Noname manusc ip No.
(will be inse ed by he edi o )
A Cha ac e iza ion o Weak P oximal No mal
S uc u e and Bes P oximi y Pai s
Abhik Diga ·Ra ael Esp´ınola Ga c´ıa ·G.
Sanka a Raju Kosu u
Recei ed: da e / Accep ed: da e
Abs ac The aim o his pape is o add ess an open p oblem gi en in [Ki k,
W. A., Shahzad, Nasee , No mal s uc u e and o bi al ixed poin condi ions, J.
Ma h. Anal. Appl. ol 463(2), (2018) 461–476]. We gi e a cha ac e iza ion o weak
p oximal no mal s uc u e using bes p oximi y pai p ope y. We also in oduce
a no ion o poin wise cyclic con ac ion w o bi s and he ein p o e he exis ence
o a bes p oximi y pai in he se ing o e lexi e Banach spaces.
Keywo ds Bes p oximi y pai s ·P oximal no mal s uc u e ·Rela i ely
nonexpansi e mappings.
Ma hema ics Subjec Classi ica ion (2010) 46E15 ·47H10 ·54H25
1 In oduc ion and P elimina ies
Le A, B be wo non-emp y subse s o a Banach space and Tbe a cyclic mapping
on A∪B(T(A)⊆B, T(B)⊆A). A pai (x, y)∈A×Bis said o be a bes p ox-
imi y pai o Ti kx−Txk=ky−T yk=d(A, B) = in {kx−yk:x∈A, y ∈B}.
The geome y o Banach spaces plays a c ucial ole o he exis ence o bes p ox-
imi y pai s. The analysis o p oximal no mal s uc u e and weak o semi-no mal
s uc u e, he p ope y UC, he p ojec ional p ope y due o Eld ed e al. ([2]),
Moosa ([4]), Suzuki e al. ([9]), G. S. Raju e al. ([7]) e c., espec i ely a e widely
used o p o e he exis ence o a bes p oximi y pai o cyclic maps. We deno e
sup{kx−yk:y∈B} o x∈Aby δ(x, B).We shall say ha he pai (A, B) is
p oximal pai i o e e y xin A( esp. in B), he e exis s yin B( esp. in A) such
Abhik Diga
Depa men o Ma hema ics, Indian Ins i u e o Technology Ropa , Punjab-140 001, India.
E-mail: [email p o ec ed]
Ra ael Esp´ınola Ga c´ıa
Depa amen o de An´alisis Ma em´a ico - IMUS, Uni e sidad de Se illa, Se illa, Spain.
E-mail: [email p o ec ed]
G. Sanka a Raju Kosu u
Depa men o Ma hema ics, Indian Ins i u e o Technology Ropa , Punjab-140 001, India.
E-mail: a[email p o ec ed]
2 Abhik Diga e al.
ha kx−yk=d(A, B).Fu he , i such a yis unique, hen (A, B) is said o be a
sha p p oximal pai ([7]). In his case we deno e yby x0.Also, (A, B) is said o be
a p oximal pa allel pai i (A, B) is sha p p oximal and B=A+h o some h∈X
([3]). I is shown in [3] ha i Xis s ic ly con ex and A, B a e weakly compac con-
ex subse s o X, hen e e y (A0, B0) is a non-emp y p oximal pa allel pai . He e
A0={x∈A: he e exis s y∈Bsuch ha kx−yk=d(A, B)}and B0={x∈B:
he e exis s y∈Asuch ha kx−yk=d(A, B)}.Also, in [7], he au ho s ha e gi en
example(s) o sha p p oximal pai which a e no pa allel. In [2], he au ho s in o-
duced a geome ical no ion called p oximal no mal s uc u e o p o e he exis ence
o a bes p oximi y pai o a ela i ely nonexpansi e mapping (kTx−T yk ≤ kx−yk
o all x∈A, y ∈B). We say (A, B) has p oximal no mal s uc u e ([2]) [ espec-
i ely, weak p oximal no mal s uc u e ([5])] i (A, B) is con ex and o any closed
bounded [ espec i ely, weakly compac ] con ex p oximal pai (H1, H2) o subse s
o (A, B) o which d(H1, H2) = d(A, B) and δ(H1, H2)> d(H1, H2), he e exis s
(x, y)∈(H1, H2) such ha δ(x, H2)< δ(H1, H2) and δ(y, H1)< δ(H1, H2).I is
well known ha e e y non-emp y closed bounded con ex pai (A, B) o a uni o mly
con ex Banach space has p oximal no mal s uc u e. In ac , e e y non-emp y
compac con ex pai (A, B) o a Banach space has p oximal no mal s uc u e. I
is p o ed (P oposi ion 3.2 in [5]) ha a bounded con ex pai has p oximal no -
mal s uc u e i and only i i doesn’ con ain any p oximal diame al sequence.
