Synasc2017 Workshops Paper 20 [632186]

MULTIVALUED SELF ALMOST LOCAL
CONTRACTIONS
Zakany Monika
September 1, 2017
1 Almost contractions, local contractions
Denition 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 some
L0such that
d(Tx;Ty )
d(x;y) +L
d(y;Tx );8x;y2X (1)
Remark 1.1. The term of almost contraction is equivalent to weak contraction,
and it was rst introduced by V. Berinde in [3].
Remark 1.2. 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 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 contractiveness of T, it is necessary to check both (1)and(2).
Remark 1.3. A strict contraction satises (1), with=aandL= 0, therefore
is an almost contraction with a unique xed point.
Other examples of almost contractions are given in [4], [5], [2], [3]. There
are many other examples of contractive conditions which implies the almost
contractiveness condition, see for example Taskovic [26], Rus [21].
We present an existence theorem 1.1, then an existence and uniqueness the-
orem 1.2, as they are presented in [3]. Their main merit is that they extend
Banach's contraction principle and Zamrescu's xed point Theorem (1972 in
[28]). They also show us a method for approximating the xed point, for whitch
both a priori and a posteriori error estimates are available.
Theorem 1.1. Let(X;d)be a complete metric space and T:X!Xa weak
(almost) contraction. Then
1.Fix(T) =fx2X:Tx=xg6=;
2. For any x02X, the Picard iteration fxng1
n=0given byxn+1=Txn
converges to some x2Fix(T);
1

3. The following estimates
d(xn;x)n
1d(x0;x1); n = 0;1;2::: (3)
d(xn;x)
1d(xn1;xn); n = 1;2::: (4)
hold, where is the constant appearing in (1).
Theorem 1.2. Let(X;d)be a complete metric space and T:X!Xbe an
almost contraction for which there exist 2(0;1)and someL10such that
d(Tx;Ty )
d(x;y) +L1
d(x;Tx );8x;y2X (5)
Then
1.Thas a unique xed point,i.e., Fix(T) =fxg;
2. For any x02X, the Picard iteration fxng1
n=0converges to x;
3. The a priori and a posteriori error estimates
d(xn;x)n
1d(x0;x1); n = 0;1;2:::
d(xn;x)
1d(xn1;xn); n = 1;2:::
hold.
4. The rate of convergence of the Picard iteration is given by
d(xn;x)
d(xn1;x); n = 1;2::: (6)
Remark 1.4. (i) Weak contractions represent a generous concept, due to var-
ious mappings satisfying the condition (1). Such examples of weak contraction
was given by V. Berinde in [3], for example it was proved that:
– any Zamrescu mapping from Theorem Z in [28] is an almost contraction;
– any quasi-contraction with 0<h<1
2is an almost contraction;
– any Kannan mapping (in [13]) is the same kind of almost contraction
(ii) There are many other examples of contractive conditions which implies the
weak contractiveness condition, see for example Taskovic [26] , Rus [21] for
some of them.
(iii) Weak contractions need not have a unique xed point, however,the weak
contractions possess other important properties, amongst which we mention
a) In the class of weak contractions a method for constructing the xed points –
i.e. the Picard iteration – is always available;
b) Moreover, for this method of approximating the xed points, both a priori
and a posteriori error estimates are available. These are very important from
a practical point of view, since they provide stopping criteria for the iterative
process;
c) Last, but not least, the weak contractive condition (1)and(2)may easily be
handled and checked in concrete applications.
2

