Acta Univ. Sapientiae, Mathematica, 10, 2 (2018) 378394 [632184]

Acta Univ. Sapientiae, Mathematica, 10, 2 (2018) 378{394
DOI: 10.2478/ausm-2018-0030
New classes of local almost contractions
M onika Z ak any
Technical University of Cluj-Napoca,
North University Center of Baia Mare, Romania
email: [anonimizat]
Abstract. Contractions represents the foundation stone of nonlinear
analysis. That is the reason why we propose to unify two dierent type
of contractions: almost contractions, introduced by V. Berinde in [2] and
local contractions (Martins da Rocha and Filipe Vailakis in [7]). These
two types of contractions operate in dierent space settings: in metric
spaces (almost contractions) and semimetric spaces (for local contrac-
tions). That new type of contraction was built up in a new space setting,
which is the pseudometric space. The main results of this paper represent
the extension for various type of operators on pseudometric spaces, such
as: generalized ALC, Ciri c-type ALC, quasi ALC, Ciri c-Reich-Rus type
ALC. We propose to study the existence and uniqueness of their xed
points, and also the continuity in their xed points, with a large number
of examples for ALC-s.
1 Introduction
First, we present the concept of almost contraction, following V. Berinde in
[2].
Denition 1 (see [2]) Let(X;d)be a metric space. T:X!Xis called almost
contraction or (;L)- contraction if there exist a constant 2(0;1) and some
L0such that
d(Tx;Ty )
d(x;y ) +L
d(y;Tx );8x;y2X: (1)
2010 Mathematics Subject Classication: 47H10,54H25
Key words and phrases: almost local contraction, coecient of contraction, xed- point
theorem
378

New classes of local almost contractions 379
Remark 1 The term of almost contraction is equivalent to weak contraction,
and it was rst introduced by V. Berinde in [2].
Because of the simmetry of the distance, the almost contraction condition
(1)includes the following dual one:
d(Tx;Ty )
d(x;y ) +L
d(x;Ty );8x;y2X; (2)
obtained from (1)by replacingd(Tx;Ty )byd(Ty;Tx) andd(x;y )byd(y;x ).
Obviously, to prove the almost contactiveness of T, it is necessary to check
both (1)and(2).
A strict contraction satises (1), with=aandL=0, therefore it is an
almost contraction with a unique xed point.
Many examples of almost contractions are given in [1]-[3]. Weak contractions
represent a generous concept, due to various mappings satisfying the condition
(1). Such examples of weak contraction was given by V. Berinde in [2].
Denition 2 [5]Let(X;d)be a metric space. Any mapping T:X!Xis
called Ciri c-Reich-Rus contraction if it is satised the condition:
d(Tx;Ty )
d(x;y ) +
[d(x;Tx ) +d(y;Ty )];8x;y2X; (3)
where;2R +and+2<1.
Proposition 1 (see [8]) Let(X;d)be a metric space. Any Ciri c-Reich-Rus
contraction,i.e., any mapping T:X!Xsatisfying the condition (3), represent
an almost contraction.
Theorem 1 A mapping satisfying the contractive condition:
there exists0h<1
2such that
d(Tx;Ty )h
maxfd(x;y );d(x;Tx );d(y;Ty );d(x;Ty );d(y;Tx )g; (4)
for allx;y2X, is a weak contraction.
An operator satisfying (4)with0<h<1 is called quasi-contraction.
Remark 2 Theorem 1prove that quasi-contractions with 0 < h <1
2are
always weak contractions. However, there exists quasi-contractions with h1
2,
presented in Example 1 by V. Berinde in [2], as it follows:

