Расстояние Wt-над метрическим пространством b- Текст научной статьи по специальности «Математика»

ЧЕБЫШЕВСКИЙ СБОРНИК

Том 25. Выпуск 1.

УДК 511.524 DOI 10.22405/2226-8383-2024-25-1-155-163

Расстояние Wt— над метрическим пространством Ь—

Э. Альмухур, М. Кусини, А. Абир, А. Манал

Альмухур Эман — факультет фундаментальных и гуманитарных наук, Частный университет прикладных наук (г. Амман, Иордания). email: е_almuhur@asu.edu.jo

Кусини Майсун — кафедра математики, Иорданский университет Аль-Зайтуна (г. Амман, Иордания).

email: М.qousiniMzuj. edu.jo

Альнана Абир — факультет математики, Университет принца Саттама бин Абдулазиза (г. Аль-Хардж, Саудовская Аравия). email: a.alnana@psau,.edu.sa

Аль-Лабади Манал — кафедра математики, Университет Петры (г. Амман, Иордания). email: Manal. allabadiMuop. edu.jo

Аннотация

В этой работе мы исследуем характеристики wt—расстояния характеристики над Ь—метрическим пространством и условия, необходимые для обеспечения наличие неподвижной точки, если позволить ,0—функции соответствующим образом. Кроме того, мы доказываем некоторые теоремы о неподвижной точке.

Ключевые олова: wt— метрика, Ь— метрика, ft— функция, неподвижная точка.

Библиография: 15 названий.

Для цитирования:

Э. Альмухур, М. Кусини, А. Алнана, М. Аль-Лабади. Расстояние Wt— над метрическим пространством Ь— // Чебышевский сборник, 2024, т. 25, выи. 1, с. 155-163.

CHEBYSHEVSKII SBORNIK Vol. 25. No. 1.

UDC 511.524 DOI 10.22405/2226-8383-2024-25-1-155-163

Wt— Distance over b— Metric Space

E. Almukhur, M. Kusini, A. Alnana, M. Al-Labadi

Almukhur Eman — department of basic science and humanities, Applied Science Private University (Amman, Jordan). e-mail: e_almuhur@asu.edu.jo

Qousini Maysoon — department of mathematics, Al-Zavtoonah University of Jordan (Amman, Jordan).

e-mail: M. qousiniMzuj. edu.jo

Alnana Abeer — department of mathematics, Prince Sattam Bin Abdulaziz University (Al-Kharj, Saudi Arabia).

e-mail: a.alnana@psau.edu.sa

Al-Labadi Manal — department of mathematics, University Of Petra (Amman, Jordan). e-mail: Manal. allahadiMuop. edu.jo

Abstract

In this paper, we examine the wt—distance characteristics over 6—metric space and the conditions required to ensure the presence of the fixed point by letting ,0—function appropriately. In addition, we prove some fixed point theorems.

Keywords: wt — metric, b— metric, ¡3— function, fixed point.

Bibliography: 15 titles.

For citation:

E. Almukhur, M. Kusini, A. Alnana, M. Al-Labadi, 2024, "On the exceptional set of a system of linear equations with prime numbers" , Chebyshevskii sbornik, vol. 25, no. 1, pp. 155-163.

1. Introduction

One of the first ideas that humans developed was the concept of distance. Distance was initially conceptualized by (Euclid). Felix Hausdorff later redefined "metric space"as the general form and more axiomatic version that was first discussed by Maurice René Frechét as "L—space."The concept of distance has since been explored, improved upon, and broadly applied in numerous contexts. In this paper, we concentrate on two of these generalizations: b-metric and wt-distance. We'll set up some notations and ideas before we begin to investigate the topic in depth. We assume that all sets and subsets examined in this paper are non-empty throughout. The function d defined on X x X to R+ is the distance function if the following axioms are satisfied for all u, v and ш in X :

(i) d(u, v) = d(v, u).

(ii) d(u, v) = 0 if и = v.

(iii) d(u,v) + d(v,w) > d(u,w) the triangle inequality as it states that the sum of a triangle's two sides is at least as large as the third side when applied to R2 with the usual metric.

