Electronic Journal of Qualitative Theory of Differential Equations 2006, No. 13, 1-9; http://www.math.u-szeged.hu/ejqtde/

New existence theorems of positive solutions for singular boundary value

problems^{∗}

Meiqiang Feng Xuemei Zhang Weigao Ge

Abstract: In this paper, some nonexistence, existence and multiplicity of positive solutions are established for a class of singular boundary value problem. The authors also obtain the relation between the existence of the solutions and the parameter λ. The arguments are based upon the fixed point index theory and the upper and lower solutions method.

1 Introduction

Consider the following second-order singular boundary value problem (BVP)

\{ \begin{matrix}\frac{1}{p(t)}(p(t)x′(t))′ + λg(t)f(x(t)) = 0,  0 < t < 1, \\ ax(0) − b lim_{t→0^{+}} p(t)x′(t) = 0, \\ cx(1) + d lim_{t→1^{−}} p(t)x′(t) = 0, \end{matrix}  (1.1)_{λ}

where a ≥ 0, b ≥ 0, c ≥ 0, d ≥ 0, ac + bc + ad > 0; λ > 0.

Differential equations with singularity arise in the fields of gas dynamics, nuclear physics, theory of boundary layer, nonlinear optics and so on. Nonlinear singular boundary value problems has become an important area of investigation in previous years; see[1-5, 7-15] and references therein.

When p(t) = 1, β = δ = 0, α = γ = 1, the BVP (1.1)_{λ} reduces to

\{ \begin{matrix}x′′(t) + λg(t)f(x(t)) = 0,  0 < t < 1, \\ x(0) = x(1) = 0, \end{matrix}  (1.2)

which is a special case of the BVP (1.2)_{λ}.

In the special cases i) f(t, x) = q(t)x^{−λ_{1}}, λ_{1} > 0, and ii) f(t, x) = q(t)x^{λ_{1}}, 0 < λ_{1} < 1, where q > 0 for t ∈ (0, 1), the existence and uniqueness of positive solutions for the BVP (1.2) as λ = 1 have been studied completely by Taliaferro in [1] with the shooting method and by Zhang [2] with the sub-super solutions method, respectively. In the special case iii)f(t, x) = q(t)g(x), q(t) is singular only at t = 0 and g(x) ≥ e^{x}, the existence of multiple positive solutions for the BVP (1.2) have been studied by Ha and Lee in [3] with the sub-super solutions method. In the special case iv) f(t, x) = q(t)g(x), q(t) is singular only at t = 0 and g(x) ∈ C(−∞, +∞), [0. + ∞), the existence of multiple positive solutions for the BVP (1.2) have been studied by Wong in [4] with the shooting method.

∗2000 Mathematics Subject Classification: 34B15.

Keywords: Positive solution; Nonexistence; Existence; Complete continuity; Singularity.

Motivated by the results mentioned above, in this paper we study the existence, multiplicity, and nonexistence of positive solutions for the BVP (1.1) by new technique(different from [3,5,11,12,13,14]) to overcome difficulties arising from the appearances of p(t) and p(t) is singular at t = 0 and t = 1. On the other hand, to the best of our knowledge, there are very few literatures considering the existence, multiplicity, and nonexistence of positive solutions for the case when p(t) is singular at t = 0 and t = 1. The arguments are based upon the fixed point index theory and the upper and lower solutions method.

Fixed point index theorems have been applied to various boundary value problems to show the existence of multiple positive solutions. An overview of such results can be found in Guo and Lakshmikantham V., [16] and in Guo and Lakshmikantham V., Liu X.Z.,[17] and in Guo, [18] and in K. Deimling, [19] and in M. Krasnoselskii, [20].

Lemma 1.1.[16,17,18,19,20] Let P be a cone of real Banach space E, Ω be a bounded open subset of E and θ ∈ Ω. Suppose A : P ∩ Ω̄ → P is a completely continuous operator, and satisfies

Ax = μx, x ∈ P ∩ ∂Ω ⟹ μ < 1.

Then i(A, P ∩ Ω, P) = 1.

Lemma 1.2.[16,17,18,19,20] Suppose A : P ∩ Ω̄ → P is a completely continuous operator, and satisfies

(1) inf_{x∈P ∩∂Ω} ||Ax|| > 0;

(2)Ax = μx, x ∈ P ∩ ∂Ω ⟹ μ ∉ (0, 1]. Then i(A, P ∩ Ω, P) = 0.

The paper is organized in the following fashion. In Section 2, we provide some necessary background. In particular, we state some properties of the Green^{,}s function associated with the BVP (1.1)_{λ}. In Section 3, the main results will be stated and proved. Finally some examples are worked out to demonstrate our main results in this section.

2 Preliminaries

Let J = [0, 1]. The basic space used in this paper is E = C[0, 1]. It is well known that E be a Banach space with the norm || · || defined by ||x|| = max_{0≤t≤1} |x(t)|. Let

S = {λ > 0|(1.1)_{λ} has at least one solution} and P = {x ∈ E|x(t) ≥ 0, t ∈ [0, 1]}. It is clear that P is a cone of E.

