Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 41, 1-12; http://www.math.u-szeged.hu/ejqtde/

Existence Theorems for Second Order

Multi-Point Boundary Value Problems

James S.W. Wong Institute of Mathematical Research, Department of Mathematics, The University of Hong Kong Pokfulam Road, Hong Kong Email: jsww@chinneyhonkwok.com

ABSTRACT

We are interested in the existence of nontrivial solutions for the second order nonlinear differential equation (E): y′′(t) = f(t, y(t)) = 0, 0 < t < 1 subject to multipoint boundary conditions at t = 1 and either Dirichlet or Neumann conditions at t = 0. Assume that f(t, y) satisfies |f(t, y)| ≤ k(t)|y| + h(t) for non-negative functions k, h ∈ L^{1}(0, 1) for all (t, y) ∈ (0, 1) × ℝ and f(t, 0) ≢ 0 for t ∈ (0, 1). We show without any additional assumption on h(t) that if ∥k∥_{1} is sufficiently small where ∥ · ∥_{1} denotes the norm of L^{1}(0, 1) then there exists at least one non-trivial solution for such boundary value problems. Our results reduce to that of Sun and Liu [11] and Sun [10] for the three point problem with Neumann boundary condition at t = 0.

Key Words: Second Order nonlinear differential equations, Multi-point boundary

value problem, Sign-changing nonlinearities EJQTDE, 2010 No. 41, p. 1 1. \underline{Introduction}

We are interested in the existence of non-trivial solutions to the second order nonlinear differential equation:

y′′ + f(t, y) = 0, 0 < t < 1, (1.1)

where f(t, y) ∈ C((0, 1) × ℝ, ℝ) satisfies

|f(t, y)| ≤ k(t)|y| + h(t)  (1.2)

with k, h ∈ L^{1}(0, 1), subject to the following non-resonant boundary conditions: (BC1) y′(0) = 0, y(1) = ⟨α, y(η)⟩ + ⟨β, y′(η)⟩

(BC2) y(0) = 0, y′(1) = ⟨α, y(η)⟩ + ⟨β, y′(η)⟩
(BC3) y(0) = 0, y(1) = ⟨α, y(η)⟩ + ⟨β, y′(η)⟩

where

⟨α, y(η)⟩ = ∑_{i=1}^{m} α_{i}y(η_{i}); ⟨β, y′(η)⟩ = ∑_{i=1}^{m} β_{i}y′(η_{i}).

Here η = (η_{1}, η_{2}, ⋯, η_{m}); 0 < η_{1} < η_{2} < ⋯ < η_{m} < 1 and α_{i} ≥ 0, β_{i} ≥ 0 for all i = 0, 1, ⋯, m. Also, y(η) = (y(η_{1}), ⋯, y(η_{m})), y′(η) = (y′(η_{1}), ⋯, y′(η_{m})) are mvectors and ⟨α, y(η)⟩ denotes usual scalar product between two vectors α and y(η) in R^{m}.

Solvability of boundary value problems (1.1) subject to boundary conditions (BC1), (BC2), (BC3) with m = 1 has been studied by Gupta [3], Ma [7], Marano [8], Ren and Ge [9] where f(t, y) is allowed to change signs subject to condition (1.2). We refer to (1.1), (BC1); (1.1), (BC2); (1.1), (BC3) as (BVP1), (BVP2), (BVP3) respectively. In recent papers by Sun and Liu [11] and Sun [10], the three-point boundary value problem subject to special cases of (BC1), was studied where they applied the Leray-Schauder nonlinear alternative theorem to prove existence of non-trivial solutions.

In [11], Sun and Liu studied the three point boundary value problem, equation (1.1) subject to the boundary condition

y′(0) = 0, y(1) = αy(η),  (1.3)

EJQTDE, 2010 No. 41, p. 2 where 0 < η < 1 and α ≠ 1. Their main result is

Theorem A (Sun and Liu [14]) Suppose that f(t, 0) ≢ 0 in [0, 1] and there exists nonnegative functions k, h ∈ L^{1}(0, 1) such that (1.2) holds. If α ≠ 1, and

(1 + |\frac{1}{1 − α}|) ∫_{0}^{1}(1 − s)k(s)ds + |\frac{α}{1 − α}| ∫_{0}^{η}(η − s)k(s)ds < 1,  (1.4)

then the boundary value problem (1.1), (1.3) has a non-trivial solution.

