Miskolc Mathematical Notes HU e-ISSN 1787-2413 [632190]

Miskolc Mathematical Notes HU e-ISSN 1787-2413
V ol. 18 (2017), No. 1, pp. 499–506 DOI: 10.18514/MMN.2017.1905
FIXED POINT THEOREMS FOR LOCAL ALMOST
CONTRACTIONS
MONIKA ZAKANY
Received 05 December, 2015
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.
2010 Mathematics Subject Classification: 47H10; 54H25
Keywords: almost local contraction, coefficient of contraction, fixed- point theorem, local con-
tractions
1. I NTRODUCTION
The concept of local contraction was presented by Martins da Rocha and Filipe
Vailakis in [4], meanwhile the almost contractive mappings was introduced by V .
Berinde in [2]. The aim of this paper is to combine this two concepts and to study
the fixed points of almost local contractions. First, we present the concept of almost
contraction, following V . Berinde in [2] (2004).
Definition 1. Let (X,D) be a metric space and TWX!Xis called almost con-
traction or.;L/ – contraction if there exist a constant 2.0;1/ and someL0such
that
d.Tx;Ty/
d.x;y/CL
d.y;Tx/;8x;y2X: (1.1)
Remark 1.The term of almost contraction is equivalent to weak contraction, and
it was first introduced by V . Berinde in [2].
Remark 2.Because of the simmetry of the distance, the almost contraction condi-
tion (1.1) includes the following dual one:
d.Tx;Ty/
d.x;y/CL
d.x;Ty/;8x;y2X (1.2)
obtained from (1.1) by replacing d(Tx,Ty) by d(Ty,Tx) and d(x,y) by d(y,x), and after
that step, changing x with y, and viceversa.
Obviously, to prove the almost contactiveness of T, it is necesarry to check both
(1.1) and (1.2).
c
2017 Miskolc University Press

500 MONIKA ZAKANY
A strict contraction satisfies (1.1), with Daand L = 0, therefore is an almost con-
traction with a unique fixed point.
The concept of local contraction was first introduced by Martins da Rocha and Filipe
Vailakis in [4] (2010)
Definition 2. Let F be a set and let DD.dj/j2Ja family of semidistances defined
on F. We letbe the weak topology on F defined by the family D.
Let r be a function from J to J. An operator TWF!Fis alocal contraction with
respect ( D,r) if, for every j, there exists j2Œ0;1/ such that
8f;g2F; dj.Tf;Tg/jdr.j/.f;g/
We present an existence theorem (Theorem 1), then an existence and uniqueness
theorem (Theorem 2), as they are presented in [2]. Their main merit is that they ex-
tend uniqueness and existence theorems for contractions to the larger class of almost
contractions. They show us a method for approximating the fixed point, for whitch
both a priori and a posteriori error estimates are available.
Theorem 1. Let (X,D) be a complete metric space and TWX!Xa weak (almost)
contraction. Then
(1)Fix.T/Dfx2X:TxDxg¤I
(2)For anyx02X, the Picard iteration fxng1
nD0given by (3) converges to some
x2Fix(T);
xnC1DTxn;nD0;1;2;


(1.3)
(3)The following estimates
d.xn;x/n
1d.x0;x1/; nD0;1;2::: (1.4)
d.xn;x/
1d.xn1;xn/; nD1;2::: (1.5)
hold, whereis the constant appearing in (1.1) .
It is possible to force the uniqueness of the fixed point of an almost contraction, see
[1] and [2], by imposing an additional contractive condition, quite similar to (1.1), as
shown by the next theorem.
Theorem 2. Let (X,d) be a complete metric space and TWX!Xbe an almost
contraction for which there exist 2.0;1) and some L10such that
d.Tx;Ty/
d.x;y/CL1
d.x;Tx/;8x;y2X (1.6)
Then
(1)T has a unique fixed point,i.e., Fix.T/Dfxg
(2)For anyx02X, the Picard iteration fxng1
nD0given by (3) converges to x;

