Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 9, 1-11; http://www.math.u-szeged.hu/ejqtde/

Singularly perturbed semilinear Neumann problem with non-normally hyperbolic critical manifold^{∗}

Róbert Vrábel’

Abstract

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

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

satisfying Neumann boundary conditions and where critical manifold is not normally hyperbolic. Our analysis relies on the method upper and lower solutions. Key words and phrases: Singular perturbation, Neumann problem, Upper and lower solutions, Fredholm integral equations. 2010 Mathematics Subject Classification: 34K10, 34K26, 45B05

1 Introduction

We will consider the singularly perturbed Neumann problem

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

The qualitative behavior of the dynamical systems near a normally hyperbolic manifold of critical points is well known (Theorem on persistence of normally hyperbolic manifold, see [2, 3, 5, 9, 12], for reference). However, the framework of the geometric singular perturbation theory is not useful for the non-hyperbolic critical manifolds, i.e. when the characteristic roots of the linearization of (1.1) along a solution u of the reduced problem ku = f(t, u) lie on the imaginary axis.

The main result (Theorem 1) is the existence of a solution y_{ϵ}(t) for ϵ belonging to a non-resonant set and an estimate of the difference between the solution y_{ϵ}(t) and a solution u(t) of the reduced problem. It is accomplished by a construction of a lower and an upper solution for the corresponding boundary value problem.

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

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}(a) ≥ 0, α′_{ϵ}(b) ≤ 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}(a) ≤ 0, β′_{ϵ}(b) ≥ 0 for every t ∈ ⟨a, b⟩. Then

Lemma 1 ([1, 8]). 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_{δ}(u) = {(t, y)| a ≤ t ≤ b, |y − u(t)| < δ} , δ is a positive constant and u ∈ C^{2} is a solution of reduced problem ku = f(t, u). Let

v_{1,ϵ}(t) = |u′(a)| \begin{matrix}cos & [\sqrt{\frac{m}{ϵ}}(b − t)] \\ \sqrt{\frac{m}{ϵ}} & sin & [\sqrt{\frac{m}{ϵ}}(b − a)] \end{matrix}

and

v_{2,ϵ}(t) = − |u′(b)| \begin{matrix}cos & [\sqrt{\frac{m}{ϵ}}(t − a)] \\ \sqrt{\frac{m}{ϵ}} & sin & [\sqrt{\frac{m}{ϵ}}(b − a)] \end{matrix}

where m = k + w (for the constant w see Theorem 1 below).

Let

J_{n}(λ) = ⟨m (\frac{b − a}{(n + 1)π − λ})^{2} , m (\frac{b − a}{nπ + λ})^{2}⟩ , n = 0, 1, 2, …,

λ > 0 is an arbitrarily small, but fixed constant and

M = {⋃ J_{n}, n = 0, 1, 2, … } .

The function v_{1,ϵ}(t) satisfies:

1. ϵv′′_{1,ϵ} + mv_{1,ϵ} = 0

2. v′_{1,ϵ}(a) = |u′(a)| , v′_{1,ϵ}(b) = 0

3. v_{1,ϵ}(t) be periodic in the variable t with the period \frac{2π\sqrt{ϵ}}{\sqrt{m}} → 0

4. v_{1,ϵ_{n}}(t) converges uniformly to 0 for every sequence {ϵ_{n}}_{n=0}^{∞} such that ϵ_{n} ∈ J_{n} and |v_{1,ϵ_{n}}(t)| ≤ \frac{\sqrt{ϵ_{n}}}{\sqrt{m} sin λ}, t ∈ ⟨a, b⟩.

The function v_{2,ϵ}(t) satisfies:

1. ϵv′′_{2,ϵ} + mv_{2,ϵ} = 0

2. v′_{2,ϵ}(a) = 0, v′_{2,ϵ}(b) = |u′(b)|

3. v_{2,ϵ}(t) be periodic in the variable t with the period \frac{2π\sqrt{ϵ}}{\sqrt{m}} → 0

4. v_{2,ϵ_{n}}(t) converges uniformly to 0 for every sequence {ϵ_{n}}_{n=0}^{∞} such that ϵ_{n} ∈ J_{n} and |v_{2,ϵ_{n}}(t)| ≤ \frac{\sqrt{ϵ_{n}}}{\sqrt{m} sin λ}, t ∈ ⟨a, b⟩.