380 M. Z ak any
Example 1 LetT: [0;1]![0;1] a mapping given by Tx=2
3forx2[0;1),
andT1=0. ThenThas the following properties:
1)Tsatises (4)withh2[2
3;1), i.e.,Tis quasi-contraction;
2)Tsatises (1), with2
3andL, i.e.,Tis also weak contraction;
3)Thas a unique xed point, x=2
3.
Since we were familiarized with the class of almost contractions, we intro-
duce the concept of local contractions, another interesting type of operators
with unexpected applications. The concept of local contraction was presented
by Martins da Rocha and Filipe Vailakis in [7].
Denition 3 (see [7]) LetFbe a set and letD= (dj)j2Jbe a family of semidis-
tances dened on F. We letbe the weak topology on Fdened by the family
D. A sequence (fn)n2Nis said to be-Cauchy if it isdj-Cauchy,8j2J. A
subsetAofFis said to be sequencially -complete if every -Cauchy sequence
inAconverges inAfor the-topology. A subset AFis said to be-bounded
ifdiamj(A)supfdj(f;g) :f;g2Agis nite for every j2J.
Letrbe a function from JtoJ. An operator T:F!Fis called local contraction
with respect (D ;r) if, for every j, there exists j2[0;1) such that
8f;g2F; dj(Tf;Tg )jdr(j)(f;g):
Denition 4 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 ofx=y,d(x;y ) =0in the metric case).
Denition 5 (see [11])
Letrbe a function from JtoJ. An operator T:F!Fis an almost local
contraction (ALC) with respect (D ;r) or(;L)- contraction, if there exist a
constant2(0;1) and someL0such that
dj(Tf;Tg )
dj(f;g) +L
dr(j)(g;Tf);8f;g2F: (5)
Theorem 2 [11] Assume that the space Fis- Hausdor, which means: for
each pairf;g2F,f6=g, there exists j2Jsuch thatdj(f;g)>0.

New classes of local almost contractions 381
IfAis a nonempty subset of F, then for each hinF, we let
dj(h;A)fdj(h;g) :g2Ag.
Consider a function r:J!Jand letT:F!Fbe an almost local contraction
with respect to (D;r). Consider a nonempty, - bounded, sequentially - com-
plete, andT- invariant subset AF.
(E)If the condition
8j2J; lim
n!1jr(j)


rn(j)diamrn+1(j)(A) =0 (6)
is satised, then the operator Tadmits a xed point finA.
(S)Moreover, if h2Fsatises
8j2J; lim
n!1jr(j)


rn(j)drn+1(j)(h;A) =0; (7)
then the sequence (Tnh)n2N is- convergent to f.
Example 2 LetX= [0;n][0;n]R2;n2N; T :X!X,
T(x;y) =(x
2;y
2)if(x;y)6= (1;0)
(0;0)if(x;y) = (1;0)
The diameter of the subset X= [0;n][0;n]R2is given by the diagonal
line of the square whose four sides have length n.
We shall use the pseudometric:
dj
(x1;y1);(x2;y2)

=jx1-x2j
e-j;8j2J; (8)
whereJis a subset of N. This is a pseudometric, but not a metric, take for
example:
dj((1;4); (1;3)) = j1-1j
e-j=0, however (1;4)6= (1;3)
In this case, we shall use the function r(j) =j
2. By applying the inequality (5)
to our mapping T, we get for all x= (x1;y1);y= (x2;y2)2X
x1
2-x2
2
e-j
jx1-x2j
e-j
2+L
x2-x1
2
e-j
2;
for allj2J, which can be write as the equivalent form
jx1-x2j
e-j
22
jx1-x2j+L
j2x2-x1j;
The last inequality became true if we take =1
22(0;1);L =40. HenceT
is an almost local contraction, with the unique xed point (0;0).
Tis continuous in the xed point, at (0;0)2Fix(T ), but is not continuous at
(1;0)=2Fix(T ):

382 M. Z ak any
Example 3 With the assumptions from the previous example and the pseu-
dometric dened by (8)wherej2J, andr(j) =j
2;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); (9)
is not valid8x;y2X, and for any 2(0;1). Indeed, (9)is equivalent with:
jx1-x2j
e-j
jx1-x2j
e-j;8j2J:
The last inequality leads us to 1, which is obviously false, considering
2[0;1). However, Tbecomes an almost local contraction if:
jx1-x2j
e-j
jx1-x2j
e-j
2+L
jx2-x1j
e-j
2
which is equivalent to : e-j
2+L. For=1
32[0;1) ,L=20andj2J;
the last inequality becomes true, i.e. Tis an almost local contraction with many
xed points:
FixT=f(x;0) :x2Rg:
In this case, we have:
8j2J; lim
n!1n+1diamrn+1(j)(A) = lim
n!11
3n+1