In [10], Sun considered a similar boundary value problem also with Neumann boundary condition at t = 0, i.e. equation (1.1) subject to

y′(0) = 0, y(1) = βy′(η),  (1.5)

and proved

Theorem B (Sun [10]) Suppose that f(t, y) satisfies the same assumptions as in Theorem A. If k(t) satisfies

2 ∫_{0}^{1}(1 − s)k(s)ds + |β| ∫_{0}^{η} k(s)ds < 1,  (1.6)

then the boundary value problem (1.1), (1.5) has a nontrivial solution.

The boundary condition at t = 1 which includes both (1.3) and (1.5) can be written as

y′(0) = 0, y(1) = αy(η) + βy′(η).  (1.7)

Condition (1.7) is a special case of (BC1) with m = 1. In this note, we prove similar results for the more general m-point problems with boundary conditions (BC1), (BC2), (BC3). We show that the methodology given in [10], [11] is equally applicable to (BVP1), (BVP2), (BVP3). The fixed point theorem required is the following (See [2; p.61], [1; p27]) :

Theorem (Schauder Fixed Point Theorem) Let T : X → X be a completely continuous mapping on a Banach space X. Suppose that there exists r > 0 such that for all x ∈ X with ∥x∥ = r, T x ≠ λx if λ > 1, then T has a fixed point in X.

EJQTDE, 2010 No. 41, p. 3 2. \underline{Integral operators with Hammerstein kernels.}

We shall represent the solutions of (BVP1), (BVP2), (BVP3) as fixed point of integral equations with kernel functions incorporating the three boundary conditions (BC1), (BC2), (BC3). We define the mapping

A_{j}y(t) = G_{j}[y](t) + C_{j}t + D_{j}, j = 1, 2, 3  (2.1)

where

G_{j}[y](t) = ∫_{0}^{1} g_{j}(t, s)f(s, y(s))ds  (2.2)

and

g_{1}(t, s) = \{ \begin{matrix}1 − s, & 0 ≤ t ≤ s ≤ 1, \\ 1 − t, & 0 ≤ s ≤ t ≤ 1; \end{matrix}  (2.3)
g_{2}(t, s) = \{ \begin{matrix}s, & 0 ≤ s ≤ t ≤ 1 \\ t, & 0 ≤ t ≤ s ≤ 1; \end{matrix}  (2.4)
g_{3}(t, s) = \{ \begin{matrix}t(1 − s), & 0 ≤ t ≤ s ≤ 1 \\ s(1 − t) & 0 ≤ s ≤ t ≤ 1. \end{matrix}  (2.4)

The Green′s functions g_{j}(t, s), j = 1, 2, 3, given in (2.3), (2.4), (2.5), arise from two point homogeneous boundary conditions, i.e. associated with boundary conditions (BC1), (BC2), (BC3) with α = β = 0. Thus we have G′_{1}(0) = G_{1}(1) = 0, G_{2}(0) = G′_{2}(1) = 0 and G_{3}(0) = G_{3}(1) = 0 upon evaluating (2.2) at t = 0 and t = 1. The constants C_{j}, D_{j}, j = 1, 2, 3 are determined from the boundary conditions (BC1), (BC2), (BC3) to be

C_{1} = 0, D_{1} = \frac{1}{1 − \overline{α}} \{⟨α, G_{1}(η)⟩ + ⟨β, G′_{1}(η)⟩\}  (2.6)
C_{2} = \frac{1}{△} \{⟨α, G_{2}(η)⟩ + ⟨β, G′_{2}(η)⟩\} , D_{2} = 0  (2.7)
C_{3} = \frac{1}{△} \{⟨α, G_{3}(η)⟩ + ⟨β, G′_{3}(η)⟩\} , D_{3} = 0  (2.8)

where △ = 1 − ⟨α, η⟩ − \overline{β}, and \overline{α} = ∑_{i=1}^{m} α_{i}, \overline{β} = ∑_{i=1}^{m} β_{i}.