A pai ({xn},{yn}) o sequences in (A, B) wi h kxn−ynk=d(A, B), n ≥1
is said o be a p oximal diame al sequence ([5]) i d(A, B)< δ({xn},{yn})
and max{lim
n→∞ d(xn+1,co ({y1, y2, ..., yn})) ,lim
n→∞ d(yn+1,co ({x1, x2, ..., xn}))}=
δ({xn},{yn}).I is easy o see ha p oximal no mal s uc u e coincides wi h weak
p oximal no mal s uc u e in e lexi e Banach spaces ([5]). Mo eo e , he ein he
au ho p o ed he exis ence o a bes p oximi y pai in he se ings o a e lexi e
Banach space. Recen ly, in [6], he au ho s posed an open p oblem o he exis-
ence o a bes p oximi y pai o a mo e gene al class o ela i ely nonexpansi e
mappings w o bi s. Also he ein he au ho s indica ed ha an a i ma i e answe
may p o ide a cha ac e iza ion o p oximal no mal s uc u e. Mo i a ed by his,
we aim o gi e a pa ial a i ma i e answe o he same. We also p o ide a cha -
ac e iza ion o weak p oximal no mal s uc u e by using he exis ence o a bes
p oximi y pai o ela i ely o bi al nonexpansi e mappings. Finally, we in oduce
he no ion o poin wise cyclic con ac ion w o bi s and p o e he minimal in a i-
an subse s o such a map ha e nondiame al poin s. This gua an ees he exis ence
o a bes p oximi y pai o such a class in he se ing o a e lexi e Banach space.
Finally, we p o e he exis ence o a bes p oximi y pai o he class o poin wise
cyclic con ac ion w o bi s.
2 Exis ence o Bes P oximi y Pai s
Le A, B be wo closed con ex subse s o a Banach space X. Le T:A∪B→
A∪Bbe a cyclic map. I Tadmi s a bes p oximi y pai , hen A06=∅ 6=B0.
Also, i Tis ela i ely nonexpansi e, hen A0∪B0is cyclically in a ian unde T
(TA0⊆B0, TB0⊆A0). The ollowing heo em is due o Eld ed e al.([2]).
A Cha ac e iza ion o Weak P oximal No mal S uc u e and Bes P oximi y Pai s 3
Theo em 1 Le (K1, K2)be a non-emp y weakly compac con ex pai in a Banach
space and suppose (K1, K2)has p oximal no mal s uc u e. Then e e y ela i ely
nonexpansi e mapping Ton A∪Bhas a bes p oximi y pai in (K1, K2).
The main ool o p o e he same is o use he geome ical no ion called “p ox-
imal no mal s uc u e” on A0∪B0. La e many au ho s es ablished he exis ence
o a bes p oximi y pai o ela i ely nonexpansi e mappings in di e en se ings
using a ian s o geome y ([3],[4],[8],[9]). In [5], Moosa in oduced poin wise el-
a i ely nonexpansi e mappings in ol ing o bi s and he ein p o ed he exis ence
o a bes p oximi y pai o such a class o mappings. Recen ly, in 2018, Ki k and
Shahzad discussed he exis ence o a bes p oximi y pai o ela i ely nonexpan-
si e mappings w o bi s and he ein hey aised he ques ion “can he assump ion
ha Tis ela i ely nonexpansi e in Theo em 1 be eplaced by he assump ion ha
Tis ela i ely nonexpansi e w o bi s?” Tis said o be a ela i ely nonexpan-
si e mapping w o bi s i kTx −T yk ≤ x(O(y)) = δ(x, {y, Ty, T 2y, ...}) o all
(x, y)∈(A, B).Using he ollowing example, we can conclude ha he answe is
nega i e o he abo e open p oblem.
