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У. ТоиаП, А. Jaid, Е1 Мо^ашакП, Замечание об общих теоремах о неподвижной точке в ограниченном метрическом пространстве, Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки, 2023, номер 2, 241-249
001: 10.14498/уБ^и1940
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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. 2, pp. 241-249 ISSN: 2310-7081 (online), 1991-8615 (print) d https://doi.org/10.14498/vsgtu1940
MSC: 47H09, 47H10
A note on common fixed point theorems in a bounded metric space
Y. Touail, A. Jaid, D. El Moutawakil
Université Sultan Moulay Slimane, Beni-Mellal, 23000, Morocco.
Abstract
In this paper, we introduce the concept of T^-contraction for a pair of commuting self-mappings and prove a common fixed point theorem for this type. Our results improve and extend many existing results in the literature. The paper also contains an application for non-linear integral equations.
Keywords: fixed point, Tp-contraction, T — a-admissible, r-distance.
Received: 5th July, 2022 / Revised: 23rd March, 2023 / Accepted: 25th May, 2023 / First online: 20th June, 2023
1. Introduction. The importance of fixed point theories lies in finding solutions for many problems in applied sciences such as physics, variational inequality, optimization, and many other problems in non-linear analysis.
In 1998, Jungck [1] introduced the concept of weakly compatible pairs of mappings, that is, the class of mappings such that they commute at their coincidence points. In recent years, several authors have obtained common fixed point results for different classes of mappings on various metric spaces, such as complete metric spaces.
In 2012, Samet et al. [2] introduced the notion of a-admissible mappings. By using this concept, the authors defined a-^-contractive mappings and proved
Differential Equations and Mathematical Physics Research Article
© 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 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: 10.14498/vsgtu1940. Authors' Details:
Youssef Touail https://orcid.org/0000-0003-3593-8253 MATIC, Faculté Polydisciplinaire de Khouribga; e-mail: youssef9touail@gmail.com; y.touail@usms.ma Amine Jaid® https://orcid.org/0000-0001-7322-2008
MATIC, Faculté Polydisciplinaire de Khouribga; e-mail: aminejaid1990@gmail.com
Driss El Moutawakil © https://orcid.org/0000-0001-7322-2008
MATIC, Faculté Polydisciplinaire de Khouribga; e-mail: d.elmoutawakil@gmail.com
a nice result for such mappings in the setting of metric spaces. Then Abdelja-wad [3] expanded the notion of «-admissibility to a pair of functions.
Recently, the authors in [4] established a common fixed point theorem without using any additional condition on the space. Namely, we assert the following theorem.
Theorem 1.1 [4]. Let (X,d) be a bounded complete metric space. Let f and g be two weakly compatible self-mappings of X .satisfying the following conditions
i) g(X) C f (X),
ii) if [d(fx,fy) - d(gx,gy)} > 0.
If the range of f or g is a S-complete subspace of X, then f and g have a unique common fixed point.
Recent works in this direction can be found in [5-11].
Note that we can find a class of weakly compatible mappings satisfying
if [d(fx,fy) - d(gx,gy)} = 0,
which have common fixed points, in this case, Theorem 1.1 does not work.
Inspired by the above facts, we prove a common fixed point theorem satisfying a new condition, named Tp-contraction, which improves Theorem 1.1, and present an example to illustrate the usability of our result.
Finally, on the basis of our main result, we study the existence of solutions for a system of differential equations.
2. Preliminaries. The purpose of this section is to explain some notions and results utilized in the paper.
Let (X, t) be a topological space and p : X x X ^ [0, to) be a function. For any e > 0 and any x G X, let Bp(x, e) = [y G X : p(x, y) < e}.
Definition 2.1 [12]. The function p is said to be r-distance if for each x G X and any neighborhood V of x, there exists e > 0 such that Bp(x,e) C V.
Definition 2.2 [12]. A sequence [xn} in a Hausdorff topological space (X,t) is a p-Cauchy if it satisfies the usual metric condition with respect to p, in other words, if lim p(xn, xm) = 0.
Definition 2.3 [12]. Let (X,t) be a topological space with a r-distance p.
1) X is S'-complete if for every p-Cauchy sequence [xn}, there exists x in X with limp(x,xn) = 0.
2) X is p-Cauchy complete if for every p-Cauchy sequence [xn}, there exists x in X such that lim xn = x with respect to t.
3) X is said to be p-bounded if sup[p(^, y) : x,y G X} < to.
