CC BY
7
DOI: 10.17516/1997-1397-2022-15-6-806-814 YflK 517.4
New Fixed Point Results on a'L-rational Contraction
L'
Mappings in 6-Metric-Like Spaces
Souheib Merad* Faycel Merghadi^
Laboratory of Mathematics, Informatics and Systems
Larbi Tebessi University Tebessa, Algeria
Taieb Hamaizia*
Laboratory of Dynamical Systems and Control, Department of Mathematics and Informatics Larbi Ben M'Hidi University Oum el Bouaghi, Algeria
Stojan RadenoviC§
Faculty of Mechanical Engineering University of Belgrade Belgrad, Serbia
Received 01.05.2022, received in revised form 11.07.2022, accepted 20.10.2022
Abstract. The purpose of this paper is to prove some new results for a^-rational contractive and cyclic a^-rational contractive mappings defined in d-complete 6-metric-like spaces. Moreover, an example is presented to illustrate the validity of our theoretical results.
Keywords: metric like-space, b-metric like-space, cyclic contractions.
Citation: S. Merad, F. Merghadi, T. Hamaizia, S.Radenovic, New Fixed Point Results on aL-rational Contraction Mappings in 6-Metric-Like Spaces, J. Sib. Fed. Univ. Math. Phys., 2022, 15(6), 806-814. DOI: 10.17516/1997-1397-2022-15-6-806-814.
1. Introduction and preliminaries
A vigorous research activity focuses on the research on fixed points for given mappings with certain contractive conditions in various abstract spaces.
Hence, there exist numerous generalizations of the concept of metric spaces; symmetric spaces, quasimetric spaces, fuzzy metric spaces, partial metric spaces, like metric spaces and this has pulled in the consideration of many researchers to obtain some fixed point theorem for mappings satisfying a different contractive condition, in particular, cyclic contractions and cyclic contractive type mappings, (we refer the reader to see [2-11,13]).
§ [email protected] © Siberian Federal University. All rights reserved
This paper points to proceed the think about of a few cyclical contractive mappings taking after many papers from this still actual subject nowadays.
Our results extend and generalize the results [11] into b-metric like-space. The obtained results extend many recent results in the literature. The following definitions and results will be needed
Definition 1 ([1]). Let X be a non empty set. A function d : X x X —>• [0, such that for all x,y,z € X, we have the following assertions :
1) d(x, y) = 0 implies x = y,
2) d(x, y) = d(y, x),
3) d(x, y) ^ d(x, z) + d(z, y).
The pair (X, d) is called a metric like-space. A metric-like d on X satisfies all conditions of a metric except that d (x,x) may be positive for some x € X.
Definition 2 ( [12]). Let X be a non empty set, s ^ 1 a fixed real number, A function d : X x X —>• [0, a mapping. Then, (X, d) is said to be b-metric like space if for all x,y,z € X, the following statements hold true :
1) d(x, y) = 0 implies x = y,
2) d(x, y) = d(y, x) ,
3) d(x, y) ^ s [d(x, z) + d(z, y)] .
Then b-metric like space is a pair (X, d),and d a b-metric-like on X.
It should be noted that the class of b-metric-like spaces is larger than the class of metric-like spaces, since a b-metric-like is a metric-like with s = 1.
Example ([17]). Let Q = {0,1, 2, 3, 4} and let
{ 5 k = T = 0
w (k,t )=! 1
- otherwise. 5
Then, (Q,w) is a b-metric-like space with a coefficient s = 5.
For more examples in metric-like and b-metric-like spaces, see [17,18].
Definition 3 ([12]). Let {xn} be a sequence in a b-metric-like space (X, d) with the coefficient s. Then :
i) The sequence {xn} is said to be convergent to x if lim d (xn,x) = d (x,x).
n^ + tt
ii) The sequence {xn} is said to be Cauchy in (X, d) if lim d (xn,xm) exists and is finite.
n,m^+tt
ii) A b-metric-like space (X,d) is d-complete if for every d-Cauchy sequence {xn} in X there exists an x € X, such that
lim d (xn,xm) = d (x,x)= lim d (xn,x) .
n,m^+tt n^+tt
Remark 1.1. The limit of the sequence need not be unique and a convergent sequence need not be a Cauchy in the context of b-metric-like.
