Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 70, 1-4; http://www.math.u-szeged.hu/ejqtde/

Three point boundary value problem for singularly perturbed semilinear differential equations

Róbert Vrábel’

Abstract

In this paper, we investigate the problem of existence and asymptotic behavior of solutions for the nonlinear boundary value problem

ϵy′′ + ky = f(t, y), t ∈ ⟨a, b⟩, k < 0, 0 < ϵ << 1

satisfying three point boundary conditions. Our analysis relies on the method of lower and upper solutions.

Key words and phrases: Singular perturbation, Boundary value problem

Mathematics Subject Classification: 34B16

This research was supported by Slovak Grant Agency, Ministry of Education of Slovak Republic under grant number 1/0068/08.

1 Introduction

We will consider the three point problem

ϵy′′ + ky = f(t, y), t ∈ ⟨a, b⟩, k < 0, 0 < ϵ << 1  (1.1)
y′(a) = 0, y(b) − y(c) = 0, a < c < b.  (1.2)

This is a singularly perturbed problem because the order of differential equation drops when ϵ becomes zero. The situation in the present case is complicated by the fact that there is an inner point in the boundary conditions, in contrast to the ”standard” boundary conditions as the Dirichlet problem, Neumann problem, Robin problem, periodic boundary value problem ([2, 3]), for example.

We apply the method of upper and lower solutions and some estimates to prove the existence of a solution for problem (1.1), (1.2) which converges uniformly to the solution of reduced problem (i.e. if we let ϵ → 0^{+} in (1.1)) on every compact set of interval ⟨a, b) for ϵ → 0^{+}. As usual, we say that α_{ϵ} ∈ C^{2}(⟨a, b⟩) is a lower solution for problem (1.1), (1.2) if ϵα′′_{ϵ}(t) + kα_{ϵ}(t) ≥ f(t, α_{ϵ}(t)) and α′_{ϵ}(0) = 0, α_{ϵ}(b) − α_{ϵ}(c) ≤ 0 for every t ∈ ⟨a, b⟩. An upper solution β_{ϵ} ∈ C^{2}(⟨a, b⟩) satisfies ϵβ′′_{ϵ}(t) + kβ_{ϵ}(t) ≤ f(t, β_{ϵ}(t)) and β′_{ϵ}(0) = 0, β_{ϵ}(b) − β_{ϵ}(c) ≥ 0 for every t ∈ ⟨a, b⟩.

Lemma 1 (cf. [1]). If α_{ϵ}, β_{ϵ} are lower and upper solutions for (1.1), (1.2) such that α_{ϵ} ≤ β_{ϵ}, then there exists solution y_{ϵ} of (1.1), (1.2) with α_{ϵ} ≤ y_{ϵ} ≤ β_{ϵ}.

Denote D(u) = {(t, y)| a ≤ t ≤ b, |y − u(t)| < d(t)} , where d(t) is the positive continuous function on ⟨a, b⟩ such that

d(t) = \{ \begin{matrix}δ & for a ≤ t ≤ b − δ \\ |u(b) − u(c)| + δ & for b − \frac{δ}{2} ≤ t ≤ b \end{matrix}

δ is a small positive constant and u ∈ C^{2} is a solution of reduced problem ku = f(t, u).

2 Existence and asymptotic behavior of solutions

Theorem 1. Let f ∈ C^{1}(D(u)) satisfies the condition

|\frac{∂f(t, y)}{∂y}| ≤ w < −k for every (t, y) ∈ D(u). (hyperbolicity condition)

