Analele Universit at ii de Vest, Timi soara [632185]

Analele Universit at ii de Vest, Timi soara
Seria Matematic a { Informatic a
Vol. , fasc. ,
Department of Mathematics and Computer Sciences
North University Center at Baia Mare, Technical University of Cluj-Napoca
Victoriei 76, 430122
Baia Mare
Romania
MULTIVALUED SELF ALMOST LOCAL
CONTRACTIONS
Monica Zakany

2 Monica Zakany An. U.V.T.
Abstract. We want to introduce a new class of contractive mappings:
the almost local contractions, starting from the almost contractions
presented by V. Berinde in [Berinde, V., Approximating xed points of
weak contractions using the Picard iteration Nonlinear Analysis Forum
9(2004) No.1, 43-53], and also from the concept of local contraction
which are presented by Martins da Rocha and Filipe Vailakis in
[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].
First of all, we present the notion of multivalued self almost contractions
with many examples.
The main results of this paper are given by the extension to the case of
multivalued self almost local contractions.
AMS Subject Classication (2000). 55M20 ; 47H10
Keywords. almost contractions,local contractions,xed
points,multivalued mappings
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 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 con-
traction, and it was rst introduced by V. Berinde in [3].
Remark 1.2. Because of the simmetry of the distance, the almost con-
traction 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) by
d(y;x), and after that step, changing xwithy, and viceversa. Obviously,

Vol. () Multivalued self ALC 3
to prove the almost contractiveness of T, it is necessary to check both (1.1)
and (1.2).
Remark 1.3. A strict contraction satises (1.1), 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 [22], Rus [18].
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 [24]). 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);
3. The following estimates
d(xn;x)n
1d(x0;x1); n = 0;1;2::: (1.3)
d(xn;x)
1d(xn1;xn); n = 1;2::: (1.4)
hold, where is the constant appearing in (1.1) .
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 (1.5)
Then

4 Monica Zakany An. U.V.T.
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::: (1.6)
Remark 1.4. (i) Weak contractions represent a generous concept, due to
various mappings satisfying the condition (1.1). 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 [24] is an almost contraction;
– any quasi-contraction with 0 <h<1
2is an almost contraction;
– any Kannan mapping (in [10]) 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 [22] , Rus [18]
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 pri-
ori and a posteriori error estimates are available. These are very important
from a practical point of view, since they provide stopping criteria for the

Vol. () Multivalued self ALC 5
iterative process;
c) Last, but not least, the weak contractive condition (1.1) and (1.2) may
easily be handled and checked in concrete applications.
(iv) The xed point xattained by the Picard iteration depends on the
initial guess x02X. Therefore, the class of weak contractions provides a
large class of weakly Picard operators.
Recall, see Rus [18],[20] 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 (1.1) 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 Rus [17] and Taskovic [22].
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 [21]) LetTbe a mapping on a metric space (X;d).
ThenTis called a generalized Berinde mapping if there exist a constant
r2[0;1)and a function bfromXinto[0;1)such that
d(Tx;Ty )r
d(x;y) +b(y)
d(y;Tx );8x;y2X (1.7)
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; (1.8)
where;2R+and+ 2 <1
Corollary 1.4. [15]. Let (X;d)be a metric space. Any Ciri c-Reich-Rus
contraction,i.e., any mapping T:X!Xsatisfying the condition (1.8) ,
represent an almost contraction.

6 Monica Zakany An. U.V.T.
Theorem 1.5. 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
(1.9)
is a weak contraction.
An operator satisfying (1.9) with 0<h< 1is called quasi-contraction.
Theorem 1.6. Any mapping satisfying the condition: there exists 0b<
1=2such that
d(Tx;Ty )b[d(x;Tx ) +d(y;Ty )];8x;y2X (1.10)
is a weak contraction.
A mapping satisfying (1.10) is called Kannan mapping.
A kind of dual of Kannan mapping is due to Chatterjea [8]. The new
contractive condition is similar to (1.10): there exists 0 c<1
2such that
d(Tx;Ty )c[d(x;Ty ) +d(y;Tx )];8x;y2X; (1.11)
Theorem 1.7. Any mapping Tsatisfying the Chatterjea contractive con-
dition, 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
3for
x2[0;1), andT1 = 0. Then Thas the following properties:
1)Tsatises (1.9) with h2[2
3;1), i.e.Tis quasi-contraction;
2)Tsatises (1.1), with 2
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 [11] (2010), here they studied the existence and
uniqueness of xed points for the local contractions.

