Nonlinea Analysis 226 (2023) 113138
Con en s lis s a ailable a ScienceDi ec
Nonlinea Analysis
www.else ie .com/loca e/na
A ixed poin index app oach o K asnosel’ski˘ı-P ecup ixed poin
heo em in cones and applica ions
Jo ge Rod íguez–López
CITMAga & Depa amen o de Es a ís ica, Análise Ma emá ica e Op imización, Uni e sidade de San iago
de Compos ela, 15782, Facul ade de Ma emá icas, Campus Vida, San iago, Spain
a i c l e i n o
A icle his o y:
Recei ed 21 June 2022
Accep ed 2 Sep embe 2022
Communica ed by Tobias We h
MSC:
47H10
47H11
45G15
34B18
35J92
Keywo ds:
Coexis ence ixed poin
ixed poin index
posi i e solu ion
Hamme s ein sys ems
p-Laplacian sys em
adial solu ion
abs ac
We p esen an al e na i e app oach o he ec o e sion o K asnosel’ski˘ı
comp ession–expansion ixed poin heo em due o P ecup, which is based on
he ixed poin index. I allows us o ob ain new gene al e sions o his ixed
poin heo em and also mul iplici y esul s. We emphasize ha all o hem a e
coexis ence ixed poin heo ems o ope a o sys ems, ha means ha e e y
componen o he ixed poin s ob ained is non- i ial. Finally, hese coexis ence
ixed poin heo ems a e applied o ob ain esul s conce ning he exis ence o
posi i e solu ions o sys ems o Hamme s ein in eg al equa ions and adially
symme ic solu ions o (p1, p2)-Laplacian sys ems.
©2022 The Au ho (s). Published by Else ie L d. This is an open access a icle unde
he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/).
1. In oduc ion
K asnosel’ski˘ı comp ession–expansion ixed poin heo em is be ween he main ools o Nonlinea Analysis
o p o ing he exis ence o non- i ial solu ions o di e en ypes o bounda y alue p oblems. Basically,
assuming cone-comp ession o cone-expansion condi ions on he bounda y o an annulus, i ensu es he
exis ence o ixed poin s o compac ope a o s de ined in cones o no med linea spaces.
In he case o sys ems, he localiza ion o he ixed poin s ob ained by means o K asnosel’ski˘ı heo em is
no gi en independen ly in each componen , so he e is no gua an ee ha all he componen s o he ixed
poin a e non- i ial, as al eady poin ed ou in [3,25,26]. This ac mo i a ed P ecup o es ablish he ec o
e sion o K asnosel’ski˘ı ixed poin heo em [25,26] (see Theo em 2.3 below), which p o ides a componen -
wise localiza ion o he ixed poin s. Thus, i gi es su icien condi ions o he exis ence o a coexis ence ixed
E-mail add ess: jo ge od iguez.lop[email p o ec ed].
h ps://doi.o g/10.1016/j.na.2022.113138
0362-546X/©2022 The Au ho (s). Published by Else ie L d. This is an open access a icle unde he CC BY license
(h p://c ea i ecommons.o g/licenses/by/4.0/).
J. Rod íguez–López Nonlinea Analysis 226 (2023) 113138
poin as coined by Lan [20], ha is, a ixed poin wi h all he componen s di e en om ze o. As a as we
know, he e a e only a ew pape s in he li e a u e which deal wi h heo e ical esul s conce ning coexis ence
ixed poin s o compac maps, see [3,15,16,20,25,26,29]. These esul s ha e di ec applica ions in popula ion
models when s udying he coexis ence o se e al compe ing species. In his con ex , he eade may ind
some use ul commen s in he pape by Dance [4].
Mo eo e , unde he assump ions o he ec o e sion o K asnosel’ski˘ı ixed poin heo em, each
componen o he compac map may ha e a di e en beha io , namely, comp ession o expansion (see
Rema k 2.1 below). To he bes o ou knowledge, ixed poin heo ems o expansi e–comp essi e maps
a e no common in he li e a u e, we e e he in e es ed eade o he esul s due o Mawhin [23] in his
di ec ion.
In his pape , we will e e o he men ioned ec o e sion o K asnosel’ski˘ı comp ession–expansion ixed
poin heo em due o P ecup as K asnosel’ski˘ı-P ecup ixed poin heo em in cones. I is well-known ha
he classical K asnosel’ski˘ı ixed poin heo em can be p o ed ia ixed poin index o compac maps, so
ou aim he e is o p esen an al e na i e app oach o K asnosel’ski˘ı-P ecup ixed poin heo em based on
his ool. I has i s own in e es since
(a) i p o ides a way o ex end K asnosel’ski˘ı-P ecup ixed poin heo em o ope a o s de ined in mo e
gene al domains;
(b) he compu a ion o he ixed poin index allows o ob ain easily new mul iplici y esul s;
(c) he p oo can be eplica ed o o he classes o maps o which a ixed poin index heo y is a ailable
(as, o ins ance, uppe semicon inuous mul i alued maps).
No e ha he ixed poin index o compac maps was also he main ool employed in o de o p o e he
coexis ence ixed poin heo ems in [15,16,20].
In addi ion, we p o e ha he ini e-dimensional e sion o K asnosel’ski˘ı-P ecup ixed poin heo em is
equi alen o Poinca ´e–Mi anda ze os heo em. This ac gi es a connec ion be ween i and classical esul s.