(n-1)2=0
This way, the existence of the xed point is assured, according to condition (E)
from Theorem 2. The continuity of Tin(0;0)2Fix(T )is valid, but we have
discontinuity in (1;1), which is not a xed point of T.
Example 4 LetXbe the set of positive functions:
X=ffjf: [0;1)![0;1)g;
which is a subset of the real functions F=ff:R!Rg.
Letdj(f;g) = jf(0) -g(0)j
e-j;8f;g2X;r(j) =j
2,8j2J:Indeed,djis a
pseudometric, but not a metric, take for example dj(x;x2) =0, butx6=x2:

New classes of local almost contractions 383
Considering the mapping Tf= jfj;8f2X, and using the inequality (1)from
the denition of almost local contractions:
jf(0) -g(0)j
e-j
jf(0) -g(0)j
e-j
2+L
jg(0) -f(0)j
e-j
2
which is equivalent to: e-j=2+L:This inequality becames true if j>0; =
1
42(0;1); L =3>0 . Hence,Tis an ALC. However, Tis not a contraction,
because the contractive condition (9)leads us again to the false assumption:
1. The mapping Thas innite number of xed points: FixT=ff2Xg=X,
by taking:
jf(x)j =f(x);8f2X;x2[0;1)
2 Main results
The main results of this paper represent the extension for various type of
operators on pseudometric spaces, such as: generalized ALC, Ciri c-type ALC,
quasi ALC, Ciri c-Reich-Rus type ALC.
a)Generalized ALC
Denition 6 Letrbe a function from JtoJ. LetAFbe a-bounded
sequencially-complete and T- invariant subset of F. A mapping T:A!Ais
called generalized almost local contraction if there exist a constant 2(0;1)
and someL0such that8x;y2X;8j2Jwe have:
dj(Tx;Ty )
dr(j)(x;y)
+L
minfdr(j)(x;Tx );dr(j)(y;Ty );dr(j)(x;Ty );dr(j)(y;Tx )g(10)
Remark 3 It is obvious that any generalized almost local contraction is an
almost contraction, i.e., it does satisfy inequality (1).
Theorem 3 LetT:A!Abe a generalized almost local contraction, i.e., a
mapping satisfying (10), and also verifying the condition (7)for the unicity
of xed point. Let Fix(T) = ffg. ThenTis continuous at f.
Proof. SinceTis a generalized almost local contraction, there exist a constant
2(0;1) and someL0such that (10) is satised. We know by Theorem 7
thatThas a unique xed point, say f.
Let fyng1
n=0be any sequence in Xconverging to f. Then by taking
y:=yn; x:=f

384 M. Z ak any
in the generalized almost local contraction condition (10), we get
dj(Tf;Tyn)
dr(j)(f;yn);n=0;1;2;


(11)
sincefis a xed point for T, we have
minfdr(j)(x;Tx );dr(j)(y;Ty );dr(j)(x;Ty );dr(j)(y;Tx )g=dr(j)(f;Tf) =0:
Now, by letting n!1 in (11), we get Tyn!Tf, which shows that Tis
continuous at f. 
b)Ciri c-type almost local contraction
Denition 7 (see Berinde, [4]) Let(X;d)be a complete metric space.
The mapping T:X!Xis called Ciri c almost contraction if there exist a
constant2[0;1) and someL0such that
d(Tx;Ty )
M(x;y ) +L
d(y;Tx );for all x,y2X; (12)
where
M(x;y ) =maxfd(x;y );d(x;Tx );d(y;Ty );d(x;Ty );d(y;Tx )g:
From the above denition the following question arises: it is possible to expand
it to the case of almost local contractions? The answer is armative and is
given by the next denition. But rst we need to remind the Lemma of Ciri c
([6]), which will be essential in proving our main results.
Lemma 1 LetTbe a quasi-contraction on Xand letnbe any positive integer.
Then, for each x2X, and all positive integers i;j;wherei;j2 f1;2;


ng
implies
d(Tix;Tjx)h
[O(x;n)];
where we denoted (A) = supfd(a;b ) :a;b2Agfor a subsetAX:
Remark 4 Observe that, by means of Lemma 1, for eachn, there exist kn
such that
d(x;Tkx) =[O(x;n)]:
Lemma 2 (see [6]) LetTbe a quasi-contraction on X.
Then the inequality
[O(x;n)]1
1-hd(x;Tkx)
holds for allx2X:

