Ci a ion: Poonguzali, G.;
P agadeeswa a , V.; De la Sen, M.
Exis ence o Bes P oximi y Poin in
O-Comple e Me ic Spaces.
Ma hema ics 2023,11, 3453.
h ps://doi.o g/10.3390/
ma h11163453
Academic Edi o s: Mi cea Balaj,
Vasile Be inde and Massimiliano
Giuli
Recei ed: 31 May 2023
Re ised: 2 Augus 2023
Accep ed: 7 Augus 2023
Published: 9 Augus 2023
Copy igh : © 2023 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
ma hema ics
A icle
Exis ence o Bes P oximi y Poin in O-Comple eMe ic Spaces
G. Poonguzali 1, V. P agadeeswa a 1,* and Manuel De la Sen 2,*
1Depa men o Ma hema ics, Am i a School o Physical Sciences, Am i a Vishwa Vidyapee ham,
Coimba o e 641112, India; [email p o ec ed]
2
Ins i u e o Resea ch and De elopmen o P ocesses IIDP, Uni e si y o he Basque Coun y, Campus o Leioa,
48940 Leioa, Bizkaia, Spain
*Co espondence: [email p o ec ed] (V.P.); [email p o ec ed] (M.D.l.S.)
Abs ac :
In his wo k, we p o e he exis ence o he bes p oximi y poin esul s o
⊥
-con ac ion
(o hogonal-con ac ion) mappings on an
O
-comple e me ic space (o hogonal-comple e me ic
space). Subsequen ly, hese exis ence esul s a e employed o es ablish he common bes p oximi y
poin esul . Finally, we p o ide sui able examples o demons a e he alidi y o ou esul s.
Keywo ds:
bes p oximi y poin ;
O
-comple e me ic space;
O
-closed se ; P-p ope y; weakly p oxi-
mally ⊥-p ese ing; ⊥-con inuous
MSC: 37C25
1. In oduc ion and P elimina ies
O e he pas 100 yea s, ixed poin heo y has been an ac i e a ea o esea ch, due
o i s signi icance in applica ions. Simul aneously, in he heo y o unc ional analysis, he
idea o p oximi y pai s o wo se s was b ie ly discussed. Many esea che s con ibu ed
hei ision on when and whe e we can ha e he bes p oximi y poin s o se s. Ano he
g oup o esea che s who we e ac i e on ixed poin esul s wan ed o analyze he case
when we do no ha e an exac solu ion o he equa ion o he o m
T(x) = x
. Resea che s
such as Ky Fan, Segal, Singh, and P olla [
1
–
3
] ha e p o ided a weal h o aluable esul s
in bes app oxima ion heo y. These indings shed ligh on si ua ions whe e ixed poin s
a e absen , and unde ce ain smoo h condi ions, we can ob ain app oxima e solu ions o
he equa ions. No ably, Ky Fan [
1
] p o ed he exis ence o he bes app oxima ion o a
con inuous unc ion on a compac con ex subse o a no med space. In a subsequen s udy
in 1989, Segal
e al. [2]
p o ed he exis ence o he bes app oxima ion o an app oxima ely
compac subse o a no med space. Fu he mo e, P olla e al. [
3
] ex ended his concep o
mul i unc ions. A ound he end o he 1990s and he s a o 2000, a g oup o esea che s
used he idea o he bes p oximi y poin o mappings, which uni ies he ixed poin
and bes app oxima ion esul s [
4
–
6
]. La e , many gene aliza ions we e made by many
esea che s; e e o [7–11].
On he o he hand, he Banach con ac ion p inciple is a signi ican ma hema ical
disco e y in ixed poin heo y. I has been expanded and applied o a ious ypes o
me ic spaces, such as semi-me ics, quasi-me ics, pseudo-me ics, uzzy me ic spaces,
and pa ial me ic spaces, among o he s (see [9,10,12–18]).