Then there exists ϵ_{0} such that for every ϵ ∈ (0, ϵ_{0}⟩ the problem (1.1), (1.2) has a unique solution satisfying the inequality

v̂_{ϵ}(t) − Cϵ ≤ y_{ϵ}(t) − (u(t) + v_{ϵ}(t)) ≤ −v̂_{ϵ}(t) + C̃\sqrt{ϵ} for u′(a) ≤ 0  (2.1)

and

v̂_{ϵ}(t) − C̃\sqrt{ϵ} ≤ y_{ϵ}(t) − (u(t) + v_{ϵ}(t)) ≤ −v̂_{ϵ}(t) + Cϵ for u′(a) ≥ 0  (2.2)

on ⟨a, b⟩ where

v_{ϵ}(t) = u′(a) · \frac{e^{\sqrt{\frac{m}{ϵ}} (t−b)} − e^{\sqrt{\frac{m}{ϵ}} (b−t)} + e^{\sqrt{\frac{m}{ϵ}} (c−t)} − e^{\sqrt{\frac{m}{ϵ}} (t−c)}}{\sqrt{\frac{m}{ϵ}} (e^{\sqrt{\frac{m}{ϵ}} (a−c)} − e^{\sqrt{\frac{m}{ϵ}} (a−b)} + e^{\sqrt{\frac{m}{ϵ}} (c−a)} − e^{\sqrt{\frac{m}{ϵ}} (b−a)})},

v̂_{ϵ}(t) = |u(b) − u(c)| · \frac{e^{\sqrt{\frac{m}{ϵ}} (t−a)} + e^{\sqrt{\frac{m}{ϵ}} (a−t)}}{(e^{\sqrt{\frac{m}{ϵ}} (a−c)} − e^{\sqrt{\frac{m}{ϵ}} (a−b)} + e^{\sqrt{\frac{m}{ϵ}} (c−a)} − e^{\sqrt{\frac{m}{ϵ}} (b−a)})},

m = −k − w and C, C̃ be the positive constants.

Proof. The functions v_{ϵ}(t) and v̂_{ϵ}(t) on ⟨a, b⟩ satisfy:

1. ϵv′′_{ϵ} − mv_{ϵ} = 0, v′_{ϵ}(a) = −u′(a), v_{ϵ}(b) − v_{ϵ}(c) = 0, v_{ϵ} ≤ 0 for u′(a) ≤ 0 and v_{ϵ} ≥ 0 for u′(a) ≥ 0

2. ϵv̂′′_{ϵ} − mv̂_{ϵ} = 0, v̂′_{ϵ}(a) = 0, v̂_{ϵ}(b) − v̂_{ϵ}(c) = −|u(b) − u(c)|, v̂_{ϵ} ≤ 0.

For u′(a) ≤ 0 we define the lower solutions by

α_{ϵ}(t) = u(t) + v_{ϵ}(t) + v̂_{ϵ}(t) − Γ_{ϵ}

and the upper solutions by

β_{ϵ}(t) = u(t) + v_{ϵ}(t) − v̂_{ϵ}(t) + Γ̃_{ϵ}

( for u′(a) ≥ 0 we proceed analogously).

Here Γ_{ϵ} = \frac{ϵΔ}{m} and Γ̃_{ϵ} = \frac{\sqrt{ϵ}Δ̃}{m} where Δ, Δ̃ be the constants which shall be defined below. α ≤ β on ⟨a, b⟩ and satisfy the boundary conditions prescribed for the lower and upper solutions of (1.1), (1.2).

Now we show that ϵα′′_{ϵ}(t) + kα_{ϵ}(t) ≥ f(t, α_{ϵ}(t)) and ϵβ′′_{ϵ}(t) + kβ_{ϵ}(t) ≤ f(t, β_{ϵ}(t)). Denote h(t, y) = f(t, y) − ky. By the Taylor theorem we obtain

h(t, α_{ϵ}(t)) = h(t, α_{ϵ}(t)) − h(t, u(t)) = \frac{∂h(t, θ_{ϵ}(t))}{∂y}(v_{ϵ}(t) + v̂_{ϵ}(t) − Γ_{ϵ})

where (t, θ_{ϵ}(t)) is a point between (t, α_{ϵ}(t)) and (t, u(t)), and (t, θ_{ϵ}(t)) ∈ D(u) for sufficiently small ϵ. Hence

ϵα′′_{ϵ}(t) − h(t, α_{ϵ}(t)) ≥ ϵu′′ + ϵv′′_{ϵ}(t) + ϵv̂′′_{ϵ}(t) − m(v_{ϵ}(t) + v̂_{ϵ}(t) − Γ_{ϵ}) ≥ −ϵ|u′′| + ϵΔ.