(iv) The xed point xattained by the Picard iteration depends on the initial
guessx02X. Therefore, the class of weak contractions provides a large class
of weakly Picard operators.
Recall, see Rus [21],[23] that an operator T:X!Xis said to be a weakly
Picard operator if the sequence fTnx0g1
n=0converges for all x02Xand the
limits are xed points of T.
(v) Condition (1)implies the so called Banach orbital condition
d(Tx;T2x)a
d(x;Tx );8x2X
studied by various authors in the context of xed point theorems, see for example
Hicks and Rhoades [11], Ivanov [12], Rus [20] and Taskovic [26].
The next Theorem show that an almost contraction is continuous at any
xed point of it, according to [1].
Theorem 1.3. Let(X;d)be a complete metric space and T:X!Xbe an
almost contraction. Then Tis continuous at p, for anyp2Fix(T).
Denition 1.2. (see [25]) Let Tbe a mapping on a metric space (X;d). Then
Tis called a generalized Berinde mapping if there exist a constant r2[0;1)and
a functionbfromXinto[0;1)such that
d(Tx;Ty )r
d(x;y) +b(y)
d(y;Tx );8x;y2X (7)
Denition 1.3. 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; (8)
where;2R +and+ 2 <1
Corollary 1.1. [18]. Let (X;d)be a metric space. Any Ciri c-Reich-Rus con-
traction,i.e., any mapping T:X!Xsatisfying the condition (8), represent an
almost contraction.
Theorem 1.4. A mapping satisfying the contractive condition:
there exists 0h<1
2such that
d(Tx;Ty )h
maxfd(x;y);d(x;Tx );d(y;Ty );d(x;Ty );d(y;Tx )g;8x;y2X
(9)
is a weak contraction.
An operator satisfying (9)with 0<h< 1is called quasi-contraction.
Theorem 1.5. Any mapping satisfying the condition: there exists 0b<1=2
such that
d(Tx;Ty )b[d(x;Tx ) +d(y;Ty )];8x;y2X (10)
is a weak contraction.
A mapping satisfying (10) is called Kannan mapping.
A kind of dual of Kannan mapping is due to Chatterjea [8]. The new con-
tractive condition is similar to (10) there exists 0 c<1
2such that
d(Tx;Ty )c[d(x;Ty ) +d(y;Tx )];8x;y2X; (11)
3

Theorem 1.6. Any mapping Tsatisfying the Chatterjea contractive condition,
i.e.: there exists 0c<1
2such that
d(Tx;Ty )c[d(x;Ty ) +d(y;Tx )];8x;y2X;
is a weak contraction.
Example 1.1. LetT: [0;1]![0;1]a mapping given by Tx=2
3forx2[0;1),
andT1 = 0 . ThenThas the following properties:
1)Tsatises (9)withh2[2
3;1), i.e.Tis quasi-contraction;
2)Tsatises (1), with2
3andL0, i.e.Tis also weak contraction;
3)Thas a unique xed point, x=2
3.
4)Tis not continuous.
The concept of local contraction was rst introduced by Martins da Rocha
and Filipe Vailakis in [14] (2010), here they studied the existence and uniqueness
of xed points for the local contractions.
Denition 1.4. LetFbe a set and letD= (dj)j2Ja family of semidistances
dened onF. We letbe the weak topology on Fdened by the family D.
Letrbe a function from JtoJ. An operator T:F!Fis a local contraction
with respect (D;r) if, for every j, there exists j2[0;1)such that
8f;g2F; d j(Tf;Tg )jdr(j)(f;g)
2 SINGLE VALUED SELF ALMOST LOCAL
CONTRACTIONS
We try to combine these two dierent type of contractive mappings: the almost
and local contractions, to study their xed points. This new type of mappings
was rst introduced in [27]
Denition 2.1. 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)
Denition 2.2. LetXbe a set and letD= (dj)j2Jbe a family of pseudometrics
dened onX. We letbe the weak topology on Xdened 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 Xconverges in Xfor the-topology.
The subsetAXis said to be -bounded if diam j(A)supfdj(x;y) :x;y2Ag
is nite for every j2J.
4