In ha line, in 2017, Go dji e al. [19,20] in oduced a new ype o me ic space called
an o hogonal me ic space and p o ed he ixed poin esul s. They also demons a ed he
applica ion o hese esul s in es ablishing he exis ence and uniqueness o solu ions o
i s -o de o dina y di e en ial equa ions, whe e he Banach con ac ion mapping p inciple
is no applicable.
Mo i a ed by he a o emen ioned esul s [
19
,
20
], in his pape , we ex end he esul s
om he ixed poin o he bes p oximi y poin o non-sel -mappings in he con ex o an
o hogonal se . Using hese exis ence esul s, we p o e a common bes p oximi y poin
Ma hema ics 2023,11, 3453. h ps://doi.o g/10.3390/ma h11163453 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2023,11, 3453 2 o 9
esul . Finally, we p o ide sui able examples o demons a e he alidi y o ou esul s,
which canno be achie ed h ough o he bes p oximi y poin echniques. Fu he mo e, in
he li e a u e o ixed poin heo y, we ha e eno mous esul s on he comple e me ic space
and pa ially o de ed me ic space, bu no many on he o hogonal me ic space.
In [
21
], he exis ence o he bes p oximi y poin s was p o ided o a map ha is a
con inuous and p oximal con ac ion, o i has o be a con ac ion map on an app oxima ely
compac se . In his pape , we p o ide he exis ence o he bes p oximi y poin o a weake
condi ion called ⊥- con inui y on an O-closed se .
Resea ch on he concep o an o hogonal space is wo h analyzing as i ep esen s a
mo e gene al space ha canno be compa ed wi h a pa ially o de ed space. The upcoming
examples will explain he necessi y o ha ing an O hogonal space.
Example 1
([
20
])
.
Conside
M=R2
.De ine
⊥
as
u⊥
i
<u
,
>=
0on
M
.Then,
(M
,
⊥)
is
an O-se , since
u= (
0, 0
)⊥
, o all
∈M
.Howe e ,
(M
,
⊥)
is no a pa ial o de se . Choose
u= (1, 0), = (0, 1), = (−1, 0);i is clea ha u ⊥ , ⊥ , bu u 6⊥ .
Example 2.
Conside
(M=R
,
≤)
. Then,
M
is a pa ially o de ed se . bu no an
O
-se wi h he
≤ ela ion, because we canno ind any u ∈M such ha u ≤p o p ≤u o all p ∈R.
Th oughou his pape , he ollowing no ions a e used:
Le Aand Bbe any wo nonemp y subse s o a me ic space X.
d(A,B):=in {d(a,b):a∈Aand b∈B},
A0={a∈A:d(a,b) = d(A,B) o some b∈B},
B0={b∈B:d(a,b) = d(A,B) o some a∈A}.
De ini ion 1.
Le
A
and
B
be any wo nonemp y subse s o a me ic space
X
. Then, a poin
p∈A
is called a bes p oximi y poin o a mapping T :A→B, i he ollowing holds ue:
d(p,Tp) = d(A,B).
De ini ion 2
([
20
])
.
Le
M6=∅
, and le
⊥⊆ M×M
be any bina y ela ion. We call
(M
,
⊥)
an
O-se (o hogonal se ) i ⊥sa is ies he ollowing condi ion:
∃u0∈M:(∀ , ⊥u0)o (∀ ,u0⊥ ).
We usually use
(M
,
⊥)
o ep esen an
O
-se . Fu he mo e, no e ha his o hogonal
ela ion is no a ansi i e ela ion.
Example 3
([
20
])
.
Take
M= [
0,
∞)
, and i
u ∈ {u
,
}
, hen
u⊥
.I is clea o see ha i
u0=0o u0=1,(M,⊥)is an o hogonal se .
De ini ion 3
([
20
])
.