In he las sec ion, we apply he heo e ical esul s ob ained in Sec ion 2 o wo di e en p oblems:
sys ems o Hamme s ein in eg al equa ions and adially symme ic solu ions o Di ichle p oblems o
(p1, p2)-Laplacian sys ems. No e ha K asnosel’ski˘ı-P ecup ixed poin heo em has been al eady employed
by se e al au ho s in o de o s udy he exis ence, localiza ion and mul iplici y o posi i e solu ions o
di e en ypes o sys ems o bounda y alue p oblems, see o ins ance [8,27,31–33]. Ou in en ion is o
emphasize he applicabili y o he new ixed poin heo ems es ablished he e and so we p esen a mul iplici y
esul o a sys em o Hamme s ein ype equa ions, which complemen s p e ious esul s in he li e a u e,
see [2,9,17,20,29] and he e e ences he ein. Mo eo e , conce ning adial solu ions o (p1, p2)-Laplacian
sys ems, ou su icien condi ions p o ide no only he exis ence o posi i e solu ions, bu also a no el
localiza ion o hem, c . [24,33]. I is wo h men ioning ha exis ence o (no necessa ily adial) solu ions o
(p1, p2)-Laplacian sys ems wi h bo h nonze o componen s was al eady s udied in [16].
2. K asnosel’ski˘
ı-P ecup ixed poin heo em in cones
Fi s , we ecall K asnosel’ski˘ı comp ession–expansion ixed poin heo em in cones [18] (see also [1,12]).
In he sequel, we need he ollowing no ions. A closed con ex subse Ko a no med linea space (X, ∥·∥)
is a cone i λ u ∈K o e e y u∈Kand o all λ≥0, and K∩(−K) = {0}. A cone Kinduces he pa ial
o de in Xgi en by u⪯ i and only i −u∈K. Mo eo e , we shall say ha u≺ i −u∈K {0}.
The ollowing no a ions will be use ul: o gi en , R ∈R+:= [0,∞), 0 < < R, we de ine
K ,R := {u∈K: < ∥u∥< R}and K ,R := {u∈K: ≤ ∥u∥ ≤ R}.
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J. Rod íguez–López Nonlinea Analysis 226 (2023) 113138
Theo em 2.1 (K asnosel’ski˘ı).Le (X, ∥·∥)be a no med linea space, Ka cone in Xand , R ∈R+,
0< < R.
Conside a compac map T:K ,R →Ksa is ying one o he ollowing condi ions:
(a)T(u)⊀ui ∥u∥= and T(u)⊁ui ∥u∥=R;
(b)T(u)⊁ui ∥u∥= and T(u)⊀ui ∥u∥=R.
Then Thas a leas a ixed poin u∈Kwi h ≤ ∥u∥ ≤ R.
Condi ion (a) in K asnosel’ski˘ı heo em is usually called a comp ession ype condi ion, whe eas au ho s
o en e e o condi ion (b) as he cone-expansion condi ion. Simila ly, in case (a) we will say ha he ope a o
Tis comp essi e, while in case (b)Tis called an expansi e ope a o .
I is well-known ha condi ions (a) and (b) can be weakened as homo opy ype condi ions. In his way, we
ha e he homo opy e sion o K asnosel’ski˘ı heo em o K asnosel’ski˘ı-Benjamin heo em, see o ins ance [1].
Theo em 2.2 (K asnosel’ski˘ı-Benjamin).Le (X, ∥·∥)be a no med linea space, Ka cone in Xand
, R ∈R+,0< < R.
Assume ha T:K ,R →Kis a compac map and he e exis s h∈K {0}such ha one o he ollowing
condi ions is sa is ied:
(a)T(u) + µ h =ui ∥u∥= and µ > 0, and T(u)=λ u i ∥u∥=Rand λ > 1;
(b)T(u)=λ u i ∥u∥= and λ > 1, and T(u) + µ h =ui ∥u∥=Rand µ > 0.
Then Thas a leas a ixed poin u∈Kwi h ≤ ∥u∥ ≤ R.
In [25,26], P ecup p oposed a comp ession–expansion ype ixed poin heo em o sys ems o ope a o s.
The main no el y is ha comp ession–expansion condi ions a e gi en in a componen -wise manne , in wha
was called he ec o e sion o K asnosel’ski˘ı ixed poin heo em. Le us ecall his esul .
Conside wo cones K1and K2o a no med linea space X, and so K:= K1×K2is a cone o X2=X×X.
Fo , R ∈R2
+, = ( 1, 2), R= (R1, R2), wi h 0 < i< Ri(i= 1,2), we deno e
(Ki) i,Ri:= {u∈Ki: i≤ ∥u∥ ≤ Ri}(i= 1,2),
K ,R := {u= (u1, u2)∈K: i≤ ∥ui∥ ≤ Ri o i= 1,2}.
Clea ly, K ,R = (K1) 1,R1×(K2) 2,R2.
The aim o he ec o e sion o K asnosel’ski˘ı heo em is o ob ain a solu ion u= (u1, u2) o he ope a o
sys em
{u1=T1(u1, u2),
u2=T2(u1, u2),
loca ed in he se K ,R, ha is, u= (u1, u2)∈Kand i≤ ∥ui∥ ≤ Ri,i= 1,2.