Denition 2.3. 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)andL0such that
dj(Tx;Ty )
dj(x;y) +L
dr(j)(y;Tx );8x;y2X (12)
Remark 2.1. The almost contractions represent a particular case of almost
local contractions, by taking (X;d)metric space instead of the pseudometrics dj
anddr(j)dened onX. Also, to obtain the almost contractions, we take in (12)
forrthe identity function, so we have r(j) =j.
Denition 2.4. The spaceXis- Hausdor if the following condition is valid:
for each pair x;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 2.1 is an existence xed point theorem for almost local contractions,
as they appear in [27].
Theorem 2.1. 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- invariant subset AX. If the condition
8j2J; lim
n!1n+1diam rn+1(j)(A) = 0 (13)
is satised, then the operator Tadmits a xed point xinA.
Proof. Letx02Xbe arbitrary andfxng1
n=0be the Picard iteration dened by
xn+1=Txn; n2N
Takex:=xn1;y:=xnin (12) to obtain
dj(Txn1;Txn)
dr(j)(xn1;xn)
which yields
dj(xn;xn+1)
dr(j)(xn1;xn);8j2J (14)
Using (12), we obtain by induction with respect to n:
dj(xn;xn+1)n
dr(j)(x0;x1); n = 0;1;2;


(15)
According to the triangle rule, by (15) we get:
dj(xn;xn+p)n(1 ++


+p1)dr(j)(x0;x1) =
=n
1(1p)
dr(j)(x0;x1); n;p2N;p6= 0
These relations show us that the sequence ( xn)n2Nisdj- Cauchy for each j2J.
The subset Ais assumed to be sequentially -complete, there exists finA
such that ( Tnx)n2Nis- convergent to x. Besides, the sequence ( Tnx)n2N
converges for the topology tox, which implies
8j2J; d j(Tx;x) = lim
n!1dj(Tx;Tn+1x):
5

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 obtain dj(Tx;x) = 0 for every j2J.
This way, we prove that Tf=f, sinceis Hausdor.
So, we prove the existence of the xed point for almost local contractions.
Remark 2.2. ForTveries (12) withL= 0, andr:J!Jthe identity
function, we nd Theorem Vailakis [14] by taking =j.
Further, for the case dj=d;8j2J, withd= metric on X, we obtain the well
known Banach contraction, with his unique xed point.
Remark 2.3. In Theorem 2.1, 2.3 the coecient of contraction 2(0;1)is
constant, but local contractions have a coecient of contraction j2[0;1)whitch
depends on j2J. Our rst goal is to extend the local almost contractions to
the most general case of j2(0;1).
One extend Denition 2.3 to the case of almost local contractions with vari-
able coecient of contraction.
Denition 2.5. Letrbe a function from JtoJ. An operator T:X!X
is called almost local contraction with respect ( D;r)or(j;Lj)- contraction, if
there exist a constant j2(0;1)and someLj0such that
dj(Tx;Ty )j
dj(x;y) +Lj
dr(j)(y;Tx );8x;y2X (16)
Theorem 2.2. With the presumptions of Theorem 2.1, if we modify the condi-
tion (13) by the following one:
8j2J; lim
n!1jr(j)


rn(j)diam rn+1(j)(A) = 0; (17)
then the operator Tadmits a xed point xinA.
The next Theorem represent an existence and uniqueness theorem for the
almost local contractions with constant coecient of contraction.
Theorem 2.3. If to the conditions of Theorem 2.1, we add:
(U) for every xed j2Jthere exists:
lim
n!1(+L)ndiam rn(j)(z;A) = 0;8x;y2X (18)
then the xed point xofTis unique.
3 Continuity of almost local contractions
This section can be regarded as an extension of V. Berinde and M. Pacurar
(2015,[1]) analysis about the continuity of almost contractions in their xed
points. The main results are given by Theorem 3.1, which give us the answer
about the continuity of local almost contractions in their xed points.
6