Conside any
O
-se
(M
,
⊥)
.Le
(un)
be any sequence, hen we say ha
(un)
is an O-sequence i
un⊥un+1o un+1⊥un o all n ∈N.
Example 4.
Le
M=R
, and de ine
u⊥
by
u ≤u o
.Take
un=
1
/n
, hen
un
is an
O-sequence, since ∀n,un⊥un+1.
De ini ion 4
([
20
])
.
Le
(X
,
⊥)
be any
O
-se . Le
A
be any subse o
X
.Then,
A
is o hogonal
closed se (O-closed se ) i , when any O-sequence xn→x, hen x ∈A.
Example 5.
Le
X= [
0,
∞)
. Choose he usual o de on
X
, hen
(X
,
≤)
is an
O
-se . Conside
A= [0, 1], hen A is an o hogonal closed se .
Ma hema ics 2023,11, 3453 3 o 9
E e y closed se is an o hogonal closed se , bu an o hogonal closed se need no be
a closed se .
Example 6. Le X = [0, 1]and p ∈(0, 1), and de ine
x⊥y⇐⇒ (x≤y≤p
x=0o he wise.
He e, choose
A= [
0,
q)
wi h
q∈(p
, 1
)
.Then,
A
is an
O
-closed se . Fu he mo e, i is no a closed
se .
De ini ion 5.
Le
(A
,
B)
be a pai o nonemp y subse s o a me ic space
(X
,
d)
. The pai
(A
,
B)
sa is ies he P-p ope y i , whene e a1,a2∈A and b1,b2∈B wi h,
d(a1,b1) = d(A,B)
d(a2,b2) = d(A,B))=⇒d(a1,a2) = d(b1,b2).
De ini ion 6
([
20
])
.
Le
(X
,
⊥
,
d)
be an o hogonal me ic space (
(X
,
⊥)
is an O-se , and
(X
,
d)
is a me ic space). Then,
T:X→X
is said o be o hogonally con inuous (o
⊥
-con inuous) in
a∈X
i , o each O-sequence
{an}n∈N
in
X
wi h
an→a
, we ha e
T(an)→T(a)
. Fu he mo e,
T is said o be ⊥-con inuous on X i T is ⊥-con inuous in each a ∈X.
E e y con inuous mapping is ⊥-con inuous, bu he con e se is no ue.
De ini ion 7
([
20
])
.
Le
(X
,
⊥
,
d)
be an o hogonal me ic space and 0
<k<
1. A mapping
T:X→X
is called an o hogonal-con ac ion (b ie ly,
⊥
-con ac ion) wi h Lipschi z cons an
k
i ,
o all x,y∈X wi h x ⊥y,
d(Tx,Ty)≤kd(x,y).
E e y con ac ion is a ⊥-con ac ion, bu he con e se is no ue.
2. Main Resul s
Now, we will p o e he lemma ha will be used o es ablish he exis ence o he bes
p oximi y poin esul s.
Lemma 1.
Le
A
be an o hogonal closed subse o an
O
-comple e me ic space
X
, hen
A
is an
O-comple e me ic space.
P oo .
Le
(xn)
be any
O
-Cauchy sequence in
A
. Then,
(xn)⊆X
. Since
X
is an
O
-comple e
me ic space, he e exis s
x∈X
such ha
xn→x
. Fu he mo e,
(xn)
is an
O
-sequence,
which con e ges o x∈X. Hence, x∈A.
De ini ion 8.
Le
A
and
B
be any wo nonemp y subse s o a me ic space
(X
,
d)
.A map
T:A→
B is said o be p oximally ⊥-p ese ing i
d(a1,Tb1) = d(A,B)
d(a2,Tb2) = d(A,B))=⇒a1⊥a2i b1⊥b2,
o all a1,a2,b1,b2∈A.
Theo em 1.