Vol. () Multivalued self ALC 7
Denition 1.4. LetFbe a set and let D= (dj)j2Ja family of semidis-
tances dened on F. We letbe the weak topology on Fdened by the
family D.
Letrbe a function from JtoJ. An operator T:F!Fis a local con-
traction 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 CONTRAC-
TIONS
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 [23]
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 let D= (dj)j2Jbe a family of pseudo-
metrics dened on X. 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.

8 Monica Zakany An. U.V.T.
Denition 2.3. Letrbe a function from JtoJ. An operator T:X!X
is called an almost local contraction (ALC) 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 (2.1)
Remark 2.1. The almost contractions represent a particular case of almost
local contractions, by taking ( X;d) metric space instead of the pseudomet-
ricsdjanddr(j)dened onX. Also, to obtain the almost contractions, we
take in (2.1) for rthe 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 contrac-
tions, as they appear in [23].
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 (2.2)
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 (2.1) to obtain
dj(Txn1;Txn)
dr(j)(xn1;xn)
which yields
dj(xn;xn+1)
dr(j)(xn1;xn);8j2J (2.3)

Vol. () Multivalued self ALC 9
Using (2.1), we obtain by induction with respect to n:
dj(xn;xn+1)n
dr(j)(x0;x1); n = 0;1;2;


(2.4)
According to the triangle rule, by (2.4) 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
finAsuch that (Tnx)n2Nis- 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 pseudo-
metricdr(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 contrac-
tions.
Remark 2.2. ForTveries (2.1) with L= 0, andr:J!Jthe identity
function, we nd Theorem Vailakis [11] 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).

10 Monica Zakany An. U.V.T.
One extend Denition 2.3 to the case of almost local contractions with
variable 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 (2.5)
Theorem 2.2. With the presumptions of Theorem 2.1, if we modify the
condition (2.2) by the following one:
8j2J; lim
n!1jr(j)


rn(j)diam rn+1(j)(A) = 0; (2.6)
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 (2.7)
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.
Theorem 3.1. LetXbe a set and D= (dj)j2Jbe a family of pseudometrics
dened on X; letT:X!Xbe an almost local contraction satisfying
condition (2.2) , soTadmits a xed point.
ThenTis continuous at f, for anyf2Fix(T).

Vol. () Multivalued self ALC 11
Proof. The mapping Tis an almost local contraction, i.e. there exist the
constants2(0;1) and some L0
dj(Tx;Ty )
dj(x;y) +L
dr(j)(y;Tx );8x;y2X (3.1)
For any sequence fyng1
n=0inXconverging to f, we takey:=yn;x:=fin
(3.1), and we get
dj(Tf;Ty n)
dj(f;yn) +L
dr(j)(yn;Tf);n= 0;1;2;::: (3.2)
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;::: (3.3)
Now by letting n!1 in (3.3) 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: (3.4)
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
continuous in (1 ;0)2X:

12 Monica Zakany An. U.V.T.
Example 3.2. With the presumptions of Example 3.1 and the pseudomet-
ric dened by (3.4) , 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); (3.5)
is not valid8x;y2X, and for any 2(0;1). Indeed, (3.5) 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= 10 andj <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
(2.1) 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.

Vol. () Multivalued self ALC 13
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 (2.1)
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 (3.5)
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 CONTRAC-
TIONS
The term of multivalued contraction was rst introduced by Nadler in [12].
The following are borrowed from Nadler [12]
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 that CB(X) form a metric space with the Pompeiu-Hausdor
distance function H. It is also known,that if ( X;d) is a complete metric

14 Monica Zakany An. U.V.T.
space then ( CB(X),H) is a complete metric space, too. (Rus [19])
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 of T. 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 multival-
ued map .
1. A pointx2Xis 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 [19], 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) (4.1)
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);L0
such that
H(Tx;Ty )
d(x;y) +L
D(y;Tx );8x;y2X (4.2)
Remark 4.1. Because of the simmetry of the distance dandH, the almost
contraction condition (4.2) includes the following dual one:
H(Tx;Ty )
d(x;y) +L
D(x;Ty );8x;y2X (4.3)