Denote ω_{0,ϵ}(t) = v_{2,ϵ}(t) − v_{1,ϵ}(t). Let ω_{1,ϵ,i}(t) be a solution of the linear problem

ϵy′′ + my = ±ϵu′′(t), i = α_{ϵ}, β_{ϵ}

with the Neumann boundary condition (1.2), where the sign + and − is considered for i = α_{ϵ} and i = β_{ϵ}, respectively. These solutions may be computed exactly

ω_{1,ϵ,i}(t) = \frac{cos [\sqrt{\frac{m}{ϵ}}(t − a)] ∫_{a}^{b} cos [\sqrt{\frac{m}{ϵ}}(b − s)] (±u′′(s)) ds}{\sqrt{\frac{m}{ϵ}} sin [\sqrt{\frac{m}{ϵ}}(b − a)]}

+ ∫_{a}^{t} \frac{sin [\sqrt{\frac{m}{ϵ}}(t − s)] (±u′′(s)) ds}{\sqrt{\frac{m}{ϵ}}}ds = O(ϵ), ϵ ∈ M.

Obviously, ω_{1,ϵ,α_{ϵ}}(t) = −ω_{1,ϵ,β_{ϵ}}(t) on ⟨a, b⟩.

Let r_{ϵ,i}(t) is a continuous solution of the Fredholm equation of the first kind

Γ(ϵ) ∫_{a}^{b} K_{ϵ}(t, s)r_{ϵ,i}(s)ds + Ω_{ϵ,i}(t) = z_{ϵ,i}(t), z_{ϵ,i}(t) ≥ 0 i = α_{ϵ}, β_{ϵ} (1.3)

where Γ(ϵ) = \frac{1}{\sqrt{\frac{m}{ϵ}} sin[\sqrt{\frac{m}{ϵ}} (b−a)]} · \frac{1}{ϵ}, Γ^{−1}(ϵ) = O(\sqrt{ϵ}), ϵ ∈ M,

Ω_{ϵ,i}(t) = ω_{0,ϵ}(t) + ω_{1,ϵ,i}(t)

and the kernel

K_{ϵ}(t, s) = \{ \begin{matrix}K_{1,ϵ}(t, s), & a ≤ s ≤ t ≤ b \\ K_{2,ϵ}(t, s), & a ≤ t ≤ s ≤ b, \end{matrix}

K_{1,ϵ}(t, s) = cos [\sqrt{\frac{m}{ϵ}}(t − a)] cos [\sqrt{\frac{m}{ϵ}}(b − s)] +

sin [\sqrt{\frac{m}{ϵ}}(b − a)] sin [\sqrt{\frac{m}{ϵ}}(t − s)]

K_{2,ϵ}(t, s) = cos [\sqrt{\frac{m}{ϵ}}(t − a)] cos [\sqrt{\frac{m}{ϵ}}(b − s)]

for ϵ ∈ M and a modulation function z_{ϵ,i}(t) is an appropriate continuous nonnegative function such that r_{ϵ,i}(t) ≤ 0.

