Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 55, 1-10; http://www.math.u-szeged.hu/ejqtde/

Multiple solutions for fourth order m-point boundary value problems with sign-changing nonlinearity ^{∗}

Haitao Li

School of Control Science and Engineering, Shandong University, Jinan, 250061, P. R. China

(E-mail address: haitaoli09@gmail.com)

Yansheng Liu

Department of Mathematics, Shandong Normal University, Jinan, 250014, P. R. China

(E-mail address: yanshliu@gmail.com)

Abstract. Using a fixed point theorem in ordered Banach spaces with lattice structure founded by Liu and Sun, this paper investigates the multiplicity of nontrivial solutions for fourth order m-point boundary value problems with sign-changing nonlinearity. Our results are new and improve on those in the literature. Keywords: Lattice; Fixed point theorem; Fourth order m-point boundary value problems; Sign-changing solution. AMS (Subject Classification): 34B16.

1 Introduction

Consider the following fourth order differential equation

x^{(4)}(t) = f(t, x(t)), t ∈ (0, 1)  (1.1)

subject to one of the following two classes of m-point boundary value conditions

\{ \begin{matrix}x(0) = 0, x(1) = ∑_{i=1}^{m−2} α_{i}x(η_{i}); \\ x′′(0) = 0, x′′(1) = ∑_{i=1}^{m−2} α_{i}x′′(η_{i}), \end{matrix}  (1.2)

and

\{ \begin{matrix}x′(0) = 0, x(1) = ∑_{i=1}^{m−2} α_{i}x(η_{i}); \\ x′′′(0) = 0, x′′(1) = ∑_{i=1}^{m−2} α_{i}x′′(η_{i}), \end{matrix}  (1.3)

∗The Project Supported by the Key Project of Chinese Ministry of Education (No:209072), and the Science & Technology Development Funds of Shandong Education Committee (J08LI10)

where f : R × R → R is a given sign-changing continuous function, m ≥ 3, η_{i} ∈ (0, 1) and α_{i} > 0 for i = 1, ⋯, m − 2 with

∑_{i=1}^{m−2} α_{i} < 1.  (1.4)

The existence of nontrivial or positive solutions of nonlinear multi-point boundary value problems (BVP, for short) for fourth order differential equations has been extensively studied and lots of excellent results have been established by using fixed point index for cone mappings, standard upper and lower solution arguments, fixed point theorems for cone mappings and so on (see [2, 8-10] and the references therein). For example, in [9], Wei and Pang studied the following fourth order differential equation

x^{(4)}(t) = f(x(t), −x′′(t)), t ∈ (0, 1)

with the boundary condition (1.2).

By means of fixed point index theory in a cone and the Leray-Schauder degree, the existence and multiplicity of nontrivial solutions are obtained.

Recently Professor Jingxian Sun advanced a new approach to compute the topological degree when the concerned operators are not cone mappings in ordered Banach spaces with lattice structure. He established some interesting results for such nonlinear operators (for details, see [3, 6, 7]). To our best knowledge, there is no paper to use this new method to study fourth order m-point boundary value problems. We try to fill this gap in the present paper.

Suppose the following conditions are satisfied throughout.

(H0) the sequence of positive solutions of the equation

sin \sqrt{s} = ∑_{i=1}^{m−2} α_{i} sin η_{i}\sqrt{s}

is 0 < λ_{1} < λ_{2} < ⋯ < λ_{n} < λ_{n+1} < · · ·.

(H0′) the sequence of positive solutions of the equation

cos \sqrt{s} = ∑_{i=1}^{m−2} α_{i} cos η_{i}\sqrt{s}

is 0 < s_{1} < s_{2} < ⋯ < s_{n} < s_{n+1} < · · ·.

(H1) lim_{x→+∞} \frac{f(t,x)}{x} = α uniformly on t ∈ [0, 1].

