AN ITERATION PROCEDURE

FOR NONLINEAR BOUNDARY CONDITIONS

Pablo Amster and María Cristina Mariani

FCEyN - Universidad de Buenos Aires

ABSTRACT

We study the general semilinear second order ordinary differential equation (1) with a nonlinear boundary condition (H). By Schauder’s Theorem we obtain solutions of the problem (1-H) under growth conditions on f and h. Moreover, we show that a solution may be constructed using an iterative procedure.

INTRODUCTION

We consider the general second order semilinear equation

(1)  u′′(t) = f(t, u, u′)

with nonlinear boundary conditions

(H)  u(0) = h_{0}(u(0), u′(0)),  u(T) = h_{T} (u(T), u′(T))

where f : [0, T] × IR^{2} → IR and h = (h_{0}, h_{T}) : IR^{2} → IR^{2} are continuous functions. In a previous paper [AMS], we have studied the particular case of the twopoint boundary value problem (i.e. the Dirichlet problem), which corresponds to constant h, proving that at least one solution exists if f grows linearly on (u, u′) , namely

|f(t, u, x)| ≤ c|(u, x)| + d for any u, x ∈ IR and t ∈ [0, T]

with slope c small enough. Furthermore, we have shown that a solution of (1-Dir) can be obtained constructively by a continuation-type method. Related cases and also periodic type and Sturm-Liouville conditions are studied in [AS], [B], [Br], [FM], [M], among other authors. Topological methods have been applied from the pioneering work of Severini [S]. Nonlinear boundary conditions are not so widely treated in the literature, although they appear frequently in different models. For example, in 1995 Rebelo and Sanchez considered the problem

u′′ + g(t, u) = 0, u(0) = f_{1}(−u′(0)), u(π) = f_{1}(u′(π))

with f_{1}^{−1} ∈ C(IR, IR) continuous and strictly increasing, which may be regarded as a mathematical model for the axial deformation of a nonlinear elastic beam [RS]. In this work, we show that some topological techniques may be applied also in this case. More precisely, we extend the results of [AMS] for the nonlinear case, showing that (1-H) can be solved under linear growth conditions for f and h. Moreover, we find solutions by a recursive method.

1. A FIXED POINT OPERATOR AND SOLVABILITY OF (1-H)

In order to find solutions of (1-H) we will define a fixed point operator on C^{1}[0, T] : recalling the Green function G associated to the second derivative ∂^{2} , i.e.

G(t, s) = \{ \begin{matrix}\frac{t(s−T)}{T}  if s ≥ t \\ \frac{(t−T)s}{T}  if s ≤ t \end{matrix}

we define K : C^{1}[0, T] → C^{1}[0, T] by

Ku(t) = φ_{u}(t) + ∫_{0}^{T} G(t, s)f(s, u, u′)ds

where

φ_{u}(t) = \frac{h_{T} (u(T), u′(T)) − h_{0}(u(0), u′(0))}{T}t + h_{0}(u(0), u′(0))

It is immediate to see that K is compact, and that any solution of (1-H) may be regarded as a fixed point of K. Moreover,

∥Ku∥_{∞} ≤ ∥φ_{u}∥_{∞} + sup_{0≤t≤T}{∥G(t, ·)∥_{p}}∥f(·, u, u′)∥_{p′}

and

∥(Ku)′∥_{∞} ≤ ∥φ′_{u}∥_{∞} + sup_{0≤t≤T}\{∥\frac{∂G}{∂t}(t, ·)∥_{p}\}∥f(·, u, u′)∥_{p′}

with \frac{1}{p} + \frac{1}{p′} = 1 . As

∥G(t, ·)∥_{p}^{p} = (\frac{t}{T} )^{p} \frac{(T − t)^{p}}{p + 1} T,

we conclude that

sup_{0≤t≤T}\{∥G(t, ·)∥_{p}\} = \frac{T}{4} (\frac{T}{p + 1})^{1/p}

In the same way,

sup_{0≤t≤T} ∥\{\frac{∂G}{∂t}(t, ·)∥_{p}\} = (\frac{T}{p + 1})^{1/p}

We will state an existence result for (1-H), assuming that f and h satisfy the linear-growth conditions

|h(x, y)| ≤ k_{h}|(x, y)| + l_{h}, |f(t, x, y)| ≤ k_{f}|(x, y)| + l_{f}

for some positive constants k_{h}, k_{f} , l_{h}, l_{f} . In this case, we will also see that p = 1 is the best choice for the previous apriori bounds for G.