Le
A
and
B
be wo nonemp y
O
-closed subse s o an
O
-comple e me ic space
(X
,
⊥
,
d)
such ha
A06=∅
.I
(A
,
B)
has he P-p ope y and also
T:A→B
sa is ies he
ollowing:
1. T is ⊥-con inuous and a ⊥-con ac ion mapping;
Ma hema ics 2023,11, 3453 4 o 9
2. T(A0)⊆B0;
3. T is p oximally ⊥-p ese ing;
4. A0is an O-se .
Then, d(u,Tu) = d(A,B), o some u ∈A.
P oo .
Since
A0
is an
O
-se , he e exis s
p∈A0
such ha
u⊥p
, o
p⊥u
o all
u∈A0
.
Wi hou loss o gene ali y, assume ha
u⊥p
. F om Condi ion 2, we ha e
Tp ∈B0
,
and hence, he e exis s
u1∈A0
such ha
d(u1
,
Tp) = d(A
,
B)
. Fu he mo e, no e ha
Tu1∈B0
, and hence,
d(u2
,
Tu1) = d(A
,
B)
. By he p oximally
⊥
-p ese ing p ope y
o
T
, we ob ain
u1⊥u2
. Applying a simila a gumen , we cons uc an
O
-sequence
u1⊥u2⊥u3⊥ ··· ⊥ u ⊥ ···
wi h
d(u +1
,
Tu ) = d(A
,
B)
o all
∈N
. Using he
P-p ope y o (A,B), we ha e d(u ,u +1) = d(Tu −1,Tu ). Conside ,
d(u ,u +1) = d(Tu −1,Tu )
≤kd(u −1,u )
.
.
.
≤k d(u0,u1). (1)
Since k<1, lim →∞k =0. Hence, lim →∞d(u ,u +1) = 0. I ,s∈Nand s< , hen
d(us,u )≤d(us,us+1) + d(us+1,us+2)···+d(u −1,u )
≤ksd(u0,u1) + ks+1d(u0,u1) + ···+k −1d(u0,u1) (by (1))
≤ks[1+k+···+k −s−1]d(u0,u1)
≤kn
1−kd(u0,u1).
As
s
,
→∞
,
d(us
,
u )→
0, which means ha
(u )
is an
O
-Cauchy sequence. He e,
A
is an
O
-closed subse o an
O
-comple e me ic space. By Lemma 1,
A
is an
O
-comple e
me ic space
(X
,
⊥
,
d)
. The e o e, he e exis s
u∗∈A
such ha
lim →∞u =u∗
. Since
T
is
⊥
-con inuous,
lim →∞Tu −1=Tu∗
, which implies
d(u
,
Tu )→d(u∗
,
Tu∗)
as
→∞
.
Hence, d(u∗,Tu∗) = d(A,B).
Theo em 2.
Le
(X
,
⊥
,
d)
be any
O
-comple e me ic space. Le
A
and
B
be wo nonemp y subse s
o X.Le T :A→B sa is y he ollowing condi ions:
1. T is ⊥-con inuous and a ⊥-con ac ion;
2. T(A0)⊆B0and (A,B)sa is y he P-p ope y;
3. T is p oximally ⊥-p ese ing;
4. The e exis s u0,u1∈A0such ha d(u1,Tu0) = d(A,B)and u0⊥u1.
Then, he e exis s an elemen u ∈A such ha d(u,Tu) = d(A,B).
P oo . By he hypo hesis, he e exis s u0and u1in A0such ha
d(u1,Tu0) = d(A,B)and u0⊥u1.
Since
u1∈A0
, his implies
Tu1∈B0
, and hence, he e exis s
u2∈A0
such ha
d(u2
,
Tu1) =
d(A
,
B)
, by he p oximally
⊥
-p ese ing condi ion o
T
, we ob ain
u1⊥u2
. P oceeding
like his, we ob ain
u1⊥u2⊥ ··· ⊥ u ⊥u +1⊥ ···
. Then,
(u )
is an
O
-sequence wi h
d(u +1,Tu ) = d(A,B) o all ∈N. Since (A,B)has he P-p ope y, we ha e
d(u ,u +1) = d(Tu −1,Tu )≤kd(u −1,u )≤k d(u0,u1).
