21st International Symposium on [632180]

SYNASC 2019
21st International Symposium on
Symbolic and Numeric Algorithms for
Scientific Computing
ALMOST LOCAL
CONTRACTIONS IN
b-PSEUDOMETRIC SPACES
Zakany Monika
supervised by
Prof. Univ. Dr. Vasile Berinde
Technical University of Cluj-Napoca
North University Center of Baia Mare
Faculty of Sciences
September 4-7, 2019

1 Introduction
In this paper, the notion of almost local contraction in a b-pseudometric space is consi-
dered. In this framework some new fixed point results are given. Many generalisations
of the concept of metric space were given by several autors, the most importants: [9],
[5], [1], [6], and recent works, amongst which we mention [3] , [4]. The concept of
b-metric space was introduced by Czerwik in [6]. Since then several publications were
studied the fixed point of single valued and multivalued operators in b-metric spaces
(see [2], [10], [6]). The starting point of this paper was the book of M. Păcurar [7], who
take an in depth look into the question of fixed points in b- metric spaces, but only for
the almost contractions.
Definition 1. (see [9]) Let Xbe a nonempty set.
A mapping db:X×X→R+is called b-metric if the following hold:
1.db(x,y) =db(y,x),∀x,y∈X;
2.db(x,z)≤b·[db(x,y) +db(y,z)],∀x,y,z∈X,
whereb≥1is a given real number.
3.db(x,y) = 0if and only if x=y;
A nonempty set Xendowed with a b-metricdb:X×X→R+is called b-metric space.
In the following, we shall define the b-pseudometric:
Definition 2. The mapping db:X×X→R+is said to be a b-pseudometric if:
1.db(x,y) =db(y,x),∀x,y∈X;
2.db(x,z)≤b·[db(x,y) +db(y,z)],∀x,y,z∈X,
whereb≥1is a given real number
3.db(x,x) = 0,∀x∈X.
Remark 1. Obviously, a metric space is a b-metric space with b= 1. However a
b-metric on Xneed not be a metric on the set X, as shown later in this paper
(Examples 1,2 ).
Remind the definition of a weakly Picard operator:
Definition 3. A mapT:X→Xis called weakly Picard operator (see [8],[9]) if the
sequence of successive approximations: {Tnx0}∞
n=0converges for all the initial points
x0∈Xand the limits are fixed points of T.
Theorem 1.1. LetXbe a set and letD= (dj)j∈Jbe a family of b- pseudometrics
defined onX, with constant b≥1. We choose a subset A⊂Xand we letτbe the weak
topology on Xdefined by the family D. Letf:A→Aan almost local contraction with
constantsθ∈[0,1
b)andL≥0.
Assume a monotonicity property valid for the operator f:
dr(j)(f,g)≤dj(f,g),∀f,g∈A,∀j∈J. (1.1)
Then:
1

(i)fis a weakly Picard operator;
(ii) If theb-pseudometric is continuous, then for any x∈Athe following error esti-
mates hold:
dj(fn(x),f∞(x))≤bθn
1−bθdj(x,f(x)),n≥1; (1.2)
dj(fn(x),f∞(x))≤bθ
1−bθdj(fn−1(x),fn(x)),n≥1; (1.3)
Proof.(i) In the beginning, we shall prove that the operator fhas at least one fixed
pointinthesubset A, i.e. thesetoffixedpointsisnonempty. Tothisend, welet x0∈A
and{xn}n≥0be the Picard iteration which starts from x0. Applying the definition of
the Picard iteration and also the definition of an ALC, we get:
dj(xn,xn+1) =dj(f(xn−1),f(xn))≤θ·dr(j)(xn−1,xn) +Ldr(j)(xn,xn),
for alln∈N. We obtain by induction with respect to n:
dj(xn,xn+1)≤θn·dr(j)(x0,x1), n = 1,2,··· (1.4)
Forn≥0,p≥1we can write:
dj(xn,xn+p)≤b[dj(xn,xn+1) +dj(xn+1,xn+p)] =
=bdj(xn,xn+1) +bdj(xn+1,xn+p)≤
≤bdj(xn,xn+1) +b2[dj(xn+1,xn+2) +dj(xn+2,xn+p))]≤···≤
≤bdj(xn,xn+1) +b2dj(xn+1,xn+2) +···+bpdj(xn+p−1,xn+p).