Theorem 1 (1-H) admits a solution in any of the following cases:

i) T ≥ 4 and k_{h} + \frac{T^{2}}{8} k_{f} < 1 . ii) T ≤ 2 and \frac{2}{T} k_{h} + \frac{T}{2} k_{f} < 1 . iii) 2 < T < 4 and (k_{f}, k_{h}) ∈ C^{o} , where C ⊂ IR^{2} is the convex hull of the points (0, 0) , (0, 1) , (\frac{2}{T} , 0) , (\frac{4(T −2)}{T^{2}}, \frac{4−T}{2} ) . \underline{Proof} From the growth conditions we obtain:

∥f(t, u, u′)∥_{p′} ≤ T^{1/p′}(k_{f}∥u∥_{1,∞} + l_{f})

and

∥φ_{u}∥_{∞} = max{|h_{0}(u(0), u′(0))|, |h_{T} (u(T), u′(T))|} ≤ k_{h}∥u∥_{1,∞} + l_{h}

and

∥φ′_{u}∥_{∞} = \frac{|h_{T} (u(T), u′(T)) − h_{0}(u(0), u′(0))|}{T} ≤ \frac{2}{T} (k_{h}∥u∥_{1,∞} + l_{h})

Then, using the previous computations, we have:

∥Ku∥_{∞} ≤ (k_{h} + k_{f} \frac{T^{2}}{4(p + 1)^{1/p}})∥u∥_{1,∞} + c

and

∥(Ku)′∥_{∞} ≤ (\frac{2}{T} k_{h} + k_{f} \frac{T}{(p + 1)^{1/p}})∥u∥_{1,∞} + c

for some constant c. Let us suppose that

( ∗_{p})  k_{h} + k_{f} \frac{T^{2}}{4(p + 1)^{1/p}} < 1,  \frac{2}{T} k_{h} + k_{f} \frac{T}{(p + 1)^{1/p}} < 1

holds. Then, for large R we conclude that K(B_{R}(0)) ⊂ B_{R}(0) and by Schauder Theorem K has a fixed point. As (p + 1)^{1/p} is nonincreasing, its maximum value is achieved for p = 1 . Hence, it will suffice to prove that ( ∗_{p}) holds for p = 1 , namely

( ∗_{1} )  k_{h} + k_{f} \frac{T^{2}}{8} < 1,  \frac{2}{T} k_{h} + k_{f} \frac{T}{2} < 1

If T ≥ 4 and k_{h} + \frac{T^{2}}{8} k_{f} < 1 , then k_{h} + \frac{T^{2}}{4} k_{f} < 1 + \frac{T^{2}}{8} k_{f} < 2 − k_{h} < \frac{T}{2} , and ( ∗_{1} ) holds. If T ≤ 2 and k_{h} + \frac{T^{2}}{4} k_{f} < \frac{T}{2} , then k_{h} + \frac{T^{2}}{8} k_{f} < \frac{T}{2} ≤ 1 and ( ∗_{1} ) holds. Finally, if 2 < T < 4 we have that ( ∗_{1} ) holds if and only if (k_{f}, k_{h}) lies in the first quadrant below the lines

k_{h} + k_{f} \frac{T^{2}}{8} = 1, k_{h} + k_{f} \frac{T^{2}}{4} = \frac{T}{2}

2. AN ITERATIVE PROCEDURE FOR (1-H)

In this section we will embed the problem (1-H) in a family of problems

(1-H)_{λ} \{ \begin{matrix}u′′(t) = λf(t, u, u′) \\ u(0) = h_{0}(u(0), u′(0)), u(T) = h_{T} (u(T), u′(T)) \end{matrix}

The aim of the method is, starting at a solution u_{0} for λ_{0} , to define recursively a sequence which will converge in C^{1}([0, T]) to a solution of (1-H)_{λ_{0}+ε} for some step ε. We recall the following result from the theory of linear operators (see [AMS]):

Lemma 2 Let L be the linear operator given by Lu = u′′ + r(t)u′ + s(t)u, with r, s ∈ L^{∞}(0, T) , s ≤ 0 . Then there exists a constant c depending only on ∥r∥_{∞} and ∥s∥_{∞} such that

∥u∥_{2,2} ≤ c∥Lu∥_{2}

for any u ∈ H^{2} ∩ H_{0}^{1}(0, T) . Moreover, the problem

\{ \begin{matrix}Lu = ψ \\ u|_{∂I} = φ \end{matrix}

is uniquely solvable in H^{2}(0, T) for any ψ ∈ L^{2}(0, T) and any boundary Dirichlet data φ.