Theorem 3.1. LetXbe a set andD= (dj)j2Jbe a family of pseudometrics
dened onX; letT:X!Xbe an almost local contraction satisfying condition
(13), soTadmits a xed point.
ThenTis continuous at f, for anyf2Fix(T).
Proof. The mapping Tis an almost local contraction, i.e. there exist the con-
stants2(0;1) and some L0
dj(Tx;Ty )
dj(x;y) +L
dr(j)(y;Tx );8x;y2X (19)
For any sequence fyng1
n=0inXconverging to f, we takey:=yn;x:=fin (19),
and we get
dj(Tf;Ty n)
dj(f;yn) +L
dr(j)(yn;Tf);n= 0;1;2;::: (20)
UsingTf=f, sincefis a xed point of T, we obtain:
dj(Tyn;Tf)
dj(f;yn) +L
dr(j)(yn;f);n= 0;1;2;::: (21)
Now by letting n! 1 in (21) we get Tyn!Tf, which shows that Tis
continuous at f.
The xed point has been chosen arbitrarily, so the proof is complete.
According to Denition 2.3, the almost local contractions are dened in a
subsetAX. In the case A=X, then an almost local contraction is actually
an usual almost contraction.
Example 3.1. 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)
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: (22)
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
contraction, with the unique xed point x= 0;y= 0.
According to Theorem 3.1, Tis continuous in (0;0)2Fix(T), but is not con-
tinuous in (1;0)2X:
Example 3.2. With the presumptions of Example 3.1 and the pseudometric
dened by (22) , 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); (23)
7

is not valid8x;y2X, and for any 2(0;1). Indeed, (23) 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
For= 1=32(0;1),L= 10andj <0, 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+1diam rn+1(j)(A) = lim
n!11
3n+1

(n1)2= 0
This way, the existence of the xed point is assured, according to condition (12)
from Theorem 2.1
Theorem 3.1 is again valid, because the continuity of Tin(0;0)2Fix(T), but
discontinuity in (1;1), which is not a xed point of T.
Example 3.3. LetXthe set of positive functions: X=ffjf: [0;1)![0;1)g
anddj(f;g) =jf(0)g(0)j
ej;8f;g2X.
Indeed,djis a pseudometric, but not a metric, take for example dj(x;x2) = 0 ,
butx6=x2
Considering the mapping Tf=jfj;8f2X, and using condition (12) 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 (23) leads
us again to the false presumption: 1. The mapping Thas innite number
of xed points: FixT =ff2Xg, by taking:
jf(x)j=f(x);8f2X;x2[0;1)
4 MULTIVALUED SELF ALMOST LOCAL CON-
TRACTIONS
The term of multivalued contraction was rst introduced by Nadler in [15]. The
following are borrowed from Nadler [15]
8

Denition 4.1. Let(X;d)be a metric space, we shall denote the family of all
nonempty bounded and closed subsets of XwithCB(X).
ForA;BX, we consider
D(A;B) =inffd(a;b) :a2A;b2Bg;the distance between AandB,
(A;B) =supfd(a;b) :a2A;b2Bg;the diameter of AandB,
H(A;B) =maxfsupfD(a;B) :a2Ag;supfD(b;A) :b2Bgg, the Pompeiu-
Hausdor metric on CB(X)induced by d.
We know thatCB(X) form a metric space with the Pompeiu-Hausdor dis-
tance function H. It is also known,that if ( X;d) is a complete metric space then
(CB(X),H) is a complete metric space, too. (Rus [22])
LetP(X) be the family of all nonempty subsets of Xand letT:X!P (X)
be a multi-valued mapping. An element x2Xwithx2T(x) is called a xed
point ofT. We shall denote Fix(T) the set of all xed points of T, i.e.,
Fix(T) =fx2X:x2T(x)g
Letf:X!Xbe a single-valued map and T:X!CB (X) be a multivalued
map .
1. A point x2Xis a xed point of f(resp.T) ifx=fx(resp.x2Tx).
The set of all xed point of f(resp.T) is denoted by F(f), (resp.F(T)).
2. A point x2Xis a coincidence point of fandTiffx2Tx.
The set of all coincidence points of fandTwill be denoted by C(f;T)
3. A point x2Xis a common xed point of fandTifx=fx2Tx:
The set of all common xed points of fandTis denoted by F(f;T)
The following lemma can be found in Rus [22], it is useful for the next theorem.
Lemma 4.1. Let(X;d)be a metric space, let A;BXandq>1.
Then, for every a2A, there exists b2Bsuch that
d(a;b)qH(A;B) (24)
Denition 4.2. Let(X;d)be a metric space and T:X!P (X)be a multi-
valued operator. Tis said to be a multi-valued weak contraction or a multi-valued
(;L)-weak contraction if there exist two constants 2(0;1);L0such that
H(Tx;Ty )
d(x;y) +L
D(y;Tx );8x;y2X (25)
Remark 4.1. Because of the simmetry of the distance dandH, the almost
contraction condition (25) includes the following dual one:
H(Tx;Ty )
d(x;y) +L
D(x;Ty );8x;y2X (26)
Obviously, to prove the almost contractiveness of T, it is necessary to check both
(25) and(26).
Theorem 4.1. (Berinde V., Berinde M. [6]) Let (X;d)be a metric space and
T:X!P(X)be a (;L)-weak contraction. Then
(1)Fix(T)6=
9