Since k<1, k →0, lim →∞d(u ,u +1) = 0.
Ma hema ics 2023,11, 3453 5 o 9
Claim: (u )is an O-Cauchy sequence. I s, ∈Nand <s, hen
d(u ,us)≤[d(u ,u +1) + ···+d(us−1,us)]
≤k d(u0,u1) + ···+ks−1d(u0,u1)
≤k
1−kd(u0,u1).
The e o e,
d(u
,
u )→
0 as
s
,
→∞
. The e o e,
(u )
is an
O
-Cauchy sequence. Hence,
lim →∞u =u∗
. Since
T
is
⊥
-con inuous,
lim →∞Tu −1=Tu∗
, which implies
d(u
,
Tu )→
d(u∗,Tu∗). The e o e, u∗is a bes p oximi y poin .
Example 7.
Conside
X:=R2
wi h
⊥
de ined as
u⊥
i
<u
,
>=
0. Now, de ine
T:
{0}×R→ {1}×Rby
T(0, x) = ((1, x/2):x∈Q∩R
(1, 0):x∈QC∩R.
He e, obse e ha
T
is
⊥
-con inuous and a
⊥
-con ac ion. I is easy o obse e ha
A0=A
and
B0=B
; he e o e,
T(A0)⊆B0
.Fu he mo e,
(A
,
B)
has he P-p ope y. I is e iden ha he
abo e map
T
sa is ies all he condi ions o Theo em 2.Clea ly,
(
0, 0
)
is he bes p oximi y poin o
T.
Theo em 3.
Le
(X
,
⊥
,
d)
be an
O
-comple e me ic space. Le
A
and
B
be wo nonemp y
O
-
closed subse s o
X
such ha
A06=∅
.Fu he mo e, assume ha
(A
,
B)
has he P-p ope y. Le
T:A→B sa is y he ollowing condi ions:
1. T is a ⊥-con ac ion mapping and p oximally ⊥-p ese ing;
2. T(A0)⊆B0;
3. I (u )is any O-sequence wi h u →u, hen u ⊥u o all ∈N;
4. A0is an O-se .
Then, he e exis s u ∈A such ha d(u,Tu) = d(A,B).
P oo .
By using he same echnique as in Theo em 2, we can cons uc an
O
-Cauchy
sequence
(u )
wi h
d(u +1
,
Tu ) = d(A
,
B)
, and he e exis s
u∈A
, such ha
u →u
. Thus,
o any
e/
2
>
0, he e exis s
N1∈N
such ha
d(u
,
u)≤e/
2, o all
≥N1
. Simila ly, o
any
e/
2
k>
0, he e exis s
N2∈N
such ha
d(us
,
u)≤e/
2
k
, whe e
k
is he con ac ion
cons an o Tand o all s≥N2. Choosing, N=max{N1,N2}, we ob ain
d(u,Tu)≤d(u,uN) + d(uN,TuN) + d(TuN,Tu)
≤e/2 +d(A,B) + kd(uN,u) (Since uN⊥u&Tis ⊥ − con ac ion)
≤e/2 +d(A,B) + e/2
≤d(A,B) + e.
Since,eis a bi a y, we can conclude ha d(u,Tu) = d(A,B).
Le us deno e he new no ion called weakly p oximally ⊥- p ese ing as ollows.
De ini ion 9. Two maps T,S:A→B a e said o be weakly p oximally ⊥- p ese ing i :
1.
Fo all
a∈A
, he e exis
1
,
2∈A
wi h
d( 1
,
Ta) = d(A
,
B)
,
d( 2
,
S 1) = d(A
,
B)
and
1⊥ 2.
2.