To apply Schauder fixed point theorem, we need to show that the operators A_{j} defined by (2.1) are completely continuous operators. Let U = {φ ∈ C[0, 1] : ∥φ∥ ≤ 1}. We need to show that the set A_{j}(U) ⊂ C[0, 1] is uniformly bounded and equicontinuous.

EJQTDE, 2010 No. 41, p. 4 Note firstly that sup_{0≤t≤1} |∫_{0}^{1} g_{j}(t, s)φ(s)ds| ≤ ∥φ∥ ≤ 1, and likewise constants C_{j}, D_{j} are bounded, so sup_{φ∈U} |A_{j}(φ)| is bounded by a constant independent of φ. To show that A_{j}φ(t) is equicontinuous, we observe

|A_{j}φ(t_{1}) − A_{j}φ(t_{2})| ≤ sup_{0≤s≤1} |g_{j}(t_{1}, s) − g_{j}(t_{2}, s)| + C_{j} |t_{1} − t_{2}|
≤ (1 + C_{j}) |t_{1} − t_{2}| .

This proves that A′_{j}s are completely continuous for j = 1, 2, 3.

Remark 2 The boundary conditions involving the derivative of a solution at some interior points in general give rise to kernels associated with the operators A_{j} in (2.1) which are discontinuous in two variables t, s. However, they are shown above to be completely continuous operators.

In [10], [11], the authors used the more customary integral operator I(t) defined by

I[y](t) = ∫_{0}^{t}(t − s)f(s, y(s))ds  (2.9)

instead of the Green’s operator G_{j}[y](t) given in (2.1).

Writing I(t) = I[y](t), G_{j}(t) = G_{j}[y](t) for short, we can relate G_{j}(t) with I(t) as follows:

G_{1}(t) = −I(t) + I(1)  (2.10)
G_{2}(t) = −I(t) + I′(1)t  (2.11)
G_{3}(t) = −I(t) + I(1)t  (2.12)

Using (2.10), (2.11), (2.12), we can rewrite the operator equations in (2.1) as follows:

A_{1}y(t) = −I(t) + \frac{1}{1 − \overline{α}} \{I(1) − ⟨α, I(η)⟩ − ⟨β, I′(η)⟩\}  (2.13)
A_{2}y(t) = −I(t) + \frac{t}{△} \{I′(1) − ⟨α, I(η)⟩ − ⟨β, I′(η)⟩\}  (2.14)
A_{3}y(t) = −I(t) + \frac{t}{△} \{I(1) − ⟨α, I(η)⟩ − ⟨β, I′(η)⟩\}  (2.15)

Results in [10], [11] can then be proved using the operator equation (2.13) for the (BVP1), i.e. (1.1), (BC1).

EJQTDE, 2010 No. 41, p. 5 3. \underline{Boundary value problem (1.1), (BC1)}

We now prove a result generalizing both Theorems A and B for the boundary value problem (BVP1).

Theorem 1 Suppose that f(t, 0) ≢ 0 in [0, 1] and condition (1.2) holds with k, h ∈ L^{1}(0, 1). If k(t) satisfies for \overline{α} ≠ 1 that

Λ_{1}(k) = Max_{0≤t≤1}G_{1}[k](t) + \frac{1}{|1 − \overline{α}|} \{⟨|α|, G_{1}[k](η)⟩ + ⟨|β|, G′_{1}[k](η)⟩\} < 1 (3.1)

where

\overline{α} = ∑_{i=1}^{m} α_{i}, |α| = (|α_{1}|, ⋯, |α_{m}|) , |β| = (|β_{1}|, ⋯, |β_{m}|)  (3.2)

and

G_{j}(t) = G_{j}[k](t) = ∫_{0}^{1} g_{j}(t, s)k(s)ds, j = 1, 2, 3  (3.3)

with g_{j}(t, s) as given in (2.3) (2.4), (2.5) then the (BVP1) has at least one non-trivial solution.