By (1.4) it follows that:
dj(xn,xn+p)≤bθndr(j)(x0,x1) +b2θn+1dr(j)(x0,x1) +···+bpθn+p−1dr(j)(x0,x1) =
=bθndr(j)(x0,x1)[1 +bθ+ (bθ)2+···+ (bθ)p−1] =
=b·1−(bθ)p
1−bθdr(j)(x0,x1)·θn, (1.5)
forn≥0,p≥1.
Remind that θ∈[0,1
b), withb≥1, it is obvious that 0≤bθ < 1, which yields from
(1.5)theconclusionthat {xn}n≥0isadj−Cauchysequenceinthe b-pseudometricspace.
This means that it is convergent with his limit denoted by
x∗= limn→∞xn. (1.6)
Applying the definition of the b-pseudometric, we get:
dj(x∗,f(x∗))≤b[dj(x∗,f(xn+1)) +dj(f(xn),f(x∗))].
After using the definition of an almost local contraction and the monotonicity property
(1.1) we obtain from the last inequality:
dj(x∗,f(x∗))≤bdj(x∗,f(xn)) +bθdr(j)(xn,x∗) +bLdr(j)(x∗,f(xn))≤
≤bdr(j)(x∗,f(xn)) +bθdr(j)(xn,x∗) +bLdr(j)(x∗,f(xn)) =
=b(1 +L)dr(j)(x∗,xn+1) +bθd(xn,x∗).
2

Having in view (1.6) and letting n→∞, it results that
dj(x∗,f(x∗)) = 0,
which means that x∗is a fixed point of f. Sofis a weakly Picard operator.
(ii)Inthesequel, remindthatthe b-pseudometriciscontinuousand 0≤bθ< 1. Letting
p→∞in (1.5) we get the a priori error estimate (1.2). By using the induction in
(1.4), we obtain:
dj(xn+k,xn+k+1)≤θk+1·dr(j)(xn−1,xn), (1.7)
for anyn,k∈N,n≥1. From that, we can write:
dj(xn,xn+p)≤bdj(xn,xn+1) +b2dj(xn+1,xn+2) +···+bpdj(xn+p−1,xn+p)≤
≤bθdj(xn−1,xn) + (bθ)2dj(xn−1,xn) +···+ (bθ)pdj(xn−1,xn) =
=bθ·1−(bθ)p
1−bθ·dj(xn−1,xn).
Again, by the continuity of the b-pseudometric and having in view that 0≤bθ< 1, by
lettingp→∞in the last inequality, it results the a posteriori estimate, namely the
second relation in (1.4).
The proof is complete.
In the sequel, our main goal is to extend the ALC-s to the case of strict almost
contractions, as a result of which we get an existence and uniqueness theorem:
Theorem 1.2. LetXbe a set and letD= (dj)j∈Jbe a family of b- pseudometrics
defined onX, with constant b≥1. We choose again a subset A⊂Xand we let τbe
the weak topology on Xdefined by the family D. Letf:A→Aa strict almost local
contraction with constants θ∈[0,1
b)andL≥0, andθu∈[0,1
b)andLu≥0.Assume a
uniqueness condition for the mapping f(see [11]), which is:
dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(x,T(x)),∀x,y∈A, (1.8)
Then:
(i)fis a Picard operator;
(ii) If theb-pseudometric is continuous, then the following error estimates hold:
dj(xn,x∗)≤bθn
1−bθdj(x0,x1),n≥1, (1.9)
dj(xn,x∗)≤bθ
1−bθdj(xn−1,xn),n≥1; (1.10)
(iii) Assume the continuity of the b-pseudometric. The rate of convergence of the
Picard iteration is given by
dj(xn,x∗)≤θudj(xn−1,x∗),n≥1, (1.11)
whereFf={x∗}.