This is an integral equation of the kernel K_{ϵ}(t, s) that is continuous on the square ⟨a, b⟩ × ⟨a, b⟩. The problem (1.3) is defined as ill-posed and, in general, may be described numerically with Tikhonov regularization ([6, 7, 10, 11]). By substituing z_{ϵ,i}(t) = r_{ϵ,i}(t) + z̃_{ϵ,i}(t), i = α_{ϵ}, β_{ϵ} into (1.3) we obtain

Γ(ϵ) ∫_{a}^{b} K_{ϵ}(t, s)r_{ϵ,i}(s)ds + Ω̃_{ϵ,i}(t) = r_{ϵ,i}(t), i = α_{ϵ}, β_{ϵ},

i.e. r_{ϵ,i}(t) is a solution of Fredholm integral equation of second kind

Γ(ϵ) ∫_{a}^{b} K_{ϵ}(t, s)y(s)ds + Ω̃_{ϵ,i}(t) = y(t), i = α_{ϵ}, β_{ϵ},  (1.4)

where Ω̃_{ϵ,i}(t) = Ω_{ϵ,i}(t) − z̃_{ϵ,i}(t) and z̃_{ϵ,i}(t) is an appropriate chosen function such that

z̃_{ϵ,i}(t) ≥ −r_{ϵ,i}(t),  (1.5)
r_{ϵ,i}(t) ≤ 0,  (1.6)

t ∈ ⟨a, b⟩, i = α_{ϵ}, β_{ϵ}.

The kernel K_{ϵ} is semiseparable ([4]), therefore the equation (1.4) can be rewritten as

y(t) = ∑_{k=1}^{3} A_{k,ϵ,a}(t) ∫_{a}^{t} B_{k,ϵ,a}(s)y(s)ds + A_{1,ϵ,b}(t) ∫_{t}^{b} B_{1,ϵ,b}(s)y(s)ds + Ω̃_{ϵ,i}(t)

where

A_{1,ϵ,a}(t) = Γ(ϵ) cos [\sqrt{\frac{m}{ϵ}}(t − a)]
A_{2,ϵ,a}(t) = Γ(ϵ) sin [\sqrt{\frac{m}{ϵ}}(b − a)] sin [\sqrt{\frac{m}{ϵ}}t]
A_{3,ϵ,a}(t) = −Γ(ϵ) sin [\sqrt{\frac{m}{ϵ}}(b − a)] cos [\sqrt{\frac{m}{ϵ}}t]
A_{1,ϵ,b}(t) = Γ(ϵ) cos [\sqrt{\frac{m}{ϵ}}(t − a)]
B_{1,ϵ,a}(s) = cos [\sqrt{\frac{m}{ϵ}}(b − s)]
B_{2,ϵ,a}(s) = cos [\sqrt{\frac{m}{ϵ}}s]
B_{3,ϵ,a}(s) = sin [\sqrt{\frac{m}{ϵ}}s]
B_{1,ϵ,b}(s) = cos [\sqrt{\frac{m}{ϵ}}(b − s)]

or

y(t) = ∑_{k=1}^{3} A_{k,ϵ,a}(t)X_{k,ϵ,a,i}(t) + A_{1,ϵ,b}(t)X_{1,ϵ,b,i}(t) + Ω̃_{ϵ,i}(t), i = α_{ϵ}, β_{ϵ} (1.7)

where

X_{k,ϵ,a,i}(t) = ∫_{a}^{t} B_{k,ϵ,a}(s)y(s)ds, X_{1,ϵ,b,i}(t) = ∫_{t}^{b} B_{1,ϵ,b}(s)y(s)ds, k = 1, 2, 3.

Multiply both sides of the integral equation (1.7) by B_{j,ϵ,a}(t) and integrate from a to t and by B_{1,ϵ,b}(t) and integrate from t to b, respectively. We obtain

X_{j,ϵ,a,i} = ∑_{k=1}^{3} ∫_{a}^{t} A_{k,ϵ,a}B_{j,ϵ,a}X_{k,ϵ,a,i}dt + ∫_{a}^{t} A_{1,ϵ,b}B_{1,ϵ,a}X_{1,ϵ,b,i}dt + ∫_{a}^{t} B_{j,ϵ,a}Ω̃_{ϵ,i}dt

X_{1,ϵ,b,i} = ∑_{k=1}^{3} ∫_{t}^{b} A_{k,ϵ,a}B_{1,ϵ,b}X_{k,ϵ,a,i}dt + ∫_{t}^{b} A_{1,ϵ,b}B_{1,ϵ,b}X_{1,ϵ,b,i}dt + ∫_{t}^{b} B_{1,ϵ,b}Ω̃_{ϵ,i}dt

j = 1, 2, 3, i = α_{ϵ}, β_{ϵ}.

