Manuscriptcmi 2017 26 2 243 248 [632192]

CREAT. MATH. INFORM.
26(2017), No. 2, 243 – 248Online version at http://creative-mathematics.ubm.ro/
Print Edition: ISSN 1584 – 286X Online Edition: ISSN 1843 – 441X
On the continuity of almostlocal contractions
ZAKANY MONIKA
ABSTRACT .This paper introduces a new class of contraction: the almost local contractions. Then, we prove
the existence and uniqueness of a fixed-point for local almost contractions in two cases: with constant and
variable coefficients of contraction.
1. I NTRODUCTION
Definition 1.1. Let (X,d) be a metric space and T:X!Xis called almost contraction or
(;L)- contraction if there exist a constant 2(0;1)and someL0such that
d(Tx;Ty )
d(x;y) +L
d(y;Tx );8x;y2X (1.1)
Remark 1.1. The term of almost contraction is equivalent to weak contraction, and it was
first introduced by V . Berinde in [3].
Remark 1.2. Because of the simmetry of the distance, the almost contraction condition
(1.1) includes the following dual one:
d(Tx;Ty )
d(x;y) +L
d(x;Ty );8x;y2X (1.2)
obtained from (1.1) by replacing d(Tx;Ty )byd(Ty;Tx )andd(x;y)byd(y;x), and after
that step, changing xwithy, and viceversa. Obviously, to prove the almost contractive-
ness ofT, it is necessary to check both (1.1) and (1.2).
The next Theorem show that an almost contraction is continuous at any fixed point of it,
according to [1].
Theorem 1.1. Let (X,d) be a complete metric space and T:X!Xbe an almost contraction.
Then T is continuous at p, for any p2Fix(T).
Example 1.1. LetT: [0;1]![0;1]a mapping given by Tx=2
3forx2[0;1), andT1 = 0 .
ThenThas the following properties:
1)Tsatisfies (2.5) with h2[2
3;1), i.e.Tis quasi-contraction;
2)Tsatisfies (1.1), with 2
3andL0, i.e.Tis also weak contraction;
3)Thas a unique fixed point, x=2
3.
4)Tis not continuous.
The concept of local contraction was first introduced by Martins da Rocha and Filipe
Vailakis in [5] (2010), here they studied the existence and uniqueness of fixed points for
the local contractions.
Definition 1.2. LetFbe a set and letD= (dj)j2Ja family of semidistances defined on F.
We letbe the weak topology on Fdefined by the family D.
Letrbe a function from JtoJ. An operator T:F!Fis alocal contraction with respect
Received: 05.12.2016. In revised form: 07.03.2017. Accepted: 14.03.2017
2010 Mathematics Subject Classification. 47H09, 47J25, 55M20, 40E99, 08-02.
Key words and phrases. Almost local contraction, coefficient of contraction, fixed- point theorem, local contractions.
243

244 Zakany Monika
(D, r) if, for every j, there exists j2[0;1)such that
8f;g2F; d j(Tf;Tg )jdr(j)(f;g)
2. A LMOST LOCAL CONTRACTIONS
We try to combine these two different type of contractive mappings: the almost and
local contractions, to study their fixed points. This new type of mappings was first intro-
duced in [12]
Definition 2.3. The mapping d(x;y) :XX!R+is said to be
a pseudometric if:
(1)d(x;y) =d(y;x)
(2)d(x;y)d(x;z) +d(z;y)
(3)x=yimpliesd(x;y) = 0
(instead of x=y,d(x;y) = 0 in the metric case)
Definition 2.4. LetXbe a set and letD= (dj)j2Jbe a family of pseudometrics defined
onX. We letbe the weak topology on Xdefined by the family D.
A sequence (xn)n2Nis said to be Cauchy if it isdj-Cauchy,8j2J.
The subset AofXis said to be sequencially -complete if every -Cauchy sequence in X
converges in Xfor the-topology.
The subset AXis said to be -bounded if diam j(A)supfdj(x;y) :x;y2Agis finite
for everyj2J.
Definition 2.5. Letrbe a function from JtoJ. An operator T:X!Xis called an almost
local contraction with respect (D,r) if, for every j, there exist the constants 2(0;1)and
L0such that
dj(Tx;Ty )
dj(x;y) +L
dr(j)(y;Tx );8x;y2X (2.3)
Remark 2.3. The almost contractions represent a particular case of almost local contracti-
ons, by taking (X;d)metric space instead of the pseudometrics djanddr(j)defined on X.
Also, to obtain the almost contractions, we take in (2.3) for r the identity function, so we
haver(j) =j.
Definition 2.6. The spaceXis- Hausdorff if the following condition is valid: for each
pairx;y2X;x6=y, there exists j2Jsuch thatdj(x;y)>0.
IfAis a nonempty subset of X, then for each zinX, we let
dj(z;A)inffdj(z;y) :y2Ag.
Theorem 2.2 is an existence fixed point theorem for almost local contractions, as they
appear in [12].
Theorem 2.2. Consider a function r:J!Jand letT:X!Xbe an almost local contraction
with respect to (D, r). Consider a nonempty, - bounded, sequentially - complete, and T- invari-
ant subsetAX. If the condition
8j2J; lim
n!1n+1diam rn+1(j)(A) = 0 (2.4)
is satisfied, then the operator Tadmits a fixed point xinA.
Proof. Letx02Xbe arbitrary andfxng1
n=0be the Picard iteration defined by
xn+1=Txn; n2N