(2) for any x02X, there exists an orbit fxng1
n=0ofTat the point x0that
converges to a xed point uofT, for which the following estimates hold:
d(xn;u)hn
1hd(x0;x1); n = 0;1;2::: (27)
d(xn;u)h
1hd(xn1;xn); n = 1;2::: (28)
for a certain constant h<1.
5 Main Results
We shall use the assumptions from the denition of almost local contractions
and we make the following notations:
Dj(A;B) =inffdj(a;b) :a2A;b2Bg,
j(A;B) =supfdj(a;b) :a2A;b2Bg,
Hj(A;B) =maxfsupfDj(a;B) :a2Ag;supfDj(b;A) :b2Bgg,
the Pompeiu-Hausdor metric on CB(X) induced by dj.
Remark 5.1. From the denition of Dj, we have the following result:
ifDj(a;B) = 0 , that implies a2B
Denition 5.1. Letrbe a function from JtoJ. An operator T:X!P(X)
is called a multivalued almost local contraction (ALC) with respect ( D;r) if, for
everyj2J, there exists the constants 2(0;1)andL0such that
Hj(Tx;Ty )
dj(x;y) +L
Dr(j)(y;Tx );8x;y2X (29)
Lemma 5.1. LetXbe a set and letD= (dj)j2Jbe a family of pseudometrics
dened onX. We letbe the weak topology on Xdened by the family D.
LetA;BXandq>1.
Then, for every j2Janda2A, there exists b2Bsuch that
dj(a;b)qHj(A;B) (30)
Proof. IfHj(A;B) = 0, then for every a2A, we have:
Hj(A;B)Dj(a;B))Dj(a;B) = 0
From that, we conclude: there exist b2Bsuch thatdj(a;b) = 0.
The inequality (30) is valid, i.e., 0 0.
IfHj(A;B)>0, then let us denote
"= (h11)H(A;B)>0 (31)
Using the denition of Hj(A;B) andDj(a;B), we conclude that for any ">0
there exists b2Bsuch that
dj(a;b)qDj(a;B) +"Hj(A;B) +" (32)
Combining (31) and (32), we get (30).
10