(H2) lim_{x→−∞} \frac{f(t,x)}{x} = β uniformly on t ∈ [0, 1].

(H3) f(t, 0) ≡ 0, lim_{x→0} \frac{f(t,x)}{x} = γ uniformly on t ∈ [0, 1].

This paper is organized as follows. In Section 2, we present some basic definitions of the lattice and some lemmas that will be used to prove the main results. In Section 3, we shall give our main results and their proofs.

2 Preliminaries

We first recall some properties of a lattice and some operators (see [3, 7]).

Let E be an ordered Banach space in which the partial ordering ≤ is induced by a cone P ⊆ E. P is called normal if there exists a constant N > 0 such that θ ≤ x ≤ y implies ∥ x ∥≤ N ∥ y ∥.

Definition 2.1.^{[7]} We call E a lattice under the partial ordering ≤, if sup{x, y} and inf{x, y} exist for arbitrary x, y ∈ E.

Definition 2.2.^{[3]} Let E be a Banach space with a cone P and A : E → E be a nonlinear operator. We say that A is a unilaterally asymptotically linear operator along P_{w} = {x ∈ E : x ≥ w, w ∈ E}, if there exists a bounded linear operator L such that

lim_{x∈P_{w}, ∥x∥→∞} \frac{∥ Ax − Lx ∥}{∥ x ∥} = 0.

L is said to be the derived operator of A along P_{w} and will be denoted by A′_{P_{w}}. Similarly, we can also define a unilaterally asymptotically linear operator along P^{w} = {x ∈ E : x ≤ w, w ∈ E}.

Remark 2.1. If w = 0 in Definition 2.2, A is a unilaterally asymptotically linear operator along P and (−P). It is remarkable that A is not assumed to be a cone mapping.

Definition 2.3.^{[7]} Let D ⊆ E and A : D → E be a nonlinear operator. A is said to be quasi-additive on a lattice, if there exists v^{∗} ∈ E such that

Ax = Ax_{+} + Ax_{−} + v^{∗}, ∀x ∈ D,

where x_{+} = x^{+} = sup{x, θ}, x_{−} = −x^{−} = − sup{−x, θ}.

The following lemma is important for us to obtain the main results.

Lemma 2.1.^{[3]} Suppose E is an ordered Banach space with a lattice structure, P is a normal cone of E, and the nonlinear operator A is quasi-additive on the lattice. Assume that

(i) A is strongly increasing on P and (−P);

(ii) both A′_{P} and A′_{(−P)} exist with r(A′_{P}) > 1 and r(A′_{−P}) > 1, and 1 is not an eigenvalue of A′_{P} and A′_{(−P)} corresponding a positive eigenvector;

(iii) Aθ = θ; the Frechet derivative A′_{θ} of A at θ is strongly positive and r(A′_{θ}) < 1; (iv) the Frechet derivative A′_{∞} of A at ∞ exists; 1 is not an eigenvalue of A′_{∞}; the sum β

of the algebraic multiplicities for all eigenvalues of A′_{∞} lying in the interval (1, ∞) is an even number.

Then A has at least three nontrivial fixed points containing one sign-changing fixed point. Let E = C[0, 1] with the norm ∥ x ∥= max_{t∈[0, 1]} | x(t) | and P = {x ∈ E : x(t) ≥ 0, t ∈

[0, 1]}. Then E is a Banach space and P is a normal cone of E. It is easy to see that E is a lattice under the partial ordering ≤ that is deduced by P.

