E
ex ac a ma hema icae Vol. 18, N´um. 2, 153 – 160 (2003)
Uppe and Lowe Solu ions o Fi s O de P oblems
wi h Nonlinea Bounda y Condi ions
Daniel F anco, Juan J. Nie o, Donal O’Regan
Depa amen o de Ma em´a ica Aplicada, Uni e sidad Nacional de Educaci´on a Dis ancia,
Apa ado de Co eos 60149, 28080-Mad id, Spain
Depa amen o de An´alisis Ma em´a ico, Facul ad de Ma em´a icas, Uni e sidad de San iago
de Compos ela, 15782-San iago de Compos ela, Spain
Depa men o Ma hema ics, Na ional Uni e si y o I eland, Galway, I eland
e-mail: d anc[email p o ec ed]d.es, [email p o ec ed], donal.o e[email p o ec ed]
AMS Subjec Class. (2000): 34B15 Recei ed Ma ch 6, 2003
1. In oduc ion
We a e in e es ed in solu ions o he nonlinea equa ion
(1) u0( ) = ( , u( )) , ∈I= [0, T], T > 0
sa is ying he condi ion
(2) g(u(0), u(T)) = 0 ,
whe e :I×R→Rand g:R2→Ra e con inuous unc ions.
I g(x, y) = x−cwi h c∈R, hen (2) is he ini ial condi ion
(3) u(0) = c .
Simila ly, i g(x, y) = x−y , hen (2) is he pe iodic condi ion
(4) u(0) = u(T).
Finally, he an ipe iodic bounda y condi ion
(5) u(0) = −u(T).
co esponds o he case g(x, y) = x+y .
153
154 d. anco, j.j. nie o, d. o’ egan
Nonlinea bounda y condi ions a e discussed in se e al pape s and he
usual hypo hesis is ha g mus be mono one noninc easing in he second
a iable (see [1, 2, 3, 9] and he e e ences he ein) o nondec easing (we
conside ed his ype o condi ion in [4, 5]).
He e we discuss a mo e gene al si ua ion since we only equi e mono onici y
in he second a iable and no mono onici y assump ions a e imposed on he
nonlinea i y . To achie e his we in oduce a new de ini ion o uppe and
lowe solu ion.
The esul s ha we p esen a e new and imp o e and complemen hose
in [1, 2, 3, 4, 5, 6, 9, 11].
I is possible o de ine he concep o subsolu ion and supe solu ion o
equa ion (1) as ollows.
De ini ion 1. We say ha a unc ion α∈C1(I) is a subsolu ion o equa-
ion (1) i
(6) α0( )≤ ( , α( )) , ∈I .
Analogously, we say ha β∈C1(I) is a supe solu ion o (1) i
(7) β0( )≥ ( , β( )) , ∈I .
In wha ollows we shall assume ha
(8) α( )≤β( ), ∈I ,
o ei he
(9) β( )≤α( ), ∈I .
Fo u, ∈C(I), u≤ we de ine he se
[u, ] = {w∈C(I) : u( )≤w( )≤ ( ), ∈I}.
O cou se, o ob ain a solu ion sa is ying some ini ial o bounda y condi-
ion and lying be ween a subsolu ion and a supe solu ion we need addi ional
condi ions. Fo example, in he pe iodic case (4) i su ices ha (see [8, 10])
α(0) ≤α(T), β(0) ≥β(T)(10)
and in he an ipe iodic case i su ices ha ( o mo e de ails see [6])
α(0) ≤ −β(T), β(0) ≥ −α(T).(11)
nonlinea bounda y condi ions 155
The pu pose o his pape is o p esen new exis ence esul s o equa ion
(1) wi h he nonlinea bounda y condi ion (2) ha includes, among o he s,
he case o he ini ial alue condi ion (3), he pe iodic condi ion (4) and he
an ipe iodic bounda y condi ion (5). To his end, we in oduce a new concep
o coupled lowe and uppe solu ions ha allow us o ob ain a solu ion in he
sec o [α, β] o [β, α] . We poin ou ha ou me hod, being new, uni ies he
ea men o many di e en i s o de p oblems.
We inish his in oduc ion wi h a lemma
Lemma 1. Le L:C(I)→C0(I)×Rbe de ined by
[Lu]( ) = µu( )−u(0) + λZ
0
u(s)ds, au(0) + bu(T)¶
whe e λ,aand ba e eal cons an s such ha
a+be−λT 6= 0,
and he e
C0(I) = {u∈C(I) : u(0) = 0}.
Then L−1exis s and i is con inuous and de ined by
[L−1(y, γ)]( ) = e−λ A+y( )−λZ
0
e−λ( −s)y(s)ds
wi h
A=γ+bλ RT
0e−λ(T−s)y(s)ds −by(T)
a+be−λT .
2. Coupled lowe and uppe solu ions
To co e di e en possibili ies o he nonlinea bounda y unc ion gwe in-
oduce he ollowing concep .
De ini ion 2. We say ha α , β ∈C1(I) a e coupled lowe and uppe
solu ions o he p oblem (1)-(2) i αis a subsolu ion and βa supe solu ion
o he equa ion (1), condi ion (8) holds, and
max {g(α(0), α(T)), g(α(0), β(T))} ≤ 0
≤min {g(β(0), β(T)), g(β(0), α(T))}.
(12)
156 d. anco, j.j. nie o, d. o’ egan
We no e ha his de ini ion gene alizes he classical concep s. Fo ins-
ance, o he pe iodic case we ob ain om (12) ha (10) holds; and o he
an ipe iodic case, (12) implies (11).
Also no e ha in some cases (pe iodic, ini ial) i is possible o de ine a
lowe o an uppe solu ion independen ly bu in o he s (an ipe iodic) i is
necessa y o de ine bo h oge he . Mo e p ecisely, i gis mono one dec easing
in he second a iable, De ini ion 2 allow us o conside a lowe and an uppe
solu ion independen ly.
Theo em 1. Assume ha α , β a e coupled lowe and uppe solu ions o
he p oblem (1)-(2). In addi ion, suppose ha he unc ions
hα(x) := g(α(0), x)
hβ(x) := g(β(0), x)
a e mono one (ei he noninc easing o nondec easing) in [α(T), β(T)].
Then he e exis s a leas one solu ion o he p oblem (1)-(2) be ween he
lowe and he uppe solu ion.
P oo . Le λ > 0 and conside he modi ied p oblem
(13) u0( ) + λu( ) = F∗( , u( )) , ∈I ,
u(0) = g∗(u(0), u(T)) ,
wi h
F∗( , u) = 


