Existence of solutions for first order ordinary. differential equations with nonlinear boundary. conditions

Transcription

1 Existence of solutions for first order ordinary differential equations with nonlinear boundary conditions Daniel Franco Departaento de Mateática Aplicada Universidad Nacional de Educación a Distancia Apartado de Correos Madrid 28080, Spain Juan J. Nieto Departaento de Análisis Mateático. Facultad de Mateáticas Universidad de Santiago de Copostela. Santiago de Copostela 15782, Spain Donal O Regan Departent of Matheatics. National University of Irel Galway, Irel Running Title: Nonlinear boundary conditions for ODEs. This is the preprint version of the paper published in Applied Matheatics Coputation 153 (2004)

2 Applied Matheatics Coputation 153 (2004) Abstract : We present an existence theore for nonlinear ordinary differential equations of first order with nonlinear boundary conditions. The result includes, for instance, the initial value proble, the final value proble, the antiperiodic boundary value proble. The ain novelty of the ethod lies in that it unifies different techniques for initial or boundary conditions. Keywords: Nonlinear ordinary differential equation, initial value proble, antiperiodic solutions, nonlinear boundary conditions, upper lower solutions. 1 Introduction We are interested in solutions of the nonlinear equation u (t) = F (t, u(t)), t I = [0, T ], T > 0 (1) satisfying the condition g(u(0), u(t )) = 0, (2) where F : [0, T ] R R g : R 2 R are continuous functions. If g(x, y) = x c with c R, then (2) is the following initial condition u(0) = c. (3) Siilarly, if g(x, y) = y c, then (2) is the final condition u(t ) = c. (4) Finally, the antiperiodic boundary condition u(0) = u(t ). (5) corresponds to the case g(x, y) = x + y.

3 Applied Matheatics Coputation 153 (2004) Initial value probles are well-known there is no doubt about their iportance. On the other h, anti-periodic solutions are perhaps not as widely discussed but there are soe papers on this subject: First order ordinary differential equations [1, 2, 3, 4, 5, 6], second order ordinary differential equations [7, 8], partial differential equations abstract differential equations [9, 10, 11, 12], antiperiodic trigonoetric polynoials [13], anti-periodic wavelets [14] periodic quasiwavelets [15], control probles [16]. In physics, antiperiodic boundary conditions are considered, for instance, in [17, 18, 19, 20]. Soe other physics works are listed in [3]. Also, extending by syetry the nonlinear function F to [ T, T ] R, any T antiperiodic solution of (1) (5) provides a solution of equation (1) satisfying the 2T periodic boundary condition u( T ) = u(t ). It is possible to define the concept of lower solution upper solution for equation (1) as follows. Definition 1 We say that a function α C 1 (I) is a subsolution of equation (1) if α (t) F (t, α(t)), t I. (6) Analogously, we say that β C 1 (I) is a supersolution of (1) if β (t) F (t, β(t)), t I. (7) In what follows we shall assue that α(t) β(t), t I, (8) define the set [α, β] = {v C(I) : α(t) v(t) β(t), t I }.

4 Applied Matheatics Coputation 153 (2004) Of course, to obtain a solution satisfying soe initial or boundary condition lying between a subsolution a supersolution we need additional conditions. For instance, in the case of initial conditions (3), one requires α(0) c β(0), (9) we have the following well-known existence result. Theore 1 (Th , [21]) Suppose that α, β C 1 (I) are subsolution supersolution of (1) respectively satisfying (8) (9). Then there exists a solution u C 1 (I) of the initial value proble (1)-(3) such that u [α, β]. Moreover, if there exists M > 0 such that F (t, v) + Mv F (t, u) + Mu, t I, α(t) v u β(t), (10) then there exist inial axial solutions of the initial proble in [α, β]. However, the situation is totally different for the antiperiodic boundary conditions (5) the requireent is now (see [3]) α(0) β(t ), β(0) α(t ). (11) The purpose of this paper is to present a new existence result for the equation (1) with the nonlinear boundary condition (2) that includes, aong others, the case of initial value condition (3), the final condition (4) the antiperiodic boundary condition (5). To this end, we introduce a new concept of coupled lower upper solutions that allow us to obtain a pair of quasisolutions in [α, β]. Then, under appropriate hypotheses, we show that there exists a solution in the sector [α, β]. We point out that our ethod, being new, unifies the treatent of different probles. 2 Coupled Lower Upper Solutions To cover different possibilities for the nonlinear boundary function g we introduce the following concept.