On the continuity of almostlocal contractions 245
Takex:=xn1;y:=xnin (2.5) to obtain
dj(Txn1;Txn)
dr(j)(xn1;xn)
which yields
dj(xn;xn+1)
dr(j)(xn1;xn);8j2J (2.5)
Using (2.5), we obtain by induction with respect to n:
dj(xn;xn+1)n
dr(j)(x0;x1); n = 0;1;2;


(2.6)
According to the triangle rule, by (2.6) we get:
dj(xn;xn+p)n(1 ++


+p1)dr(j)(x0;x1) = (2.7)
=n
1(1p)
dr(j)(x0;x1); n;p2N;p6= 0 (2.8)
Conditions (2.7), (2.8) show us that the sequence (xn)n2Nisdj- Cauchy for each j2J. The
subsetAis assumed to be sequentially -complete, there exists finAsuch that (Tnx)n2N
is- convergent to x. Besides, the sequence (Tnx)n2Nconverges for the topology tox,
which implies
8j2J; d j(Tx;x) = lim
n!1dj(Tx;Tn+1x):
Recall that the operator Tis an almost local contraction with respect to (D;r). From that,
we have
8j2J; d j(Tx;x)jlim
n!1dr(j)(x;Tnx):
The convergence for the - topology implies convergence for the pseudometric dr(j), we
obtaindj(Tx;x) = 0 for everyj2J.
This way, we prove that Tf=f, sinceis Hausdorff.
So, we prove the existence of the fixed point for almost local contractions. 
Remark 2.4. ForTverifies (2.3) with L= 0, andr:J!Jthe identity function, we find
Theorem Vailakis [5] by taking =j.
Further, for the case dj=d;8j2J, withd=metric onX, we obtain the well known
Banach contraction, with his unique fixed point.
Remark 2.5. In Theorem 2.2, the coefficient of contraction 2(0;1)is constant, but local
contractions have a coefficient of contraction j2[0;1)whitch depends on j2J. Our first
goal is to extend the local almost contractions to the most general case of j2(0;1).
The next Theorem represent an existence and uniqueness theorem for the almost local
contractions with constant coefficient of contraction.
Theorem 2.3. If to the conditions of Theorem 2.2, we add:
(U)for every fixed j2Jthere exists:
lim
n!1(+L)ndiam rn(j)(z;A) = 0;8x;y2X (2.9)
then the fixed point xofTis unique.
Proof. Suppose, by contradiction, there are two different fixed points xandyofT. Then
for every fixed j2J we have:
0<dj(x;y) =dj(Tx;Ty)dr(j)(x;y) +Ldr(j)(y;Tx) =
= (+L)
dr(j)(x;y)


 (+L)ndrn(j)(x;y)
(+L)ndiam rn(j)(z;A)