FIXED POINT THEOREMS FOR LOCAL ALMOST CONTRACTIONS 501
(3)The a priori and a posteriori error estimates
d.xn;x/n
1d.x0;x1/; nD0;1;2:::
d.xn;x/
1d.xn1;xn/; nD1;2:::
hold.
(4)The rate of convergence of the Picard iteration is given by
d.xn;x/
d.xn1;x/; nD1;2::: (1.7)
The existence and uniqueness of fixed points for the local contractions was studied
in [4].
Theorem 3. Assume that the space F is - Hausdorff, which means: for each pair
f,g2F , f¤g, there exists j2J such thatdj.f;g/>0 .
Consider a function rWJ!Jand letTWF!Fbe a local contraction with re-
spect to ( D,r). Consider a nonempty, - bounded, sequentially - complete, and T-
invariant subset AF.
(1)(E) If the condition
8j2J; lim
n!1jr.j/


rnC1.j/diamrnC1.j/.A/D0 (1.8)
is satisfied, then the operator T admits a fixed point fin A.
(2)(S) Moreover, if h2F satisfies
8j2J; lim
n!1jr.j/


rnC1.j/drnC1.j/.h;A/D0 (1.9)
then the sequence .Tnh/n2N is-convergent to f.
If A is a nonempty subset of F , then for each h in F , we let
dj.h;A/inffdj.h;g/Wg2Ag.
Remark 3.There exists other interpretations of the local contractions, different
from Martins da Rocha and Filipe Vailakis in [4], for example the definition given by
Fang Jin-xuan in [3]:
Definition 3. A distribution function is a function FWŒ1;1!Œ0;1 which is
left continuous on R, non-decreasing and F.1/D0;F.1/D1.
We will denote by
the family of all distribution functions on Œ1;1. H is a
special element of
defined by
H.t/D0;if t0 and 1;if t>0

502 MONIKA ZAKANY
If X is a nonempty set, FWXX!
is called a probabilistic distance on X and
F.x;y/ is usually denoted by Fxy.
Definition 4 (Schweizer and Sklar [5]) .The ordered pair (X,F) is called a probab-
ilistic metric space (shortly PM-space) if X is a nonempty set and F is a probabilistic
distance satisfying the following conditions: for all x;y;´2Xandt;s>0 ,
(1) (FM-0)Fxy.t/D1,xDy
(2) (FM-1)Fxy.0/D0
(3) (FM-2)FxyDFyx
(4) (FM-3)Fx´.t/D1;F´y.s/D1)Fxy.tCs/D1.
The ordered triple .X;F;/is called Menger space if .X;F/ is a PM-space,
is a t- norm and the following condition is also satisfied: for all x;y;´;2X
andt;s>0
(5) (FM-4)Fxy.tCs/Fx´.t/F´y.s/.
Definition 5. The PM- space (X,F) is said to be "- chainable if for given ">0
and anyx;y2Xthere is a finite set of points in X: xDx0;x1;


xnDysuch that
Fxi1;xi."/D1; iD1;2;


n
We denote by Tthe topology on X and it is called the .";/ topology of.X;F;
/
. Thus we can induce the concept of T- Cauchy sequence and also T- complete
sequence.
Theorem 4 (The Fixed Point Theorem of One-valued Local Contraction Map-
pings, Fang [3]) .Let.X;F;
/ be an"- chainable and T- complete Menger space.
Let the mapping TWX!Xsatisfy the following condition: there exists 2.0;1/ so
that for each 2.0;/ there is a function .t/satisfying the condition such that
FTx;Ty./.t/Fx;y.t/ (1.10)
forFx;y."/¤0andFx;y.t/>1.
Then T has a unique fixed point xin X andTnx0!xfor anyx02X
Now, we shall try to combine these two different type of contractive mappings: the
almost and local contractions, to study their fixed points.
2. A LMOST LOCAL CONTRACTIONS
Definition 6. The mapping d.x;y/WXX!RCis said to be
a pseudometric if:
(1)d.x;y/Dd.y;x/
(2)d.x;y/d.x;´/Cd.´;y/
(3)xDyimpliesd.x;y/D0
(instead ofxDy,d.x;y/D0in the metric case)

