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

E extracta mathematicae Vol. 18, N´um. 2, 153 – 160 (2003) Upper and Lower Solutions for First Order Problems with Nonlinear Boundary Conditions Daniel Franco, Juan J. Nieto, Donal O’Regan Departamento de Matem´atica Aplicada, Universidad Nacional de Educaci´on a Distancia, Apartado de Correos 60149, 28080-Madrid, Spain Departamento de An´alisis Matem´atico, Facultad de Matem´aticas, Universidad de Santiago de Compostela, 15782-Santiago de Compostela, Spain Department of Mathematics, National University of Ireland, Galway, Ireland e-mail: dfranc[email protected]d.es, [email protected], donal.ore[email protected] AMS Subject Class. (2000): 34B15 Received March 6, 2003 1. Introduction We are interested in solutions of the nonlinear equation (1) u0(t) = f(t, u(t)) , t ∈I= [0, T], T > 0 satisfying the condition (2) g(u(0), u(T)) = 0 , where f:I×R→Rand g:R2→Rare continuous functions. If g(x, y) = x−cwith c∈R,then (2) is the initial condition (3) u(0) = c . Similarly, if g(x, y) = x−y , then (2) is the periodic condition (4) u(0) = u(T). Finally, the antiperiodic boundary condition (5) u(0) = −u(T). corresponds to the case g(x, y) = x+y . 153 154 d. franco, j.j. nieto, d. o’regan Nonlinear boundary conditions are discussed in several papers and the usual hypothesis is that g must be monotone nonincreasing in the second variable (see [1, 2, 3, 9] and the references therein) or nondecreasing (we considered this type of condition in [4, 5]). Here we discuss a more general situation since we only require monotonicity in the second variable and no monotonicity assumptions are imposed on the nonlinearity f. To achieve this we introduce a new definition of upper and lower solution. The results that we present are new and improve and complement those in [1, 2, 3, 4, 5, 6, 9, 11]. It is possible to define the concept of subsolution and supersolution for equation (1) as follows. Definition 1. We say that a function α∈C1(I) is a subsolution of equation (1) if (6) α0(t)≤f(t, α(t)) , t ∈I . Analogously, we say that β∈C1(I) is a supersolution of (1) if (7) β0(t)≥f(t, β(t)) , t ∈I . In what follows we shall assume that (8) α(t)≤β(t), t ∈I , or either (9) β(t)≤α(t), t ∈I . For u, v ∈C(I), u≤vwe define the set [u, v] = {w∈C(I) : u(t)≤w(t)≤v(t), t ∈I}. Of course, to obtain a solution satisfying some initial or boundary condition and lying between a subsolution and a supersolution we need additional conditions. For example, in the periodic case (4) it suffices that (see [8, 10]) α(0) ≤α(T), β(0) ≥β(T)(10) and in the antiperiodic case it suffices that (for more details see [6]) α(0) ≤ −β(T), β(0) ≥ −α(T).(11) nonlinear boundary conditions 155 The purpose of this paper is to present new existence results for equation (1) with the nonlinear boundary condition (2) that includes, among others, the case of the initial value condition (3), the periodic condition (4) and the antiperiodic boundary condition (5). To this end, we introduce a new concept of coupled lower and upper solutions that allow us to obtain a solution in the sector [α, β] or [β, α] . We point out that our method, being new, unifies the treatment of many different first order problems. We finish this introduction with a lemma Lemma 1. Let L:C(I)→C0(I)×Rbe defined by [Lu](t) = µu(t)−u(0) + λZt 0 u(s)ds, au(0) + bu(T)¶ where λ,aand bare real constants such that a+be−λT 6= 0, and here C0(I) = {u∈C(I) : u(0) = 0}. Then L−1exists and it is continuous and defined by [L−1(y, γ)](t) = e−λtA+y(t)−λZt 0 e−λ(t−s)y(s)ds with A=γ+bλ RT 0e−λ(T−s)y(s)ds −by(T) a+be−λT . 