Lemma 1. Let {xn} be a sequence on a complete b-metric space (X,d) with s ^ 1 such that lim d (x„,x„+i) =0.
n^ + tt
If lim d (xn,xm) = 0, there exist e > 0 and two sequences {mk}+:tt1 , {nk}+tt of positive
n,m^+tt
integers with nk > mk > k such that
e
d (xnk , xmk ) ^ e,d (xmk , xnk-1 ) < e, ~2 ^ limk^+ SUp d (xnk —1, xmk —1) ^ se,
s2
ee
-limk^+tt sup d (xnk-i,xmk) < e, -2 lim^+tt sup d (i„t ,xmk -1) < es2. s s2
Definition 4 ( [13,14]). Let (X,d) be a b-metric-like space with the coefficient s. A sequence {xn} is called 0-d-Cauchy sequence if lim d (xn,xm) = 0. The space (X,d) is said to be 0-
d-complete if every 0-d-Cauchy sequence in X converges to a point x £ X such that d (x, x) = 0.
Lemma 2 ([15,16]). Let {xn} be a sequence in a b-metric-like space (X,d) with the coefficient s ^ 1 such that
d(xn, xn+1) ^ qd(xn—1, xn) ,
for some q £ [0,1) and n £ N then {xn} is a d-Cauchy sequence in (X, d) such that
lim d (xn, xm) = 0.
n,m^+w
Definition 5 ( [17]). Let X be a non empty set. A mapping T : X ^ X is said to be an a-admissible mapping if a(x,y) ^ 1 implies a(Tx,Ty) ^ 1 for all x,y £ X and a : X x X ^ [0, .
Further T called a-continuous on X if lim xn = x implies lim Txn = Tx for any sequence
n^+w n^+w
{xn} for Y which a
(xn,xn+1 ) > 1, n £ N.
Moreover, [17] present some new fixed point theorems for a aL cyclic rational contraction self-mapping on complete metric-like spaces, subsequently the authors [11] found some doubts about some defintions and theorems, and give a new formula of it as follow:
Let denote Q the class of all function * : [0, ^ [0, satisfying the following
condition:
i) * non-decreasing and continuous;
ii) lim *n (t) = 0 for all t > 0.
n^+w
Definition 6. Let (X, d) be a b-metric-like space, p £ N, B1, B2,..., Bp be d-closed subsets of X, Y = B1 U • • • U Bp and a : Y x Y ^ [0, to) be a mapping. We say that T : Y x Y ^ Y is cyclic a£-rational contractive mapping if :
1) T (Bi) C Bi+1, i = 1, 2,...,p, where Bp+1 = Bx.
2) for any x £ Bi and y £ Bi+1, i = 1, 2,.. .,p, where Bp+1 = B1 and a (x, Tx) a (y, Ty) ^ 1, holds
* (d (Tx, Ty)) < * (M (x, y)) - LM (x, y), (1)
where * £ Q, L £ (0,1) and
M (x y)-max {d (x y) d (x Ty) d (x >Tx) d (y Ty) d (y Ty) [1+ d (x ,Tx)] d (x ,Ty)+ d (y, Tx) } Md{x,y) = max jd (x,y) , 2g , 1+ d (x,y) ' 1+ d (x,y) ' 4S / '
2. Main results
This part is devoted to define a cyclic a£-rational contractive mapping, and some new fixed point results through this contractive on the setting of complete b-metric-like spaces are presented.
Theorem 2.1. Let(X,d) be a d-complete b-metric like space and a : X x X ^ [0, +to) be a mapping. Assume that T : X ^ X is an aL-contractive mapping satisfying the following assertions:
(i) T is an a-admissible mapping,
(ii) a (x0,Tx0) ^ 1 for an element x0 in X,
(iii) T is a-continuous, or;
ib
(iv) if {xn} is a sequence in X such that a (xn, xn+1) ^ 1 for all n £ N and xn ^ x as n ^ then a (xn,Txn) ^ 1. Then T admits a fixed point in X.
Moreover, if
(v) a (x, x) ^ 1, whenever x £ Fix (T), then T admits a unique fixe point.