Theo em 2.3 (K asnosel’ski˘ı-P ecup).Le (X, ∥·∥)be a no med linea space, K1and K2 wo cones in X
and , R ∈R2
+, = ( 1, 2),R= (R1, R2), wi h 0< i< Ri(i= 1,2).
Assume ha T= (T1, T2) : K ,R →Kis a compac map and o each i∈ {1,2} he e exis s hi∈Ki {0}
such ha one o he ollowing condi ions is sa is ied in K ,R:
(a)Ti(u) + µ hi=uii ∥ui∥= iand µ > 0, and Ti(u)=λ uii ∥ui∥=Riand λ > 1;
(b)Ti(u)=λ uii ∥ui∥= iand λ > 1, and Ti(u) + µ hi=uii ∥ui∥=Riand µ > 0.
Then Thas a leas a ixed poin u= (u1, u2)∈Kwi h i≤ ∥ui∥ ≤ Ri(i= 1,2).
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J. Rod íguez–López Nonlinea Analysis 226 (2023) 113138
Rema k 2.1. As al eady poin ed ou in [25,26], he ope a o Tmay exhibi a di e en beha io
(comp ession o expansion) in each componen . Mo e exac ly, he ollowing op ions a e possible:
(i) bo h ope a o s T1and T2a e comp essi e;
(ii) bo h ope a o s T1and T2a e expansi e;
(iii) one o he ope a o s T1o T2is comp essi e, while he o he one is expansi e.
Le us ecall b ie ly he main ideas o he p oo o Theo em 2.3 gi en in [25,26], which is essen ially di ided
in o wo cases: (1) bo h ope a o s T1and T2a e comp essi e; (2) one o he ope a o s is expansi e. The i s
case elies on Schaude ixed poin heo em. In he second case, he ixed poin p oblem is educed o an
equi alen one in which bo h ope a o s sa is y he comp ession ype condi ion and so he exis ence o a ixed
poin is ensu ed by he o me case.
No e ha he same p oo due o P ecup emains alid o he ollowing n-dimensional ec o e sion o
K asnosel’ski˘ı ixed poin heo em o an ope a o T= (T1, T2, . . . , Tn) de ined in Xn.
Theo em 2.4. Le (X, ∥·∥)be a no med linea space, K1, . . . , Kncones in X,K:= K1×···×Kn, , R ∈Rn
+,
= ( 1, . . . , n),R= (R1, . . . , Rn), wi h 0< i< Ri(i= 1, . . . , n), and K ,R := {u= (u1, . . . , un)∈K:
i≤ ∥ui∥ ≤ Ri o i= 1, . . . , n}.
Assume ha T= (T1, . . . , Tn) : K ,R →Kis a compac map and o each i∈ {1, . . . , n} he e exis s
hi∈Ki {0}such ha one o he ollowing condi ions is sa is ied in K ,R:
(a)Ti(u) + µ hi=uii ∥ui∥= iand µ > 0, and Ti(u)=λ uii ∥ui∥=Riand λ > 1;
(b)Ti(u)=λ uii ∥ui∥= iand λ > 1, and Ti(u) + µ hi=uii ∥ui∥=Riand µ > 0.
Then Thas a leas a ixed poin u= (u1, . . . , un)∈Kwi h i≤ ∥ui∥ ≤ Ri(i= 1, . . . , n).
We highligh ha ou p oo he e is comple ely di e en o ha due o P ecup, since i is based on ixed
poin index heo y independen ly o he possibili y (i)–(iii) in Rema k 2.1. In pa icula , ou app oach does
no equi e o u n he expansi e ope a o s in o comp essi e ones.
2.1. Fixed poin index compu a ion
Fi s , le us ecall some o he use ul p ope ies o he ixed poin index o compac maps. Fo mo e
de ails, we e e he eade o [1,5,12] (see also [14]).
P oposi ion 2.1. Le Pbe a cone o a no med linea space, U⊂Pbe a bounded ela i ely open se and
T:U→Pbe a compac map such ha Thas no ixed poin s on he bounda y o U(deno ed by ∂ U). Then
he ixed poin index o Tin Po e U,iP(T, U), has he ollowing p ope ies:
1. (Addi i i y) Le Ube he disjoin union o wo open se s U1and U2. I 0∈ (I−T)(U (U1∪U2)), hen
iP(T, U) = iP(T, U1) + iP(T, U2).
2. (Exis ence) I iP(T, U)= 0, hen he e exis s u∈Usuch ha u=Tu.
3. (Homo opy in a iance) I H:U×[0,1] →Pis a compac homo opy and 0∈ (I−H)(∂ U ×[0,1]), hen
iP(H(·,0), U) = iP(H(·,1), U).
4. (No maliza ion) I Tis a cons an map wi h T(u) = u0 o e e y u∈U, hen
iP(T, U) = {1,i u0∈U,
0,i u0∈ U.
Mo eo e , we ha e he ollowing condi ions conce ning he compu a ion o he ixed poin index.
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J. Rod íguez–López Nonlinea Analysis 226 (2023) 113138
P oposi ion 2.2. Le Ube a bounded ela i ely open subse o a cone Psuch ha 0∈Uand T:U→P
be a compac map.
(a) I T(u)=λ u o all u∈∂ U and all λ≥1, hen iP(T, U)=1.
(b) I he e exis s h∈P {0}such ha T(u) + λ h =u o e e y λ≥0and all u∈∂ U, hen iP(T, U)=0.