Theorem 5.1. With the assumptions of Denition 5.1, let T:X!P (X)be
a multivalued ALC. Then we have:
(1)Fix(T)6=
(2) for any x02X, there exists an orbit fxng1
n=0ofTat the point x0that
converges to a xed point uofT, for which the following estimates hold:
dj(xn;u)hn
1hdj(x0;x1); n = 0;1;2::: (33)
dj(xn;u)h
1hdj(xn1;xn); n = 1;2::: (34)
for a certain constant h<1.
Proof. We consider q>1, letx02Xandx12Tx0. IfHj(Tx0;Tx 1) = 0, that
means from the denition of DjandHj:
0 =Hj(Tx0;Tx 1)Dj(x1;Tx 1) (35)
and that is possible only if Dj(x1;Tx 1) = 0, from here, we conclude x12Tx1,
which leads us to the conclusion Fix(T)6=.
LetHj(Tx0;Tx 1)6= 0. According to Lemma 5.1, there exists x22Tx1such
that
dj(x1;x2)qHj(Tx0;Tx 1) (36)
By (29) we have
dj(x1;x2)q[
dj(x0;x1) +L
Dr(j)(x1;Tx 0)] =q
dj(x0;x1):
sincex12Tx0,Dr(j)(x1;Tx 0) = 0.
We takeq>1 such that
h=q< 1
and we obtain dj(x1;x2)<h
dj(x0;x1).
IfHj(Tx1;Tx 2) = 0 thenDj(x2;Tx 2) = 0, that means x22Tx2using Remark
5.1.
LetHj(Tx1;Tx 2)6= 0. Again, using Lemma 5.1, there exists x32Tx2such
that
dj(x2;x3)qh
dj(x1;x2) (37)
This way, we obtain an orbit fxng1
n=0ofTat the point x0satisfying
dj(xn;xn+1)h
dj(xn1;xn); n = 1;2;::: (38)
By (38), we inductively obtain
dj(xn;xn+1)hndj(x0;x1) (39)
and, respectively,
dj(xn+k;xn+k+1)hk+1dj(xn1;xn); k2N (40)
Using the inequality (39), we obtain
dj(xn;xn+p)hn(1hp)
1hdj(x0;x1); n;p2N (41)
11

Recall 0< h < 1, conditions (40),(41) show us that the sequence ( xn)n2Nis
dj-Cauchy for each j, which shows that fxng1
n=0is a Cauchy sequence. That
meansfxng1
n=0is convergent with the limit u:
u= lim
n!1xn (42)
After simple computations, we get:
Dr(j)(u;Tu )Dr(j)(u;xn+1)+Dr(j)(xn+1;Tu)dr(j)(u;xn+1)+Hr(j)(Txn;Tu)
which by (29) yields
Dr(j)(u;Tu )dr(j)(u;xn+1) +dr(j)(xn;u) +L
Dr(j)(u;Tx n) (43)
Lettingn!1 and using the fact that xn+12Txnimplies by (42),
Dr(j)(u;Tx n)!0, asn!1 . We get
Dr(j)(u;Tu ) = 0
SinceTuis closed, this implies u2Tu.
We letp!1 in (41) to obtain (33). Using (40), we get
d(xn;xn+p)h(1hp)
1hd(xn1;xn); p2N;n1 (44)
and letting p!1 in (44), we obtain (34). The proof is complete.
The next Theorem shows that any multivalued ALC is continuous at the
xed point.
Theorem 5.2. With the assumptions of Denition 5.1, let T:X!P (X)be
a multivalued ALC, i.e., a mapping for which there exists the constants
2(0;1)andL0such that, for every j2J, the next inequality is valid:
Hj(Tx;Ty )
dj(x;y) +L
Dr(j)(y;Tx );8x;y2X (45)
ThenFix(T)6=and for any p2Fix(T),Tis continuous at p.
Proof. The rst part of the conclusion follows by Theorem 5.1.
Letfyng1
n=0be any sequence in Xconverging to the xed point p. Then by
takingy:=ynandx:=pin the multivalued ALC condition (45), we get
dj(Tp;Ty n)
dj(p;yn) +L
Dr(j)(yn;Tp);n= 0;1;2;::: (46)
Using the denition of Dr(j)(yn;Tp), we know that is the smallest distance
betweenynand any element from Tp, take for example p2Tp. Now, we have
the following inequalities:
Dr(j)(yn;Tp)Dj(yn;Tp)dj(yn;p)
By replacing Dr(j)(yn;Tp) from (46) with dj(yn;p), we get:
dj(Tyn;Tp)(+L)
dj(yn;p);n= 0;1;2;::: (47)
Now, by letting n!1 in (47) we get Tyn!Tpasn!1 , that means: Tis
continuous at p.
12