5 Applied Matheatics Coputation 153 (2004) Definition 2 We say that α, β C 1 (I) are coupled lower upper solutions for the proble (1)-(2) if α is a subsolution β a supersolution for the equation (1), condition (8) holds, g(α(0), β(t )) 0 g(β(0), α(t )). (12) Definition 3 We say that v, w C 1 (I) are coupled quasisolutions for the proble (1)-(2) if v w are solutions of the equation (1), α(t) v(t) w(t) β(t), t I (13) g(v(0), w(t )) = 0 = g(w(0), v(t )). (14) Theore 2 Assue that α, β are coupled lower upper solutions for the proble (1)-(2) such that (10) holds. In addition, suppose that there exists > 0 such that for any x, x [α(0), β(0)] with x x y, y [α(t ), β(t )] with y y one has that g(x, y) x g(x, y) x, (15) g(x, y) g(x, y ). (16) Then there exist v, w coupled quasisolutions of the proble (1)-(2). Proof For η, ξ [α, β], consider the functions ϕ(t) = ax {η(t), ξ(t)} φ(t) = in {η(t), ξ(t)} the initial value probles v (t) + Mv(t) = F (t, φ(t)) + Mφ(t), t I ; v(0) = φ(0) g(φ(0), ϕ(t )), (17) w g(ϕ(0), φ(t )) (t)+mw(t) = F (t, ϕ(t))+mϕ(t), t I ; w(0) = ϕ(0). (18) These probles have a unique solution since they are linear. Therefore, we can define the operator B : [α, β] [α, β] C(I) C(I) C(I) C(I), B(η, ξ) = (v, w), (19)

6 Applied Matheatics Coputation 153 (2004) where v, w are the solutions of (17)-(18). It is easy to see that B is copact by a direct application of Ascoli-Arzelà s theore. We now show that (13) is valid. Indeed, let z = v α. Taking into account (10), we have for any t I that z (t) + Mz(t) F (t, φ(t)) + Mφ(t) F (t, α(t)) Mα(t) 0. Now, using conditions (12), (15) (16) we get z(0) = φ(0) g(φ(0), ϕ(t )) α(0) φ(0) α(0) g(φ(0), β(t )) g(α(0), β(t )) g(φ(0), β(t )) g(α(0), β(t )) φ(0) α(0) + [φ(0) α(0)] φ(0) α(0) + = 0. Thus, we conclude that z 0 on I, α v on I. Analogously, one can show that v β on I that w [α, β]. To show that v w on I, set z = w v so that = z (t) + Mz(t) = F (t, ϕ(t)) + Mϕ(t) F (t, φ(t)) Mφ(t) 0, t I, g(φ(0), ϕ(t )) g(ϕ(0), φ(t )) z(0) = ϕ(0) φ(0) + = g(φ(0), ϕ(t )) g(φ(0), φ(t )) + g(φ(0), φ(t )) g(ϕ(0), φ(t )) ϕ(0) φ(0) + ϕ(0) φ(0) + [ϕ(0) φ(0)] + g(φ(0), ϕ(t )) g(φ(0), φ(t )) This allows us to conclude that z 0 on I that v w on I. Hence, B : [α, β] [α, β] [α, β] [α, β] 0. is continuous copact, by Schauder s fixed point theore, B has a fixed point, i.e., there exist (v, w) [α, β] [α, β] such that B(v, w) = (v, w)

7 Applied Matheatics Coputation 153 (2004) v w (by an arguent siilar to the one above). Now, by (19) we have that v, w satisfy v (t) + Mv(t) = F (t, v(t)) + Mv(t), t I ; v(0) = v(0) w (t) + Mw(t) = F (t, w(t)) + Mw(t), t I ; w(0) = w(0) g(v(0), w(t )) g(w(0), v(t )) As a result, v w are coupled quasisolutions of (1)-(2). This concludes the proof.,. If v = w, then u = v = w is a solution of the proble (1)-(2). Condition (15) eans that g(x, y) x is onotone nonincreasing in x [α(0), β(0)]. The condition (16) eans g is nondecreasing in the second variable. In the case that g is decreasing in y, we could change g by g obtaining the sae boundary condition (16) would be valid. This is the case of the periodic boundary conditions u(0) = u(t ), (20) corresponding to g(x, y) = x y. This does not verify (16) one would be tepted to write g(x, y) = x + y, but unfortunately we have probles as we will see in section 6. We also observe that, for exaple, the function g corresponding to the case of initial condition, antiperiodic boundary conditions satisfy (15) (16). 3 Initial Final Value Probles Now consider the following condition of initial type h(u(0)) = 0 (21) where h : R R is continuous. Of course, when h(x) = x c we have the initial condition (3).