Lemma 2.1 [12]. Let (xn) be a sequence in a Hausdorff topological space (X,t) with a t-distance p and x,y G X, then
1) if [®n} C R+ a sequence converging to 0 such that p(x,xn) ^ an for all n G N, then [xn} converges to x with respect to the topology t;
2) p(x, y) = 0 implies x = y;
3) if lim p(x, xn) = 0 and lim p(y, xn) = 0, then x = y.
Definition 2.4 [12]. ^ is the class of all functions ^ : [0, +to) ^ [0, +to) satisfying:
i) ^ is nondecreasing;
ii) lim ^n(t) = 0, for all t e [0, x>).
Definition 2.5 [2]. Let (X, d) be a metric space, T : X ^ X and a : X xX ^ R+ be two given mappings. Then, T is called an a-admissible mapping if
a(x, y) ^ 1 a(Tx, Ty) ^ 1 for all x,y e X.
Lemma 2.2 [4]. Let (X,d) be a metric space and p : XxX ^ R+ be a function defined by
p(x,y) = ed(x'y) — 1.
Then p is a Td-distance on X, where Td is the metric topology.
Lemma 2.3 [4]. Let (X,d) be a bounded metric space. Then the function p defined in Lemma 2.2 is a bounded t-distance.
Lemma 2.4 [4]. Let (X,d) be a complete metric space. Then the function p defined in Lemma 2.2 is a S-complete t—distance.
3. Main results. We start our work by introducing the notion of T-a-admissible for a pair of self-mappings f and g on a metric space X.
Definition 3.1. Let f, g be two self-mappings of a bounded metric space (X, d) and a : XxX ^ R+ be a function. (f,g) is said to be a pair of T-a-admissibility if fg = gf and
a(x,y) ^ 1 a(gx,gy) ^ 1 and a(fx,fy) ^ 1, for all x,y e X.
Theorem 3.1. Let (X,t) be a Hausdorff topological space with a t-distance p. Suppose that X is p-bounded and f (X) is S-complete. Let f and g be two self-mappings of X such that
i) g(X) C f (X);
ii) (f,g) is a pair of T-a-admissibility;
iii) a(x,gx) ^ 1 for all x e X;
iv) there exists x0 e X such that a(x0, gnx0) ^ 1 and a(x,gnx0) = 0, for all x e X and n e N;
v) a(x, y)p(gx, gy) ^ ^(p(fx, fy)), for all x,y e X, where ^ e
Then f and g have a common fixed point.
Proof. Let x0 e X such that a(x0,gnx0) ^ 1. Since g(X) C f(X), then there exist x\, x2 e X such that g(x0) = f (x{) = x2, continuing this process, we can choose xn e X such that x2n+2 = fx2n+\ = gx2n for any n e N.
Now, consider the sequences {yn} = {x2n}, {zn} = {x2n+i} and {tn} = {x^n}.
Let n, m e N, since (f,g) is T-a-admissible, we obtain a(x2n,x2n+2m) ^ 1 and we have
p(fzn,fzn+m) = p(fX2n+1, fX2n+2m+l) = p(gX2n, gX2n+2m) ^ ^ a(X2n, X2n+2m)p(gX2n, gX2n+2m) ^ ^ ^(p(fX2n, fX2n+2m)) ^
< i>2n(p(f 2nX0, f 2nX2m)) < i>2n(M),
where M = sup[p(^,y) : x,y G X}. As lim^n(M) = 0, so the sequence [fzn} is a p-Cauchy sequence. Since f (X) is S'-complete, there exists u G X such that lim p(fu,fzn) = 0.
By the same argument, it is easy to prove that [fyn}, [ftn} are p-Cauchy sequences, which leads to limp(fu, fzn) = limp(fu, fyn) = limp(fu, ftn).
On the other hand we have
a(u, yn)p(gu, fzn) = a(u, yn)p(gu, fx2n+i) = = a(u,x2n)p(gu,gx2n) < < p(fu,fX2n) = p(fu, fyn).
Since a(u, yn) = a(u, gnx0) = 0, we have
limp(gu, fzn) = limp(fu, fyn) = limp(fu, fzn) = 0.
By Lemma 2.1, we conclude that fu = gu.
Suppose that p(gu,ggu) = 0, the assumption that a(u,gu) ^ 1 implies
p(gu,ggu) ^ a(u,gu)p(gu,ggu) ^
^ ^(p(fu,ggu)) < p(gu,ggu)
which is a contradiction.