Proof. Since f(t, 0) ≢ 0, we note from (1.2) and (2.10) that

sup_{0≤t≤1} G_{1}[h](t) = ∫_{0}^{1}(1 − s)h(s)ds
≥ ∫_{0}^{1}(1 − s)|f(s, 0)|ds > 0,

so by (3.1), we have Λ(h) > 0. Condition (3.1) now permits us to define r > 0 by

r = Λ_{1}(h)(1 − Λ_{1}(k))^{−1} and Ω_{r} = {y(t) ∈ C[0, 1] : ∥y∥ < r}.

Now suppose that there exists y_{0} ∈ ∂Ω_{r}, i.e. ∥y_{0}∥ = r, and A_{1}y_{0} = λy_{0} for some λ > 1. Using (3.1), we obtain from (2.1), (2.2), (2.6) and (1.2) that the operator A_{1} satisfies

∥A_{1}y_{0}∥ ≤ Λ_{1}(k)∥y_{0}∥ + Λ_{1}(h),

or

λ∥y_{0}∥ ≤ Λ_{1}(k)∥y_{0}∥ + Λ_{1}(h).  (3.4)

EJQTDE, 2010 No. 41, p. 6 Substituting r = Λ_{1}(h)(1 − Λ_{1}(k))^{−1} for ∥y_{0}∥ in (3.4), we find λr ≤ r which contradicts the assumption that λ > 1. Thus by Schauder’s Fixed point theorem, A_{1} has a fixed point in \overline{Ω}_{r} which is not the identically zero function because of f(t, 0) ≢ 0. This completes the proof.

Theorem 2 Under the same assumptions as in Theorem 1, if k(t) satisfies for \overline{α} ≠ 1 that

Γ_{1}(k) = (+\frac{1}{|1 − \overline{α}|}) I[k](1) + \frac{1}{|1 − \overline{α}|} \{⟨|α|, I[k](η)⟩ + ⟨|β|, I′[k](η)⟩\} < 1  (3.5)

then the (BVP1) has at least one non-trivial solution where I[k](t) and I′[k](t) are defined like (2.9) by

I[k](t) = ∫_{0}^{1}(t − s)k(s)ds, I′[k](t) = ∫_{0}^{t} k(s)ds.  (3.6)

Proof. We use the integral representation (2.13) for the operator A_{1}. Since f(t, 0) ≢ 0 in [0, 1], we also have Γ_{1}(h) > 0 by (1.2). Using (3.5) we define the positive constant r_{1} > 0 by

r_{1} = Γ_{1}(h)(1 − Γ_{1}(k))^{−1}, Ω_{r_{1}} = {y ∈ C[0, 1] : ∥y∥ < r_{1}}  (3.7)

To apply the Schauder Fixed Point Theorem, we suppose that there exists \overline{y} ∈ ∂Ω_{r_{1}} = \{y ∈ \overline{Ω}_{r_{1}} : ∥y∥ = r_{1}\} such that A_{1}\overline{y} = \overline{λ}\overline{y} for some \overline{λ} > 1. Now apply (1.2), (3.5) to the integral representation given by (2.13), and obtain by (3.7)

λr_{1} = ∥A_{1}\overline{y}∥ ≤ Γ_{1}(k)∥\overline{y}∥ + Γ_{1}(h) ≤ Γ_{1}(h)(1 − Γ_{1}(k))^{−1} = r_{1}

which contradicts the assumption that λ > 1. Now Schauder′s Fixed point theroem shows that there exists ŷ ∈ \overline{Ω}_{r_{1}} such A_{1}ŷ = ŷ. Since f(t, 0) ≢ 0, so ŷ cannot be the identically zero solution. This complets of the proof.

Corollary 1 Suppose that f(t, y) satisfies the assumptions of Theorem 1. If k ∈ L^{1}(0, 1) satisfies either

Λ̂_{1}(k) = (1 + |\frac{α}{1 − α}|) ∫_{0}^{1}(1 − s)k(s)ds + \frac{|β|}{|1 − α|} ∫_{0}^{η}(s)ds < 1  (3.8)

EJQTDE, 2010 No. 41, p. 7 or

Γ̂_{1}(k) = (1 + |\frac{1}{1 − α}|) ∫_{0}^{1}(1 − s)k(s)ds + \frac{|α|}{|1 − α|} ∫_{0}^{η}(η − s)(s)ds
+ \frac{|β|}{|1 − α|} ∫_{0}^{η} k(s)ds < 1,  (3.9)

then the three-point boundary value problem (1.1), (1.7) has at least one non-trivial solution.