Assuming that u_{0} is a solution of (1-H)_{λ_{0}} we define the sequence {u_{n}} ⊂ H^{2}(0, T) by the problems:

\{ \begin{matrix}\frac{u′′_{n+1}}{λ_{0}+ε} = \frac{∂f}{∂u′} (t, u_{n}, u′_{n})(u_{n+1} − u_{n})′ + \frac{∂f}{∂u}(t, u_{n}, u′_{n})(u_{n+1} − u_{n}) + f(t, u_{n}, u′_{n}) \\ u_{n+1}(0) = h_{0}(u_{n}(0), u′_{n}(0)), u_{n+1}(T) = h_{T} (u_{n}(T), u′_{n}(T)) \end{matrix}

As a basic assumption, we will suppose that f is C^{2} with respect to u, u′, and \frac{∂f}{∂u}(t, u_{0}(t), u′_{0}(t)) ≥ 0 . We remark that if {u_{n}} is well defined and u_{n} → u for the C^{1} -norm, then u is a solution of (1-H)_{λ_{0}+ε}. Moreover, if \frac{∂f}{∂u}(t, u_{n}(t), u′_{n}(t)) ≥ 0 , from lemma 2 we conclude that u_{n+1} is well defined. In this case, for z_{n} = u_{n+1} − u_{n} we have:

L_{n}z_{n} := z′′_{n} − (λ_{0} + ε)[r_{n}(t)z′_{n} + s_{n}(t)z_{n}] = (λ_{0} + ε)R_{n}

with r_{n}(t) = \frac{∂f}{∂u′} (t, u_{n}, u′_{n}) , s_{n}(t) = \frac{∂f}{∂u}(t, u_{n}, u′_{n}) and R_{n} the Taylor remainder

R_{n}(t) = \frac{1}{2} [\frac{∂^{2}f}{∂u^{2}} (t, ξ)z_{n−1}^{2} + 2\frac{∂^{2}f}{∂u∂u′} (t, ξ)z_{n−1}z′_{n−1} + \frac{∂^{2}f}{∂u^{′2}} (t, ξ)(z′_{n−1})^{2}]

for some mean value ξ ∈ L^{∞}([0, T], IR^{2}) . Writing

φ_{n}(t) = m_{n}t + h_{0}(u_{n}(0), u′_{n}(0)) − h_{0}(u_{n−1}(0), u′_{n−1}(0))

where the slope m_{n} is given by

m_{n} = \frac{h_{T} |_{(u_{n−1}(T),u′_{n−1}(T))}^{(u_{n}(T),u′_{n}(T))} − h_{0}|_{(u_{n−1}(0),u′_{n−1}(0))}^{(u_{n}(0),u′_{n}(0))}}{T}

we obtain from lemma 2 and the imbedding H^{2}(0, T) ↪ C^{1}([0, T]) :

∥z_{n} − φ_{n}∥_{1,∞} ≤ c_{n}∥L_{n}(z_{n} − φ_{n})∥_{2} ≤ c_{n}(λ_{0} + ε) (∥R_{n}∥_{2} + ∥r_{n}φ′_{n} + s_{n}φ_{n}∥_{2})

for some constant c_{n} depending only on ∥r_{n}∥_{∞} and ∥s_{n}∥_{∞}. Thus, if u_{n−1}, u_{n} ∈ B_{R}(u_{0}) we have that

∥z_{n}∥_{1,∞} ≤ (1 + \overline{c})∥φ_{n}∥_{1,∞} + c∥z_{n−1}∥_{1,∞}^{2}

where the constants \overline{c}, c can be choosen depending only on R. Furthermore, if h is Lipschitz with constant k_{h} then ∥φ_{n}∥_{1,∞} ≤ k_{h} max\{1, \frac{2}{T} \}∥z_{n−1}∥_{1,∞}. Hence,

∥z_{n}∥_{1,∞} ≤ b∥z_{n−1}∥_{1,∞} + c∥z_{n−1}∥_{1,∞}^{2}

for b = (1 + \overline{c})k_{h} max\{1, \frac{2}{T} \}. Moreover,

z′′_{0} − (λ_{0} + ε) [\frac{∂f}{∂u′} (t, u_{0}, u′_{0})z′_{0} + \frac{∂f}{∂u} (t, u_{0}, u′_{0})z_{0}] = εf(t, u_{0}, u′_{0})

and as z_{0}|_{∂I} = 0 , we obtain:

∥z_{0}∥_{1,∞} ≤ εc_{0}∥f(·, u_{0}, u′_{0})∥_{2}

Thus we have:

Theorem 3 With the previous notations, let us assume that u_{0} is a solution of (1-H)_{λ_{0}} , and

i) \frac{∂f}{∂u}(t, x, y) ≥ 0 for (x, y) ∈ K_{R} = B_{R}(u_{0}([0, T]) × u′_{0}([0, T])) ii) h is Lipschitz on K_{R} with constant k_{h} < \frac{min\{1, \frac{T}{2} \}}{(1+\overline{c})} . Then the sequence {u_{n}} is well defined and converges in B_{R}(u_{0}) ⊂ C^{1}([0, T]) for any step ε such that