Example 1 Le A={x∈R:−2≤x≤ −1}, B ={x∈R: 1 ≤x≤2}.De ine
T(x) = (−x, i x∈A;
−1−x
2,i x∈B.
Le y∈B. Fo any n, T2ny= 1 + 2n−1−1
2n−1+y
2n= 2 −1
2n−1+y
2nand T2n+1y=
−1−T2ny
2=−2 + 1
2n−y
2n+1 .Now, o any x∈A, y ∈B, kTx −Tyk=
(−x)−−1−y
2≤2−x= x(O(y)) .Simila ly, o x∈B, y ∈Aone can
ha e kT x −T yk ≤ x(O(y)) .Thus Tis ela i ely nonexpansi e w o bi s, bu T
doesn’ ha e any bes p oximi y pai . u
I has o be obse ed ha a cyclic map Ton A∪B ha sa is ies kTx −T yk ≤
x(O(y)) does no gua an ee A0∪B0is cyclically in a ian unde T. Hence, i
is no easonable o expec he exis ence o a bes p oximi y pai o such a map
T. To o e come his, we ede ine he ela i ely nonexpansi e mappings w o bi s.
Fo x∈A∪B, we deno e {T2nx:n∈N∪ {0}} by O2(x).
De ini ion 1 Le A, B be wo non-emp y subse s o a Banach space X. A cyclic
map T:A∪B→A∪Bis said o be a ela i ely o bi al nonexpansi e mapping i
(i) kTx −Tyk=d(A, B) i kx−yk=d(A, B) o x∈A, y ∈B;
(ii) o all x∈A, y ∈B, kTx −Tyk ≤ min{ xO2(y), yO2(x)}.
I is wo h men ioning ha a ela i ely o bi al nonexpani e mapping is no nec-
essa ily ela i ely nonexpansi e.
Example 2 Le A={(0, x)∈R2: 0 ≤x≤1}, B ={(1, y)∈R2: 0 ≤y≤1}and
T:A∪B→A∪Bbe de ined by
x∈A, T(x) = ((1,x
4) i x≥1
2;
(1,x
2) i x < 1
2.
y∈B, T(y) = ((0,y
4) i y≥1
2;
(0,y
2) i y < 1
2.
4 Abhik Diga e al.
We see ha Tis no ela i ely nonexpansi e bu i is a ela i ely o bi al nonex-
pansi e mapping. u
Le (A, B) be a non-emp y sha p p oximal pai in a Banach space Xand le T
be a ela i ely o bi al nonexpansi e mapping on A∪B. Then i is easy o see
ha (A0, B0) is cyclically in a ian unde Tand Tx0= (T x)0.The pai (A, B) is
said o sa is y he weak bes p oximi y pai p ope y (WBPP) i e e y ela i ely
o bi al nonexpansi e mapping on A∪Bhas a bes p oximi y pai . The ollowing
heo em ensu es ha e e y non-emp y weakly compac con ex pai o subse s o
a s ic ly con ex Banach space sa is ies he WBPP. The ollowing heo em is in
a way di e en han Theo em 2.6 o [5]. Fo he sake he comple eness, we p o e
he same he e.
Theo em 2 Le A, B be wo non-emp y weakly compac con ex subse s o a s ic ly
con ex Banach space X. I (A, B)is ha ing weak p oximal no mal s uc u e, hen
(A, B)has WBPP.
P oo Wi hou loss o gene ali y we may assume ha A0=Aand B0=B.
Le Tbe a ela i ely o bi al nonexpansi e mapping o A∪B. Le Fdeno e he
collec ion o non-emp y closed bounded con ex p oximal pai (E1, E2) o subse s
o (A0, B0) wi h (E1, E2) cyclically in a ian unde Tand d(E1, E2) = d(A, B).
F6=∅,since (A0, B0)∈F.By Zo n’s Lemma Fhas a minimal elemen unde
he se inclusion o de “ ⊆”, say, (F1, F2).I (F1, F2) is a single on pai , we
ha e δ(F1, F2) = d(A, B),i.e., Thas a bes p oximi y pai . Suppose (F1, F2) is
no single on. By weak p oximal no mal s uc u e, he e exis s (x1, y1)∈(F1, F2)
such ha m1=δ(x1, F2)< δ(F1, F2) and m2=δ(y1, F1)< δ(F1, F2).Se m=
max{m1, m2}. De ine
L1={x∈F1:δ(x, F2)≤m}
L2={y∈F2:δ(y, F1)≤m}.