New classes of local almost contractions 385
Denition 8 Under the assumptions of denition 5, the operator T:A!A
is called Ciri c-type almost local contraction with respect ( D;r) if, for every
j2J, there exist the constants 2[0;1) andL0such that
dj(Tf;Tg )
Mr(j)(f;g) +L
dr(j)(g;Tf); for all f,g2A; (13)
where
Mr(j)(f;g) =max
dr(j)(f;g);dr(j)(f;Tf);dr(j)(g;Tg);dr(j)(f;Tg );dr(j)(g;Tf)
:
Remark 5 Although this class is more wide than the one of almost local con-
tractions, similar conclusions can be stated as in the case of almost local con-
tractions, as it follows:
Theorem 4 Consider a function r:J!J, let a nonempty, - bounded, se-
quentially- complete, and T- invariant subset AXand letT:A!Abe
Ciri c- type almost local contraction with respect to ( D,r). Then
1.Thas a xed point,i.e., Fix(T ) = fx2X:Tx=xg6=;
2.For anyx0=x2A, the Picard iteration fxng1
n=0converges to x2Fix(T );
3.The following a priori estimate is available:
dj(xn;x)n
(1-)2dj(x;Tx ); n =1;2::: (14)
Proof. For the conclusion of the Theorem, we have to prove that Thas at
least a xed point in the subset AX. To this end, let x2Abe arbitrary,
and let fxng1
n=0be the Picard iteration dened by xn+1=Txn; n2Nwith
x0=x.
Takex:=xn-1;y:=xnin (13) to obtain
dj(xn;xn+1) =dj(Txn-1;Txn)
Mr(j)(xn-1;xn);
sincedj(xn;Txn-1) =dj(Txn-1;Txn-1) =0. Continuing in this manner, for
n1, by Lemma 1we have
dj(Tnx;Tn+1x) =dj(TTn-1x;T2Tn-1x)
[O(Tn-1x;2)]:
By using Remark 4, we can easily conclude: there exist a positive integer
k12f1;2g such that
[O(Tn-1x;2)] =dj(Tn-1x;Tk1Tn-1x)

386 M. Z ak any
and therefore
dj(xn;xn+1)
dj(Tn-1x;Tk1Tn-1x):
By using once again Lemma 1, we obtain, for n2,
dj(Tn-1x;Tk1Tn-1x) =dj(TTn-2x;Tk1+1Tn-2x)

[O(Tn-2x;k1+1)]
[O(Tn-2x;3)]:
Continuing in this manner, we get
dj(Tnx;Tn+1x)
[O(Tn-1x;2)]2
[O(Tn-2x;3)]:
By applying repeatedly the last inequality, we get
dj(Tnx;Tn+1x)
[O(Tn-1x;2)]


n
[O(x;n +1)]: (15)
At this point, by Lemma 2, we obtain
[O(x;n +1)][O(x;1)]1
1-dj(x;Tx );
which by (15) yields
dj(Tnx;Tn+1x)n
1-dj(x;Tx ): (16)
The last inequality and the triangle inequality can be merged to obtain the
following estimate:
dj(Tnx;Tn+px)n
1-
1-p
1-dj(x;Tx ): (17)
Let us remind the fact that 01, then, by using (17), we can conclude
that fxng1
n=0is a Cauchy sequence. The subset Ais assumed to be sequentially
-complete, there exists xinAsuch that fxngis- convergent to x. After
simple computations involving the triangular inequality and the Denition
(13), we get
dj(x;Tx)dj(x;xn+1) +dj(xn+1;Tx)
=dj(Tn+1x;x) +dj(Tnx;Tx)
dj(Tn+1x;x) +maxfdj(Tnx;u);dj(Tnx;Tn+1x);dj(x;Tx);
dj(Tnx;Tx);dj(Tn+1x;x)g+ +L
dj(x;Txn)