A non-empty set X together with a function d is a metric space. We short (X, d) bv X.

The space (X, d) is complete if none of its points are missed from its inside or boundary. For example, the sequence (xn)"=1 in the metric space X is complete if Vn £ N ^k £ Z : Vx1, x2 > k, d(x1, x2) < n.

On the other side, the set of rational numbers is not complete since we cannot construct a Cauchv sequence of rational numbers that converges to a rational number.

The hyperbolic metric space, introduced by Mikhael Gromov [3], is defined as: X is d—hyperbolic iff all x,y,z,u £ X we have:

min((x, y)„, (y, z)^) — d < (x, z)M......(1)

If (1) is satisfied Vw0 a fixed base point and Vx,y,z £ X, then it is satisfied for all with a constant 2d.

A space (X, d) is called a pseudometric if Vx,y £ X, for x = y, one may have d(x,y) = 0. This notion is introduced by Duro Kurepa [4]. Typically, every metric space is pseudometric. The pseudometric topology is generated by open balls defined as Br (a) = [x £ X : d(a,x) < r}.

A space (X, d) is said to be a v—generalized metric space [5] if V x = y in X, we have the following:

i) d(x, y) = 0.

ii) d(x, y) = d(y, x) .

iii) d(x, y) < d(x, zi) + d(zi,z2) + ... + d(zv ,y) Vzi = Z2 = ... = zv £ X.

If X is a non-empty set, then the partial metric is the function p : X x X R

such that Va, b,c £ X, the following conditions hold:

i) a = b iff p(a, a) = p(a, b) = p(b, b)

ii) p(a, a) = p(a, b)

iii) p(a, b) = p(b, a)

iv) p(a, b) < p(a, c) + p(z, b) — p(c, c)

The partial metric space is the couple (X, p).

In 1998, Czerwik [1] and Bakhtin [2] introduced the extension b—metric space. The metric space (X, d) is a 6—metric space over the constant k if the following hold Vx, y,z £ X :

(i) d(x, y) = d(y, x).

(ii) d(x, y) = 0 iff x = y.

(iii) If the relaxed triangle inequality hols for some constant k > 1 :

d(x, z) < k[d(x, y) + d(y, z)}.

We see that any b—metric space is unquestionably a metric space under the scenario where k = 1. So, this idea is less strong than the concept of metric space.

2. b—MetricSpace

Lemma 1. If (X,d) is a b— metric space, then for then natural number n and (x0,x1, ...,xn) £ Xn+1, we have

d(xo,Xn) < £n=i2k%+1d(Xl,Xl+i)+ kn-1d(xn-i,Xn)......(2)

In Euclidean space, the convergence of a sequence [xn}'^)=1 to the point x is defined as [6]: if Ve > 0, 3n £ N : Vn > N, d(xn, x^) < e.

Such concept in topology is defined as: the sequence [xn}"=1 converges to the point xiiU open set containing x, 3n £ N :Vn>N,xn £ U.

Both of these concepts are valid and equivalent in metric spaces.

Definition 1. For the sequence [xn}"=1 in th e space (X, d) and a sub set A in X [6]:

(i) [xn}"=1 converges to x if limn d(xn,x) = 0.

(ii) [xn}"=1 is Cauchy iflimn sup[d(xn,xm)} = 0 Vrn > n.

(in) {x^^i is complete if every Cauchy sequence converges.

(iv) A is closed if for any convergent sequence {xn}c^=l C A, limra^(£ra) g A.

(v) A is bounded if sup{d(x,y)} < x> Vx,y G A.

Lemma 2. The sequence {xn}c^=l in the b—metric space (X,d) is Cauchy [7] if 3m G g [0,1 ] : d(xn+i,xn+2) < md(xn,xn+i) Vn G N.

Note that every 6—metric space is metrizable, even though not all v—generalized metric spaces are metrizable. As a result, we observe that definition 2.2 above leaves no opportunity for ambiguity.