Differentiating these equations and taking into consideration the definition of X_{j,ϵ,a}, X_{1,ϵ,b} we obtain the boundary value problem for the system of linear differential equations

X′_{j,ϵ,a,i} = ∑_{k=1}^{3} A_{k,ϵ,a}B_{j,ϵ,a}X_{k,ϵ,a,i} + A_{1,ϵ,b}B_{1,ϵ,a}X_{k,ϵ,b,i} + B_{j,ϵ,a}Ω̃_{ϵ,i} (1.8)

X′_{1,ϵ,b,i} = − ∑_{k=1}^{3} A_{k,ϵ,a}B_{1,ϵ,b}X_{k,ϵ,a,i} − A_{1,ϵ,b}B_{1,ϵ,b}X_{1,ϵ,b,i} − B_{1,ϵ,b}Ω̃_{ϵ,i} (1.9)

X_{j,ϵ,a,i}(a) = 0, X_{1,ϵ,b,i}(b) = 0  (1.10)

j = 1, 2, 3, i = α_{ϵ}, β_{ϵ} or in the block matrix notation

X′ = ( \begin{matrix}P_{1,ϵ}(t) & P_{3,ϵ}(t) \\ P_{2,ϵ}(t) & P_{4,ϵ}(t) \end{matrix} ) X + D_{ϵ,i}(t)

where

X = (X_{1,ϵ,a,i}(t), X_{2,ϵ,a,i}(t), X_{3,ϵ,a,i}(t), X_{1,ϵ,b,i}(t))^{T} ,

P_{1,ϵ}(t) = ( \begin{matrix}A_{1,ϵ,a}(t)B_{1,ϵ,a}(t) & A_{2,ϵ,a}(t)B_{1,ϵ,a}(t) & A_{3,ϵ,a}(t)B_{1,ϵ,a}(t) \\ A_{1,ϵ,a}(t)B_{2,ϵ,a}(t) & A_{2,ϵ,a}(t)B_{2,ϵ,a}(t) & A_{3,ϵ,a}(t)B_{2,ϵ,a}(t) \\ A_{1,ϵ,a}(t)B_{3,ϵ,a}(t) & A_{2,ϵ,a}(t)B_{3,ϵ,a}(t) & A_{3,ϵ,a}(t)B_{3,ϵ,a}(t) \end{matrix} ) ,

P_{2,ϵ}(t) = − ( A_{1,ϵ,a}(t)B_{1,ϵ,b}(t) A_{2,ϵ,a}(t)B_{1,ϵ,b}(t) A_{3,ϵ,a}(t)B_{1,ϵ,b}(t) ) ,

P_{3,ϵ}(t) = ( \begin{matrix}A_{1,ϵ,b}(t)B_{1,ϵ,a}(t) \\ A_{1,ϵ,b}(t)B_{2,ϵ,a}(t) \\ A_{1,ϵ,b}(t)B_{3,ϵ,a}(t) \end{matrix} ) , P_{4,ϵ}(t) = − (A_{1,ϵ,b}(t)B_{1,ϵ,b}(t))

and

D_{ϵ,i}(t) = Ω̃_{ϵ,i}(t) (B_{1,ϵ,a}(t), B_{2,ϵ,a}(t), B_{3,ϵ,a}(t), −B_{1,ϵ,b}(t))^{T} ,

i = α_{ϵ}, β_{ϵ}. Thus,

r_{ϵ,i}(t) = r_{ϵ,i} (z̃_{ϵ,i}(t))
= ∑_{k=1}^{3} A_{k,ϵ,a}(t)X_{k,ϵ,a,i}(t) + A_{1,ϵ,b}(t)X_{1,ϵ,b,i}(t) + Ω̃_{ϵ,i}(t) (1.11)

where X is a solution of the linear boundary value problem (1.8), (1.9), (1.10). The conditions (1.5), (1.6) we may write in the form