246 Zakany Monika
Now, letting n!1 , we obtained a contradiction with condition (2.9), i.e. the fixed point
is unique. 
3. M AIN RESULTS
This paper can be regarded as an extension of V . Berinde and M. P ˘acurar (2015, [1])
analysis about the continuity of almost contractions in their fixed points. The main results
of this paper are given by Theorem 5, which give us the answer about the continuity of
local almost contractions in their fixed points.
Theorem 3.4. LetXbe a set andD= (dj)j2Jbe a family of pseudometrics defined on X; let
T:X!Xbe an almost local contraction satisfying condition (2.3), so Tadmits a fixed point.
ThenTis continuous at f, for anyf2Fix(T).
Proof. The mapping Tis an almost local contraction, i.e. there exist the constants 2(0;1)
and someL0
dj(Tx;Ty )
dj(x;y) +L
dr(j)(y;Tx );8x;y2X (3.10)
For any sequence fyng1
n=0inXconverging to f, we takey:=yn;x:=fin (3.10), and we
get
dj(Tf;Ty n)
dj(f;yn) +L
dr(j)(yn;Tf);n= 0;1;2;::: (3.11)
UsingTf=f, sincefis a fixed point of T, we obtain:
dj(Tyn;Tf)
dj(f;yn) +L
dr(j)(yn;f);n= 0;1;2;::: (3.12)
Now by letting n!1 in (3.12) we get Tyn!Tf, which shows that Tis continuous at f.
The fixed point has been chosen arbitrarily, so the proof is complete. 
According to Definition 2.4, the almost local contractions are defined in a subset AX.
In the caseA=X, then an almost local contraction is actually an usual almost contraction.
Example 3.2. LetX= [1;n][1;n]R2,T:X!X,
T(x;y) =(x=2;y=2) if (x,y)6= (1;0)
(0;0) if (x,y) = (1;0)(3.13)
The diameter of the subset X= [1;n][1;n]R2is given by the diagonal line of the
square with (n1)side.
We shall use the pseudometric:
dj((x1;y1);(x2;y2)) =jx1x2j
ej;8j2Q: (3.14)
This is a pseudometric, but not a metric, take for example:
dj((1;4);(1;3)) =j11j
ej= 0, however (1;4)6= (1;3)
In this case, the mapping Tis contraction, which implies that is an almost local con-
traction, with the unique fixed point x= 0;y= 0.
According to Theorem 3.4, Tis continuous in (0;0)2Fix(T), but is not continuous in
(1;0)2X:
Example 3.3. With the presumptions of Example 3.2 and the pseudometric defined by
(3.13), we get another example for almost local contractions.
Considering T:X!X,
T(x;y) =(x;y) if (x,y)6= (1;1)
(0;0) if (x,y) = (1;1)
Tis not a contraction because the contractive condition:
dj(Tx;Ty )
dj(x;y); (3.15)

On the continuity of almostlocal contractions 247
is not valid8x;y2X, and for any 2(0;1). Indeed, (3.14) is equivalent with:
jx1x2j
ej
jx1x2j
ej
The last inequality leads us to 1, which is obviously false, considering 2(0;1).
However,Tbecomes an almost local contraction if:
jx1x2j
ej
jx1x2j
ej+L
jx2x1j
ej
2
which is equivalent to : ej=2
ej=2+L
(1)
ej=2L (3.16)
For= 1=32(0;1),L= 10andj <0, the (3.15) inequality becomes true, i.e. Tis an
almost local contraction with many fixed points:
FixT =f(x;0) :x2Rg
In this case, we have:
8j2J; lim
n!1n+1diam rn+1(j)(A) = lim
n!1(1=3)n+1
(n1)2= 0
This way, the existence of the fixed point is assured, according to condition (2.4) from
Theorem 2.2.
Theorem 3.4 is again valid, because the continuity of Tin(0;0)2Fix(T), but discontinu-
ous in (1;1), which is not a fixed point of T.
Example 3.4. LetXthe set of positive functions: X=ffjf: [0;1)![0;1)g
anddj(f;g) =jf(0)g(0)j
ej;8f;g2X.
djis indeed a pseudometric, but not a metric, take for example dj(x;x2) = 0 , butx6=x2
Considering the mapping Tf=jfj;8f2X, and using condition (2.3) for almost local
contractions:
jf(0)g(0)j
ej
jf(0)g(0)j
ej+L
jg(0)f(0)j
ej
2
which is equivalent to: ej=2
ej=2+L
This inequality becames true if j <0;  =1
32(0;1);L= 3>0
However,Tis also not a contraction, because the contractive condition (3.14) leads us
again to the false presumption: 1. The mapping Thas infinite number of fixed points:
FixT =ff2Xg, by taking:
jf(x)j=f(x);8f2X;x2[0;1)
4. C ONCLUSIONS
This paper analyse the continuity of contractions in their fixed points (if they exists)
initiated by V . Berinde and M. P ˘acurar in [1]. The main extension of Theorem 1.1 in [1] to
the more general class of almost local contractions is given by Theorem 5, with constant
coefficient of contraction. The case of variable coefficient of contraction is an interesting
open problem.
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248 Zakany Monika
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DEPARTMENT OF MATHEMATICS AND COMPUTATIONAL SCIENCE
NORTH UNIVERSITY OF BAIA MARE
VICTORIEI 76, 430122 B AIA MARE, ROMANIA
E-mail address :zakanymoni@yahoo.com