( , β( )) + λβ( ),i β( )< u
( , u) + λu, i α( )≤u≤β( )
( , α( )) + λα( ),i u < α( ),
and
g∗(x, y) = p(0, x)−g(p(0, x), p(T, y))
and
p( , x) = max {α( ),min {x, β( )}} .
No e ha i uis a solu ion o (13) be ween αand β, hen uis a solu ion
o (1)-(2).
We de ine he mappings
L:C(I)→C0(I)×R
and
N:C(I)→C0(I)×R
nonlinea bounda y condi ions 157
by
[Lu]( ) = µu( )−u(0) + λZ
0
u(s)ds, u(0)¶
and
[Nu]( ) = µZ
0
F∗( , u( )), g∗(u(0), u(T))¶.
Clea ly Nis con inuous and compac (by he A zel´a-Ascoli heo em). Also
om Lemma 1 wi h a= 1 and b= 0, L−1exis s and is con inuous.
On he o he hand, sol ing (13) is equi alen o ind a ixed poin o
L−1N:C(I)→C(I).
Now, Schaude ’s ixed poin heo em gua an ees he exis ence o a leas a
ixed poin since L−1Nis con inuous and compac .
I emains o show ha usa is ies
α( )≤u( )≤β( ), ∈[0, T].
Assume ha u−βa ains a posi i e maximum on [0, T ] a s0. We shall
conside h ee cases:
Case 1. s0∈(0, T].
Then he e exis s τ∈(0, s0) such ha
0≤u( )−β( )≤u(s0)−β(s0), o all ∈[τ, s0].
This yields a con adic ion, since
β(s0)−β(τ)≤u(s0)−u(τ) = Zs0
τ
[ (s, β(s)) −λ(u(s)−β(s))]ds
<Zs0
τ
β0(s)ds =β(s0)−β(τ).
Case 2. s0= 0 and hβmono one noninc easing.
Then 0 < u(0) −β(0) and om (12) we ob ain ha
g(α(0), α(T)) ≤0≤g(β(0), β(T)).
Now we ha e
u(0) = g∗(u(0), u(T)) = β(0) −g(β(0), p(T, u(T)))
≤β(0) −g(β(0), β(T)) ≤β(0)

158 d. anco, j.j. nie o, d. o’ egan
which con adic s 0 < u(0) −β(0).
Case 3. s0= 0 and hβmono one nondec easing.
In his case we ha e 0 < u(0) −β(0) and
g(α(0), β(T)) ≤0≤g(β(0), α(T)).
Now we ge he con adic ion
u(0) = g∗(u(0), u(T)) = β(0) −g(β(0), p(T, u(T)))
≤β(0) −g(β(0), α(T)) ≤β(0).
Consequen ly, u( )≤β( ) o all ∈I. Simila ly, one can show ha α≤u
on I.
3. Coupled lowe and uppe solu ions in e e se o de
Now we conside he case when (9) holds.
De ini ion 3. We say ha α , β ∈C1(I) a e coupled lowe and uppe
solu ions o he p oblem (1)-(2) in e e se o de i αis a subsolu ion and β
a supe solu ion o he equa ion (1), condi ion (9) holds, and
max {g(α(0), α(T)), g(β(0), α(T))} ≤ 0
≤min {g(β(0), β(T)), g(α(0), β(T))}.
(14)
Theo em 2. Assume ha α , β a e coupled lowe and uppe solu ions in
e e se o de o he p oblem (1)-(2). In addi ion, suppose ha he unc ions
hα(x) := g(x, α(T))
hβ(x) := g(x, β(T))
a e mono one (ei he noninc easing o nondec easing) in [β(0), α(0)].
Then he e exis s a leas one solu ion o he p oblem (1)-(2) in [β, α].
P oo . Le λ > 0 and conside he modi ied p oblem
u0( )−λu( ) = F∗( , u( )) , ∈I ,
u(T) = g∗(u(0), u(T)) ,
nonlinea bounda y condi ions 159
wi h
F∗( , u) = 


( , β( )) −λβ( ),i β( )< u
( , u)−λu, i β( )≤u≤α( )
( , α( )) −λα( ),i u > α( ),
and
g∗(x, y) = p(T, x) + g(p(0, x), p(T, y))
and
p( , x) = max {β( ),min {x, α( )}} .
Now he p oo is analogous o he p oo o Theo em 1 using Lemma 1 wi h
a= 0 and b= 1.
I is possible o eplace con inuous by L1–Ca a h´eodo y, and (6) and
(7) by he in eg al condi ions in [7] and he esul s in his pape a e also ue.
Acknowledgemen s
Fi s and second au ho s we e suppo ed in pa by Minis e io de
Ciencia y Tecnolog´ıa (Spain) and FEDER, p ojec BFM2001-3884-C02-
01. The i s au ho was suppo ed in pa by “Plan de P omoci´on de la
In es igaci´on en la UNED”.
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