New classes of local almost contractions 387
Continuing in this manner, we obtain
dj(x;Tx)dj(Tn+1x;x) +
[dj(Tnx;u) +dj(Tnx;Tn+1x)
+dj(x;Tx) +dj(Tn+1x;x)] +L
dj(x;Txn):
These relations leads us to the following inequalities:
dj(x;Tx)1
1-[(1+)dj(Tn+1x;x)
+ (+L)dj(x;Txn) +dj(Tnx;Tn+1x)]:(18)
Lettingn!1 in (18) we obtain
dj(x;Tx) =0;
which means that xis a xed point of T. The estimate (14) can be obtained
from (16) by letting p!1.
This completes the proof. 
Remark 6 1)Theorem 4represent a very important extension of Banach's
xed point theorem, Kannan's xed point theorem, Chatterjea's xed point the-
orem, Zamrescu's xed point theorem, as well as of many other related results
obtained on the base of similar contractive conditions. These xed point theo-
rems mentioned before ensures the uniqueness of the xed point, but the Ciri c
type almost local contraction need not have a unique xed point.
2)Let us remind (see Rus [9], [10]) that an operator T:X!Xis said to be
a weakly Picard operator (WPO) if the sequence fTnx0g1
n=0converges for all
x02Xand the limits are xed point of T. The main merit of Theorem 4is
the very large class of Weakly Picard operators assured by using it.
The uniqueness of the xed point of a Ciri c type almost local contraction can
be assured by imposing an additional contractive condition, quite similar to
(13), according to the next theorem.
Theorem 5 With the assumptions of Theorem 4, letT:A!Abe a Ciri c
type almost local contraction with the additional inequality, which actually
means the monotonicity of the pseudometric:
dr(j)(f;g)dj(f;g);8f;g2A;8j2J: (19)
If the mapping Tsatises the supplementary condition: there exist the con-
stants2[0;1) and someL10such that
dj(Tf;Tg )
dr(j)(f;g) +L1
dr(j)(f;Tf);for all f,g2A;8j2J; (20)
then

388 M. Z ak any
1)Thas a unique xed point, i.e., Fix(T ) = ffg;
2)The Picard iteration fxng1
n=0given byxn+1=Txn; n2Nconverges tof,
for anyx02A;
3)The a priori error estimate (14) holds;
4)The rate of the convergence of the Picard iteration is given by
dj(xn;f)
dr(j)(xn-1;f); n =1;2;:::;8j2J (21)
Proof. 1) Suppose, by contradiction, there are two distinct xed points f
andgofT. Then, by using (20), and condition (19) for every xed j2Jwith
f:=f;g:=gwe get:
dj(f;g)
dr(j)(f;g)
dj(f;g),(1-)
dj(f;g)0;
which is obviously a contradiction with dj(f;g)>0. So, we prove the unique-
ness of the xed point.
The proof for 2) and 3) is quite similar to the proof from the Theorem 4.
4) At this point, letting g:=xn;f:=fin (20), it results the rate of conver-
gence given by (21). The proof is complete. 
The contractive conditions (13) and (20) can be merged to maintain the
unicity of the xed point, stated by the next theorem.
Theorem 6 Under the assumptions of denition 8, letT:A!Abe a map-
ping for which there exist the constants 2[0;1) and someL0such that
for allf;g2Aand8j2J
dj(Tf;Tg )
Mr(j)(f;g)
+L
minfdr(j)(f;Tf);dr(j)(g;Tg);dr(j)(f;Tg );dr(j)(g;Tf)g;(22)
where
Mr(j)(f;g) =maxfdr(j)(f;g);dr(j)(f;Tf);dr(j)(g;Tg);dr(j)(f;Tg );dr(j)(g;Tf)g:
Then
1.Thas a unique xed point,i.e., Fix(T ) = ffg;
2.The Picard iteration fxng1
n=0given byxn+1=Txn; n2Nconverges tof,
for anyx02A;