−z̃_{ϵ,i}(t) ≤ r_{ϵ,i}(t) ≤ 0, i = α_{ϵ}, β_{ϵ}  (1.12)

or

0 ≤ ∑_{k=1}^{3} A_{k,ϵ,a}(t)X_{k,ϵ,a,i}(t) + A_{1,ϵ,b}(t)X_{1,ϵ,b,i}(t) + Ω_{ϵ,i}(t) ≤ z̃_{ϵ,i}(t). (1.13)

Remark 1. The matrix

( \begin{matrix}P_{1,ϵ}(t) & P_{3,ϵ}(t) \\ P_{2,ϵ}(t) & P_{4,ϵ}(t) \end{matrix} )

of the system is periodic with period p tendings to 0 for ϵ → 0^{+}, ϵ ∈ M and using the Floquet theory, then the solution of the linear homogeneous system

X′ = ( \begin{matrix}P_{1,ϵ}(t) & P_{3,ϵ}(t) \\ P_{2,ϵ}(t) & P_{4,ϵ}(t) \end{matrix} ) X

can be written as X_{hom,ϵ}(t) = p_{ϵ}(t)e^{Θ_{ϵ}t} where p_{ϵ}(t) is a periodic function and a matrix Θ_{ϵ} is time independent. This fact is instructive for the numerical description and the computer simulation of the system (1.8), (1.9).

Remark 2. The condition (1.13) is the fundamental assumption for existence of the barrier functions α_{ϵ}, β_{ϵ} for proving Theorem 1.

Now let v_{c,ϵ,i}(t) be a solution of Neumann boundary value problem (1.2) for Diff. Eq.

ϵy′′ + my = r_{ϵ,i}(t), i = α_{ϵ}, β_{ϵ}  (1.14)

i.e.

v_{c,ϵ,i}(t) = \frac{cos [\sqrt{\frac{m}{ϵ}}(t − a)] ∫_{a}^{b} cos [\sqrt{\frac{m}{ϵ}}(b − s)] \frac{r_{ϵ,i}(s)}{ϵ}ds}{\sqrt{\frac{m}{ϵ}} sin [\sqrt{\frac{m}{ϵ}}(b − a)]}

+ ∫_{a}^{t} \frac{sin [\sqrt{\frac{m}{ϵ}}(t − s)] \frac{r_{ϵ,i}(s)}{ϵ}ds}{\sqrt{\frac{m}{ϵ}}}ds = O (r_{ϵ,i}(t)) , ϵ ∈ M.

As follows from (1.3), the functions v_{c,ϵ,i}(t) must appear in the region as illustrated in Figure 1.1.

Now we may state the main result of this article.

Figure 1.1: The region for v_{c,ϵ,i}(t)

2 Main result

Theorem 1. (A1) Let z̃_{ϵ,i}(t), ϵ ∈ (0, ϵ_{0}] ∩ M, i = α_{ϵ}, β_{ϵ} be the continuous functions such

that (1.13) holds. (A2) Let f ∈ C^{1}(D_{δ}(u)) satisfies the condition

|\frac{∂f(t, y)}{∂y}| ≤ w < k for every (t, y) ∈ D_{δ}(u)

(nonhyperbolicity condition)

where δ ≥ max {ω_{0,ϵ}(t) + ω_{1,ϵ,i}(t) + v_{c,ϵ,i}(t) : i = α_{ϵ}, β_{ϵ}; t ∈ ⟨a, b⟩; ϵ ∈ (0, ϵ_{0}] ∩ M} . Then the problem (1.1), (1.2) has for ϵ ∈ (0, ϵ_{0}] ∩ M a solution satisfying the inequality −ω_{0,ϵ}(t) − ω_{1,ϵ,α_{ϵ}}(t) − v_{c,ϵ,α_{ϵ}}(t) ≤ y_{ϵ}(t) − u(t) ≤ ω_{0,ϵ}(t) + ω_{1,ϵ,β_{ϵ}}(t) + v_{c,ϵ,β_{ϵ}}(t) on ⟨a, b⟩.