In he sequel, le (X, ∥·∥X) and (Y, ∥·∥Y) be no med linea spaces, K1⊂X,K2⊂Y wo cones and
K:= K1×K2 he co esponding cone o X×Y. When no con usion may occu , bo h no ms ∥·∥Xand ∥·∥Y
will be simply deno ed by ∥·∥.
Now, we p esen a echnical esul which will be c ucial in he compu a ion o he ixed poin index in he
main esul s o his sec ion. I was al eady p o en in [29, Lemma 2.3], bu we include he e he p oo again
o he eade ’s con enience.
Lemma 2.1. Le Uand Vbe bounded ela i ely open subse s o K1and K2, espec i ely, such ha 0∈U.
Assume ha T:U×V→K,T= (T1, T2), is a compac map and he e exis s h∈K2 {0}such ha
T1(u, )=λ u o u∈∂K1U, ∈Vand λ≥1; (2.1)
T2(u, ) + µ h = o u∈U, ∈∂K2Vand µ≥0.(2.2)
Then iK(T, U ×V)=0.
P oo . Conside he homo opy H:U×V×[0,1] →Kgi en by
H((u, ), )=( T1(u, ), T2(u, ) + (1 − )µ0h),
wi h µ0>0 big enough such ha =T2(0, )+µ0h o all ∈V. No e ha he exis ence o such a posi i e
numbe µ0is gua an eed since Vis bounded and Tis compac .
Assump ions (2.1) and (2.2) gua an ee ha he homo opy unc ion Hhas no ixed poin s on ∂K(U×V).
The e o e, by he homo opy in a iance p ope y o he ixed poin index, we ha e ha
iK(H(·,0), U ×V) = iK(H(·,1), U ×V) = iK(T, U ×V).(2.3)
On he o he hand, o = 0, he map H((u, ),0) = (0, T2(u, )+µ0h) has no ixed poin s in U×V. Indeed,
i (u, )∈U×Vis such a ixed poin , hen u= 0 and =T2(0, )+µ0h, a con adic ion wi h he hypo hesis
abou µ0. Hence, iK(H(·,0), U ×V) = 0 and so he conclusion ollows om (2.3).□
Rema k 2.2. Ob iously, he oles ha play T1and T2in he s a emen o Lemma 2.1, gi en by assump ions
(2.1) and (2.2), a e in e changeable.
Fo , R ∈R2
+, 0 < i< Ri(i= 1,2), ixed, ou aim is o compu e he ixed poin index o a compac
ope a o T= (T1, T2) : K ,R →Kin he ela i ely open se
K ,R := {u= (u1, u2)∈K: i<∥ui∥< Ri o i= 1,2}
unde he condi ions o K asnosel’ski˘ı-P ecup ixed poin heo em. Ob iously, we need o assume also ha
Thas no ixed poin s on he bounda y o K ,R in o de o ha e he ixed poin index well-de ined o e his
se .
In he sequel, we will also use he ollowing no a ions:
(Ki) i={u∈Ki:∥u∥< i},
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J. Rod íguez–López Nonlinea Analysis 226 (2023) 113138
(Ki) i={u∈Ki:∥u∥ ≤ i}(i= 1,2),
K ={u= (u1, u2)∈K:∥ui∥< i o i= 1,2},
K ={u= (u1, u2)∈K:∥ui∥ ≤ i o i= 1,2}.
To compu e he ixed poin index o e K ,R, we need o ex end he de ini ion o T o he se KR. A key
ing edien in ou pu pose is he de ini ion o a e ac ion om KRin o K ,R. To do so, we use ha (Ki) i,Ri
is a e ac o (Ki)Ri(i= 1,2), see [7, Example 3]. Indeed, we ha e he e ac ion ρi: (Ki)Ri→(Ki) i,Ri
de ined as
ρi(ui) = ⎧
⎪
⎨
⎪
⎩
i
ui+ ( i−∥ui∥)2hi
ui+ ( i−∥ui∥)2hi
,i ∥ui∥< i,
ui,i i≤ ∥ui∥ ≤ Ri,
(2.4)
whe e hi∈Ki {0}is ixed. No e ha ρiis well-de ined:
ui+ ( i−∥ui∥)2hi
= 0 o all ui∈(Ki) i.
O he wise, −ui= ( i−∥ui∥)2hi∈Ki, wha oge he wi h ui∈Kiimplies ui= 0, om he de ini ion o
cone. Taking ui= 0, we ha e
2
ihi
>0 since i>0 and hi∈Ki {0}. Mo eo e , i is clea ha ρiis
con inuous and ρi(ui) = ui o all ui∈(Ki) i,Ri.
Now we a e in a posi ion o compu e he ixed poin index o e he se K ,R. Fi s , we s udy he case in
which bo h ope a o s a e comp essi e.
Theo em 2.5. Assume ha T= (T1, T2) : K ,R →Kis a compac map and o each i∈ {1,2} he e exis s
hi∈Ki {0}such ha he ollowing condi ions a e sa is ied in K ,R:
(i) Ti(u) + µ hi=uii ∥ui∥= iand µ≥0;
(ii) Ti(u)=λ uii ∥ui∥=Riand λ≥1.
Then
iK(T, K ,R)=1.