Using the same method as in [9], we can easily convert BVP (1.1) and (1.2) into the following operator equation

x(t) = (L_{1}^{2}F x)(t),  (2.1)

where the operators F and L_{1} are defined by

(F x)(t) = f(t, x(t)), ∀t ∈ [0, 1], x ∈ E;  (2.2)
(L_{1}x)(t) = ∫_{0}^{1} H_{1}(t, s)x(s)ds, ∀t ∈ [0, 1], x ∈ E;  (2.3)

where

H_{1}(t, s) = G_{1}(t, s) + \frac{∑_{i=1}^{m−2} α_{i}G(η_{i}, s)}{1 − ∑_{i=1}^{m−2} α_{i}η_{i}}t,

G_{1}(t, s) = \{ \begin{matrix}(1 − t)s, 0 ≤ s ≤ t ≤ 1; \\ (1 − s)t, 0 ≤ t ≤ s ≤ 1. \end{matrix}

Similarly, we can convert BVP (1.1) and (1.3) into the following operator equation

x(t) = (L_{2}^{2}F x)(t),  (2.4)

where

(L_{2}x)(t) = ∫_{0}^{1} H_{2}(t, s)x(s)ds, ∀t ∈ [0, 1], x ∈ E,  (2.5)

H_{2}(t, s) = G_{2}(t, s) + \frac{∑_{i=1}^{m−2} α_{i}G(η_{i}, s)}{1 − ∑_{i=1}^{m−2} α_{i}η_{i}},

G_{2}(t, s) = \{ \begin{matrix}1 − t, 0 ≤ s ≤ t ≤ 1; \\ 1 − s, 0 ≤ t ≤ s ≤ 1. \end{matrix}

Define

A_{1} = L_{1}^{2}F, A_{2} = L_{2}^{2}F,  (2.6)

then the following lemma is obvious.

Lemma 2.2. x(t) is a solution of the BVP (1.1) and (1.2) (BVP (1.1) and (1.3)) if and only if x(t) is a solution of the operator equation

x(t) = (A_{1}x)(t) (x(t) = (A_{2}x)(t)).

Lemma 2.3. (i) L_{i}^{2} : E → E(i = 1, 2) is a completely continuous linear operator; (ii) F : E → E is quasi-additive on the lattice; (iii) A_{i} = L_{i}^{2}F (i = 1, 2) is quasi-additive on the lattice; (iv) the sequences of all eigenvalues of the operators L_{1}^{2} and L_{2}^{2} are \{\frac{1}{λ_{n}^{2}} , n = 1, 2, ⋯\} and

\{\frac{1}{s_{n}^{2}} , n = 1, 2, ⋯\}, respectively, where λ_{n} and s_{n} are respectively defined by (H0) and (H0′), and the algebraic multiplicities of \frac{1}{λ_{n}^{2}} and \frac{1}{s_{n}^{2}} are 1.

(v) r(L_{1}^{2}) = \frac{1}{λ_{1}^{2}} , r(L_{2}^{2}) = \frac{1}{s_{1}^{2}} , where r(L_{i}) is the spectral radius of the operator L_{i} (i = 1, 2).

Proof. The proof of (i)-(iii) is obvious. Now we start to prove the conclusions (iv) and (v). Let μ be a positive eigenvalue of the linear operator L_{1}^{2}, and y ∈ E \ {θ} be an eigenfunction

corresponding to the eigenvalue μ. Then we have

\{ \begin{matrix}μy^{(4)} = y, t ∈ [0, 1]; \\ y(0) = 0, y(1) = ∑_{i=1}^{m−2} α_{i}y(η_{i}); \\ y′′(0) = 0, y′′(1) = ∑_{i=1}^{m−2} α_{i}y′′(η_{i}). \end{matrix}  (2.7)

Define D = \frac{d}{dt}, L = μD^{4} − 1; then there exist two constants r_{1}, r_{2} such that

Ly = μ(D^{2} + r_{1})(D^{2} + r_{2})y.

By properties of differential operators, if (2.7) has a nonzero solution, then there exists r_{s}, s ∈ \{1, 2\} such that r_{s} = λ_{k}, k ∈ N. In this case, sin t\sqrt{λ_{k}} is a nonzero solution of (2.7). Substituting this solution into (2.7), we have

μλ_{k}^{2} − 1 = 0.