8 Applied Matheatics Coputation 153 (2004) Corollary 1 Suppose that α, β are coupled lower upper solutions of (1)- (21) such that (10) holds. Then there exists at least one solution between the lower the upper solution. Proof By Theore 2, we have 0 = g(v(0), w(t )) = h(v(0)), 0 = g(w(0), v(t )) = h(w(0)). As a result, v, w are solutions of equation (1) satisfying h(v(0)) = h(w(0)) = 0. For the condition we obtain a siilar result. h(u(t )) = 0 (22) Corollary 2 Suppose that α, β are coupled lower upper solutions of (1)- (22) such that (10) holds. Then there exists at least one solution between the α β. Proof We now have that 0 = g(v(0), w(t )) = h(w(t )) 0 = g(w(0), v(t )) = h(v(t )). Consequently, h(w(t )) = 0 = h(v(t )). 4 AntiPeriodic Boundary Value Proble For the antiperiodic proble the situation is different. Theore 3 Suppose that α, β are coupled lower upper solutions of (1)- (5) such that (10) holds. Moreover, suppose that for U, V [α, β] with U(t) < V (t), t I, we have that T 0 F (s, V (s)) F (s, U(s)) V (s) U(s) ds 0. (23) Then there exists at least one solution between the lower the upper solution.

9 Applied Matheatics Coputation 153 (2004) Proof Let z = w v note z 0 on I. Then, we obtain that z(0) = z(t ), for t I we have that z (t) = F (t, w(t)) F (t, v(t)) Mz(t). As a result, d dt [emt z(t)] 0, t I. (24) If z(t) > 0 for each t I then z (t) = ζ(t)z(t), t I, where, obviously, Therefore, ζ(t) = F (t, w(t)) F (t, v(t)) z(t) t z(t) = z(0) e ζ(s) ds 0, T ζ(s) ds z(0) = z(t ) = z(0)e 0. which contradicts (23). Hence, we can suppose that there exists t 0 I with z(t 0 ) = 0. This iplies (see (24)) that z(t) 0, t [0, t 0 ]. In particular, z(0) = 0 z(t ) = 0. As a result fro (24) we have that z(t) 0, t I, w = v is a solution. Reark: Note that if F is a C 1 function, then (23) is equivalent to T 0 F (s, ζ(s)) ds 0, u for any ζ [α, β]. This is the usual condition in the literature (see forula (3.5) in [2]). 5 General Case We now consider the general nonlinear boundary condition (2).

10 Applied Matheatics Coputation 153 (2004) Theore 4 Let α, β be coupled lower upper solutions of (1)-(2) respectively such that (10), (15) (16) hold. Moreover, suppose that there exist nonnegative constants, such that for every x, x [α(0), β(0)], y, y [α(t ), β(t )], x < x, y < y the following growth conditions are satisfied g(x, y) g(x, y) x x, 0 g(x, y ) g(x, y) y y for U, V [α, β] with U(t) < V (t), t I, we have that, (25) T c c F (s,v (s)) F (s,u(s)) e 0 V (s) U(s) ds, (26) for any c [, ], c [0, ]. Then there exists at least one solution in the sector [α, β]. Proof Let z = w v. Then, 0 = g(v(0), w(t )) = g(w(0), v(t )), we get g(v(0), w(t )) g(v(0), v(t )) = g(w(0), v(t )) g(v(0), v(t )). But now, using (25), we can write, g(w(0), v(t )) g(v(0), v(t )) = cz(0), g(v(0), w(t )) g(v(0), v(t )) = c z(t ), for soe c [, ], c [0, ]. Hence, cz(0) = g(v(0), v(t )) = c z(t ). As in the proofs of the previous Theores, we have that d dt [emt z(t)] 0, t I. Again, as in the proof of Theore 3, suppose that z(t) > 0, t I. Then, t z(t) = z(0)e ζ(s) ds F (t, w(t)) F (t, v(t)) 0, ζ(s) =, t I, z(t)

11 Applied Matheatics Coputation 153 (2004) we get the contradiction T cz(0) = c z(t ) = c z(0)e ζ(s) ds 0. As a result, there exists at least one point t 0 I such that z(t 0 ) = 0. Fro this, we conclude that z 0 on I, w = v. Reark: Note that if g is a C 1 function, then (25) is valid. Reark: In this last result, we could consider other boundary conditions such as, for exaple, g(x, y) = x + y 2 g(x, y) = ax + by, a, b R,, or g(x, y) = x + h(y) where h : R R is a nondecreasing function. Reark: For the periodic boundary condition (20) we could still try to apply Theore 2 4 with g(x, y) = x + y, but we do not obtain any interesting result. Indeed, (8) (12) iply that α(0) = β(0), α(t ) = β(t ). This eans that α = β is already a solution. Acknowledgeent: D.G.E.S.I.C. (Spain), project PB First second authors were supported in part by

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