Upper and lower solutions for first order problems with nonlinear boundary conditions

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. 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