Hence, \{\frac{1}{λ_{k}^{2}} , k = 1, 2, ⋯\} is the sequence of all eigenvalues of the operator L_{1}^{2}. Then μ is one of the values

\frac{1}{λ_{1}^{2}} > \frac{1}{λ_{2}^{2}} > ⋯ > \frac{1}{λ_{n}^{2}} ⋯

and the eigenfunction corresponding to the eigenvalue \frac{1}{λ_{n}^{2}} is

y_{n}(t) = C sin t\sqrt{λ_{n}}, t ∈ [0, 1],

where C is a nonzero constant. By the ordinary method, we can show that any two eigenfunctions corresponding to the same eigenvalue \frac{1}{λ_{n}^{2}} are merely nonzero constant multiples of each other. Consequently,

dim ker(\frac{1}{λ_{n}^{2}} I − L_{1}^{2}) = 1.  (2.8)

Now we show that

ker(\frac{1}{λ_{n}^{2}} I − L_{1}^{2}) = ker(\frac{1}{λ_{n}^{2}} I − L_{1}^{2})^{2}.  (2.9)

Obviously, we need only show that

ker(\frac{1}{λ_{n}^{2}} I − L_{1}^{2})^{2} ⊆ ker(\frac{1}{λ_{n}^{2}} I − L_{1}^{2}).  (2.10)

For any y ∈ ker(\frac{1}{λ_{n}^{2}} I − L_{1}^{2})^{2}, (I − λ_{n}^{2}L_{1}^{2})y is an eigenfunction of the linear operator L_{1}^{2} corresponding to the eigenvalue \frac{1}{λ_{n}^{2}} if (I − λ_{n}^{2}L_{1}^{2})y ≠ θ. Then there exists a nonzero constant γ such that

(I − λ_{n}^{2}L_{1}^{2})y = γ sin t\sqrt{λ_{n}}, t ∈ [0, 1].

By direct computation, we have

\{ \begin{matrix}y^{(4)} = λ_{n}^{2}y + γλ_{n}^{2} sin t\sqrt{λ_{n}}, t ∈ [0, 1]; \\ y(0) = 0, y(1) = ∑_{i=1}^{m−2} α_{i}y(η_{i}); \\ y′′(0) = 0, y′′(1) = ∑_{i=1}^{m−2} α_{i}y′′(η_{i}). \end{matrix}  (2.11)

It is easy to see that the general solution of (2.11) is of the form

y(t) = C_{1} cos t\sqrt{λ_{n}} +C_{2} sin t\sqrt{λ_{n}} +C_{3} exp(t\sqrt{λ_{n}})+C_{4} exp(\sqrt{−tλ_{n}})+ \frac{γ\sqrt{λ_{n}}}{4} t cos t\sqrt{λ_{n}}, t ∈ [0, 1]

where C_{1}, C_{2}, C_{3}, C_{4} are nonzero constants.

Applying the boundary conditions, we obtain that

\{ \begin{matrix}C_{1} = 0, \\ C_{3} + C_{4} = 0, \\ C_{3}F + ^{\frac{γ\sqrt{λ_{n}}}{4}} G = 0, \\ C_{3}λ_{n}F − λ_{n}^{\frac{γ\sqrt{λ_{n}}}{4}} G = 0, \end{matrix}  (2.12)

where

F = e^{\sqrt{λ_{n}}} − e^{−\sqrt{λ_{n}}} − ∑_{i=1}^{m−2} α_{i}(exp(\sqrt{λ_{n}η_{i}}) − exp(−\sqrt{λ_{n}η_{i}})) > 0,  (2.13)
G = cos \sqrt{λ_{n}} − ∑_{i=1}^{m−2} α_{i}η_{i} cos η_{i}\sqrt{λ_{n}}.  (2.14)

If G ≠ 0, then C_{3} = ^{\frac{γ\sqrt{λ_{n}}}{4}} = 0, which is a contradiction to γ ≠ 0, and y(t) = C_{2} sin t\sqrt{λ_{n}} ∈ ker(\frac{1}{λ_{n}^{2}} I − L_{1}^{2}). So (2.10) holds, which means (2.9) also holds.