Proof. We define the lower solutions by

α_{ϵ}(t) = u(t) − (ω_{0,ϵ}(t) + ω_{1,ϵ,α_{ϵ}}(t) + v_{c,ϵ,α_{ϵ}}(t))

and the upper solutions by β_{ϵ}(t) = u(t) + (ω_{0,ϵ}(t) + ω_{1,ϵ,β_{ϵ}}(t) + v_{c,ϵ,β_{ϵ}}(t)) .

EJQTDE, 2010 No. 9, p. 7

After simple algebraic manipulation we obtain

ω_{0,ϵ}(t) + ω_{1,ϵ,i}(t) + v_{c,ϵ,i}(t) = z_{ϵ,i}(t) ≥ 0, i = α_{ϵ}, β_{ϵ}

on ⟨a, b⟩. The functions α_{ϵ}, β_{ϵ} satisfy the boundary conditions prescribed for the lower and upper solutions of (1.1), (1.2) and α_{ϵ}(t) ≤ β_{ϵ}(t) on ⟨a, b⟩. Now we show that

ϵα′′_{ϵ}(t) + kα_{ϵ}(t) ≥ f(t, α_{ϵ}(t))  (2.1)

and

ϵβ′′_{ϵ}(t) + kβ_{ϵ}(t) ≤ f(t, β_{ϵ}(t)).  (2.2)

Denote h(t, y) = f(t, y) − ky. From the assumption (A2) on the function f(t, y) we have

−m ≤ \frac{∂h(t, y)}{∂y} ≤ 2w − m < 0

in D_{δ}(u). By the Taylor theorem we obtain

ϵα′′_{ϵ}(t) − h(t, α_{ϵ}(t)) = ϵα′′_{ϵ}(t) − [h(t, α_{ϵ}(t)) − h(t, u(t))]
= ϵu′′(t) − ϵω0,ϵ^{′′}(t) − ϵω1,ϵ,α_{ϵ}^{′′}(t) − ϵvc,ϵ,α_{ϵ}^{′′}(t)
− \frac{∂h (t, θ_{ϵ}(t))}{∂y} (−ω_{0,ϵ}(t) − ω_{1,ϵ,α_{ϵ}}(t) − v_{c,ϵ,α_{ϵ}}(t))
≥ ϵu′′(t) − ϵω0,ϵ^{′′}(t) − ϵω1,ϵ,α_{ϵ}^{′′}(t) − ϵvc,ϵ,α_{ϵ}^{′′}(t)
+ (−m) (ω_{0,ϵ}(t) + ω_{1,ϵ,α_{ϵ}}(t) + v_{c,ϵ,α_{ϵ}}(t))
= −ϵvc,ϵ,α_{ϵ}^{′′}(t) − mv_{c,ϵ,α_{ϵ}}(t) = −r_{ϵ,α_{ϵ}}(t).

From the condition (1.6) is −r_{ϵ,α_{ϵ}}(t) ≥ 0 therefore ϵα′′_{ϵ}(t) − h(t, α_{ϵ}(t)) ≥ 0 on ⟨a, b⟩. The inequality for β_{ϵ}(t) :

h(t, β_{ϵ}(t)) − ϵβ′′_{ϵ}(t) = \frac{∂h (t, θ̃_{ϵ}(t))}{∂y} (ω_{0,ϵ}(t) + ω_{1,ϵ,β_{ϵ}}(t) + v_{c,ϵ,β_{ϵ}}(t))
− ϵu′′(t) − ϵω0,ϵ^{′′}(t) − ϵω1,ϵ,β_{ϵ}^{′′}(t) − ϵvc,ϵ,β_{ϵ}^{′′}(t)
≥ (−m) (ω_{0,ϵ}(t) + ω_{1,ϵ,β_{ϵ}}(t) + v_{c,ϵ,β_{ϵ}}(t))
− ϵu′′(t) − ϵω0,ϵ^{′′}(t) − ϵω1,ϵ,β_{ϵ}^{′′}(t) − ϵvc,ϵ,β_{ϵ}^{′′}(t)
= −ϵvc,ϵ,β_{ϵ}^{′′}(t) − mv_{c,ϵ,β_{ϵ}}(t) = −r_{ϵ,β_{ϵ}}(t) ≥ 0