Proof. Let start with define the sequence xn = Tnx0, where x0 is the given point for which a (x0, Tx0) > 1. Since T is an a-admissible mapping, we get that
a (xi, Txi) = a (Txo, TTx 1) ^ 1.
Continuing this process, we get a (xn, Txn) > 1 for all n £ N, and so,
a (xn, Txn) a (xn-i, Txn-i) ^ 1 for all n £ N.
If xn = xn-1 for some n £ N, xn-1 is a fixed point of T. Therefore, assume that xn-1 = xn for all n £ N. Hence, we have that
d (xn-1,xn) > 0 for all n £ N.
In order to prove that the sequence {xn} is a d-Cauchy sequence. According to (1), we get
< max {d (xn-i,xn) ,d (xri,xri+i)} Hence, we get
^ (d (x„,x„+i)) < ^ (max {d (x„_i,x„) ,d (xn,xn+i)}) - L max {d (x„_i,x„) ,d (xn,xn+i)} . If
^ (d (xn, xn+i)) < ^ (M (xn-i, xn)) - LM (
xn — i,xn) ,
(2)
where
M (xn—i,xn)
max {d (xn—i ,xn) ,d (xn,xn+i)} = d (xn,xn+i) for some n £ N,
we have
^(d (xn,xn+i)) < ^(d (x„,x„+i)) - Ld (
xn,xn+i ) ,
0 < -Ld (xn,xn+i).
Which contradiction. Hence, we get
d(xn,xn+i) ^ d(xn,xn+i) •
So, there exists
lim d (xn, xn+1) = dk > 0.
n^+w
Letting n ^ in (2), we obtain that
lim * (d (xn,xn+1)) < lim [* (d (xn,xn+1)) - Ld (xn,xn+1)],
n^+w n^+w
*(dk) < * (dk) - Ldk.
Thus
lim d (xn,xn+1)=0.
n^+w
Now, if lim d (xn,xm) = 0, we have sequences {mk} and {nk} such that
n,m^+w
lim d (xrik ,xmk )= e > 0.
k^+w
Let u = xnk and v = xmk in (1), we get
* (d (xnk+1, xmfc+1)) < * (M (x nk , xmk )) — LM (xnk , xmk) , (4)
where
d (xnk , xmk ) d (xmk , xmk + 1)
d (xnk ? ) 7 d ? xnk + 1 ) ?
M (xnk, xmk ) = max <
1 + d (xnk ,xnk+1) d (xmk ? xmk + 1 ) t1 + d (xnk ? xnk + 1 )] 1 + d (xnk ? xmk ) d (xnk ? xmk + 1) + d (xmk ? xnk+i )
4s
ma« e. —. 0. 0. —1
I ' 2s' ' ' 4s J
So, as n ^ in (4) ? we have
^ (e) < ^ (e) - Le
which is a contradiction.
Hence, the sequence{xn} is a cauchy and
lim d (xn? xm) = 0.
This means that there exists a unique point x* G X such that
d (x*?x*)= lim d (xn?x*) = lim d (xn?xm) = 0.
Now, we will proof that x* is fixed point of T i.e., Tx* = x*? thus is clear if T is a-continuous. Further, suppose that for any sequence xn in X and for all n G N?if a (xn?xn+1 ) > 1 and lim xn = x* ? then
a (x*?Tx*) > 1.
Let d (x*? Tx*) > 0. Since
a (xn?Txn) a (x*?Tx*) ^ 1.
According to the given contractive condition, we have
where
^(d (Tx, Tx*)) < ^(M (x,x*)) - LM (x,x*),
d (xn, x*) d (x*, Tx*)
d (xn, x*) ,d (x*, xn+i), ■
1 + d (xn,xn+i)
M (x, x*) = max <
d (x*,Tx*) [1 + d (xn, xn+i)]
1 + d (xn,xn+i) ''
d (xn, Tx ) + d (x , xn+i ) 4s
d (xn, x*) ,d (x*, xn+i),
< max
d (xn, x*) d (x*, Tx*)
1 + d (xn,xn+i) '
d (x*, Tx*), s [d (xn,x*) + d (x*, Tx*)] + d (x*,xn+i)
4S
^ maxjo, 0, 0,d (x*,Tx* ) ,d (x ^ ) } = d (x*,Tx*) as n ^ +œ.