If G = 0, then

cos \sqrt{λ_{n}} = ∑_{i=1}^{m−2} α_{i}η_{i} cos η_{i}\sqrt{λ_{n}}.

By the Schwarz inequality, we obtain

1 − sin^{2} \sqrt{λ_{n}} = ( ∑_{i=1}^{m−2} α_{i}η_{i} cos η_{i}\sqrt{λ_{n}})^{2}
≤ ( ∑_{i=1}^{m−2} η_{i}^{2})( ∑_{i=1}^{m−2} α_{i}^{2} cos^{2} η_{i}\sqrt{λ_{n}})
= ( ∑_{i=1}^{m−2} η_{i}^{2})( ∑_{i=1}^{m−2} α_{i}^{2}) − ( ∑_{i=1}^{m−2} η_{i}^{2})( ∑_{i=1}^{m−2} α_{i}^{2} sin^{2} η_{i}\sqrt{λ_{n}}).

Applying the condition sin \sqrt{λ_{n}} = ∑_{i=1}^{m−2} α_{i} sin η_{i}\sqrt{λ_{n}}, we obtain

1 ≤ ( ∑_{i=1}^{m−2} η_{i}^{2})( ∑_{i=1}^{m−2} α_{i}^{2}) + ( ∑_{i=1}^{m−2} α_{i} sin η_{i}\sqrt{λ_{n}})^{2} − ( ∑_{i=1}^{m−2} η_{i}^{2})( ∑_{i=1}^{m−2} α_{i}^{2} sin^{2} η_{i}\sqrt{λ_{n}})
= ( ∑_{i=1}^{m−2} η_{i}^{2})( ∑_{i=1}^{m−2} α_{i}^{2}) + [1 − ( ∑_{i=1}^{m−2} η_{i}^{2})]( ∑_{i=1}^{m−2} α_{i}^{2} sin^{2} η_{i}\sqrt{λ_{n}})

+ ∑_{i≠j} α_{i}α_{j} sin η_{i}\sqrt{λ_{n}} sin η_{j}\sqrt{λ_{n}}

≤ ( ∑_{i=1}^{m−2} η_{i}^{2})( ∑_{i=1}^{m−2} α_{i}^{2}) + [1 − ( ∑_{i=1}^{m−2} η_{i}^{2})]( ∑_{i=1}^{m−2} α_{i}^{2}) + ∑_{i≠j} α_{i}α_{j}
= ( ∑_{i=1}^{m−2} α_{i})^{2},

which is a contradiction to ∑_{i=1}^{m−2} α_{i} < 1. Thus, (2.9) holds. It follows from (2.8) and (2.9) that the algebraic multiplicity of the eigenvalue \frac{1}{λ_{n}^{2}} is 1.

Similarly, we can show that the sequence of all eigenvalues of the operator L_{2}^{2} is \{\frac{1}{s_{n}^{2}} , n = 1, 2, ⋯\}, and the algebraic multiplicity of \frac{1}{s_{n}^{2}} is 1.

By the definition of the spectral radius, we have

r(L_{1}^{2}) = sup_{λ∈\{_{\frac{1}{λ^{2}_{n}}} , n=1,2,···\}} | λ |= \frac{1}{λ_{1}^{2}} ,

r(L_{2}^{2}) = sup_{λ∈\{_{\frac{1}{s^{2}_{n}}} , n=1,2,···\}} | λ |= \frac{1}{s_{1}^{2}} .