L16=∅, L26=∅,since x1∈L1, y1∈L2.Being closed subse o a weakly compac
subse , L1, L2a e weakly compac . To see L1is con ex, le a, b ∈L1.Fo any
λ∈[0,1],
δ(λa + (1 −λ)b, F2)≤λδ(a, F2) + (1 −λ)δ(b, F2)≤λm + (1 −λ)m=m. Hence
we can conclude ha (L1, L2) is a con ex pai . Le ∈F2.Suppose he unique
bes app oxima ion o an elemen z∈A∪Bis deno ed by z0. Then




x1+y0
1
2− 



≤1
2kx1− k+
y0
1− 

=1
2kx1− k+
y1− 0

≤1
2[δ(x1, F2) + δ(y1, F1)]
≤m.
Since ∈F2is a bi a y, δx1+y0
1
2, F2≤m. Hence, x1+y0
1
2∈L1.Simila ly,
x0
1+y1
2∈L2.Mo eo e , 


x1+y0
1
2−x0
1+y1
2

=d(A, B).Hence, d(L1, L2) = d(A, B).
To see (L1, L2) is a p oximal pai , le x∈L1.Then x∈F1and hence x0∈F2.
A Cha ac e iza ion o Weak P oximal No mal S uc u e and Bes P oximi y Pai s 5
The e o e δ(x0, F1) = δ(x, F2)≤m. Thus x0∈L2.I in e s (L1, L2) is a p oximal
pai . In ac , L2={x0∈F2:x∈L1}.
Nex , le x∈L1, ∈F2.Then, kTx −T k ≤ xO2( )=δx, O2( )≤
δ(x, F2)≤m. I ollows ha T(F2)⊆B(Tx;m)∩F1=: F0
1.Simila ly, T(F1)⊆
BTx0;m∩F2=: F0
2.Clea ly, (F0
1, F0
2)∈F.By minimali y, F0
1=F1, F0
2=F2.
Then F1⊆B(Tx;m) and F2⊆B(Tx0;m).Fo any u∈F1,ku−Txk ≤ m, hence,
δ(Tx, F1)≤m. The e o e, Tx ∈L2.Hence, T(L1)⊆L2.Fu he , i y∈L2, hen
y0∈L1.This implies (Ty)0=T y0∈L2.Thus T y ∈L1.As y∈L2is a bi a y, we
ha e T(L2)⊆L1.Hence, (L1, L2)∈F.Fo z∈L1, w ∈L2,kz−wk ≤ δ(z, F2)≤
m < δ(F1, F2).This in e s ha δ(L1, L2)< δ(F1, F2).This con adic s he mini-
mali y o (F1, F2).u
Le Tbe a cyclic map on A∪B. We say ha he pai (A, B) has a p oximal
nondiame al pai i he e exis s (x, y)∈A×Bsuch ha max{δ(x, B), δ(y, A)}<
δ(A, B) whene e d(A, B)< δ(A, B).A simila echnique can be used o ob ain
he ollowing:
Theo em 3 Le (A, B)be a non-emp y closed bounded con ex p oximal pai o
subse s o a Banach space and le Tbe a ela i ely o bi al nonexpansi e mapping
on A∪B. I Thas a non-emp y closed bounded con ex p oximal minimal cyclically
in a ian pai (A, B)ha ing a nondiame al pai hen Thas a bes p oximi y pai .
Example 3 Le A, B and Tas in he Example 2. I is easy o see ha ((0,0),(1,0))
is a bes p oximi y pai . u
3 Cha ac e iza ion o weak p oximal no mal s uc u e
Le (A, B) be a non-emp y bounded con ex p oximal pai o a Banach space X. A
non-cons an pai o sequences ({xn},{yn}) o (A, B) is said o be a p oximal di-
ame al sequence ([5]) i kxn−ynk=d(A, B) o e e y n∈Nand δ({xn},{yn}) =
lim
n→∞ d(xn+1,co ({y1, y2, ..., yn})) = lim
n→∞ d(yn+1,co ({x1, x2, ..., xn})) such ha
d(A, B)< δ({xn},{yn}).I has o be obse ed ha i d(A, B) = 0, hen he p oxi-
mal diame al sequence u ns ou o be a diame al sequence in A∩Bin he sense
o B odski˘ı and Mil0man ([1]). The ollowing esul is discussed in [5].