Fo all
a∈A
, he e exis
w1
,
w2∈A
wi h
d(w1
,
Sa) = d(A
,
B)
,
d(w2
,
Tw1) = d(A
,
B)
and w1⊥w2.
Theo em 4.
Le
A
and
B
be wo nonemp y
O
-closed subse s o an
O
-comple e me ic space
(X
,
⊥
,
d)
wi h
A06=∅
,and also, assume ha
(A
,
B)
has he P-p ope y. Le
T
,
S:A→B
be wo
non-sel -mappings sa is ying he ollowing condi ions:
Ma hema ics 2023,11, 3453 6 o 9
1. (T,S)is weakly p oximally ⊥-p ese ing;
2. T o S is ⊥-con inuous;
3. Fo all u, wi h u ⊥ ,d(Tu,S )≤kd(u, ) o some k ∈[0, 1),
4. I any O-sequence (un)con e ges, hen un⊥u o all n,whe e u =limn→∞un.
Then, he e exis s u ∈A such ha d(u,Tu) = d(u,Su) = d(A,B).
P oo .
Since
A06=∅
, choose any
u0∈A0
. Applying
T
on
u0
, hen
Tu0∈B0
. As
(T
,
S)
is weakly p oximally
⊥
-p ese ing, we ha e
d(u1
,
Tu0) = d(A
,
B)
,
d(Tu2
,
Su1) = d(A
,
B)
,
and
u1⊥u2
. Con inuing he same way using he weakly p oximally
⊥
-p ese ing con-
di ion o
(T
,
S)
, we can cons uc an
O
-sequence
(u )
wi h
d(u2 +1
,
Tu2 ) = d(A
,
B)
,
d(u2 +2
,
Su2 +1)= d(A
,
B)
and
u +1⊥u2 +2
. Now, i is ime o ou usual echnique
o p o ing his (u ) o be a Cauchy sequence. Fo ha , obse e
d(u2 +1,u2 +2) = d(Tu2 ,Su2 +1)(By P-P ope y)
≤kd(u2 ,u2 +1)
=kd(Tu2 −1,Su2 )
≤k2d(u2 −1,u2 )
.
.
.
≤k2 +1d(u0,u1).
Since
k<
1,
k2 +1→
0, his implies
lim →∞d(u2 +1
,
u2 +2) =
0. Now, o
,
s∈N
wi h
s> , we ha e
d(u ,us)≤d(u ,u +1) + d(u +1,u +2) + ···+d(us−1,us)
≤k d(u0,u1) + k +1d(u0,u1) + ···ks−1d(u0,u1)
≤k [1+k+k2+···+ks− −1]d(u0,u1).
By he abo e inequali y, i is e iden ha
(u )
is an
O
-Cauchy sequence. Since ou space
is
O
-comple e,
(u )
con e ges, say
u
, which implies
u ⊥u
o all
∈N
. Wi hou loss o
gene ali y, assume ha
T
is
⊥
-con inuous, hen i is easy o conclude ha
d(u2 +1
,
Tu2 )→
d(u
,
Tu)
. Fu he mo e, no e ha
d(u
,
Tu) = d(A
,
B)
. Thus,
u
is he bes p oximi y poin
o T.
Nex , ou claim is o show ha
u
is he bes p oximi y poin o
S
. By he con e gence o
(u )
, o
e/
2
>
0, he e exis s
N1∈N
, such ha
d(u
,
u)≤e/
2 o all
≥N1
; u he mo e,
o
e/
2
k>
0, he e exis s
N2∈N
, such ha
d(u
,
u)≤e/
2 o all
≥N2
. By choosing
N=max{N1,N2}, conside
d(u,Su)≤d(u,u2N+1) + d(u2N+1,Tu2N) + d(Tu2N,Su)
≤e/2 +d(u2N+1,Tu2N) + kd(u2N,u)
≤e/2 +d(u2N+1,Tu2N) + e/2
≤e+d(u2N+1,Tu2N).