If we choose a constant Δ such that Δ ≥ |u′′(t)|, t ∈ ⟨a, b⟩ then ϵα′′_{ϵ}(t) ≥ h(t, α_{ϵ}(t)) in ⟨a, b⟩. The inequality for β_{ϵ}(t)) :

h(t, β_{ϵ}(t)) − ϵβ′′_{ϵ}(t) = \frac{∂h(t, θ̃_{ϵ}(t))}{∂y}(v_{ϵ}(t) − v̂_{ϵ}(t) + Γ̃_{ϵ}) − ϵβ′′_{ϵ}(t) =

\frac{∂h(t, θ̃_{ϵ}(t))}{∂y}(v_{ϵ}(t) − v̂_{ϵ}(t) + Γ̃_{ϵ}) − ϵ(u′′ + v′′_{ϵ}(t) − v̂′′_{ϵ}(t)) ≥

\frac{∂h(t, θ̃_{ϵ}(t))}{∂y}v_{ϵ}(t) + mΓ̃_{ϵ} − ϵu′′ − ϵv′′_{ϵ}(t) =

(\frac{∂h(t, θ̃_{ϵ}(t))}{∂y} − m) v_{ϵ}(t) + mΓ̃_{ϵ} − ϵu′′.

Let L = max\{|u′′(t)| |t ∈ ⟨a, b⟩\} and denote ṽ_{ϵ} = \frac{v_{ϵ}}{\sqrt{ϵ}}. Then ϵβ′′_{ϵ}(t) ≤ h(t, β_{ϵ}(t)) if

mΓ̃_{ϵ} − ϵL ≥ (\frac{∂h(t, θ̃_{ϵ}(t))}{∂y} − m) |v_{ϵ}(t)|

i. e.

\sqrt{ϵ} (Δ̃ − \sqrt{ϵ}L) ≥ (\frac{∂h(t, θ̃_{ϵ}(t))}{∂y} − m) \sqrt{ϵ}|ṽ_{ϵ}(t)|

Δ̃ ≥ \sqrt{ϵ}L + (\frac{∂h(t, θ̃_{ϵ}(t))}{∂y} − m) |ṽ_{ϵ}(t)|.

Thus, from the inequalities m ≤ \frac{∂h(t,θ̃_{ϵ}(t))}{∂y} ≤ m + 2w in D(u) and v_{ϵ}(t) ≤ 0 follows that it is sufficient to choose a constant Δ̃ such that

Δ̃ ≥ \sqrt{ϵ}L + 2w \frac{|u′(a)|}{\sqrt{m}}.

The existence of a solution for (1.1), (1.2) satisfying the above inequality follows from Lemma 1.

Remark 1. Theorem 1 implies that y_{ϵ}(t) = u(t) + O(\sqrt{ϵ}) on every compact subset of ⟨a, b) and lim_{ϵ→0^{+}} y_{ϵ}(b) = u(c). Boundary layer effect occurs at the point b in the case,

when u(c) ≠ u(b).

References

[1] J. Mawhin: Points fixes, points critiques et problemes aux limites, Semin. Math. Sup. no. 92, Presses Univ. Montreal, 1985.

[2] R. Vrábel’: Asymptotic behavior of T-periodic solutions of singularly perturbed second-order differential equation, Mathematica Bohemica 121 (1996), 73-76.

[3] R. Vrábel’: Semilinear singular perturbation, Nonlinear analysis, TMA. - ISSN 0362-546X. - Vol. 25, No. 1 (1995), 17-26.

(Received December 17, 2008)

Dr. Róbert Vrábel’ Institute of Applied Informatics, Automation and Mathematics Faculty of Materials Science and Technology Hajdoczyho 1 917 24 Trnava SLOVAKIA

Email address: robert.vrabel@stuba.sk