Theo em 4 A bounded con ex pai (A, B)o a Banach space Xhas p oximal
no mal s uc u e i and only i i does no con ain a p oximal diame al sequence.
Le (A, B) be a non-emp y weakly compac con ex sha p p oximal pai o
subse s o a Banach space ha ing WBPP. Suppose (A, B) does no ha e weak
p oximal no mal s uc u e. Then by Theo em 4, (A, B) has a p oximal diame al
sequence, say, ({xn},{yn}).Consequen ly, lim
n→∞ d(xn+1,co ({y1, y2, ..., yn})) =
δ({xn},{yn}) = lim
n→∞ d(yn+1,co ({x1, x2, ..., xn})) .Since, (A, B) is weakly com-
pac , he e exis s a subsequence ({xnk},{ynk}) o ({xn},{yn}) which is weakly
con e gen . I is easy o see ha he sequence ({xnk},{ynk}) is a p oximal di-
ame al subsequence. Hence, wi hou loss o any gene ali y, we may assume ha
he sequence ({xn},{yn}) is p oximal diame al and weakly con e gen . Now,
H= co ({x1, x2, ...}), K = co ({y1, y2, ...}) a e weakly compac con ex subse s o
A, B espec i ely. De ine T:H∪K→H∪Kby

6 Abhik Diga e al.
T(x) = (y1,i x /∈ {xn:n∈N}
yn+1,i x=xn o some n∈N;
T(y) = (x1,i y /∈ {yn:n∈N}
xn+1,i y=yn o some n∈N.
Clea ly, δ(H, K) = δ({xn},{yn}) and lim
n→∞ kxn−zk=δ(H, K) = lim
n→∞ kyn− k
o any z∈K, ∈H. Hence, xO2(y)=δ(H, K) o each x∈H, y ∈K. Now,
kTx −Tyk ≤ δ(H, K) = xO2(y) o each x∈H, y ∈K.
Also, i (x, y)∈H×Kwi h kx−yk=d(H, K), hen kTx −Tyk=d(H, K).
The e o e Tis a ela i ely o bi al nonexpansi e mapping. As (A, B) is a sha p
p oximal pai , hen so is (H, K) and Tdoes no ha e any bes p oximi y pai .
Thus we ha e he ollowing:
P oposi ion 1 Le A, B be wo non-emp y weakly compac con ex subse s o a
Banach space X. I (A, B)is a sha p p oximal pai and (A, B)has WBPP, hen
(A, B)has weak p oximal no mal s uc u e.
By Theo em 2 and P oposi ion 1 we ha e he ollowing cha ac e iza ion:
Theo em 5 Le A, B be wo non-emp y weakly compac con ex subse s o a s ic ly
con ex Banach space X. Then (A, B)has weak p oximal no mal s uc u e i and
only i e e y ela i ely o bi al nonexpansi e mapping T:A∪B→A∪Bhas a
bes p oximi y pai .
4 Poin wise Cyclic Con ac ion w O bi s
Le (A, B) be a pai o subse s o a no med linea space. A cyclic map Ton
A∪Bis said o be a p oximal poin wise con ac ion i o any x∈A, he e exis s
α(x)∈[0,1) such ha kTx −T yk ≤ α(x)kx−yk([10]). La e many au ho s
ob ained he exis ence o a bes p oximi y pai o ce ain ypes o poin wise
cyclic con ac ions ([8], [11], [12]). Now we in oduce he no ion o poin wise cyclic
con ac ion w o bi s and p o e he exis ence o a bes p oximi y pai o such a
map. Ou esul is a gene aliza ion o he main esul s gi en in he a o emen ioned
a icles.