Let CB(X) = {F с X : F = ф closed and bounded} Mid v x g X , then vA, В с X, if d(x,A) = inf{d(x,y) : у G A}, then the Hausdorff metric space or Pompeiu-Hausdorff distance (H, d) [8] is defined bv

H(A, B) = max{sup{d(£, В) : x G A}, sup{d(y, A) : у G B}.

Remark 1. Define the function f : N ^ N u {0} by f (n) = —[— iog2 n]......(3)

If n G N and (x0,xi, ...,xn) < k?d(xi,xi+i) and the following hold:

(i) f (2n) = f (n) + 1

(ii) f(n + 1) G{f(n),f(n) + 1} (Hi) f is non-decreasing.

Lemma 3. If {xn}%=i is a sequence in the b—metric space (X,d), and r G [0,1) : d(xn+i,xn+2) < r.d(xn,xn+i) Vn G N Леn {xn}c^'=i is a Cauchy sequence [6].

Доказательство. If r = 0, the result holds. If 0 < r < 1, then for some s G N kr2s < 1. Define the function f be defined as (3). For n,m G N : m <n <m + 2, by remark 2.4 we have: d(xm, Xn) < kf (n~m). Zn=l d(xi, Xi+i) <ks.EП=Ш d(xi,X2)

<ks-Z Zrn rl~ id(xl,xn) < ksrmA

where A =

Now, m + 2s<n and v = [ы] )S0

d(xm,Xn) < EГ= 0 ki+ld(xm+24, xm+(i+i)2°) + kvd(xm+V2° ,Xn) <Y1V 0 k,i+s+i rm+i2s A + ku+s rm+v+2s д

< rm.AY^V о ki+s+i _rm+i2s

< Гт.А^2^ fci+s+i _rm+i2s

< rmA

Thus, {xn}^= i is a Cauchy sequence. □

Теорема 1. If (X,d) is a b—metric space and the function g : N U{0} ^ [0, <x>) defined as:

g(n) =0 if n = 0 and g(n) = (2n — 2f(n))kf (n) + (2f(n) — n)kf (■n)-l Vn G N g is strictly increasing.

g(n)= kg [n] + g(n — [n])......(4)

0 <g(n — 1) — g(n — 2) < g(n) — g(n — 1)......(5)

g(n) < k(g(k)+ g(n — k))......(6)

Vk G N and 2 < n where к <n [6].

Теорема 2. If (X, d) is a complete b—metric space and f : X ^ CB(X), and 3r £ [0, |] such that Ух, у £ X,

H(f (x),f (у)) < rd(x,y)......(7)

we have the following:

(i) [1] 3z £ X : z £ z £ f (z).

(ii) ¡6] 3e > 0 .• d(x,y) < e. (Hi) [6] 3x £ X : d(x, f (x)) < e.

Доказательство. If p = Чг £ (0,1), then Ух, у £ X and и £ f (x) with d(x, y) < e, 3v £ f (y) : d(u, v) < pd(x, y).

Then, if {un}^ i £ X : d(ui,f (ui)) < d(ui,u2) < e so, un+i £ f (un) and d(un+i,un+2) < pd(un,un+i). Уп £ N and by lemma 2.3, {un1 is Cauchv. But, X is complete, so {un1 ^ z in X and d(z, f (z)) < iimra^ k(d(z, un+i) + d(un+i, f (z))) = к limra^ d(z,f (z))

< к iimra^ H(f (un), f (z))

< kp iimra^ d(un,z)) = 0.

Hence, f (z) is closed and z £ f (z). □

Corollary 1. If (X, d) is a complete b—metric space and

f : X ^ CB(X). Define a bijective fu,nction I : [0, те) ^ [0,1) such that Ух, у £ X we have H(f (x),f (y)) < l(d(x,y))d(x,y)) and

iima^i+o sup 1(a) < 1 yt £ [0, те), then 3z £ X : z £ f (z).

Доказательство. iimsupa^0 1(a) < 1, so we can choose e > 0 and r £ [0,1) : l(t) < r yt £ [0,6).

Now, for t £ [0, те), we define h : [0, те) ^ (0,1) by h(t) = . Let {xn}^=i £ X : xn+i £ f (xn) andУn £ N : d(xn+i,xn+2) < h( d(xn+i,xn+2)). d(xn+i,xn+2)

Typically, yt £ [0, те), we have h(t) < 1 and { d(xn+i,xn+2)}c^=0 is non-increasing.