where (t, θ_{ϵ}(t)) is a point between (t, α_{ϵ}(t)) and (t, u(t)), (t, θ_{ϵ}(t)) ∈ D_{δ}(u). Analogously, (t, θ̃_{ϵ}(t)) is a point between (t, u(t)) and (t, β_{ϵ}(t)), (t, θ̃_{ϵ}(t)) ∈ D_{δ}(u) for ϵ ∈ (0, ϵ_{0}] ∩ M. The existence of a solution for (1.1), (1.2) satisfying the inequality above follows from Lemma 1.

Remark 3. We note, that if there exists the solution of (1.3) such that r_{ϵ,i}(t) = O(ϵ^{ν}), ν > 0 then for every sequence{ϵ_{n}} , ϵ_{n} ∈ (0, ϵ_{0}] ∩ M, ϵ_{n} ∈ J_{n} we have

|y_{ϵ_{n}}(t) − u(t)| ≤ (|u′(a)| + |u′(b)|) O (\sqrt{ϵ_{n}}) + M_{u′′}O (ϵ_{n}) + O (ϵ_{n}^{ν}) ,

M_{u′′} = max {|u′′(t)| , t ∈ ⟨a, b⟩} on ⟨a, b⟩.

Remark 4. In the trivial case, when u(t) = c =const is ω_{0,ϵ}(t) = ω_{1,ϵ,i}(t) =^{id} 0,

r_{ϵ,i}(t) =^{id} 0, i = α_{ϵ}, β_{ϵ} and

|y_{ϵ}(t) − u(t)| ≤ 0

i.e. y_{ϵ}(t) = u(t) on ⟨a, b⟩.

Example 1. Consider nonlinear problem (1.1), (1.2) with f(t, y) = y^{2} + g(t), i.e.

ϵy′′ + ky = y^{2} + g(t), t ∈ ⟨a, b⟩, k > 0, 0 < ϵ << 1
y′(a) = 0, y′(b) = 0.

For 0 ≤ g(t) < \frac{k^{2}}{4} on ⟨a, b⟩ the solution

u(t) = \frac{1}{2} (k − \sqrt{k^{2} − 4g(t)})

of the reduced problem ku = u^{2} + g(t) satisfies the assumption (A2) of Theorem 1. Let z̃_{ϵ,i}(t), ϵ ∈ (0, ϵ_{0}] ∩ M, i = α_{ϵ}, β_{ϵ} are the functions satisfying (1.13) (the assumption (A1)).

Thus, according to Theorem 1 above, there is for ϵ ∈ (0, ϵ_{0}]∩M a solution y_{ϵ}(t) of the considered boundary value problem satisfying the inequality

−ω_{0,ϵ}(t) − ω_{1,ϵ,α_{ϵ}}(t) − v_{c,ϵ,α_{ϵ}}(t) ≤ y_{ϵ}(t) − u(t) ≤ ω_{0,ϵ}(t) + ω_{1,ϵ,β_{ϵ}}(t) + v_{c,ϵ,β_{ϵ}}(t)

on ⟨a, b⟩.

3 Generalization of the assumption (A1)

The assumption of nonnegativity of z_{ϵ,i}(t) in (1.3) and the condition (1.12) may be generalized in the following sense. Denote

I_{+,ϵ,i} = {t ∈ ⟨a, b⟩ : z_{ϵ,i}(t) ≥ 0} , i = α_{ϵ}, β_{ϵ}

and

I_{−,ϵ,i} = {t ∈ ⟨a, b⟩ : z_{ϵ,i}(t) ≤ 0} , i = α_{ϵ}, β_{ϵ}.