Vol. () Multivalued self ALC 15
Obviously, to prove the almost contractiveness of T, it is necessary to check
both (4.2) and (4.3).
Theorem 4.2. (Berinde V., Berinde M. [6]) Let (X;d)be a metric space
andT:X!P(X)be a (;L)-weak contraction. Then
(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:
d(xn;u)hn
1hd(x0;x1); n = 0;1;2::: (4.4)
d(xn;u)h
1hd(xn1;xn); n = 1;2::: (4.5)
for a certain constant h<1.
5 Main Results
We shall use the assumptions from the denition of almost local contrac-
tions 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 every j2J, there exists the constants 2(0;1)andL0
such that
Hj(Tx;Ty )
dj(x;y) +L
Dr(j)(y;Tx );8x;y2X (5.1)

16 Monica Zakany An. U.V.T.
Lemma 5.1. LetXbe a set and let D= (dj)j2Jbe a family of pseudo-
metrics dened on X. 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) (5.2)
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 (5.2) is valid, i.e., 0 0.
IfHj(A;B)>0, then let us denote
"= (h11)H(A;B)>0 (5.3)
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) +" (5.4)
Combining (5.3) and (5.4), we get (5.2).
Theorem 5.2. 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::: (5.5)
dj(xn;u)h
1hdj(xn1;xn); n = 1;2::: (5.6)
for a certain constant h<1.

Vol. () Multivalued self ALC 17
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) (5.7)
and that is possible only if Dj(x1;Tx 1) = 0, from here, we conclude x12
Tx1, which leads us to the conclusion Fix(T)6=.
LetHj(Tx0;Tx 1)6= 0. According to Lemma 5.1, there exists x22Tx1
such that
dj(x1;x2)qHj(Tx0;Tx 1) (5.8)
By (5.1) 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 then Dj(x2;Tx 2) = 0, that means x22Tx2using
Remark 5.1.
LetHj(Tx1;Tx 2)6= 0. Again, using Lemma 5.1, there exists x32Tx2
such that
dj(x2;x3)qh
dj(x1;x2) (5.9)
This way, we obtain an orbit fxng1
n=0ofTat the point x0satisfying
dj(xn;xn+1)h
dj(xn1;xn); n = 1;2;::: (5.10)
By (5.10), we inductively obtain
dj(xn;xn+1)hndj(x0;x1) (5.11)
and, respectively,
dj(xn+k;xn+k+1)hk+1dj(xn1;xn); k2N (5.12)

18 Monica Zakany An. U.V.T.
Using the inequality (5.11), we obtain
dj(xn;xn+p)hn(1hp)
1hdj(x0;x1); n;p2N (5.13)
Recall 0< h < 1, conditions (5.12),(5.13) show us that the sequence
(xn)n2Nisdj-Cauchy for each j, which shows that fxng1
n=0is a Cauchy
sequence. That means fxng1
n=0is convergent with the limit u:
u= lim
n!1xn (5.14)
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 (5.1) yields
Dr(j)(u;Tu )dr(j)(u;xn+1) +dr(j)(xn;u) +L
Dr(j)(u;Tx n) (5.15)
Lettingn!1 and using the fact that xn+12Txnimplies by (5.14),
Dr(j)(u;Tx n)!0, asn!1 . We get
Dr(j)(u;Tu ) = 0
SinceTuis closed, this implies u2Tu.
We letp!1 in (5.13) to obtain (5.5). Using (5.12), we get
d(xn;xn+p)h(1hp)
1hd(xn1;xn); p2N;n1 (5.16)
and letting p!1 in (5.16), we obtain (5.6). The proof is complete.
The next Theorem shows that any multivalued ALC is continuous at the
xed point.
Theorem 5.3. 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 (5.17)
ThenFix(T)6=and for any p2Fix(T),Tis continuous at p.

Vol. () Multivalued self ALC 19
Proof. The rst part of the conclusion follows by Theorem 5.2.
Letfyng1
n=0be any sequence in Xconverging to the xed point p. Then
by takingy:=ynandx:=pin the multivalued ALC condition (5.17), we
get
dj(Tp;Ty n)
dj(p;yn) +L
Dr(j)(yn;Tp);n= 0;1;2;::: (5.18)
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 (5.18) with dj(yn;p), we get:
dj(Tyn;Tp)(+L)
dj(yn;p);n= 0;1;2;::: (5.19)
Now, by letting n!1 in (5.19) we get Tyn!Tpasn!1 , that means:
Tis continuous at p.
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Monica Zakany
Baia Mare, str.Simion Barnutiu nr.24, Romania
E-mail: zakanymoni@yahoo.com