So, { d(xn+i, xn+2)}^=0 converges to some point ft £ [0, те).

Since iim supQ,^/3+0 ft(a) < 1 and h(ft) < 1, 3p £ [0,1) and 5 > 0 :

У a £ [ft, ft + we have h(a) < p.

d(xu,xv< ft + 5 for some v < n £ N

d(xn+i,xn+2) < (h(d(xn,xn+i).d(xn,xn+i)) < pd(xn,xn+i)

Then, d(xn, xn+i) = 0 and sо d(xn, xn+i) < d(xn, f (xn)) < e. □

Definition 2. (i) The function ц, : [0, те) ^ [0, те) is called auxiliary distance, (ii) If ц, is a non-decreasing auxiliary distance function such that

Vй(t) =0 Уt £ [0, те), then ц, is a comparison [9] if it, is continuous at t = 0, and p(t) <t У1 > 0.

(Hi) If r £ [1, те) and there exist positive integers k0, s £ (0,1) and a convergent se ries Ylfi uk with Uk > 0;

rk+i^k+i(t) < srk/лк(t) + Uk Ук0 < к, then У1 £ [0, те), the monotonic auxiliary distance function ц, is called b—comparison [10].

We denote the set of all b—comparison functions by B.

Lemma 4. If ^ is a b—comparison function, then J2kL0 rk(t) convergent, increasing and continuous at t = 0 Уt £ [0, те) [9].

Remark 2. Each b—comparison function is comparison [11].

Definition 3. (i) The function f : X ^ X is ft—orbital admissible where ft : X x X ^ [0, <x>) if Vu e X

1 < P(u,f (U)) ^ 1 < ft(f (u),f2(u)) ......(8)

(ii) IfVu,v e X, we have 1 < P(u,v) and 1 < P(v,f (v)) ^ 1 < P(v,f (v)) ......(9)

(Hi) If (8) and (9) are fulfilled, then f is called triangular ft—orbital addmissible [12].

Lemma 5. If f : (X, d, a) ^ (X, d, a) is a triangular ft—orbital admissible function and 3v0 e X

• 1 < ft(vo, f (vo)), then 1 < P(vn, vm) and f (vn) = vn+\ Vn,m e N [12]

—

fixed point [13].

3. Distance over b—Metric Space

Definition 4. The metric d : X x X ^ [0, <x>) is a wt—distance оver (X, d, a) if the following hold [11]:

(i) Vu,v,w G X, a—weighted triangle inequality d(v,w) < a[d(u,v) + d(v,w)\ holds

(ii) If vn ^ v in X and d(v,.) : X ^ [0, ж) such th at d(u,v) < liminfn^^ ad(u,vn) Vv G X, then d is a—lower semicontinuous.

(Hi) Ve> 0 > 0 .• if d(u,v) < 5 and d(v,w) < 5, then d(u,v) < e.

Lemma 6. Ifp : X x X ^ [0, ж) be a wt—distance о ver (X, d, a) and the sequences (an) , (bn) in X and (un) , (vn) in [0, ж) converging to 0,then [11]:

i) d is a wt—distance о ver (X, d, a).

ii) if p(an, bn) < kn and p(bn, c) < un, Vn G N, then a = c.

Hi) ifp(an,bn) < kn and p(bn, c) < un, Vn G N, then (bn) converges to c.

Теорема 4. Let p be a wt—distance о ver (X, d, a) an d f : X ^ X, then:

i) f is continuous.

ii) f is triangular /3—orbital admissible, ii) 3a0 G X such th at 1 < @(a0, f (a0). Hi) Vu G X, 1 < P(u, f (u))

iv) Vu G X with 1 < P (u, f (u)) such th at и = f (u), we have inf {p(u,v) + p(u,f (f))} > 0; then f has a fixed point.

Доказательство, i) If a0 g X and a sequence {an}^c=i is given bv an = fn(a0) , then 3b0 G N : f (аь0) = аь0 + аь0 is a fixed point of the function f. □

Corollary 2. If conditions of 3.3 hold and for r,s G Fix(f) we have 1 < fi(r, s), then r = s.