P oo . Conside he e ac ion ρ:KR→K ,R de ined as ρ(u1, u2)=(ρ1(u1), ρ2(u2)), whe e he unc ions
ρia e gi en by (2.4),i= 1,2. Now, de ine he auxilia y map N= (N1, N2) : KR→Kas ollows
N(u) := (T◦ρ)(u).(2.5)
Clea ly, Nis a compac ope a o , N(u) = T(u) o e e y u∈K ,R and o each i∈ {1,2} he ollowing
condi ions hold in KR:
(i∗)Ni(u) + µ hi=uii ∥ui∥= iand µ≥0;
(ii∗)Ni(u)=λ uii ∥ui∥=Riand λ≥1.
No e ha (i∗) implies ha N(u) + µ h =u o all u∈∂K and µ≥0 (whe e h= (h1, h2)), so
P oposi ion 2.2 yields
iK(N, K )=0.
Simila ly, condi ion (ii∗) in conjunc ion wi h P oposi ion 2.2 gua an ee ha
iK(N, KR)=1.
Mo eo e , by Lemma 2.1,
iK(N, (K1)R1×(K2) 2) = iK(N, (K1) 1×(K2)R2)=0.
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J. Rod íguez–López Nonlinea Analysis 226 (2023) 113138
By he addi i i y p ope y o he ixed poin index,
iK(N, (K1) 1,R1×(K2) 2) = iK(N, (K1)R1×(K2) 2)−iK(N, K )=0,
so we ob ain
iK(N, K ,R) = iK(N, KR)−iK(N, (K1) 1,R1×(K2) 2)−iK(N, (K1) 1×(K2)R2) = 1.
Finally, since T=Non he se K ,R, we ha e iK(T, K ,R) = iK(N, K ,R) = 1. □
Nex , le us conside he case in which we ha e comp ession o one ope a o and expansion o he o he
one.
Theo em 2.6. Assume ha T= (T1, T2) : K ,R →Kis a compac map and o each i∈ {1,2} he e exis s
hi∈Ki {0}such ha he ollowing condi ions a e sa is ied in K ,R:
(i) T1(u) + µ h1=u1i ∥u1∥= 1and µ≥0, and T1(u)=λ u1i ∥u1∥=R1and λ≥1;
(ii) T2(u) + µ h2=u2i ∥u2∥=R2and µ≥0, and T2(u)=λ u2i ∥u2∥= 2and λ≥1.
Then
iK(T, K ,R) = −1.
P oo . Conside he map N:KR→Kde ined as in (2.5).ByP oposi ion 2.2,
iK(N, (K1)R1×(K2) 2)=1, iK(N, (K1) 1×(K2)R2)=0.
Mo eo e , Lemma 2.1 yields
iK(N, K ) = iK(N, KR)=0.
Hence, i ollows om he addi i i y p ope y o he index ha
iK(N, (K1) 1,R1×(K2) 2) = iK(N, (K1)R1×(K2) 2)−iK(N, K )=1,
and so
iK(N, K ,R) = iK(N, KR)−iK(N, (K1) 1×(K2)R2)−iK(N, (K1) 1,R1×(K2) 2) = −1.
Then, iK(T, K ,R) = −1. □
Rema k 2.3. The s a emen o Theo em 2.6 co esponds o he case in which T1is comp essi e and T2is
expansi e. Clea ly, he same conclusion can be ob ained i T1is expansi e and T2is comp essi e.
Finally, we deal wi h he case in which bo h T1and T2a e expansi e.
Theo em 2.7. Assume ha T= (T1, T2) : K ,R →Kis a compac map and o each i∈ {1,2} he e exis s
hi∈Ki {0}such ha he ollowing condi ions a e sa is ied in K ,R:
(i) Ti(u) + µ hi=uii ∥ui∥=Riand µ≥0;
(ii) Ti(u)=λ uii ∥ui∥= iand λ≥1.
Then
iK(T, K ,R)=1.
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J. Rod íguez–López Nonlinea Analysis 226 (2023) 113138
P oo . Conside again he map N:KR→Kde ined as in (2.5). Now, by P oposi ion 2.2,
iK(N, KR)=0, iK(N, K )=1.
Mo eo e , acco ding wi h Lemma 2.1,
iK(N, (K1) 1×(K2)R2) = iK(N, (K1)R1×(K2) 2)=0.
As a consequence o he addi i i y p ope y o he index, we deduce ha
iK(N, (K1) 1×(K2) 2,R2) = iK(N, (K1) 1×(K2)R2)−iK(N, K ) = −1.
The e o e,
iK(N, K ,R) = iK(N, KR)−iK(N, (K1) 1×(K2) 2,R2)−iK(N, (K1)R1×(K2) 2)=1.
In conclusion, iK(T, K ,R) = 1, as wished. □
Rema k 2.4. The compu a ion o he ixed poin index gi en by Theo ems 2.5–2.7 is independen o
ha ob ained in [29, Theo em 2.8]. Indeed, in [29], he assump ions on Tand, in pa icula , he homo opy
condi ions (i)–(ii) we e imposed in he whole se KRins ead o i s subse K ,R, as he e.
As a s aigh o wa d consequence o he compu a ion o he ixed poin index p o ided by Theo ems 2.5–
2.7, we ha e an al e na i e e sion o Theo em 2.3.
Theo em 2.8. Assume ha T= (T1, T2) : K ,R →Kis a compac map and o each i∈ {1,2} he e exis s
hi∈Ki {0}such ha one o he ollowing condi ions is sa is ied in K ,R:
(a)Ti(u) + µ hi=uii ∥ui∥= iand µ≥0, and Ti(u)=λ uii ∥ui∥=Riand λ≥1;
(b)Ti(u)=λ uii ∥ui∥= iand λ≥1, and Ti(u) + µ hi=uii ∥ui∥=Riand µ≥0.