We ob ain
d(u
,
Su)≤d(A
,
B) + e
. I is easy o conclude ha
d(u
,
Su) = d(A
,
B)
, since
e
is
a bi a y. Hence, d(u,Tu) = d(u,Su) = d(A,B).
Till now, in he li e a u e on hew bes p oximi y poin , he exis ence o a common
bes p oximi y poin in me ic spaces o pa ially o de ed me ic spaces equi es a s onge
condi ion called he con inui y o a map o he app oxima e compac ness o a se . In he
ollowing example, one can easily obse e ha
T
is no a con inuous map. Ne e heless, a
common bes p oximi y poin exis s.
Ma hema ics 2023,11, 3453 7 o 9
Example 8.
Conside
X=R2
wi h
⊥
de ined as
(u1
,
u2)⊥( 1
,
2)
, i
u1≤ 1
and
u2≤ 2
.
Fu he mo e, choose
d(u
,
) = |u1− 1|+|u2− 2|
.Then,
(X
,
⊥
,
d)
is an
O
-comple e me ic
space. Le us conside
A:={(
0,
a):a∈R}
and
B:={(
1,
b):b∈R}
.Then,
d(A
,
B) =
1.
Now, de ine
T:A→B
by
T(
0,
a) = ((1, −a/2):a∈Q∩R
(1, −a/4):a∈QC∩R
and
S:A→B
as
S(
0,
b) =
(1, −b/4).We a e now eady o e i y he condi ions o Theo em 4.
Condi ion 1. (T,S)is weakly p oximally ⊥-p ese ing:
Le u ∈A, hen u = (0, u1), whe e u1∈R.
Case (i): I
u1∈Q∩R
, hen
Tu = (
1,
−u1/
2
)
.I is easy o see ha , i we ake
=
(0, −u1/2)and w = (0, −u1/8), hen d(u,T ) = d(A,B) = d( ,Sw)and also ⊥w.
Case (ii): I
u1∈QC∩R
, hen
Tu = (
1,
−u1/
4
)
.I is easy o see ha , i we ake
=
(
0,
−u1/
4
)
and
w= (
0,
−u1/
16
)
.Then,
d(u
,
T ) = d(A
,
B) = d(
,
Sw)
and also
⊥w
.
Simila ly, o all
u∈A
,we can ind
w
,
w0∈A
wi h
d(w
,
Su) = d(A
,
B)
,
d(w0
,
Tw) = d(A
,
B)
,
which also implies w ⊥w0.
Condi ion 2. T o S is ⊥-con inuous:
He e,
S
is a con inuous unc ion, and hence,
S
is
⊥
-con inuous. Fu he mo e, obse e ha
T
is no
⊥
-con inuous, since
O
-sequence
xn= (
0,
−
1
−√2/n)
con e ges o
x= (
0,
−
1
)
.Howe e ,
T(xn) = 1, −(−1−√2/n)
4!con e ges o (1, 1/4), which is no equal o Tx = (1, 1/2).
Condi ion 3.
I
u⊥
, hen
d(Tu
,
S )≤kd(u
,
)
o some
k∈[
0, 1
)
. Le
u= (
0,
u1)
,
=
(0, 1)∈A.
Case (i): I u1∈Q, hen
d(Tu,S ) = d((1, −u1/2),(1, − 1/4))
=|−u1/2 + 1/4|
≤ |−u1/2 + 1/2|(Since u1≤ 1)
≤1
2d(u, ).
Case (ii): I u1∈QC, hen
d(Tu,S ) = d((1, −u1/4),(1, − 1/4))
=|−u1/4 + 1/4|
≤1
4d(u, )
≤1
2d(u, ).
By choosing k =1/2, i is e iden ha , o all x ⊥y,d(Tu,S )≤d(u, ).