2. Coupled lower and upper solutions To cover different possibilities for the nonlinear boundary function gwe introduce the following concept. Definition 2. We say that α , β ∈C1(I) are coupled lower and upper solutions for the problem (1)-(2) if αis a subsolution and βa supersolution for the equation (1), condition (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. franco, j.j. nieto, d. o’regan We note that this definition generalizes the classical concepts. For instance, for the periodic case we obtain from (12) that (10) holds; and for the antiperiodic case, (12) implies (11). Also note that in some cases (periodic, initial) it is possible to define a lower or an upper solution independently but in others (antiperiodic) it is necessary to define both together. More precisely, if gis monotone decreasing in the second variable, Definition 2 allow us to consider a lower and an upper solution independently. Theorem 1. Assume that α , β are coupled lower and upper solutions for the problem (1)-(2). In addition, suppose that the functions hα(x) := g(α(0), x) hβ(x) := g(β(0), x) are monotone (either nonincreasing or nondecreasing) in [α(T), β(T)]. Then there exists at least one solution of the problem (1)-(2) between the lower and the upper solution. Proof. Let λ > 0 and consider the modified problem (13) u0(t) + λu(t) = F∗(t, u(t)) , t ∈I , u(0) = g∗(u(0), u(T)) , with F∗(t, u) =    f(t, β(t)) + λβ(t),if β(t)< u f(t, u) + λu, if α(t)≤u≤β(t) f(t, α(t)) + λα(t),if u < α(t), and g∗(x, y) = p(0, x)−g(p(0, x), p(T, y)) and p(t, x) = max {α(t),min {x, β(t)}} . Note that if uis a solution of (13) between αand β, then uis a solution of (1)-(2). We define the mappings L:C(I)→C0(I)×R and N:C(I)→C0(I)×R nonlinear boundary conditions 157 by [Lu](t) = µu(t)−u(0) + λZt 0 u(s)ds, u(0)¶ and [Nu](t) = µZt 0 F∗(t, u(t)), g∗(u(0), u(T))¶. Clearly Nis continuous and compact (by the Arzel´a-Ascoli theorem). Also from Lemma 1 with a= 1 and b= 0, L−1exists and is continuous. On the other hand, solving (13) is equivalent to find a fixed point of L−1N:C(I)→C(I). Now, Schauder’s fixed point theorem guarantees the existence of at least a fixed point since L−1Nis continuous and compact. It remains to show that usatisfies α(t)≤u(t)≤β(t), t ∈[0, T]. Assume that u−βattains a positive maximum on [0, T ] at s0. We shall consider three cases: Case 1. s0∈(0, T]. Then there exists τ∈(0, s0) such that 0≤u(t)−β(t)≤u(s0)−β(s0),for all t∈[τ, s0]. This yields a contradiction, since β(s0)−β(τ)≤u(s0)−u(τ) = Zs0 τ [f(s, β(s)) −λ(u(s)−β(s))]ds <Zs0 τ β0(s)ds =β(s0)−β(τ). Case 2. s0= 0 and hβmonotone nonincreasing. Then 0 < u(0) −β(0) and from (12) we obtain that g(α(0), α(T)) ≤0≤g(β(0), β(T)). Now we have 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. franco, j.j. nieto, d. o’regan which contradicts 0 < u(0) −β(0). Case 3. s0= 0 and hβmonotone nondecreasing. In this case we have 0 < u(0) −β(0) and g(α(0), β(T)) ≤0≤g(β(0), α(T)). Now we get the contradiction 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). Consequently, u(t)≤β(t) for all t∈I. Similarly, one can show that α≤u on I. 3. Coupled lower and upper solutions in reverse order Now we consider the case when (9) holds. Definition 3. We say that α , β ∈C1(I) are coupled lower and upper solutions for the problem (1)-(2) in reverse order if αis a subsolution and β a supersolution for the equation (1), condition (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) Theorem 2. Assume that α , β are coupled lower and upper solutions in reverse order for the problem (1)-(2). In addition, suppose that the functions hα(x) := g(x, α(T)) hβ(x) := g(x, β(T)) are monotone (either nonincreasing or nondecreasing) in [β(0), α(0)]. Then there exists at least one solution of the problem (1)-(2) in [β, α]. Proof. Let λ > 0 and consider the modified problem u0(t)−λu(t) = F∗(t, u(t)) , t ∈I , u(T) = g∗(u(0), u(T)) , nonlinear boundary conditions 159 with F∗(t, u) =    f(t, β(t)) −λβ(t),if β(t)< u f(t, u)−λu, if β(t)≤u≤α(t) f(t, α(t)) −λα(t),if u > α(t), and g∗(x, y) = p(T, x) + g(p(0, x), p(T, y)) and p(t, x) = max {β(t),min {x, α(t)}} . Now the proof is analogous to the proof of Theorem 1 using Lemma 1 with a= 0 and b= 1. It is possible to replace fcontinuous by f L1–Carath´eodory, and (6) and (7) by the integral conditions in [7] and the results in this paper are also true. 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