3

Proof.(i) The first part of the conclusion of the theorem, namely the existence of
the fixed point is assured by Theorem 1.1. In order to prove the uniqueness of the
fixed point, suppose that fhas two different fixed points x∗,y∗∈A. Then, applying
the monotonicity condition (1.1) for the b- pseudometric and the uniqueness condition
(1.8), we can write that:
dj(f(x∗,f(y∗))≤θudr(j)(x∗,y∗) +Ludr(j)(x∗,f(x∗))
≤θudj(x∗,y∗) +Ludj(x∗,f(x∗)),
which means that dj(x∗,y∗)≤θudj(x∗,y∗).
As0≤θu<1, we get the obvious contradiction dj(x∗,y∗)<dj(x∗,y∗).
It results that Ff={x∗},sofis a Picard operator.
(ii) The a priori and a posteriori estimates (1.9), (1.10) follows by Theorem 1.1.
(iii) From (1.8) we obtain:
dj(f(x∗),f(xn−1))≤θudr(j)(x∗,xn−1) +Ludr(j)(x∗,f(x∗))
≤θudj(x∗,xn−1) +Ludj(x∗,f(x∗)),
which means:
dj(xn,x∗)≤θudj(xn−1,x∗),n≥1.
Next, we present two examples of almost local contractions in b-pseudometric
spaces:
Example 1. LetX={−1,0,1}×{− 1,0,1}⊂R2. We shall use the b-pseudometric:
dj((x1,y1),(x2,y2)) = 0, ifx1=x2, dj((x1,y1),(x2,y2)) =e−j, if|x1−x2|=1,
dj((x1,y1),(x2,y2)) =b·e−j, if|x1−x2|= 2, whereb≥2andJis a subset of N.
This is a pseudometric, but not a metric, take for example:
dj((1,−1),(1,0)) = 0, however (1,−1)/negationslash= (1,0).
In this case, we shall use the function r(j) =j, wherej∈J.
Considering T:X→X,
T(x,y) =/braceleftBigg
(x,−y)if (x,y)/negationslash= (1,1)
(0,0)if (x,y) = (1,1)
Tis not a contraction because the contractive condition:
dj(Tx,Ty )≤θ·dj(x,y), (1.12)
is not valid∀x,y∈X, and for any θ∈[0,1). Indeed, (1.12)is equivalent with:
|x1−x2|·e−j≤θ·|x1−x2|·e−j,∀j∈J
The last inequality leads us to 1≤θ, which is obviously false, considering θ∈[0,1).
However,Tbecomes an almost local contraction if:
|x1−x2|·e−j≤θ·|x1−x2|·e−j
2+L·|x2−x1|·e−j
2
which is equivalent to : e−j
2≤θ+L.
Forθ=1
4∈[0,1),L= 2≥0andj >0, the last inequality becomes true, i.e. Tis
an almost local b-contraction with three fixed points:
FixT ={(−1,0),(0,0),(1,0)}.
4

Example 2. LetX={0,1,2} ⊂R. Define the b-pseudometric: dj(x,y) = 0, if
x=y,dj(x,y) =e−j, if|x−y|= 1,dj(x,y) =b·e−j, if|x−y|= 2,where
b≥2andJis a subset of N. We shall use the function r(j) =j
2, wherej∈J.
Considering T:X→X,
T(x,y) =/braceleftBigg
(x
2,y
2)if (x,y)/negationslash= (1,0)
(0,0)if (x,y) = (1,0)
By applying the inequality (1.13)to our mapping T, we get for all x= (x1,y1),y=
(x2,y2)∈X
/vextendsingle/vextendsingle/vextendsinglex1
2−x2
2/vextendsingle/vextendsingle/vextendsingle·e−j≤θ·|x1−x2|·e−j
2+L·/vextendsingle/vextendsingle/vextendsinglex2−x1
2/vextendsingle/vextendsingle/vextendsingle·e−j
2,
for allj∈J, which can be write as the equivalent form
|x1−x2|·e−j
2≤2θ·|x1−x2|+L·|2×2−x1|,
The last inequality became true if we take θ=1
2∈(0,1),L= 4≥0.