FIXED POINT THEOREMS FOR LOCAL ALMOST CONTRACTIONS 503
Definition 7. Let X be a set and let DD.dj/j2Jbe a family of pseudometrics
defined on X. We let be the weak topology on X defined by the family D.
A sequence.xn/n2Nis said to beCauchy if it isdj-Cauchy,8j2J.
The subset A of X is said to be sequencially -complete if every -Cauchy sequence
in X converges in X for the -topology.
The subset AX is said to be -bounded ifdiamj.A/supfdj.x;y/Wx;y2Agis
finite for every j2J.
Definition 8. Let r be a function from J to J. An operator TWX!Xis called an
almost local contraction with respect ( D,r) if, for every j, there exist the constants
2.0;1/ andL0such that
dj.Tx;Ty/
dj.x;y/CL
dr.j/.y;Tx/;8x;y2X (2.1)
Remark 4.The almost contractions represent a particular case of almost local con-
tractions, by taking (X,d) metric space instead of the pseudometrics djanddr.j/
defined on X. Also, to obtain the almost contractions, we take in (1.1) for r the iden-
tity function, so we have r(j)=j.
Definition 9. The space X is - Hausdorff if the following condition is valid: for
each pair x,y2X, x¤y, there exists j2J such thatdj.x;y/>0 .
If A is a nonempty subset of X, then for each z in X, we let
dj.´;A/inffdj.´;y/Wy2Ag.
Theorem 5 is an existence fixed point theorem for almost local contractions.
Theorem 5. Consider a function rWJ!Jand letTWX!Xbe an almost local
contraction with respect to ( D,r). Consider a nonempty, - bounded, sequentially -
complete, and T- invariant subset AX. If the condition
8j2J; lim
n!1nC1diamrnC1.j/.A/D0 (2.2)
is satisfied, then the operator T admits a fixed point xin A.
Proof. Letx02Xbe arbitrary andfxng1
nD0be the Picard iteration defined by
xnC1DTxn; n2N
TakexWDxn1;yWDxnin (2.1) to obtain
dj.Txn1;Txn/
dr.j/.xn1;xn/
which yields
dj.xn;xnC1/
dr.j/.xn1;xn/;8j2J (2.3)
Using (2.3), we obtain by induction with respect to n:
dj.xn;xnC1/n
dr.j/.x0;x1/; nD0;1;2;


(2.4)

504 MONIKA ZAKANY
According to the triangle rule, by (2.4) we get:
dj.xn;xnCp/n.1CC


Cp1/dr.j/.x0;x1/D (2.5)
Dn
1.1p/
dr.j/.x0;x1/; n;p2N;p¤0 (2.6)
Conditions (2.5) show us that the sequence .xn/n2Nisdj- Cauchy for each j. The
subset A is assumed to be sequentially -complete, there exists fin A such that
.Tnx/n2Nis- convergent to x. Besides, the sequence .Tnx/n2Nconverges for
the topologytox, which implies
8j2J; dj.Tx;x/Dlim
n!1dj.Tx;TnC1x/:
Recall that the operator T is an almost local contraction with respect to ( D,r). From
that, we have
8j2J; dj.Tx;x/jlim
n!1dr.j/.x;Tnx/:
The convergence for the - topology implies convergence for the pseudometric dr.j/,
we obtaindj.Tx;x/D0for everyj2J.
This way, we prove that TfDf, sinceis Hausdorff.
So, we prove the existence of the fixed point for almost local contractions. 
Remark 5.For T verifies (2.1) with L = 0, we find Theorem Vailakis [4] by taking
jD.
Further, for the case djDd;8j2J, with d = metric on X, we obtain the well known
Banach contraction, with his unique fixed point.
Remark 6.In Theorem 5, the coefficient of contraction 2.0;1/ is constant, but
local contractions have a coefficient of contraction j2Œ0;1/ whitch depends on j2
J. 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 6. If to the conditions of Theorem 5, we add:
(U) for every fixed j2Jthere exists:
lim
n!1.CL/ndiamrn.j/.´;A/D0;8x;y2X (2.7)
then the fixed point xof T is unique.
Proof. Suppose, by contradiction, there are two different fixed points xandy
of T. Then for every fixed j 2J we have:
0<dj.x;y/Ddj.Tx;Ty/dr.j/.x;y/CLdr.j/.y;Tx/D