Hence ggu = gu and fgu = gfu = ggu = gu, it follows that that gu is a common fixed point of f and g. □
For f = Idx, &(x,y) = 1 and ^(t) = kt, where k G [0,1) in Theorem 3.1, we
get.
Corollary 3.1 [12]. Let (X,t) be a Hausdorff topological space with a t-distance p. Suppose that X is p-bounded and S-complete. Let g be a self-mapping of X, if there exist k G [0,1) such that
p(gx,gy) < kp(x,y),
for all x,y G X. Then g has a fixed point.
Now, we introduce the notion of pair of -contraction.
Definition 3.2. Let f, g be two self-mappings of a bounded metric space (X, d), (f,g) is said to be a pair of Tp-contraction if fg = gf, 0(x,gx) ^ 0 and
if {d(fx, fy) - d(gx,gy) + @(x,y)} > 0,
where ft : X xX ^ R is a function satisfying
P(x,y) ^ 0 ft(gx,gy) ^ 0 and @(fx,fy) ^ 0,
for all x,y G X.
Our first result is the following.
Theorem 3.2. Let (f,g) be a pair of Tp-contraction of a bounded complete metric space (X, d) such that
i) g(X) C f (X);
ii) there exists x0 e X such that ft(xo, gnx0) ^ 0, for all n e N;
iii) ft(a,b) ^ inf {d(fx,fy) — d(gx,gy)+ ft(x,y)\, for all a,b e X.
Then f and g have a common fixed point.
Proof. Since (f,g) is a pair of Tp-contraction, so there exists a function ft : XxX ^ R such that
inf {d(fx, fy) — d(gx,gy)+ ft(x,y)} > 0.
We put
7 = inf {d(fx,fy) — d(gx,gy)+ ft(x,y)}, which implies that for all x = y e X, we have
d(gx, gy) — ft(x, y) < d(fx, fy) — j.
Thus
a(x,y)ed(9X'9y) < ked(fxJy), where k = e-1 < 1 and a(x,y) = e-l3(x'y\ Then, it follows from (iii) that
a(x,y)p(gx,gy) < kp(fx,fy),
for all x, y e X, with p(x, y) = ed(x'y) — 1 is the t-distance defined in Lemma 2.2.
Finally, we deduce from Lemmas 2.2, 2.3, 2.4 and Theorem 3.1 that f and g have a common fixed point. □
Corollary 3.2 [5]. Let g : X ^ X be a mapping of a bounded complete metric space (X, d) such that
inf {d(x, y) — d(gx, gy)} > 0.
Then g has a fixed point.
Example. Let X = {0,1, 2} endowed with the discrete metric
7/ \ { 0, if x = y,
d(x,y) = { I, if a = y.
Define self-mappings g and f on X by
g0 = g1= g2 = 2, f 0 = f 1 = 0, f 2 = 2, and a function ft : X x X ^ R by
' 1, if x,y e {0,1},
^ I 0, otherwise.
It is clear that
ft(x,y) < 0 ß(gx,gy) < 0 and ft(fxjy) < 0,
and
fgx = g fx = 2,
for all x,y G X. Also,
g(X) = {2} c f (X) = {0,2},
ß(x,gx) ^ 0, ß(2,gn2) ^ 0,
for all x G X and n G N,
d(fx, fy) - d(gx, gy) + ß(x,y) = l,
for all x = y G X.
Then g and f satisfy all conditions of Theorem 3.2 and have the common fixed point 2.
Remark. Note that, in the class of commuting mappings, Theorem 3.2 is a real extension of Theorem 1.1, indeed:
d(f0,f l) - d(g0,g1) = 0.
4. Application. In this section, we will prove the existence of a common solution for two nonlinear integral equations
x(t) = J k^s, J k(u,x(u^jdu)ds, t G [0, r], (1)
x(t)= [ k(s,x(s))ds, t G [0, t], (2)
Jo
where x G C[0,r], the space of all continuous functions from [0,r] into R, with t > 0. K : [0, t] xR ^ R is a continuous mapping. Let X = C [0, t] be endowed by the metric
d(x,y) = sup lx(t) - y(t)l te[o, t]
Define the mappings f,g : X ^ X as follows
ft / rs
gx(t) = y k^s, J k(u,x(u))d^jds, t G [0,r], (3)
fx(t)= Î k(s,x(s))ds, t G [0, t]. (4)
Jo
Hence, equations (1) and (2) have a common solution if and only if the mappings f and g have a common fixed point.