HenceTis an almost local b-contraction, with the unique fixed point (0,0).
In the sequel, we shall make a comparison to other type of contractive conditions
inb-pseudometric spaces.
Lemma 1. In ab-pseudometric space any Kannan-type ALC with constant k∈[0,1
2b)
is an almost local contraction (see [11]), i.e., satisfies the inequality:
dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(y,Tx ),∀x,y∈A (1.13)
withθ=kb
1−kbandL=2kb
1−kb.
Proof.Supposef:A→Ais a Kannan-type ALC with constant k∈[0,1
2b). This
means:
dj(f(x),f(y))≤k[dr(j)(x,f(x)) +dr(j)(y,f(y))],∀x,y∈A. (1.14)
Applying the definition of the b-pseudometric and the monotonicity property, we can
write:
dj(f(x),f(y))≤kbdr(j)(x,y) +kbdr(j)(y,f(x)) +kbdr(j)(y,f(x)) +kbdr(j)(f(x),f(y))≤
≤kbdj(x,y) +kbdj(y,f(x)) +kbdj(y,f(x)) +kbdj(f(x),f(y))
for allx,y∈A.
After simple computations we get:
dj(f(x),f(y))≤kb
1−kbdj(x,y) +2kb
1−kbdj(y,f(x)),
for anyx,y∈A. The last inequality shows that fis an almost local contraction, i.e.
it satisfies (1.13) with θ=kb
1−kb∈[0,1)andL=2kb
1−kb≥0.
This completes the proof of the Lemma.
5

Lemma2. In ab-pseudometric space any Kannan-type ALC with constant k∈[0,1
b(b+1))
is a strict almost local contraction with θ=kb
1−kbandL=2kb
1−kband, respectively
θu=kb2
1−kbandLu=k(1+b2)
1−kb.
Proof.Having in view that b≥1, the condition k∈[0,1
b(b+1))impliesk <1
2b, which
means the conclusion of Lemma1 is valid. Furthermore, from (1.14), it results:
dj(f(x),f(y))≤kdr(j)(x,f(x)) +kbdr(j)(f(y),f(x)) +kbdr(j)(f(x),y)≤
≤kdr(j)(x,f(x)) +kbdr(j)(f(y),f(x)) +kb2dr(j)(f(x),x) +kb2dr(j)(x,y)
From that we have:
dj(f(x),f(y))≤kb2
1−kbdj(x,y) +k(b2+ 1)
1−kbdj(x,f(x)),
for anyx,y∈A, i.e.fis a strict ALC with θu=kb2
1−kbandLu=k(1+b2)
1−kb.
The following two Lemmas refers to the Ćirić-Reich-Rus ALC-s.
Lemma 3. In ab-pseudometric space any Ćirić-Reich-Rus -type ALC with constants
α,β∈R+such thatα+ 2bβ < 1is an almost local contraction with θ=α+bβ
1−bβand
L=2bβ
1−bβ.
Proof.Denotef:A→Aa Ćirić-Reich-Rus -type ALC, where Ais a subset of the
b-pseudometric space X. Letα,β∈R+,α+ 2bβ < 1, such that
dj(f(x),f(y))≤αdj(x,y) +β[dj(x,f(x)) +dj(y,f(y))], (1.15)
for anyx,y∈A.
At this point, we can write:
dj(f(x),f(y))≤αdj(x,y) +bβdj(x,y) +bβdj(y,f(x)) +
+bβdj(y,f(x)) +bβdj(f(x),f(y)),
which implies
dj(f(x),f(y))≤α+bβ
1−bβdj(x,y) +2bβ
1−bβdj(y,f(x)),
for everyx,y∈A, i.e.,fsatisfies (1.13) with θ=α+bβ
1−bβ∈[0,1)andL=2bβ
1−bβ≥0.This
completes the proof.