Доказательство. Let r,s g Fix(f) such th at r = s, then by 3.3 we have p(r,s) < p(r,s)p(f (r),f (s))

< is)

< ц.(р(г, s))

< p(r, s)

which is a contradiction. Therefore, f has a unique fixed point. □

Corollary 3. If conditions of 3.3 hold and p : X x X ^ [0, ж) is a wt—distance on (X *,d,a), Vu,v G X, Ve > 0 > 0 such th at e < y(d(u,v)) < e + 5 implies

0 < &(P(u, v))p(f (u), f (v), e) for some a G £ and ^ G P where y(t) < | Vt > 0, then a function f has a fixed point.

Доказательство. Let [vn}n=i be a sequence defined as vn = fn(v0) Уп e N. If vn = vn-i and because / is a triangular ^—orbital admissible, we have 1 < @(v

0 < a (P(u,v)p(f (u),f (v)),e) <€ — P(u,v)p(f (U),f (v)) < p,(d(u, v) — ¡3(u, v)p(f (u), f (v), f (v)) Then У и = v we get,

P(u,v)p(f (u),f (v)) < p, (p(u,v)) <p(u,v)......(10)

Considering и = vn-i and v = vn,

Hence, (10) holds and p(u, v)is a decreasing sequence that converges to the positive real number I. □

Corollary 4. If conditions of 3.4 hold and r,s e Fix(f ) such that

i) 1 < P (r, s)

ii) Уе > 0 > 0 such that e < p,(d(r, s)) < e + 5, then, r = s

Доказательство. Suppose that r = s in Fix(f ) such that 1 < fi(r, s)

By (ii) and (10) we get a contradiction, hence f has a unique fixed point. □

Теорема 5 (15). If (X, <) is a partially ordered set such that У(х,у) and (z,t) e X x X, 3 (a,b) e X x X such that

a < x , z and b < y , t, and if (X, p) is a complete partial metric space,

g : X x X ^ X is a function with the mixed monotone property on X .

Assuming that for some a e X , ф e Ф we have

a(p(g(x, y),g(a, b)) < a(ap(x, a) + fip(y, b)) — ф(ар(х, a) + fip(y, a))

У a + P < 1, if 3xo,yo e X such that x0 < g(x0,y0) and g(yo, x0) < y0

then, 3x, y e X such that g has a coupled fixed point, that is; g(x, y) = x and g(y, x) = y.

СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ

1. Czerwik S. Nonlinear set-valued contraction mappings in b— metric spaces // Atti del Seminario Matematico e Fisico dell'Universita di Modena, 46, (1998), 263-276.

2. Bakhtin I.A. The contraction mapping principle in quasimetric spaces // Journal of Functional Analysis, 1989, 30, pp. 26-37.

3. Gromov M. Groups of polynomial growth and expanding maps // Publications Mathématiques de l'Institut des Hautes Études Scientifiques. 53, 1981, pp. 53-73. doi:10.1007/BF02698687. MR 0623534. S2CID 121512559. Zbl 0474.20018.

4. Kurepa, D.J. Tableaux ramifiés d'ensembles, espaces pseudodistaciés // Comptes Rendus de l'Académie des Sciences . Paris. 198, 1934, pp. 1563-1565.

5. Branciari A. A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces // Publicationes Mathematicae Debrecen, 57, 2000, pp. 31-37. MR1771669.

6. Suzuki T. Journal of Inequalities and Applications, 256, 2017, pp. 1-11. http://dx.doi.org/ 10.1186/sl3660-017-1528-3.

7. Singh SL, Czerwik S, Krôl, К and Singh. A Coincidences and fixed points of hybrid contractions. Tamsui Oxford Journal of Information and Mathematical Sciences. 24, pp. 401-416.