New classes of local almost contractions 389
3.The a priori error estimate (14) holds.
Particular case
1. The famous Ciri c' s xed point theorem for single valued mappings given
in [6] can be obtain from Theorems 4, 6, 5 by taking L=L1=0and
consideringrthe identity mapping: r(j) =j. The Ciri c' s contractive con-
dition represent one of the most general metrical condition that provide a
unique xed point by means of Picard iteration. Despite this observation,
the contractive condition given for Ciri c-type almost local contraction (in
(13)) possess a very high level of generalisation. Note that the xed point
could be approximated by means of Picard iteration, just like in the case
ofCiri c' s xed point theorem, although the uniqueness of the xed point
is not ensured by using (13).
2. If the maximum from Theorem 6 becomes:
max
dr(j)(f;g);dr(j)(f;Tf);dr(j)(g;Tg);dr(j)(f;Tg );dr(j)(g;Tf)
=dj(f;g);
for allf;g2A, then we can easily obtain Theorem 2(E) from Theorem 4.
Also, by Theorem 5we obtain Theorem 2(U) (see Zakany,[11]).
In the light of the above informations about the Ciri c-type ALC-s, it is
natural to extend it to the Ciri c-type strict almost local contractions.
Denition 9 LetXbe a set and letD= (dj)j2Jbe a family of pseudometrics
dened onX. In order to underline the local character of these type of contrac-
tions, we let AXa subset ofX. We letbe the weak topology on Xdened
by the familyD. Letrbe a function from JtoJ. The operator T:A!A
is called Ciri c-type strict almost local contraction with respect (D;r)if it si-
multaneously satises conditions (Ci-ALC) and(ALC -U), with some real
constantsC2[0;1),LC0andu2[0;1),Lu0, respectively.
(Ci-ALC)dj(Tf;Tg )C
Mr(j)(f;g) +LC
dr(j)(g;Tf); for all f,g2A;
for everyj2J, where
Mr(j)(f;g) =max
dr(j)(f;g);dr(j)(f;Tf);dr(j)(g;Tg);dr(j)(f;Tg );dr(j)(g;Tf)
:
(ALC -U)dj(Tf;Tg )u
dr(j)(f;g)+Lu
dr(j)(f;Tf);for all f,g2A;8j2J;
We end with a few examples that have an illustrative role. They presents
Ciri c' type almost local contractions, without having unique xed point.

390 M. Z ak any
Example 5 LetAbe the set of positive functions A=ffjf: [0;1)![0;1)g,
which is the subset of all real functions X=ff:R!Rg;AX:
We shall use the pseudometric:
dj(f;g) = jf(0) -g(0)j
j;8j2J;JN;8f;g2A:
Indeed,djis a pseudometric, but not a metric, take for example dj(x3;x2) =0,
butx36=x2:Considering the mapping Tf= jfj;8f2A,r(j) =j+1. Note
that the restrictive condition (19) is also veried. By using condition (5)for
almost local contractions:
jf(0) -g(0)j
j
jf(0) -g(0)j
(j+1) +L
jg(0) -f(0)j
(j+1)
which is equivalent to: j(+L)(j+1):This inequality becames true if
j>1;  =1
52(0;1); L =3>0; andj
j-12(1;2). Hence, Tis an almost local
contraction. However, Tis not a contraction, because the contractive condition
d(Tx;Ty )
d(x;y )
leads us to the false assumption: 1.
The mapTisCiri c-type almost local contraction, because
Mr(j)(f;g) = jf(0) -g(0)j
(j-1);
and from (13) we have the equivalent form
jf(0) -g(0)j
j
jf(0) -g(0)j
(j-1) +L
jf(0) -f(0)j
(j-1):
Again, we get the inequality j(+L)(j-1). The mapping Thas innite
number of xed points: FixT=ff2Ag=A, by taking:
jf(x)j =f(x);8f2A; x2[0;1):
In fact, the uniqueness condition (20) is not valid, having in view the equivalent
form:
jf(0) -g(0)j
j
jf(0) -g(0)j
(j-1) +L1
jf(0) -f(0)j
(j-1);
which leads us to the contradiction j(j-1), i.e. the mapping Tnot satisfy
the uniqueness condition (20).
In fact, not even (22) is satised, by computing Mr(j)(f;g) = jf(0)-g(0)j
(j-1)
andminfdr(j)(f;Tf);dr(j)(g;Tg);dr(j)(f;Tg );dr(j)(g;Tf)g = jf(0) -g(0)j
(j-1)
(sincej>1):By replacing these values in (22), we get
jf(0) -g(0)j
j
jf(0) -g(0)j
(j-1) +L
jf(0) -f(0)j
(j-1);
which also lead to the previous contradiction.

