21st International Symposium on [632179]

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 paperwork, almost local contractions were introduced in a pseudometric space
setting. The framework shall be the b-pseudometric spaces, providing some interesting
new results. The generalisation of metric and respectively metric space was given by
many autors, the most importants: [8], [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], [9], [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 [8]) Let Xbe a nonempty set.
A mapping db:X×X→R+is called b-metric if the following hold:
1.db(x,y) = 0if and only if x=y;
2.db(x,y) =db(y,x),∀x,y∈X;
3.db(x,z)≤b·[db(x,y) +db(y,z)],∀x,y,z∈X,
whereb≥1is a given real number
A nonempty set Xendowed with a b-metricdb:X×X→R+is called b-metric space.
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:
(i)fis a weakly Picard operator;
(ii) for any x∈Xthe following error estimates 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.In the beginning, we shall prove that the operator fhas at least one fixed point
inX, i.e. the set of fixed points is nonempty. To this end, we let x0∈Xand{xn}n≥0
be 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),
1

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) +b2[dj(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∗).
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.
In the sequel, remind that 0≤bθ < 1and 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).
2

But0≤bθ< 1, lettingp→∞inthelastinequality, itresultstheaposterioriestimate,
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 [10]), 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) 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) the rate of convergence of the Picard iteration is given by
dj(xn,x∗)≤θudj(xn−1,x∗),n≥1, (1.11)
whereFf={x∗}.
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∗∈X. 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.
3

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 [10]), i.e., satisfies the inequality:
dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(y,Tx ),∀x,y∈A (1.12)
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. (1.13)
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 (1.12) with θ=kb
1−kb∈[0,1)andL=2kb
1−kb≥0.
This completes the proof of the Lemma.
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 (1.13), 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.
4

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))], (1.14)
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 (1.12) 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 (1.14), 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),
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. (1.15)
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 (1.12) with θ=cb2
1−cb∈[0,1)andL=c(b2+1)
1−cb≥0.
Now, the proof is complete.
5

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 (1.15) 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).
In the sequel, 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
.
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))},
(1.16)
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).
6

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 (1.12) 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
(1.12) and (1.17), namely:
dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(x,Ty ),∀x,y∈A, (1.17)
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 (1.17) is satisfied with θ= 0andL=hb
1−hb.
From that, the almost local contraction condition (1.12) 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 (1.12) is
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 (1.17) 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 (1.12)
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
(1.12) is verified.
The proof is complete.
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