Let there exist the functions z̃_{ϵ,i}(t) such that

r_{ϵ,i}(t) ≤ 0 on I_{+,ϵ,i}, i = α_{ϵ}, β_{ϵ}  (3.1)

and

r_{ϵ,i}(t) ≤ 2wz_{ϵ,i}(t) on I_{−,ϵ,i}, i = α_{ϵ}, β_{ϵ}  (3.2)

and

v_{c,ϵ,α_{ϵ}}(t) + v_{c,ϵ,β_{ϵ}}(t) ≥ −2ω_{0,ϵ}(t) on I_{−,ϵ,α_{ϵ}} ∪ I_{−,ϵ,β_{ϵ}}  (3.3)

where r_{ϵ,i}(t) is from (1.11) and z_{ϵ,i}(t) = r_{ϵ,i}(t) + z̃_{ϵ,i}(t), i = α_{ϵ}, β_{ϵ}. Taking into consideration the fact that

ω_{0,ϵ}(t) + ω_{1,ϵ,i}(t) + v_{c,ϵ,i}(t) = z_{ϵ,i}(t) ≤ 0 on I_{−,ϵ,i}, i = α_{ϵ}, β_{ϵ}, (3.4)

for the required inequality (2.1) for α_{ϵ}(t) on the interval I_{−,ϵ,α_{ϵ}} (in the case of the inequality for β_{ϵ}(t) i.e. (2.2) on I_{−,ϵ,β_{ϵ}}, we proceed analogously) we obtain

ϵα′′_{ϵ}(t) − h(t, α_{ϵ}(t)) = ϵu′′(t) − ϵω0,ϵ^{′′}(t) − ϵω1,ϵ,α_{ϵ}^{′′}(t) − ϵvc,ϵ,α_{ϵ}^{′′}(t)
− \frac{∂h (t, θ_{ϵ}(t))}{∂y} (−ω_{0,ϵ}(t) − ω_{1,ϵ,α_{ϵ}}(t) − v_{c,ϵ,α_{ϵ}}(t))
≥ ϵu′′(t) − ϵω0,ϵ^{′′}(t) − ϵω1,ϵ,α_{ϵ}^{′′}(t) − ϵvc,ϵ,α_{ϵ}^{′′}(t)
+ (−m + 2w) (ω_{0,ϵ}(t) + ω_{1,ϵ,α_{ϵ}}(t) + v_{c,ϵ,α_{ϵ}}(t))
= −r_{ϵ,α_{ϵ}}(t) + 2w (ω_{0,ϵ}(t) + ω_{1,ϵ,α_{ϵ}}(t) + v_{c,ϵ,α_{ϵ}}(t)) .

From (3.2) and (3.4), −r_{ϵ,α_{ϵ}}(t) + 2w (ω_{0,ϵ}(t) + ω_{1,ϵ,α_{ϵ}}(t) + v_{c,ϵ,α_{ϵ}}(t)) ≥ 0 for t ∈ I_{−,ϵ,α_{ϵ}}. The condition (3.3) guarantees that α_{ϵ}(t) ≤ β_{ϵ}(t) on ⟨a, b⟩. Hence, Theorem 1 holds.

From (3.2), we get

(1 − 2w)r_{ϵ,i}(t) ≤ 2wz̃_{ϵ,i}(t) ≤ −2wr_{ϵ,i}(t)  (3.5)

and we may generalize the assumption (A1) as follows.

(A1′) Let z̃_{ϵ,i}(t), i = α_{ϵ}, β_{ϵ} be the continuous functions such that

[(1.12)] ∨ [(3.5) ∧ (v_{c,ϵ,α_{ϵ}}(t) + v_{c,ϵ,β_{ϵ}}(t) ≥ −2ω_{0,ϵ}(t)) ]

on ⟨a, b⟩, ϵ ∈ (0, ϵ_{0}] ∩ M holds.

References

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(Received August 26, 2009)

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