Now, letting the limit in for n ^ we get
^(d (x*,Tx*)) < ^(d (x*,Tx*)) - Ld (x*,Tx*)
which is a contradiction again. This means that d (x* ,Tx*) = 0, that is, we prouve that Tx* = x*.
Finally, to show the uniqueness of the fixed point of the map T, suppose that u,v (u = v) are two fixed point of T.
Then,we get d (u, v) > 0, a (u, u) ^ 1, a (v, v) ^ 1. Further, since
a (u, u) a (v, v) ^ 1,
we obtain where
^ (d (u, v)) < ^ (M (u, v)) - LM (u, v),
d (v,u) d (u,v) d (v,v)
d (u, v) ,
M (u, v) = max
max < d (u, v)
2s ' 1 + d (u, u) d (v, v) [1 + d (u, u)]
1 + d (u, v) d (u, v) + d (v, u)
4s
d (v, u) d ( v, u)
2s
0, 0,
(v,u) 1 2s J
d (u, v) .
Hence
^(d (u,v)) < ^(d (u,v)) - Ld (u,v)
which is a contradiction. This finishe the proof. □
Remark 2.1. It is useful to notice that the The case s = 1 means that (X,d) is actually a complete metric-like space and we get the results of [11].
Example. Let X = R be a 5-metric-like space with constant s = 4. Define the function d : R2 [0, by d (x,y) = (|x| + |y|)3 . It is clear that (X, d) is a complete 5-metric-like space. Suppose that
and
Bi = (-œ, 0], B2 = [0, +œ)
Y = Bi U B2.
Define T : Y ^ Y and a : Y x Y ^ [0, +œ) by
Tx =
if x G (-œ, 1) if x G [-1, 0]
and a (x, y) = |
\x\ + \y\ + 1, if x,y g [-1; 1] 0, otherwise
7
if x G [0,1]
if x G (1, +œ)
Also, define ^ : [0, +œ) ^ [0, +œ) by ^ (t) = 11 and L = 1. Clearly, T (Bi) C B2 and
2 6
T (B2) C Bi.
Let x G Bi, y G B2 and a (x, Tx) a (y, Ty) > 1. If x G [-1,1] or y G [-1,1], then a (x, Tx) = 0 or (y,Ty) = 0. That is, a (x,Tx) a (y,Ty) = 0 which is a contradiction. Hence x G Bi, y G B2 and x,y G [-1,1].
This implies that x G [-1, 0] or y G [1, 0]. Then
3
y 1
+
*(d (Tx, Ty)) = -
fx y2 \
r 6 + - t)
x || 3 1 y2
6 + 7
(\x\3 + |y2|3)
(\x\3 + M3) , since y G [1, 0] .
^ 2
< 3 (|x
< 1(\x|3 +
< 3 (|x| + |y|)3
= 3d (x,y)=*(Md (x,y)) — LMd (x,y) .
Then T is a cyclic «¿-rational contractive mapping. It is clear that a (0, T0) > 1 and so the condition (ii) of Theorem 2.1 is satisfied.
If a (x,y) ^ 1, then x,y G [—1,1] which implies that a (Tx,Ty) ^ 1, that is, T is an a-admissible mapping.
Let {xn} be a sequence in X such that, a (xn, Txn) > 1 and xn ^ x as n ^ Then, we must have xn G [—1,1] and so, x G [—1,1], that is a (x,Tx) > 1. Hence, all the conditions of Theorem 2.1 hold and T has a fixed point x = 0 G Bi n B2.
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Новые результаты с фиксированной точкой для aL-рациональных отображений притяжения в 6-метрических пространствах
Сухейб Мерад Файсел Мергади
Университет Ларби Тебесси Тебесса, Алжир
Тайеб Хамаизия
Университет Ларби Бен М'Хиди Ум-эль-Буаги, Алжир
Стоян Раденович
Белградский университет Белград, Сербия
Аннотация. Целью данной статьи является доказательство некоторых новых результатов для а^-рациональных сжимающих и циклических а^-рациональных стягивающих отображений, определенных в (¿-полные 6-метрикоподобные пространства. Кроме того, приведен пример, иллюстрирующий справедливость наших теоретических результатов.
Ключевые слова: метрическое подобное пространство, Ь-метрическое подобное пространство, циклические сокращения.