FIXED POINT THEOREMS FOR LOCAL ALMOST CONTRACTIONS 505
D.CL/
dr.j/.x;y/


.CL/ndrn.j/.x;y/
.CL/ndiamrn.j/.´;A/
Now, letting n!1 , we obtained a contradiction with condition (2.7), i.e. the fixed
point is unique. 
Remark. If the function r is the identity ( i.e., r.j/Dj), then the operator T
is said to be a 0- local contraction and, in that case, conditions (2.1) and (2.7) are
automatically satisfied. In particular, if a fixed point exists, it is unique on the whole
space X.
We extend the Definition 8. to the case of almost local contractions with variable
coefficient of contraction.
Definition 10. Let r be a function from J to J. An operator TWX!Xis called
almost local contraction with respect ( D,r) or.j;Lj/- contraction, if there exist a
constantj2.0;1/ and someLj0such that
dj.Tx;Ty/j
dj.x;y/CLj
dr.j/.y;Tx/;8x;y2X (2.8)
Theorem 7. With the presumptions of Theorem 7, if we modify the condition (2.2)
by the following one:
8j2J; lim
n!1jr.j/


rn.j/diamrnC1.j/.A/D0; (2.9)
then the operator T admits a fixed point xin A.
Proof. Let x an element in A. From the definition of almost local contraction T,
for every pair of integers q>n>0 , we have
dj.Tqx;Tnx/
jdr.j/.Tq1x;Tn1x/CLjdr.j/.Tn1x;Tqx/
jŒr.j/dr2.j/.Tq2x;Tn2x/CLjdr2.j/.Tn2x;Tq1x/C
CLjŒr.j/dr2.j/.Tn2x;Tq1x/CLr.j/dr2.j/.Tq1x;Tn1x/D
DjŒr.j/dr2.j/.Tq2x;Tn2x/CLj.jCr.j//dr2.j/.Tn2x;Tq1x/C
CLjLr.j/dr2.j/.Tq1x;Tn1x/



jr.j/


rn1.j/drn.j/.Tqnx;x/
C


CLjLr.j/


Lrn1.j/drn.j/.Tqnx;x/
Since A is T- invariant, Tqngbelongs to A, which yields
dj.Tqx;Tnx/jr.j/


rn1.j/diamrn.j/.A/:
That means: the sequence .Tnx/n2Nisdj-Cauchy for each j. The subset A is as-
sumed to be sequentially -complete, there exists xin A such that .Tnx/n2Nis-

506 MONIKA ZAKANY
convergent to x. Besides, the sequence .Tnx/n2Nconverges for the topology to
x, which implies
8j2J; dj.Tx;x/Dlim
n!1dj.Tx;TnC1x/:
Recall that the operator T is an almost local contraction with respect to ( D,r). From
that, we have
8j2J; dj.Tx;x/jlim
n!1dr.j/.x;Tnx/CLjdr.j/.Tnx;Tx/:
The convergence for the - topology implies convergence for the pseudodistance
dr.j/, we obtaindj.Tx;x/D0for everyj2J. This way, we prove that TxDx
sinceis Hausdorff. So, we prove the existence of the fixed point. 
REFERENCES
[1] V . Berinde, “On the approximation of fixed points of weak contractive mappings,” Carpathian J.
Math. , vol. 19, no. 1, pp. 7–22, 2003.
[2] V . Berinde, “Approximating fixed points of weak contractions using the picard iteration,” Nonlinear
Analysis Forum , vol. 9, no. 1, pp. 43–53, 2004.
[3] F. Jin-xuan, “Fixed point theorems of local contraction mappings on menger spaces,” Applied Math-
ematics and Mechanics , vol. 12, no. 4, April 1991, doi: 10.1007/BF02020399.
[4] F. Martins-da Rocha and V . Yiannis, “Existence and uniqueness of a fixed point for local contrac-
tions,” Econometrica , vol. 78, no. 3, pp. 1127–1141, May 2010, doi: 10.3982/ECTA7920.
[5] B. Schweizer, A. Sklar, and E. Thorp, “The metrization of statistical metric spaces,” Pacific J.
Math. , vol. 10, pp. 673–675, 1960.
Author’s address
Monika Zakany
Department of Mathematics and Computational Science, North University of Baia Mare, str. Vic-
toriei 76, 430122 Baia Mare, Romania,, Tel. +40-262-276059; (Fax: 275368)
E-mail address: vberinde@ubm.ro, zakanymoni@yahoo.com