Then Thas a leas a ixed poin u= (u1, u2)∈Kwi h i<∥ui∥< Ri(i= 1,2).
P oo . By Theo ems 2.5–2.7, we ha e ha
iK(T, K ,R) = ±1,
and so he exis ence p ope y o he ixed poin index ensu es ha Thas a leas a ixed poin in K ,R.□
2.2. O he e sions o K asnosel’ski˘ı -P ecup ixed poin heo em: di e en domains
A e he p e ious compu a ion o he ixed poin index o To e he se K ,R, we can hink o p o ing
simila esul s o ope a o s Tde ined in o he egions di e en om K ,R. In his way, we inc ease he ange
o applicabili y o he o iginal K asnosel’ski˘ı-P ecup ixed poin heo em.
Fo each i∈ {1,2}, le φi:Ki→R+be a con inuous conca e unc ional on Ki, ha is, φiis a con inuous
unc ion and
φi(λ u + (1 −λ) )≥λ φi(u) + (1 −λ)φi( ), o all u, ∈Ki, λ ∈[0,1].
Then, o , R ∈R2
+, 0 < i< Ri(i= 1,2), ixed, conside he se s
Kφ
,R := {u= (u1, u2)∈K: i< φi(ui) and ∥ui∥< Ri o i= 1,2},
Kφ
,R := {u= (u1, u2)∈K: i≤φi(ui) and ∥ui∥ ≤ Ri o i= 1,2}.
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J. Rod íguez–López Nonlinea Analysis 226 (2023) 113138
This ype o se s has been al eady conside ed by Legge and Williams [21] in he con ex o hei celeb a ed
ixed poin heo em. No e ha Kφ
,R is a closed con ex se . Hence, by Dugundji ex ension heo em (see [6,
Theo em 4.1] o [12]), he subse Kφ
,R is a e ac o K.
Wi h his in mind, one can easily es ablish al e na i e e sions o Theo ems 2.5–2.7. We sum up hem in
he ollowing esul .
Theo em 2.9. Assume ha he e exis con inuous conca e unc ionals φi:Ki→R+such ha φi(u)≤ ∥u∥
o all u∈Ki(i= 1,2), he se Kφ
,R is nonemp y, T= (T1, T2) : Kφ
,R →Kis a compac map and o each
i∈ {1,2} he e exis s hi∈Ki {0}such ha one o he ollowing condi ions is sa is ied in Kφ
,R:
(i) Ti(u) + µ hi=uii φi(ui) = iand µ≥0, and Ti(u)=λ uii ∥ui∥=Riand λ≥1;
(ii) Ti(u)=λ uii φi(ui) = iand λ≥1, and Ti(u) + µ hi=uii ∥ui∥=Riand µ≥0.
Then
iK(T, K φ
,R)=(−1)k,
whe e k= 0 i bo h T1and T2a e comp essi e, k= 1 i one o he ope a o s T1o T2is comp essi e and he
o he one is expansi e and k= 2 i bo h ope a o s a e expansi e.
P oo . Conside a e ac ion ρ:K→Kφ
,R and de ine he map N:K→Kas ollows:
N(u) := (T◦ρ)(u).
We in oduce he ollowing use ul no a ion:
(Ki)φi
i={u∈Ki:φi(u)< i}and (Ki)φi
i={u∈Ki:φi(u)≤ i}.
Clea ly, ∂(Ki)φi
i⊂ {u∈Ki:φi(u) = i}(i= 1,2) and, mo eo e ,
Kφ
,R =((K1)R1 (K1)φ1
1)×((K2)R2 (K2)φ2
2),
so now he p oo ollows as hose o Theo ems 2.5–2.7, eplacing ∥·∥ wi h φi(·) whe e needed. □
Rema k 2.5. The p e ious ixed poin index compu a ion emains ue o an ope a o Tde ined in a much
mo e gene al domain o ype (U1 V1)×(U2 V2), whe e o each i∈ {1,2}, one has 0 ∈Vi⊂Vi⊂Ui,Ui
and Via e bounded and ela i ely open se s in Kiand Ui Viis a e ac o Ui. Obse e ha , in pa icula ,
Ui Viis a e ac o Uip o ided ha ∂ Viis a e ac o Vi, wha allows Ui o be an a bi a y bounded
open se la ge enough. I can be immedia ely deduced ha his is he case o Vi= (Ki) i. Indeed, o a
ixed hi∈Ki {0}, one can de ine he e ac ion ρi:Vi→∂ Vias
ρi(ui) = i
ui+ ( i−∥ui∥)2hi
ui+ ( i−∥ui∥)2hi
.
No e ha in his case he se Uineeds no be he in e sec ion o a ball wi h he cone Ki, which enla ges he
applicabili y o Theo ems 2.5–2.7.
In his con ex , condi ions (i) and (ii) abo e can be w i en in he ollowing way:
(i) Ti(u) + µ hi=uii ui∈∂ Viand µ≥0, and Ti(u)=λ uii ui∈∂ Uiand λ≥1;
(ii) Ti(u)=λ uii ui∈∂ Viand λ≥1, and Ti(u) + µ hi=uii ui∈∂ Uiand µ≥0;
and hey mus be sa is ied o e he se (U1 V1)×(U2 V2).