Condi ion 4. I (xn)is an O-sequence wi h xn→x, hen xn⊥x o all n:
Since
(xn)
is an
O
-sequence, we ha e
xn= (
0,
an)≤xn+1= (
0,
an+1)
, which implies
an≤an+1
.Hence,
(xn)
is a mono onically inc easing sequence, which con e ges o he sup emum,
say x := (0, a).I is clea ha xn⊥x o all n ∈N.Fu he mo e, i is easy o obse e ha (A,B)
has he P-p ope y. He e, u∗= (0, 0)sa is ies d(u∗,Tu∗) = d(u∗,Su∗) = d(A,B).
Theo em 5.
Le
A
and
B
be wo nonemp y closed subse s o an
O
-comple e me ic space
(X
,
⊥
,
d)
wi h
A06=∅
,and also, assume ha
(A
,
B)
has he P-p ope y. Le
T
,
S:A→B
be wo
non-sel -mappings sa is ying he ollowing condi ions:
1. (T,S)is weakly p oximally ⊥-p ese ing;
2. T o S is ⊥-con inuous;
3. Fo all u, wi h u ⊥ ,d(Tu,S )≤kd(u, ) o some k ∈[0, 1);
Ma hema ics 2023,11, 3453 8 o 9
4. I u is a bes p oximi y poin o ei he T o S, hen u ⊥u.
Then, he e exis s u ∈A such ha d(u,Tu) = d(u,Su) = d(A,B).
P oo .
Following he same echnique ha we used in Theo em 4, we can easily cons uc
he
O
-Cauchy sequence
(un)
such ha
d(u2n+1
,
Tu2n) = d(A
,
B)
, and
d(u2n+1
,
Su2n+2) =
d(A
,
B)
. As usual,
O
-comple eness p o ides he con e gence o
(un)
, ha is he e exis s
u∈A
such ha
un→u
. Wi hou loss o gene ali y, assume ha
S
is
⊥
-con inuous, hen
i is easy o conclude ha
d(u2n+1
,
Su2n+2)→d(u
,
Su)
. Fu he mo e, no e ha
d(u
,
Su) =
d(A,B). Hence, uis he bes p oximi y poin o S; hus u⊥u. Conside
d(u,Tu)≤d(u,Su) + d(Su,Tu)
≤d(u,Su) + kd(u,u)
≤d(u,Su).
Simila ly, conside
d(u,Su)≤d(u,Tu) + d(Tu,Su)
≤d(u,Tu) + kd(u,u)
≤d(u,Tu).
Hence, d(u,Tu) = d(u,Su), which means ha d(u,Tu) = d(u,Su) = d(A,B).
3. Conclusions
The ixed poin and bes p oximi y poin esul s ensu e he exis ence o solu ions o
many p oblems in non-linea analysis. In ou pape , we ha e gi en he exis ence o he
bes p oximi y poin and common bes p oximi y poin in a mo e gene al me ic space
called he
O
-me ic space, which ails o sa is y he ansi i i y condi ion. Fu he mo e, we
p o ided an example whe e ou map ails o be con inuous and ails o be a con ac ion;
s ill, we can ind he bes p oximi y poin and common bes p oximi y poin s.
Au ho Con ibu ions:
Concep ualiza ion, G.P. and V.P.; me hodology, G.P., V.P. and M.D.l.S.; alida-
ion, G.P. and V.P.; w i ing—o iginal d a p epa a ion, G.P., V.P. and M.D.l.S.; w i ing— e iew and
edi ing, G.P., V.P. and M.D.l.S.; unding acquisi ion, M.D.l.S. All au ho s ha e ead and ag eed o he
published e sion o he manusc ip .
Funding:
This wo k has been pa ially unded by he Basque Go e nmen h ough G an IT1207-19
and G an IT1155-22.
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen : No applicable.
Con lic s o In e es : The au ho s decla e ha hey ha e no compe ing in e es s.
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