Proof. From (2.10), we have G_{1}[k](η) = −I[k](η) + I[k](1) so |α|G_{1}[k](η) ≤ |α|I[k](1). Using this in (3.1), we obtain (3.8). Next we note that (3.9) is simply (3.5) with m = 1. This completes the proof.

Remark 3 Condition (3.9) reduces to (1.4) when β = 0 and it becomes (1.6) when α = 0. Thus Corollary 1 includes both Theorem A and B. Condition (3.8) is sharper than condition (1.6) when α = 0 where the “2” can be replaced by “1”, so Corollary 1 improves upon Theorem B. When β = 0, conditions (3.8) and (3.9) are not strictly comparable because their values depend on α and η.

4. \underline{Boundary value problem (BVP2), (BVP3)}

We now use the integral representations (2.1), (2.2), (2.4) and (2.1), (2.2), (2.5) and state analogues of Theorems 1 and 2 for (BVP2),(BVP3).

Theorem 3 Suppose that f(t, 0) ≢ 0 in [0, 1] and condition (1.2) holds with k, h ∈ L^{1}(0, 1). If k(t) satisfies for △ = 1 − ⟨α, η⟩ + \overline{β} ≠ 0

Λ_{2}(k) = Max_{0≤t≤1} G_{2}[k](t) + \frac{1}{△}\{⟨|α|, G_{2}[k](η)⟩ + ⟨|β|, G′_{2}[k](η)⟩\} < 1,  (4.1)

where G_{2}[k](t) is given by (3.3), then the (BVP2) has at least one non-trivial solution.

Theorem 4 Under the same assumptions as in Theorem 3, if k(t) satisfies

Λ_{3}(k) = Max_{0≤t≤1} G_{3}[k](t) + \frac{1}{△}\{⟨|α|, G_{3}[k](η)⟩ + ⟨|β|, G′_{3}[k](η)|⟩\} < 1,  (4.2)

where G_{3}[k](t) is defined by (3.3), then the (BVP3) has at least one non-trivial solution. EJQTDE, 2010 No. 41, p. 8

Likewise we use representations (2.14), (2.15) for operators A_{2}, A_{3} in terms of I[y](t) as defined by (2.9) and can prove the following results for (BVP2), (BVP3).

Theorem 5 Under the same assumptions of Theorem 3, if k(t) satisfies

Γ_{2}(k) = I[k](1) + \frac{1}{△}\{I′[k](1) + ⟨|α|, I[k](η)⟩ + ⟨|β|, I′[k](η)⟩\} < 1,  (4.3)

then the boundary value problem (BVP2) has at least one non-trivial solution.

Theorem 6 Under the same assumptions of Theorem 5, if k(t) satisfies

Γ_{3}(k) = I[k] (1 + \frac{1}{△}) + \frac{1}{△}\{⟨|α|, I[k](η)⟩ + ⟨|β|, I′[k](η)⟩\} < 1,  (4.4)

then the boundary value problem (BVP3) has at least one non-trivial solution.

The proofs of Theorems 3, 4, 5, 6 are similar to those given for Theorem 1 and 2 and we shall not repeat them here.

Remark 4 Denote K_{1} = I[k](1), K_{2} = I′[k](1). We can give upper bounds of Γ_{1}(k), Γ_{2}(k), Γ_{3}(k) in terms of K_{1}, K_{2} as follows

Γ_{1}(k) ≤ K_{1} \{1 + \frac{1}{|1 − α|}(1 + |α̂|)\} + K_{2} \frac{|β̂|}{|1 − α|},

Γ_{2}(k) ≤ K_{1} (1 + \frac{1}{△}|α̂|) + K_{2} (1 + \frac{1}{△}|β̂|) ,

Γ_{3}(k) ≤ K_{1} \{1 + \frac{1}{△}(1 + |α̂|)\} + K_{2} \frac{|β̂|}{△},

where |α̂| = ∑_{i=1}^{m} |α_{i}|, |β̂| = ∑_{i=1}^{m} |β_{i}|. This provides a convenient method to establish existence of a non-trivial solution for (BVP1), (BVP2), (BVP3).

