 0 = g0. Therefore f, g are ^-continuous mappings.

Clearly, the mappings f,g satisfy the condition 1) of Theorem 2. On the other hand, let < be a function defined by <fi(t) = 31/4, for all t > 0. Let x,y e X such that x ± y, we obtain x = 0 or y = 0. Case 1: If x = 0, we have

d(f 0,gy) = ^ < 3d(0, y) ^<(d(0, y)). Case 2: If = 0, we have

x 3

d(fx, g0) = ix ^ -d(x, 0) < <(d(x, 0)).

Then, all assumptions of Theorem 2 are satisfied and 1 = f 1 = g1 is the common fixed point.

Now, we introduce a new definition named weakly-±-preserving self-mapping: Definition 6. Let (X, ±) be an O-set. A mapping T : X ^ X is said to be weakly-^-preserving if Tx ± Ty or Ty ± Tx whenever x ± y.

Remark 2. It is clear that a ^-preserving mapping is a weakly-±-preserving mapping, but in general the converse is not true.

Example 3. Let X = [0,1], define the function Tx = 1 — x, x e X. Define a binary relation IcX x X by

x ±y ^^ x ^ y.

Therefore, (X, ±) is an O-set with the orthogonal element xo = 0.

We have 0 ± 1, T0 = 1 and T 1 = 0, thus T1 ± T0 but T0 ± T1 does not hold.

Thus, the mapping T is weakly-±-preserving, but not ^-preserving.

By taking g = f in our main theorem, we obtain a new generalization of Theorem 1.

Theorem 3. Let (X, ±,d) be an O-complete metric space and T be a self-mapping on X which is weakly-^-preserving and ^-continuous. If there exists e $ such that for all x,y e X, we have

x ± y implies d(Tx,Ty) ^ <fi(d(x,y)).

Then, T has a unique fixed point.

3. Application. In this section, we will prove the existence of a common solution for the two differential equations:

IV(t) = k(t,x(t)), t e I =[0,d],d e (1, +to); (3)

I x(1) = a, a ^ 2, ()

and

{^ (i) — k(i n + f k(n i c T — i0 ft] fid (1

(4)

x'(t) = , a + J^ k(u,x(u)duj, t £ I = [0 , d], 6 £ (1 ,+<x); ^x(1) = a, a ^ 2,

where x £ C(I), the space of all continuous functions from I into R and k : I xR ^ R is a continuous mapping.

Let X = [x £ C(I)/x(t) ^ 1} endowed by the metric

d(x,y) = sup lx(t) - y(t) I. tei

Define the mappings f, g : X ^ X, as follows:

fx(t) = a + J k(s,x(s))ds, (5)

and

gx(t) = a + k(^s,a + k(u,x(u))d^jds, (6)

for all t £ I.

Hence, equations (3) and (4) have a common solution if and only if the mappings f and g have a common fixed point.

Theorem 4. Let f, g : X ^ X be the mappings defined by (5) and (6). Assuming that the following conditions are satisfied:

1) k(t, x) ^ 0 for all x ^ 0 and t £ I;

2) there exists h < 1 such that for all x,y £ X, we have

h

Ik( • ,fx) - k( • ,y)I < — |x - yl, (7)

for any x,y £ C(I), with xy ^ y or xy ^ x.

Then, the differential equations (3) and (4) have a positive common solution. Proof. Let x,y £ X. Define an orthogonal relation ± on X by

x ± y ^^ x(t)y(t) ^ y(t) or x(t)y(t) ^ x(t), for all t £ I. (8)

It is clear that (X, ±, d) is a O-complete metric space.

Let x, y e C(I) be such that x ±y, since fx(t), gy(t) ^ 2, then fx(t)gy(t) ^ gy(t) and gx(t) fy(t) ^ f y(t), which means that condition 1) of Theorem 2 holds. Also, from the definitions of f and g, we see that f, g are ^-continuous.

On the other hand, we will show that the contraction 2) of Theorem 2 is satisfied.

By considering (7) and (8), we obtain

\gx(t) — fy(t)\ ^J \k(s,fx(s)) — k(s, y(s))\ds <

f° h

^J \x(s) — y(s)\rls ^ hd(x, y).

So

d(gx,f y) < <(d(x, y)).

where <(t) = ht, with h < 1.

Finally, we conclude by Theorem 2 that the differential equations (3) and (4) have a positive common solution. □

Remark 3. In the above theorem, the function x0(t) = 2 for all t e I is an orthogonal element.