Ob iously, om Theo em 2.9 i ollows immedia ely a new ixed poin heo em in he line o Theo em 2.3.
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J. Rod íguez–López Nonlinea Analysis 226 (2023) 113138
64 <∥ 1∥∞<522,2<∥ 2∥∞<512,
1
4≤ ∥w1∥∞≤64,2<∥w2∥∞<512.
We emphasize ha he second componen o all he h ee solu ions is si ua ed in he same egion, so he
mul iplici y is ob ained due o we a e able o localize hei i s componen in dis inc se s.
3.2. Radial solu ions o (p1, p2)-Laplacian sys ems
In his sec ion, we conside he exis ence o posi i e adial solu ions o he (p1, p2)-Laplacian sys em
−∆p1u= 1(u, ) in B,
−∆p2 = 2(u, ) in B,
u=0= on ∂B,
(3.13)
whe e ∆pu= di (∥∇u∥p−2∇u),Bis he uni open ball in Rncen e ed a o igin, p1, p2> n ≥2 and
1, 2:R2
+→R+a e con inuous and nondec easing unc ions ( ha is, i (u1, 1),(u2, 2)∈R2
+wi h u1≤u2
and 1≤ 2, hen i(u1, 1)≤ i(u2, 2) o i= 1,2).
Se ing, as usual, =∥x∥,u(x) = u1( ) and (x) = u2( ), he Di ichle sys em (3.13) is educed o he
ollowing sys em o o dina y di e en ial equa ions wi h mixed bounda y condi ions
−[ n−1ϕp1(u′
1)]′= n−1 1(u1, u2) in (0,1),
−[ n−1ϕp2(u′
2)]′= n−1 2(u1, u2) in (0,1),
u′
1(0) = u1(1) = 0 = u′
2(0) = u2(1),
(3.14)
whe e ϕp( ) := | |p−2 is he p-Laplacian homeomo phism. We will look o posi i e solu ions o (3.14), ha
is, adially symme ic solu ions o (3.13).
A Ha nack ype inequali y has been es ablished in [28] o p oblem
−[ n−1ϕp( ′)]′= n−1h( , ) in (0,1), ′(0) = (1) = 0,
in e ms o he ene ge ic no m. By using H¨olde inequali y, one can de i e a Ha nack ype inequali y in
e ms o he usual max-no m, see [13]. The esul can be summa ized as ollows.
Lemma 3.1. Le p > n. E e y unc ion ∈C1[0,1] wi h n−1ϕp( ′)∈C1[0,1] and [ n−1ϕp( ′)]′≤0on
[0,1] sa is ies ha ′≤0on [0,1]. I , in addi ion, − 1−n[ n−1ϕp( ′)]′is noninc easing on (0,1], hen
( )≥p−n
p−1(1 − ) n
p−1∥ ∥∞, ∈[0,1] .
Le us conside he ollowing cones in he space o con inuous unc ions C(I), wi h I:= [0,1],
Ki={ ∈ C(I) : ≥0 on I, is noninc easing on Iand min
∈[a,b] ( )≥ci∥ ∥∞}(i= 1,2),
whe e [a, b]⊂(0,1) and ci:= pi−n
pi−1(1 −b)an
pi−1,i= 1,2. As be o e, we de ine he cone K:= K1×K2in
he p oduc space.
We will look o posi i e solu ions o p oblem (3.14) as ixed poin s o he ope a o T= (T1, T2) : K→K
de ined as
Ti(u1, u2)( ) = ∫1
ϕ−1
pi(1
sn−1∫s
0
τn−1 i(u1(τ), u2(τ)) dτ)ds, (i= 1,2).(3.15)
The ope a o Tis well-de ined, ha is, i maps he cone Kin o i sel . Indeed, ake u= (u1, u2)∈K
and le us show ha i:= Ti(u)∈Ki, o i= 1,2. Clea ly, i∈ C(I) and i≥0 on I, since iis
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J. Rod íguez–López Nonlinea Analysis 226 (2023) 113138
con inuous and nonnega i e. Mo eo e , [ n−1ϕpi( ′
i)]′≤0 on I, so iis noninc easing on I, as a consequence
o Lemma 3.1. On he o he hand, u1and u2a e noninc easing and iis nondec easing, which implies ha
he map ↦→ i(u1( ), u2( )) is noninc easing on Iand hen so is − 1−n[ n−1ϕpi( ′
i)]′= i(u1( ), u2( )).
Hence, again by Lemma 3.1, one has ha min ∈[a,b] i( )≥ci∥ i∥∞. In conclusion, i∈Ki, as desi ed.
I is a ou ine o check ha Tis comple ely con inuous.
Now, le us de ine a con inuous conca e unc ional on Ki,φi:Ki→R+, as ollows
φi( ) = min
∈[a,b] ( ), i = 1,2.
We in end o apply Theo em 2.10 in o de o ob ain su icien condi ions o he exis ence o ixed poin s o
Tloca ed in a se o he o m Kφ
,R. No e ha his se can also be w i en as Kφ
,R = (U1 V1)×(U2 V2),
whe e
Vi={u∈Ki: min
∈[a,b]u( )< i}, Ui={u∈Ki:∥u∥∞< Ri}(i= 1,2).
The bounded open se s Viwe e in oduced by Lan in [19] and la e employed by se e al au ho s, see [14]
and he e e ences he ein.