New classes of local almost contractions 391
Example 6 By taking the mapping from Example 4, with a small modica-
tion, which is: let Xbe the set of positive functions
X=ffjf: [0;1)![0;1)g;
which is a subset of the real functions F=ff:R!Rg.
Letdj(f;g) = jf(x0) -g(x0)j
ej;8f;g2X;r(j) =j
2,8j2Z.
We can conclude in the same manner that Tis also a Ciri c type almost local
contraction, i.e., it satisfy the contractive condition (13).
Indeed, we have Mr(j)(f;g) = jf(x0) -g(x0)j
ej
2. This way, the condition (13)
became the contractive condition for almost local contractions (5).
By considering L=0in the denition 8ofCiri c-type almost local contraction,
we get a new type of ALC, that is the quasi-almost local contraction.
c)Quasi-almost local contractions
Denition 10 Under the assumptions of denition 5, the operator
T:A!Ais called quasi-almost local contraction with respect (D;r)if, for
everyj2J, there exist the constant 2[0;1) such that
dj(Tf;Tg )
Mr(j)(f;g);for all f,g2A; (23)
where
Mr(j)(f;g) =maxfdr(j)(f;g);dr(j)(f;Tf);dr(j)(g;Tg);dr(j)(f;Tg );dr(j)(g;Tf)g:
Theorem 7 Consider a function r:J!J, let a nonempty, - bounded, se-
quentially- complete, and T- invariant subset AXand letT:A!Abe
quasi-almost local contraction with respect to (D;r):
Then
1.Thas a xed point,i.e., Fix(T ) = fx2X:Tx=xg6=;
2.For anyx0=x2A, the Picard iteration fxng1
n=0converges to x2Fix(T );
3.The following a priori estimate is available:
dj(xn;x)n
(1-)2dj(x;Tx ); n =1;2;::: (24)

392 M. Z ak any
Proof. Obviously, we have to follow the steps from the proof of Theorem
4, with the only dierence that the constant L=0, as in the case of quasi
ALC-s. 
The uniqueness of the xed point is also assured by imposing an additional
condition, just like in the class of Ciri c-type almost local contraction, as it
follows.
Theorem 8 With the assumptions of Theorem 4, letT:A!Abe a quasi-
almost local contraction with the additional inequality:
dr(j)(f;g)dj(f;g);8f;g2A;8j2J: (25)
If the mapping Tsatises the supplementary condition: there exist the con-
stants2[0;1) such that
dj(Tf;Tg )
dr(j)(f;g) +L1
dr(j)(f;Tf);for all f,g2A;8j2J; (26)
then
1.Thas a unique xed point,i.e., Fix(T ) = ffg;
2.The Picard iteration fxng1
n=0given byxn+1=Txn; n2Nconverges tof,
for anyx02A;
3.The a priori error estimate (14) holds;
4.The rate of the convergence of the Picard iteration is given by
dj(xn;f)
dr(j)(xn-1;f); n =1;2;:::;8j2J (27)
d)Ciri c-Reich-Rus type almost local contraction
Denition 11 Under the assumptions of denition 5, the operator
T:A!Ais called Ciri c-Reich-Rus type almost local contraction with respect
(D;r) if the mapping T:A!Asatisfying the condition
dj(Tf;Tg )
dr(j)(f;g) +L
[dr(j)(f;Tf) +dr(j)(g;Tg)]; (28)
for allf;ginA, where;L2R +and+2L<1
Theorem 9 If the pseudometric dsatisfy the condition:
dr(j)(f;g)< dj(f;g);8j2J;8f;g2A, then any Ciri c- Reich- Rus type
almost local contraction, i.e. any mapping T:A!Asatisfying the condition
(28) withL6=1is an almost local contraction.

New classes of local almost contractions 393
Proof. Using condition (28) and the triangle rule, we get
dj(Tf;Tg )
dr(j)(f;g) +L
[dr(j)(f;Tf) +dr(j)(g;Tg)]

dr(j)(f;g) +L
[dr(j)(g;Tf)
+dr(j)(Tf;Tg ) +dr(j)(f;g) +dr(j)(g;Tf)]
The condition for the pseudometric leads us to:
dj(f;g)>dr(j)(f;g);
dj(Tf;Tg )>dr(j)(Tf;Tg );
dj(g;Tf)>dr(j)(g;Tf)
From this point, we get after simple computations:
(1-L)
dj(Tf;Tg )(+L)
dj(f;g) +2L
dr(j)(g;Tf) (29)
and which implies
dj(Tf;Tg )+L
1-L
dj(f;g) +2L
1-L
dr(j)(g;Tf);8f;g2A (30)
Considering;L2R +and+2L<1 , the inequality (28) holds, with
+L
1-L2(0;1) and2L
1-L0. Therefore, any Ciri c-Reich-Rus type almost local
contraction with the condition for the pseudometric, is an almost local con-
traction. 
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Received: June 13, 2017