Competing interests. On behalf of all authors, the corresponding author states that there is no conflict of interest.

Authors' contributions and responsibilities. Each author has participated in the development of the concept of the article and in the writing of the manuscript. The authors are absolutely responsible for submitting the final manuscript in print. Each author has approved the final version of the manuscript. Data availability. No data were used to support this study.

References

1. Banach S. Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales, Fund. Math., 1922, vol. 3, no. 1, pp. 133-181. http://eudml.org/doc/213289.

2. Gordji M. E., Rameani M., De La Sen M., Cho Y.J. On orthogonal sets and Banach fixed point theorem, Fixed Point Theory, 2017, vol.18, no. 2, pp. 569-578. DOI: https://doi. org/10.24193/fpt-ro.2017.2.45.

3. Baghani H., Eshaghi Gordji M., Ramezani M. Orthogonal sets: The axiom of choice and proof of a fixed point theorem, J. Fixed Point Theory Appl., 2016, vol. 18, pp. 465-477. DOI: https://doi.org/10.1007/s11784-016-0297-9.

4. Khalehoghli S., Rahimi H., Eshaghi Gordji M. Fixed point theorems in ^-metric spaces with applications, AIMS Mathematics, 2020, vol. 5, no. 4, pp. 3125-3137. DOI: https://doi.org/ 10.3934/math.2020201.

5. Touail Y., El Moutawakil D. Fixed point theorems on orthogonal complete metric spaces with an application, Int. J. Nonl. Anal. Appl., 2021, vol.12, no. 2, pp. 1801-1809. DOI: https://doi.org/10.22075/ijnaa.2021.23033.2464.

6. Jaid J., Touail Y., El Moutawakil D. On ^^-contraction: New types of fixed points and common fixed points in Banach spaces, Asian-European J. Math., 2023, vol.16, no. 11, 2350198. DOI: https://doi.org/10.1142/S179355712350198X.

7. Touail Y., El Moutawakil D. ew common fixed point theorems for contractive self mappings and an application to nonlinear differential equations, Int. J. Nonlinear Anal. Appl., 2021, vol. 12, no. 1, pp. 903-911. DOI: https://doi.org/10.22075/IJNAA.2021.21318.2245.

8. Touail Y., El Moutawakil D. Some new common fixed point theorems for contractive selfmappings with applications, Asian-European J. Math., 2022, vol.15, no. 4, 2250080. DOI: https://doi.org/10.1142/S1793557122500802.

9. Touail Y., Jaid A., El Moutawakil D. A note on common fixed point theorems in a bounded metric space, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2023, vol.27, no. 2, pp. 241-249. EDN: ZXSBPZ. DOI: https:// doi.org/10.14498/vsgtu1940.

Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки. 2023. Т. 27, № 4. С. 737-744_

ISSN: 2310-7081 (online), 1991-8615 (print) d https://doi.org/10.14498/vsgtu1998

EDN: KPNGNF

УДК 517.988.523

Новая общая теорема о неподвижной точке в ортогональных метрических пространствах и ее приложение

Y. Touail, A. Jaid, D. El Moutawakil

Университет Султана Мулая Слимана, Бени-Меллаль, 23000, Марокко.

Аннотация

Доказывается общий результат о неподвижной точке для самоотображений на ортогональных полных метрических пространствах, которые не обязательно полны. В качестве приложения полученного результата найдено существование решений двух дифференциальных уравнений.

Ключевые слова: общая неподвижная точка, ортогональное метрическое пространство.

Получение: 28 января 2023 г. / Исправление: 17 ноября 2023 г. / Принятие: 13 декабря 2023 г. / Публикация онлайн: 25 декабря 2023 г.

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Touail Y., Jaid A., El Moutawakil D. A new common fixed point theorem on orthogonal metric spaces and an application, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2023, vol. 27, no. 4, pp. 737-744. EDN: KPNGNF. DOI: 10.14498/vsgtu1998.

Сведения об авторах

Youssef Touail https://orcid.org/0000-0003-3593-8253 MATIC, Faculte Polydisciplinaire de Khouribga; e-mail: [email protected]; [email protected] Amine Jaid® https://orcid.org/0000-0001-7322-2008

MATIC, Faculte Polydisciplinaire de Khouribga; e-mail: [email protected]

Driss El Moutawakil © https://orcid.org/0000-0001-7322-2008

MATIC, Faculte Polydisciplinaire de Khouribga; e-mail: [email protected]