5. \underline{Discussion}

We illustrate our results with examples in three point boundary value problems and begin with two examples discussed in [10], [11].

\underline{Example 1} Consider the boundary value problem

(E1)\{ \begin{matrix}y′′ + c\sqrt{t}(1 + y^{4})^{−1}y^{3} − sin t = 0,  0 < t < 1 \\ y′(0) = 0,  y(1) = 2y(\frac{1}{2}), c > 0, \end{matrix}

EJQTDE, 2010 No. 41, p. 9 which was discussed in [11, Example 3] with c = 1 and was shown to possess at least one non-trivial solution. Here f(t, y) = c\sqrt{t}y^{3}(1 + y^{4})^{−1} so |f(t, y)| ≤ k(t)|x| + h(t) with k(t) = \frac{c}{2} \sqrt{t} and h(t) = sin t. Apply Corollary 1 with β = 0, α = 2 and η = \frac{1}{2}, we find c < 60(16 + 7\sqrt{2})^{−1}, so in particular (E1) has a non-trivial solution for c = 2.

\underline{Example 2} Consider the boundary value problem

(E2)\{ \begin{matrix}y′′ + (t − t^{2})|y| sin y − t^{2}y + t^{3} − 2 sin t = 0, 0 < t < 1, \\ y′(0) = 0, y(1) = αy(\frac{1}{2}), +βy′(\frac{1}{2}). \end{matrix}

This example was studied in [10; Example 4.1] with α = 0, β = 4. Here f(t, y) satisfies (1.2) with k(t) = t, h(t) = t^{3} + 2 sin t. Apply (3.9) in Corollary 1, we find |β| < 16/3 the same as from Theorem B. However, using (3.8) in Corollary 1 with α = 0, we obtain |β| < 20/3 which ensures the existence of a nontrivial solution of (E2). When β = 0, (3.8) requires

\frac{1}{6} (1 + \frac{|α|}{|1 − α|}) + \frac{1}{2} \frac{1}{|1 − α|} < 1.

Solving the above inequality, we require α ∉ [\frac{1}{3}, \frac{1}{2}] for the existence of a non-trivial solution of (E2).

\underline{Example 3} Consider the boundary value problem

(E3)\{ \begin{matrix}y′′ + \frac{σ}{\sqrt{t}}(|y| sin y + t^{2}) + \frac{cos t}{\sqrt{t}} = 0, 0 < t < 1, \\ y(0) = 0, y′(1) = \frac{3}{10}y(\frac{1}{3}) + \frac{1}{10}y′(\frac{1}{3}). \end{matrix}

where σ > 0 and (1.2) is satisfied with k(t) = σt^{− \frac{1}{2}} and h(t) = t^{− \frac{1}{2}} + σ. The boundary value problem is a special case of (BVP2) and we can apply Theorem 3 to compute Λ_{2}(k) as defined by (4.1). Here △ = 4/5.

Max_{0≤t≤1} G_{2}[k](t) = Max_{0≤t≤1} ∫_{0}^{1} g_{2}(t, s)k(s)ds ≤ σ ∫_{0}^{1} \sqrt{s}ds = \frac{2σ}{3} ,

G_{2}[k] (\frac{1}{3}) = (\frac{2}{3} − \frac{4}{9\sqrt{3}}) σ, G′_{2}[k] (\frac{1}{3}) = (2 − \frac{2}{\sqrt{3}}) σ,

hence Λ_{2}(k) ≤ (\frac{7}{6} − \frac{5\sqrt{3}}{36}) σ = 0.9261σ < 1. In particular when σ = 1, the boundary value problem (E3) has a non-trivial solution.