The proof of this Lemma is complete. □ Lemma 2.4. Let A_{i} and L_{i}^{2} (i = 1, 2) be defined by (2.1) - (2.6). Then

(i) (A_{i})′_{P} = αL_{i}^{2} (i = 1, 2) if f satisfies (H1); (ii) (A_{i})′_{(−P)} = βL_{i}^{2} (i = 1, 2) if f satisfies (H2); (iii) (A_{i})′_{θ} = γL_{i}^{2} (i = 1, 2) if f satisfies (H3). Proof. We only prove the conclusion (i), the proofs of conclusions (ii) and (iii) are similar. Suppose that f satisfies (H1). Then there exists R > 0 such that for a given ε > 0,

| f(t, x) − αx |≤ εx, t ∈ [0, 1], x > R.

Set M_{R} = max_{t∈[0, 1], 0≤x≤R} | f(t, x) |. Then

| (F x)(t) − αx(t) | = | f(t, x(t)) − αx(t) |
≤ M_{R} + αR + ε ∥ x ∥, ∀x ∈ P, ∥ x ∥≥ R,

and hence

∥ A_{i}x − αL_{i}^{2}x ∥ = ∥ L_{i}^{2}(F x) − αL_{i}^{2}x ∥
≤ ∥ L_{i}^{2} ∥ (M_{R} + αR + ε ∥ x ∥), (i = 1, 2),

which means

lim inf_{x∈P, ∥x∥→∞} \frac{∥ A_{i}x − αL_{i}^{2}x ∥}{∥ x ∥} ≤ ε ∥ L_{i}^{2} ∥, (i = 1, 2).

Therefore, (A_{i})′_{P} = αL_{i}^{2} (i = 1, 2). □

3 Main Results

In order to consider the existence of multiple solutions for the BVP (1.1) and (1.2) and BVP (1.1) and (1.3), let us introduce another ordered Banach space.

Let e_{1,i} be the first normalized eigenfunction of L_{i}^{2} corresponding to its first eigenvalue; then e_{1,i}(t) > 0, ∀t ∈ (0, 1) and ∥ e_{1,i} ∥= 1, (i = 1, 2).

Let

E_{e,i} = {x ∈ E : ∃μ > 0, −μe_{1,i}(t) ≤ x(t) ≤ μe_{1,i}(t), t ∈ [0, 1]}, (i = 1, 2).

By [1] and [3], we know that E_{e,i} is an ordered Banach space, P_{e,i} = P ⋂ E_{e,i} (i = 1, 2) is a normal solid cone in E_{e,i}, and L_{i}^{2} : E → E_{e,i} is a linear completely continuous operator satisfying L_{i}^{2}(P \ {θ}) ⊆ intP_{e,i} = {x ∈ P_{e,i} : ∃α > 0, β > 0, αe_{1,i}(t) ≤ x(t) ≤ βe_{1,i}(t), t ∈ [0, 1]}, (i = 1, 2).

Now we are ready to give our main results.

Theorem 3.1. Suppose that f satisfies (H1) -(H3). In addition, suppose

(i) f(t, x) is strictly increasing in x;

(ii) there exist an even number n_{1} and a positive integer n_{2}, such that

λ_{n_{1}}^{2} < α < λ_{n_{1}+1}^{2}, λ_{n_{2}}^{2} < β < λ_{n_{2}+1}^{2};  (3.1)

(iii) 0 < γ < λ_{1}^{2}.

Then BVP (1.1) and (1.2) has at least three nontrivial solutions containing a sign-changing solution.

Proof. From Lemma 2.4 we know that

(A_{1})′_{θ} = γL_{1}^{2}, (A_{1})′_{P} = αL_{1}^{2}, (A_{1})′_{(−P)} = βL_{1}^{2}.

Notice that P_{e,1} = P ⋂ E_{e,1} ⊆ P implies (A_{1})′_{P_{e,1}} = αL_{1}^{2} and (A_{1})′_{(−P_{e,1})} = βL_{1}^{2}.

Using a method similar to the one used in the proof of Lemma 2.4, it is not difficult to prove that (A_{1})′_{∞} = αL_{1}^{2}.