De ini ion 2 A cyclic map Ton a non-emp y pai (A, B) o subse s o a Banach
space is said o be poin wise cyclic con ac ion w o bi s i i sa is ies
(i) kTx −Tyk=d(A, B) whene e kx−yk=d(A, B) o (x, y)∈A×B;
(ii) o each (x, w)∈(A, B) he e exis s α(x), α(w)∈(0,1) such ha
kTx −Tyk ≤ α(x) xO2(y)+ (1 −α(x)) d(A, B) o all y∈B, and
kTw −Tuk ≤ α(w) wO2(u)+ (1 −α(w)) d(A, B) o all u∈A.
I is easy o see ha e e y poin wise cyclic con ac ion mapping w o bi s is
ela i ely o bi al nonexpansi e.
A Cha ac e iza ion o Weak P oximal No mal S uc u e and Bes P oximi y Pai s 7
Theo em 6 Suppose (A, B)is a weakly compac , con ex pai o a s ic ly con ex
Banach space Xand T:A∪B→A∪Bis a poin wise cyclic con ac ion w
o bi s. Then Thas a bes p oximi y pai .
P oo Le Fdeno e he collec ion o all non-emp y p oximal closed con ex subse s
(H1, H2) o (A0, B0) such ha TH1⊆H2, T H2⊆H1and d(H1, H2) = d(A, B).
Since A0∪B0∈F,we ha e F6=∅. By Zo n’s lemma, Fhas a minimal, say,
(K1, K2) wi h espec o he pa ial o de “ ⊆”.Le (x, y)∈(K1, K2) be such
ha kx−yk=d(K1, K2) = d(A, B).I δ(x, K2) = d(A, B), hen d(A, B) =
d(K1, K2)≤ kx−Txk ≤ δ(x, K2) = d(A, B).This in e s kx−Txk=d(A, B).
Since, Tis poin wise cyclic con ac ion w o bi s, we ha e kT x−T2xk=d(A, B).
The e o e, (x, Tx) is a bes p oximi y pai . Simila ly, i δ(y, K1) = d(A, B), hen
(y, T y) is a bes p oximi y pai . Hence, we may assume ha δ(x, K2)> d(A, B)
and δ(y, K1)> d(A, B). De ine
Kx={z∈K1:kz−Txk ≤ α(x)δ(x, K2) + (1 −α(x)) d(A, B)};
Ky={w∈K2:kw−Tyk ≤ α(y)δ(y, K1) + (1 −α(x)) d(A, B)}.
He e we ha e
kTx −Tyk=d(A, B) = α(x)d(A, B) + (1 −α(x))d(A, B)
< α(x)δ(x, K2) + (1 −α(x))d(A, B).
Then (Ty, T x)∈(Kx, Ky) and hence Kx6=∅ 6=Ky.I is easy o see ha (Kx, Ky)
is con ex. I {un}∞
n=1 ⊆Kxis a sequence con e ges o u∈Xweakly, hen u∈K1.
Now, ku−Txk ≤ lim in {kun−Txk:n∈N} ≤ α(x)δ(x, K2) + (1 −α(x))d(A, B).
Then u∈Kxand hence Kxis closed. Fu he , o any z∈Kx,kTz −T yk ≤
α(y) yO2(z)+(1−α(y))d(A, B)≤α(y)δ(y, K1)+(1−α(y))d(A, B). This implies
ha Tz ∈Ky.Hence, TKx⊆Ky. Simila ly, T Ky⊆Kx. The e o e, (Kx, Ky)∈
F.By minimali y, Kx=K1, Ky=K2. Now, o any w∈K2,kw−T yk ≤
α(y)δ(y, K1) + (1 −α(y))d(A, B)< δ(y, K1)≤δ(K1, K2).Hence, δ(Ty, K2)<
δ(K1, K2).Simila ly, δ(Tx, K1)< δ(K1, K2).Thus (K1, K2) has a p oximal non-
diame al pai . By Theo em 3, T has a bes p oximi y pai . u
Acknowledgemen s The au ho s would like o hank he e e ee o he aluable commen s
and sugges ions. The esea ch o Ra ael Esp´ınola Ga c´ıa was suppo ed by Minis e io de Cien-
cia, Inno aci´on y Uni e sidades del Gobie no de Espa˜na h ough he g an PGC2018-098474-
B-C21, Plan Es a al 2017-2020 Gene aci´on Conocimien o - P oyec os I+D+i and P oyec o de
Excelencia de la Jun a de Andaluci´on FQM-127.
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