EJQTDE, 2010 No. 41, p. 10 \underline{Example 4} Consider the three point BVP

(E4)\{ \begin{matrix}y′′ + \frac{2ty^{2}e^{−y}}{t^{2}+y^{2}} + 3 sin^{2} t − cos e^{t} = 0, 0 < t < 1 \\ y(0) = 0, y(1) = 4y (\frac{1}{2}) + βy′ (\frac{1}{2}) . \end{matrix}

A similar equation in (E4) was discussed in [10; Example 4.5] as a special case of (BVP1). The boundary value problem (E4) is a special case of (BVP3). Here (1.2) is satisfied with k(t) ≡ 1, h(t) = 3 sin^{2} t + cos e^{t}. We now apply Theorem 4 and compute Λ_{3}(k) as given in (4.2). Observe that

Max_{0≤t≤1} G_{3}[k](t) : Max_{0≤t≤1} ∫_{0}^{1} g_{3}(t, s)k(s)ds = ∫_{0}^{1} s(1 − s)ds = \frac{1}{6},

G_{3}[k] (\frac{1}{2}) = \frac{5}{16}, G′_{3}[k] (\frac{1}{2}) = 0. Using these in (4.2), we find |\frac{1}{β+1} | < \frac{2}{3}, alternatively β ∉ [−\frac{5}{2}, \frac{1}{2}], which shows that the boundary value problem (E4) has a non-trivial solution, when β ≥ \frac{1}{2}.

We close our discussion with several additional remarks:

1. The condition that \overline{α} ≠ 1 for (BVP1) and ⟨α, η⟩ + \overline{β} ≠ 1 for (BVP2), (BVP3) are known as non-resonance conditions. These conditions ensure that the constants C_{j}, D_{j} in (2.6), (2.7), (2.8) can be determined by requiring A_{j}y(t), as defined by the operator equation (2.1) (2.2), to satisfy the boundary conditions (BC1), (BC2), (BC3).

2. Consider the simple three point boundary value problem

(E5)  y′′ + y = 1,  y′(0) = 0, y(1) = βy′ (\frac{1}{2}) ,

a special case of (BVP1). With k(t) ≡ 1 in (1.6), Theorem B is not applicable. We can use (3.8) in Corollary 1 and find Λ̂_{1}(k) = \frac{1}{2}(1 + |β|) < 1, or |β| < 1. However, (E5) admits an exact unique solution y(t) = 1−(cos 1+β sin \frac{1}{2})^{−1} cost for all β ≠ cos 1/ sin \frac{1}{2}. This shows that conditions (3.8) and (3.9) are not the best possible.

3. We refer the reader to the papers by Liu and Yu [5] which discussed similar problem in resonant cases where ∥k∥ is also required to be small as compared with the value 1. There are also recent papers by Han and Wu [4], and Liu, Liu and Wu [6] which dealt with sign-changing nonlinearities like condition (1.2) by comparison with the smallest positive eigenvalue of certain associated linear boundary value problem.

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REFERENCES

[1] R. F. Brown, A topological introduction to nonlinear analysis, Birkhauser, Boston 1993.

[2] J. Dugundji and A. Granas, Fixed point theory, vol I, Monografie Matematyczne vol 61, Warsaw 1982.

[3] C. P. Gupta, Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equations, J. Math. Anal. Appl., 168(1992), 540-551.

[4] G. Han and Y. Wu, Nontrivial solutions of m-point boundary value problems with sign-changing nonlinear terms, J. Math. Anal. Appl., 325(2007), 1327-1338.

[5] B. Liu and Y. Yu, Solvability of Multi-point boundary value problems at resonance III, Appl. Math. Comput., 129(2004), 119-143.

[6] L. Liu, B. Liu and Y. Wu, Nontrivial solutions of m-point boundary value problems for singular second order differential equations with a sign-changing nonlinear term, J. Comput. & Appl. Math., 224(2009), 373-382.

[7] R. Ma, Existence of theorem for a second order three-point boundary value problem, J. Math. Anal. Appl., 212(1997), 430-442.

[8] S. A. Marano, A remark on a second order 3-point boundary value problem, J. Math. Anal. Appl., 183(1994), 518-522.

[9] J. Ren and W. Ge, Positive solutions for three point boundary value problems with sign-changing nonlinearities, Appl. Math. Letters, 17(2004), 451-458.

[10] Y. P. Sun, Nontrivial solutions for a three-point boundary value problem, Electronic J. Diff. Equ., 2004 (2004), 1-10.

[11] Y. P. Sun and L. Liu, Solvability for a nonlinear second order three point boundary value problem, J. Math. Anal. Appl., 296(2004), 265-275.

(Received December 14, 2009)

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