( ∗)  εc_{0}∥f(·, u_{0}, u′_{0})∥_{2} < \frac{R(1 − b)}{(1 + cR)}

\underline{Proof} From the previous computations and ( ∗), it follows that ∥z_{0}∥_{1,∞} ≤ R or, equivalently, u_{1} ∈ B_{R}(u_{0}) . This proves that u_{2} is well defined. Moreover, as

εc_{0}∥f(·, u_{0}, u′_{0})∥_{2} < R[1 − (b + cεc_{0}∥f(·, u_{0}, u′_{0})∥_{2})]

we conclude that b+c∥z_{0}∥_{1,∞} < 1 and ∥z_{1}∥_{1,∞} ≤ (b+c∥z_{0}∥_{1,∞})∥z_{0}∥_{1,∞} < R. Inductively, we see that the sequence {u_{n}} is well defined, and

∥z_{n}∥_{1,∞} ≤ (b + c∥z_{0}∥_{1,∞})^{n}∥z_{0}∥_{1,∞}

Hence

∑_{n=0}^{∞} ∥z_{n}∥_{1,∞} ≤ \frac{∥z_{0}∥_{1,∞}}{1 − (b + c∥z_{0}∥_{1,∞})} ≤ R,

which implies that {u_{n}} is a Cauchy sequence with lim_{n→∞} u_{n} = u ∈ B_{R}(u_{0}) .

Furthermore, if f and its first and second order derivatives with respect to u, u′ are bounded in [0, T] × IR^{2} , with \frac{∂f}{∂u} ≥ 0 the step ε can be choosen independently of u_{0} . Hence we have:

Corollary 4 Let us assume that u_{0} is a solution of (1-H)_{λ_{0}} , and

i) f , ∂f and ∂^{2}f are bounded, and \frac{∂f}{∂u} ≥ 0 in [0, T] × IR^{2} . ii) h is Lipschitz with constant k_{h} small enough. Then there exists a sequence λ_{0} < λ_{1} < ... < λ_{n} = 1 such that a solution of (1-H)_{λ_{j}} can be constructed recursively.

Remark: By Theorem 1, as |h(x, y)| ≤ k_{h}|(x, y)| + |h(0, 0)| we conclude that (1-H)_{0} is solvable for k_{h} < min\{1, \frac{T}{2} \}.

REFERENCES

[AMS] P.Amster, M.C.Mariani, J.Sabia: A boundary value problem for a semilinear second order ODE. To appear in Revista de la Unión Matemática Argentina.

[AS] S.Ahmad, J.Salazar: On existence of periodic solutions for nonlinearly perturbed conservative systems. Differential Equations, pp. 103-114, Academic Press, Orlando, FL (1980)

[B] P.W.Bates: Solutions of nonlinear elliptic systems with meshed spectra, J.Nonlinear Anal. 4 (1980), 1023-1030

[Br] K.J.Brown, S.S.Lin: Periodically perturbed conservative systems and a global inverse function theorem. J.Nonlinear Anal. 4 (1980), 193-201

[FM] A.Fonda, J.Mawhin: Iterative and Variational Methods for the Solvability of Some Semilinear Equations in Hilbert Spaces. Journal of Differential Equations, Vol 98, No. 2 (1992).

[M] J.Mawhin: Boundary value problems for nonlinear ordinary differential equations: from successive approximations to topology. Recherches de mathématique (1998), Inst. de Math Pure et Apliquée, Univ.Cath. de Louvain. Prepublication

[P] E.Picard: Sur l’application des méthodes d’approximations succesives à l’étude de certaines équations différentielles ordinaires, J.Math. Pures Appl. 9 (1893), 217-271.

[RS] C.Rebelo and L.Sanchez: Existence and multiplicity for an O.D.E. with nonlinear boundary conditions. Differential Equations and Dynamical Systems, Vol 3, Number 4, October 1995, pp.383-396.

[S] C.Severini: Sopra gli integrali delle equazione differenziali del secondo ordine con valori prestabiliti in due punti dati, Atti R. Acc. Torino 40 (1904-5), 1035-40.

P.Amster and M. C. Mariani Universidad de Buenos Aires CONICET

Address for correspondence: P.Amster and M. C. Mariani, Depto. de Matemática, Facultad de Ciencias Exactas y Naturales, UBA Pab. I, Ciudad Universitaria - (1428) Buenos Aires, Argentina

E-mail: pamster@dm.uba.ar - mcmarian@dm.uba.ar