Lemma 4. In ab-pseudometric space any Ćirić-Reich-Rus -type ALC with constants
α,β∈R+such thatα+b(b+ 1)β <1is a strict almost local contraction with θ=α+bβ
1−bβ
andL=2bβ
1−bβ≥0, and respectively, θu=α+b2β
1−bβandLu=β(b2+1)
1−bβ.
Proof.Asb≥1, assumption α+b(b+ 1)β < 1impliesα+ 2bβ < 1, therefore the
conclusions of Lemma 3 holds. Furthermore, according to (1.15), we obtain:
dj(f(x),f(y))≤αdj(x,y) +βdj(x,f(x)) +βdj(f(y),y)≤
≤αdj(x,y) +βdj(x,f(x)) +bβdj(f(y),f(x)) +b2βdj(f(x),x) +b2βdj(x,y),
6

so
dj(f(x),f(y))≤α+b2β
1−bβdj(x,y) +β(b2+ 1)
1−bβdj(x,f(x)),
for everyx,y∈A, i.e.,fsatisfies (1.8) with θu=α+b2β
1−bβ∈[0,1)andLu=β(b2+1)
1−bβ,which
means that fis a strict almost local contraction.
Lemma 5. In ab-pseudometric space, any Chatterjea-type ALC with constant
c∈[0,1
b(b+1))is an almost local contraction with θ=cb2
1−cbandL=c(b2+1)
1−cb.
Proof.LetAa subset of the b-pseudometric space Xand letf:A→Aa Chatterjea-
type ALC with c∈[0,1
b(b+1))such that
dj(f(x),f(y))≤c[dj(x,f(y)) +dj(y,f(x))],for anyx,y∈A. (1.16)
From that, we can write:
dj(f(x),f(y))≤cdj(f(y),x) +cdj(y,f(x))≤
≤cbdj(f(y),f(x)) +cbdj(f(x),x) +cdj(y,f(x))≤
≤cbdj(f(x),f(y)) +cb2dj(x,y) +cb2dj(y,f(x)) + +cdj(y,f(x)),
so
dj(f(x),f(y))≤cb2
1−cbdj(x,y) +c(b2+ 1)
1−cbdj(y,f(x)),
for anyx,y∈A, that is,fsatisfies (1.13) with θ=cb2
1−cb∈[0,1)andL=c(b2+1)
1−cb≥0.
Now, the proof is complete.
Lemma 6. In ab-pseudometric space, any Chatterjea-type ALC with constant c∈
[0,1
b(b+1))is a strict almost local contraction with θ=cb2
1−cbandL=c(b2+1)
1−cband, respec-
tively,θu=cb
1−cbandLu=2cb
1−cb.
Proof.The conclusions of Lemma 5 holds and from (1.16) we have that:
dj(f(x),f(y))≤cbdj(x,f(y)) +cbdj(f(x),f(y)) +cbdj(y,x) +cbdj(x,f(x)),
therefore
dj(f(x),f(y))≤cb
1−cbdj(x,y) +2cb
1−cbdj(x,f(x)),
for anyx,y∈A, that is,fsatisfies (1.8) with θu=cb
1−cbandLu=2cb
1−cb≥0.
Having in view that c∈[0,1
b(b+1))andb≥1, clearly, it results that c <1
2b, which
meansθu∈[0,1).
At this point, it is natural to study the Zamfirescu-type ALC-s in this new space
setting.
Lemma 7. In ab-pseudometric space, any Zamfirescu-type ALC with constants α∈
[0,1),k∈[0,1
2b)andc∈[0,1
b(b+1))is an almost local contraction with
θ= max/braceleftBig
α,bk
1−bk,b2c
1−bc/bracerightBig
andL= max/braceleftBig
2bk
1−bk,(b2+1)c
1−bc/bracerightBig
.
7

Lemma 8. In ab-pseudometric space, any Zamfirescu-type ALC with constants α∈
[0,1),k∈[0,1
b(b+1))andc∈[0,1
2b)satisfies the inequality (1.8)with
θu= max/braceleftBig
α,b2k
1−bk,bc
1−bc/bracerightBig
andLu= max/braceleftBig(b2+1)k
1−bk,2bc
1−bc/bracerightBig
.