By condition (i) and the fact that L_{1}^{2}(P \ {θ}) ⊆ intP_{e,1}, we know that A_{1} is strongly increasing and hence the condition (i) of Lemma 2.1 is satisfied.

The condition (ii) shows that 1 is not an eigenvalue of (A_{1})′_{P} and (A_{1})′_{(−P)}, and by Lemma 2.3, we have

r((A_{1})′_{P}) = \frac{α}{λ_{1}^{2}} > 1, r((A_{1})′_{(−P)}) = \frac{β}{λ_{1}^{2}} > 1.

Hence, condition (ii) of Lemma 2.1 is also satisfied.

Similarly, we can show that (A_{1})′_{θ} is strongly positive, so condition (iii) of Lemma 2.1 is satisfied.

Since f satisfies (H1) and (H2), by condition (iii) we find that condition (iv) of Lemma 2.1 is satisfied.

Consequently, Lemmas 2.1 and 2.2 guarantee that the conclusion is valid. □ By the method used in the proof of Theorem 3.1, it is easy to show the following theorem. Theorem 3.2. Suppose that f satisfies (H1) -(H3). In addition, suppose

(i) f(t, x) is strictly increasing in x; (ii) there exist an even number n_{1} and a positive integer n_{2}, such that

s_{n_{1}}^{2} < α < s_{n_{1}+1}^{2}, s_{n_{2}}^{2} < β < s_{n_{2}+1}^{2};  (3.2)

(iii) 0 < γ < s_{1}^{2}. Then BVP (1.1) and (1.3) has at least three nontrivial solutions containing a sign-changing

solution.

Example 3.1 Consider the following fourth order differential equation

\{ \begin{matrix}x^{(4)}(t) = f(t, x(t)), t ∈ (0, 1); \\ x′(0) = 0, x(1) = α_{1}x(η_{1}); \\ x′′′(0) = 0, x′′(1) = α_{1}x′′(η_{1}), \end{matrix}  (3.3)

where

f(t, x) = \{ \begin{matrix}inf_{t∈[0,1]}[400x(1 + t)], & x ≥ 10; \\ \frac{7998}{20−π} (x − \frac{π}{2} ) + 1, & \frac{π}{2} ≤ x < 10; \\ sin x, & −\frac{π}{2} ≤ x ≤ \frac{π}{2} ; \\ \frac{38}{20+π} (x + \frac{π}{2} ) − 1, & −10 ≤ x < −\frac{π}{2} ; \\ sup_{t∈[0,1]}[2tx], & x < −10, \end{matrix}

and α_{1} = \frac{\sqrt{3}}{3}, η_{1} = \frac{1}{2}. It is easy to see f(t, x) is strictly increasing in x. By simple calculations, we can show s_{1} = \frac{π^{2}}{9} , s_{2} = (2π−2 arccos \frac{\sqrt{3}}{3})^{2}, s_{3} = (2π+2 arccos \frac{\sqrt{3}}{3})^{2}, s_{4} = \frac{121π^{2}}{9} , ⋯, s_{4k+1} = (4kπ + \frac{π}{3} )^{2}, s_{4k+2} = (4kπ + 2π − 2 arccos \frac{\sqrt{3}}{3})^{2}, s_{4k+3} = (4kπ + 2π + 2 arccos \frac{\sqrt{3}}{3})^{2}, s_{4k+4} = (4kπ + \frac{11π}{3} )^{2}, k ∈ N. And α = 400, β = 2, γ = 1, so s_{2}^{2} < α < s_{3}^{2}, s_{1}^{2} < β < s_{2}^{2}, 0 < γ < s_{1}^{2}. Thus, by Theorem 3.2, BVP (3.3) has at least three nontrivial solutions containing a signchanging solution.

Acknowledgements The authors thank the anonymous referee for his or her valuable suggestions for the improve-

ment of this paper.

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(Received December 22, 2009)