Theorem 4.1. Let g,f : X ^ X be the mappings defined by (3) and (4) and assume the following condition is satisfied.
There exist M > 0 and a function 9 : X xX ^ R such that for all x,y £ X with x = y, we have:
0(x,y) > 0 \k(t,x(t)) - k(t,y(t))\^ - (|x(t) - y(t)\- M),
- (5)
d(x,y) < 0 \ k{t,x{t)) - k(t,y(t)) \ < - \ x{t) - y{t) \
T
and
- for all x, y £ X, d(x,y) ^ 0 implies d(gx,gy) ^ 0 and d(fx, fy) ^ 0,
- there exists x0 £ X such that d(x0,gnx0) ^ 0 for all n £ N,
- d(x,gx) ^ 0 for all x £ X.
Then the functional equations (1) and (2) have a common solution.
Proof. It is easy to see that fg(x) = gf (x) for all x £ X and g(X) c f (X). Let x = y £ X and t £ [0, t]. We discuss two cases. Case 1. If d(x,y) ^ 0, we have
y k^s, J k(u,x(u))d^jds — J k^s, J k(u,y(u))d^jds
\gx(t) — gx(t)\ =
10 / Jo \ J0
/ k(u,x(u))du — k(u,y(u))du 00
<
< '-i T Jo
—Mjds ^
< d(fx, fy) — M.
Then d(gx,gy) < d(fx, fy) — M. Case 2. If d(x, y) < 0, we have
\gx(t) — gx(t)\ =
k(u,x(u))d^jds — J k^s, J k(u,y(u))d^jds
-< -
I kis, k(u,x\ 0 V Jo /Jo
■t rt
ds < d(fxjy).
<
T Jo
(/ k(u,x(u))du — J k(u,y(u))du^j
So d(gx,gy) < d(fx,fy).
Now, define ß : XxX ^ R by
ß(xy) = ( 0 lf °(X,y) > 0, KV ,yj I M, otherwise.
Then, by (5) we have
pf {d(fx, fy) - d(gx,gy)+ @(x,y)} > 0.
If we have ft(x,y) ^ 0, from the definition of ft we obtain that ft(gx,gy) ^ 0 and P(fx, fy) ^ 0, and ft(xo,gnxo) ^ 0 for all n £ N.
Also, we have ft(x,gx) ^ 0 for all x £ X, so the pair (f,g) is a -contraction. Moreover, we have for all a, b £ X
P(a,b) < inf {d(fx,fy) - d(gx,gy) + @(x,y)}.
x=y£X
Finally, we conclude by Theorem 3.2 that the functional equations (1) and (2) have a common solution. □
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
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Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки. 2023. Т. 27, № 2. С. 241-249 ISSN: 2310-7081 (online), 1991-8615 (print) d https://doi.org/10.14498/vsgtu1940
EDN: ZXSBPZ
УДК 517.988.523
Замечание об общих теоремах о неподвижной точке в ограниченном метрическом пространстве
Y. Touail, A. Jaid, D. El Moutawakil
Университет Султана Мулая Слимана, Бени-Меллаль, 23000, Марокко.
Аннотация
Вводится концепция Тр-сжатия для пары коммутирующих самопреобразований и доказывается общая теорема о неподвижной точке для этого типа. Полученные результаты улучшают и обобщают многие известные в литературе результаты. В качестве приложения полученных результатов приводится доказательство существования общего решения для двух нелинейных интегральных уравнений.
Ключевые слова: неподвижная точка, Тр-сжатие, Т — а-допустимость, т-расстояние.
Получение: 5 июля 2022 г. / Исправление: 23 марта 2023 г. / Принятие: 25 мая 2023 г. / Публикация онлайн: 20 июня 2023 г.
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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: 10.14498/vsgtu1940.
Сведения об авторах
Youssef Touail https://orcid.org/0000-0003-3593-8253 MATIC, Faculte Polydisciplinaire de Khouribga; e-mail: youssef9touail@gmail.com; y.touail@usms.ma Amine Jaid® https://orcid.org/0000-0001-7322-2008
MATIC, Faculte Polydisciplinaire de Khouribga; e-mail: aminejaid1990@gmail.com
Driss El Moutawakil © https://orcid.org/0000-0001-7322-2008
MATIC, Faculte Polydisciplinaire de Khouribga; e-mail: d.elmoutawakil@gmail.com