Lemma 9. In ab-pseudometric space, any Zamfirescu-type ALC with constants α∈
[0,1),k∈[0,1
b(b+1))andc∈[0,1
b(b+1))is a strict almost local contraction with
θ= max/braceleftBig
α,bk
1−bk,b2c
1−bc/bracerightBig
andL= max/braceleftBig
2bk
1−bk,(b2+1)c
1−bc/bracerightBig
and, respectively,
θu= max/braceleftBig
α,b2k
1−bk,bc
1−bc/bracerightBig
andLu= max/braceleftBig(b2+1)k
1−bk,2bc
1−bc/bracerightBig
.
Our next goal is to study the case of quasi- almost local contractions.
Lemma 10. In ab-pseudometric space, any quasi- ALC with constant
h∈[0,1
b(b+1))is an almost local contraction with θ=b2h
1−bhandL=b2h
1−bh.
Proof.Letf:A→Abe a quasi- ALC with constant h∈[0,1
b(b+1))such that
dj(f(x),f(y))≤hmax{dr(j)(x,y),dr(j)(x,f(x)),dr(j)(y,f(y)),dr(j)(x,f(y)),dr(j)(y,f(x))},
(1.17)
for allx,y∈A. We shall use the notation
Mr(j)(x,y) = max{dr(j)(x,y),dr(j)(x,f(x)),dr(j)(y,f(y)),dr(j)(x,f(y)),dr(j)(y,f(x))}.
and we distinguish five different cases:
I.Mr(j)(x,y) =dj(x,y). We can write
dj(f(x),f(y))≤hdr(j)(x,y).
II.Mr(j)(x,y) =dj(x,f(x)).Then
dj(f(x),f(y))≤hdr(j)(x,f(x))≤hbdr(j)(x,y) +hbdr(j)(y,f(x)),
therefore (1.13) is satisfied with θ=hb∈[0,1)andL=hb≥0.
Remind that ALC-s are not symmetric operators, so we need to verify both conditions
(1.13) and (1.18), namely:
dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(x,Ty ),∀x,y∈A, (1.18)
in order to prove that an operator fis an ALC. Then, for any x,y∈Awe can write:
dj(f(x),f(y))≤hdr(j)(x,f(x))≤hbdr(j)(x,f(y)) +hbdr(j)(y,f(x)),
it results
dj(f(x),f(y))≤hb
1−hbdr(j)(x,f(y)),
which means that (1.18) is satisfied with θ= 0andL=hb
1−hb.
From that, the almost local contraction condition (1.13) is verified with:
θ= max{hb,0}=hbandL= max{hb,hb
1−hb}=hb
1−hb.
III. IfMr(j)(x,y) =dj(y,f(y)),in a similar manner to case II., it results that (1.13) is
8

fulfilled with θ=hbandL=hb
1−hb.
IV. IfMr(j)(x,y) =dj(x,f(y)),then we can write
dj(f(x),f(y))≤hdr(j)(x,f(y)).
This means that (1.18) holds with θ= 0andL=h.
By using the b-pseudometric property, it results
dj(f(x),f(y))≤hdr(j)(x,f(y))≤
≤bhdr(j)(f(y),f(x)) +b2hdr(j)(f(x),y) +b2hdr(j)(y,x),
so
dj(f(x),f(y))≤b2h
1−hbdr(j)(x,y) +b2h
1−hbdr(j)(y,f(x)).
From that, we may state that for the previously chosen xandy, the ALC (1.13)
condition is verified with θ= max{b2h
1−bh,0}=b2h
1−bh∈[0,1),L= max{h,b2h
1−bh}=b2h
1−bh.
V.Mr(j)(x,y) =dj(y,f(x)).This case is quite similar to case IV.
These five cases leads us to the conclusion that for any x,y∈A, the ALC condition
(1.13) is verified.
The proof is complete.
References
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