References
[1] Berinde, V., P acurar, M. Fixed points and continuity of almost contrac-
tions Carpathian J. Math. 19 (2003) No. 1, 7-22
[2] Berinde, V., On the approximation of xed points of weak contractive map-
pings Carpathian J. Math. 19 (2003) No. 1, 7-22
[3] Berinde, V., Approximating xed points of weak contractions using the
Picard iteration Nonlinear Analysis Forum 9(2004) No.1, 43-53
[4] Berinde, V., On the convergence of the Ishikawa iteration in the class of
quasi operators Acta Math. Univ. Comenianae 73 (2004), 119-126
[5] Berinde, V., A convergence theorem for some mean value xed point it-
erations in the class of quasi contractive operators Demonstr. Math. 38
(2005), 177-184
[6] Berinde, V.,Berinde M., On a general class of multi-valued weakly Picard
mappings J. Math. Anal. Appl.No.326(2007), 772-782
[7] Berinde, V., P acurar, M. Fixed points theorems for nonself single-valued
almost contractions Fixed Point Theory, 14(2013), No. 2, 301-312
[8] Chatterjea SK, Fixed-point theorems C.R. Acad. Bulgare Sci.
1972;25:727{730.
[9] Fang Jin-xuan, FIXED POINT THEOREMS OF LOCAL CONTRAC-
TION MAPPINGS ON MENGER SPACES, Applied Mathematics and
Mechanics, (English Edition, Vol. 12, No. 4, Apr. 1991)Published by
SUT,Shanghai, China
[10] G. V. R. Babu, M. L. Sandhya, M. V. R. Kameswari, A note on a xed
point theorem of Berinde on weak contractions, Carpathian J. Math., 24
(2008), No.1
[11] Hicks, T.L. and Rhoades, B.E. , A Banach type xed point theorem, Math.
Japonica 24 (3) (1979) 327-330
[12] Ivanov, A.A., Fixed points of metric space mappings (in Russian), Isle-
dovaniia po topologii. II, Akademia Nauk, Moskva, 1976, pp. 5-102
[13] Kannan, R., Some results on xed points, Bull. Calcutta Math. Soc. 10
(1968) 71-76
[14] Martins-da-Rocha,Filipe, Vailakis, Yiannis, Existence and uniqueness of a
xed point for local contractions , Econometrica, vol.78, No.3 (May, 2010)
1127-1141
[15] Nadler, S.B., Multi-valued contraction mappings , Pacic J. Math.
30,1969, 475-488
[16] Osilike, M.O., Stability of the Ishikawa Iteration method for quasi-
contractive maps , Indian J. Pure Appl. Math. 28 (9) (1997) 1251-1265
13

[17] P acurar M ad alina, Berinde,V., Two new general classes of multi-valued
weakly Picard mappings (submitted) Amer. Math. Soc. 196 (1974) 161-
176
[18] P acurar M ad alina, Iterative Methods for Fixed Point Approximation. Cluj-
Napoca:Editura Risoprint;2009.
[19] Rus, I.A., Metrical xed point theorems , Univ. of Cluj-Napoca, 1979
[20] Rus, I.A., Principles and Applications of the Fixed Point Theory , (in Ro-
manian) Editura Dacia, Cluj-Napoca, 1979
[21] Rus, I.A., Generalized Contractions Seminar on Fixed Point Theory 3
1983, 1-130
[22] Rus,I.A., Generalized Contractions and Applications, Cluj University
Press, Cluj-Napoca, 2001
[23] Rus, I.A., Picard operator and applications, Babes-Bolyai Univ., 1996
[24] Schweizer, B., A. Sklar and E. Thorp, The metrization of statistical metric
spaces, Pacic J. Math., 10 (1960), 673 – 675.
[25] Suzuki, Tomonari, Fixed Point Theorems For Berinde Mappings, Bull.
Kyushu Inst. Tech., Pure Appl. Math. No. 58, 2011, pp. 13{19
[26] Taskovic M. Osnove teorije ksne tacke , (Fundamental Elements of Fixed
Point Theory) Matematicka Biblioteka 50, Beograd, 1986
[27] Zakany, Monika, Fixed Point Theorems For Local Almost Contractions,
Miskolc Mathematical Notes,vol.18, no.1 (2017), 499-506
[28] Zamrescu, T., Fix point theorems in metric spaces Arch. Math. (Basel),
23 (1972), 292-298
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