8. Rockafellar R, Tyrrell; Wets, Roger J-B. Variational Analysis // Springer-Verlag, 2005, p. 117.

9. Berinde, V. Generalized contractions in quasimetric spaces // Semin. Fixed Point Theory, 3, 1993, pp. 3-9.

10. Rus, I.A. Generalized Contractions and Applications // Cluj University Press: Clui-Napoca, Romania, 2001.

11. Karapmar E, Chifu C. Results in wt-Distance over 6-Metric Spaces // SIGMA Mathematics, 220(8), 2020, pp. 1-10.

12. Popescu, O. Some new fixed point theorems for fi — Geraghtv contractive type maps in metric spaces // Fixed Point Theory and Applications, 190 p. (2014). https://doi.org/10.1186/1687-1812-2014-190.

13. Khojasteh, F.; Shukla, S.; Radenovi C, S. A new approach to the study of fixed point theorems via simulation functions // Filomat, 29, 2015, pp. 1189-1194.

14. Avdi, H. Fixed point results for weakly contractive mappings in ordered partial metric spaces // Journal of Advanced Mathematical Studies, 4(2), 2011, pp. 1-12.

15. Al-Sharif, S., Al-Khaleel, M., Khandaqji, M. Coupled Fixed Point Theorems for Nonlinear Contractions in Partial Metric Spaces // International Journal of Mathematics and Mathematical Sciences, 2012, pp. 1-12.

REFERENCES

1. Czerwik S. 1998, "Nonlinear set-valued contraction mappings in b—metric spaces", Atti del Seminario Matematico e Fisico deWUniversita di Modena, 46, pp. 263-276.

2. Bakhtin, I.A., 1989. "The contraction mapping principle in quasimetric spaces", Journal of Functional Analysis, 30, pp. 26-37.

3. Gromov M.,1981. "Groups of polynomial growth and expanding maps", Publications Mathématiques de l'Institut des Hautes Etudes Scientifiques, 53: pp. 53-73. doi:10.1007/BF02698687. MR 0623534. S2CID 121512559. Zbl 0474.20018.

4. Kurepa, D.J., 1934. "Tableaux ramifiés d'ensembles, espaces pseudodistaciés", Comptes Rendus de l'Académie des Sciences . Paris, 198, pp. 1563-1565.

5. Branciari A., 2000. "A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces", Publicationes Mathematicae Debrecen, 57, pp. 31-37. MR1771669.

6. Suzuki T.,2017. "Journal of Inequalities and Applications", 256, pp. 1-11. http://dx.doi.org/ 10.1186/sl3660-017-1528-3.

7. Singh SL, Czerwik S, Krôl, K. k, Singh, 2008. "A Coincidences and fixed points of hybrid contractions", Tamsui Oxford Journal of Information and Mathematical Sciences, 24, pp. 401416.

8. Rockafellar R, Tyrrell; Wets, Roger J-B, 2005. "Variational Analysis", Springer-Verlag, p. 117.

9. Berinde, V., 1993. '"Generalized contractions in quasimetric spaces", Semin. Fixed Point Theory, 3, pp. 3-9.

10. Rus, I.A.,2001. "Generalized Contractions and Applications", Cluj University Press: Clui-Napoca, Romania.

11. Karapmar E, Chifu С., 2020. "Results in -wi-Distance over 6-Metric Spaces", SIGMA Mathematics, 220(8), pp. HO.

12. Popescu, O., 2014. "Some new fixed point theorems for ft—Geraghtv contractive type maps in metric spaces", Fixed Point Theory and Applications, 190 p.. https://doi.org/10.1186/1687-1812-2014-190.

13. Khojasteh, F., Shukla, S., Radenovi, C. S., 2015. "A new approach to the study of fixed point theorems via simulation functions", Filomat, 29, pp. 1189-1194.

14. Avdi, H.,2011. "Fixed point results for weakly contractive mappings in ordered partial metric spaces", Journal of Advanced Mathematical Studies, 4(2), pp. 1-12.

15. Al-Sharif, S., Al-Khaleel, M., Khandaqji, M., 2012. "Coupled Fixed Point Theorems for Nonlinear Contractions in Partial Metric Spaces", International Journal of Mathematics and Mathematical Sciences, pp. 1-12.

Получено: 18.04.2023 Принято в печать: 21.03.2024