Theo em 3.3. Assume ha he e exis αi, βi>0wi h βi/ci< αi,i= 1,2, such ha
i(β1, β2)>βpi−1
i
(b−a)an−1(1 −b)pi−1, i(α1, α2)< αpi−1
i(i= 1,2).(3.16)
Then he sys em (3.14) has a leas one posi i e solu ion (u1, u2)∈Ksuch ha
βi< φi(ui)and ∥ui∥∞< αi, i = 1,2.
P oo . Conside he ope a o T= (T1, T2) : Kφ
,R →Kde ined as in (3.15), wi h i=βiand Ri=αi,
i= 1,2. Le us check ha i ul ills he assump ions o Theo em 2.10.
Fi s , ix i∈ {1,2}and ake u= (u1, u2)∈Kφ
,R wi h φi(ui) = i. Then (u1( ), u2( )) ≥( 1, 2) o all
∈[a, b] and hus, by he mono onici y assump ion on i, we ha e ha i(u1( ), u2( )) ≥ i( 1, 2) o all
∈[a, b]. Hence, o ∈[a, b],
Ti(u)( )≥∫1
b
ϕ−1
pi(1
sn−1∫s
0
τn−1 i(u1(τ), u2(τ)) dτ)ds
≥∫1
b
ϕ−1
pi(1
sn−1∫b
a
τn−1 i(u1(τ), u2(τ)) dτ)ds
≥∫1
b
ϕ−1
pi(1
sn−1∫b
a
τn−1 i( 1, 2)dτ)ds
≥(1 −b)ϕ−1
pi((b−a)an−1 i( 1, 2))> i,
whe e he las inequali y ollows om (3.16). This clea ly implies ha Ti(u) + µ1
1
1=uii u∈Kφ
,R wi h
φi(ui) = iand µ≥0.
Suppose now ha u∈Kφ
,R wi h ∥ui∥=Ri o some i∈ {1,2}. Then i(u1( ), u2( )) ≤ i(R1, R2) o
e e y ∈Iand so, by (3.16), we ha e
∥Ti(u)∥∞≤∫1
0
ϕ−1
pi(1
sn−1∫s
0
τn−1 i(u1(τ), u2(τ)) dτ)ds ≤ϕ−1
pi( i(R1, R2)) < R1.
The e o e, al e na i e (a) in Theo em 2.10 holds and hence we each he hesis. □
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J. Rod íguez–López Nonlinea Analysis 226 (2023) 113138
Rema k 3.2. Assump ion (3.16) is gua an eed by he ollowing asymp o ic condi ions: o e e y i∈ {1,2},
lim
ui→0
i(u1, u2)
upi−1
i
= +∞and lim
ui→∞
i(u1, u2)
upi−1
i
= 0
uni o mly wi h espec o uj,j=i.
In his case, i is said ha bo h unc ions 1and 2a e supe linea a 0 and sublinea a in ini y wi h
espec o ϕp1and ϕp2, espec i ely.
Rema k 3.3. Unde he assump ions o Theo em 3.3, bo h ope a o s T1and T2a e comp essi e. No ice
ha he beha io s comp essi e–expansi e and expansi e–expansi e a e also possible:
1. (Comp essi e–expansi e) Assume ha he e exis αi, βi>0, i= 1,2, wi h β1/c1< α1and α2< β2,
such ha
1(β1, c2α2)>βp1−1
1
(b−a)an−1(1 −b)p1−1, 1(α1, β2/c2)< αp1−1
1,
2(β1, β2)>βp2−1
2
(b−a)an−1(1 −b)p2−1, 2(α1, α2)< αp2−1
2.
Then he ope a o T= (T1, T2) de ined as in (3.15) has a leas one ixed poin in (U1 V1)×(U2 V2),
whe e V1={u∈K1:φ1(u)< β1}, U1={u∈K1:∥u∥∞< α1},
V2={u∈K2:∥u∥∞< α2}, U2={u∈K2:φ2(u)< β2}.
In his case, he ope a o T1is comp essi e and T2is expansi e on (U1 V1)×(U2 V2).
2. (Expansi e–expansi e) Assume ha he e exis αi, βi>0, wi h αi< βi,i= 1,2, such ha
1(β1, c2α2)>βp1−1
1
(b−a)an−1(1 −b)p1−1, 1(α1, β2/c2)< αp1−1
1,
2(c1α1, β2)>βp2−1
2
(b−a)an−1(1 −b)p2−1, 2(β1/c1, α2)< αp2−1
2.
Then he ope a o T= (T1, T2) has a leas one ixed poin in (U1 V1)×(U2 V2), whe e
Vi={u∈Ki:∥u∥∞< αi}, Ui={u∈Ki:φi(u)< βi}(i= 1,2).
No e ha bo h ope a o s T1and T2a e o expansi e ype on (U1 V1)×(U2 V2).
The conclusion can be ob ained essen ially as in he p oo o Theo em 3.3, aking in o accoun Rema k 2.5.
Acknowledgmen s
Jo ge Rod ´ıguez–L´opez was pa ially suppo ed by Xun a de Galicia (Spain), p ojec ED431C 2019/02
and AEI, Spain and FEDER, g an PID2020-113275GB-I00. The au ho hanks he e e ee o use ul
commen s which led o he imp o emen o his pape and o he sugges ed addi ional e e ences.
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