The following assumptions will stand throughout this paper: (H_{1}) p ∈ C((0, 1), (0, +∞)) and 0 < ∫_{0}^{1} \frac{dt}{p(t)} < +∞; (H_{2}) g ∈ C((0, 1), (0, +∞)) and 0 < ∫_{0}^{1} G(s, s)p(s)g(s)ds < +∞; (H_{3}) f ∈ C([0, +∞), (0, +∞)) is nondecreasing and there exist δ̄ > 0, m ≥ 2 such that f(x) > δ̄x^{m}, x ∈ (0, +∞).

Let G(t, s) be Green^{,}s function of the following BVP

\{ \begin{matrix}\frac{1}{p(t)}(p(t)x′(t))′ = 0,  0 < t < 1, \\ ax(0) − b lim_{t→0^{+}} p(t)x′(t) = 0, \\ cx(1) + d lim_{t→1^{−}} p(t)x′(t) = 0. \end{matrix}

Then G(t, s) is defined by

G(t, s) = \frac{1}{Δ} \{ \begin{matrix}(b + a ∫_{0}^{s} \frac{dr}{p(r)})(d + c ∫_{t}^{1} \frac{dr}{p(r)}), if 0 ≤ s ≤ t ≤ 1, \\ (b + a ∫_{0}^{t} \frac{dr}{p(r)})(d + c ∫_{s}^{1} \frac{dr}{p(r)}), if 0 ≤ t ≤ s ≤ 1, \end{matrix}  (2.1)

where Δ = ad + ac ∫_{0}^{1} \frac{dr}{p(r)} + bc. It is easy to prove that G(t, s) has the following properties. Property 2.1. For all t, s ∈ [0, 1] we have

G(t, s) ≤ G(s, s) ≤ \frac{1}{Δ}(b + a ∫_{0}^{1} \frac{dr}{p(r)})(d + c ∫_{0}^{1} \frac{dr}{p(r)}) < +∞.  (2.2)

Property 2.2. For all t, s ∈ J_{θ} = [θ, 1 − θ], θ ∈ (0, \frac{1}{2}) we have

G(t, s) ≥ \frac{1}{Δ}(b + a ∫_{0}^{θ} \frac{dr}{p(r)})(d + c ∫_{1−θ}^{1} \frac{dr}{p(r)}) > 0.  (2.3)

Property 2.3. For all t ∈ J_{θ}, s ∈ [0, 1] we have

G(t, s) ≥ σ_{0}G(s, s),  (2.4)

where

σ_{0} = min \{\frac{b+a ∫_{0}^{θ} \frac{dr}{p(r)}}{b+a ∫_{0}^{1} \frac{dr}{p(r)}} , \frac{d+c ∫_{1−θ}^{1} \frac{dr}{p(r)}}{d+c ∫_{0}^{1} \frac{dr}{p(r)}} \} .  (2.5)

It is easy to see that 0 < σ_{0} < 1. Definition 2.1. Letting x(t) ∈ C[0, 1] ∩ C^{1}(0, 1), we say x(t) is a lower solution for the BVP (1.1)_{λ} if x(t) satisfies :

\{ \begin{matrix}−\frac{1}{p(t)}(p(t)x′(t))′ ≤ λg(t)f(x(t)),  0 < t < 1, \\ ax(0) − b lim_{t→0^{+}} p(t)x′(t) ≤ 0, \\ cx(1) + d lim_{t→1^{−}} p(t)x′(t) ≤ 0. \end{matrix}

Definition 2.2. Letting y(t) ∈ C[0, 1] ∩ C^{1}(0, 1), we say y(t) is an upper solution for the BVP (1.1)_{λ} if y(t) satisfies :

\{ \begin{matrix}−\frac{1}{p(t)}(p(t)y′(t))′ ≥ λg(t)f(y(t)),  0 < t < 1, \\ ay(0) − b lim_{t→0^{+}} p(t)y′(t) ≥ 0, \\ cy(1) + d lim_{t→1^{−}} p(t)y′(t) ≥ 0. \end{matrix}

Firstly, we consider the following BVP:

\{ \begin{matrix}\frac{1}{p(t)}(p(t)x′(t))′ + λg(t)f(x(t)) = 0,  0 < t < 1, \\ ax(0) − b lim_{t→0^{+}} p(t)x′(t) = 0, \\ cx(1) + d lim_{t→1^{−}} p(t)x′(t) = ρ ≥ 0. \end{matrix}  (2.6)_{λ}^{ρ}

Define T_{λ}^{ρ} : E → E by

T_{λ}^{ρ}x(t) = ∫_{0}^{1} G(t, s)λp(s)g(s)f(x(s))ds + ρh(t),  (2.7)

where h(t) = \frac{1}{△}(b + a ∫_{0}^{t} \frac{dr}{p(r)}). It is not difficult to see that 0 < h(t) ≤ 1 for c ≥ 1.

It is easy to obtain the following Lemma 2.1 by (2.7). Lemma 2.1. Let (H_{1}) − (H_{3}) be satisfied. Then the BVP (1.1)_{λ} has a solution x if and only if x is a fixed point of T_{λ}^{0}. Proof. It is easy to prove Lemma 2.1 by calculation.

In order to prove the following results we define a cone by

Q = {x ∈ C[0, 1]|x(t) ≥ 0, min_{t∈J_{θ}} x(t) ≥ σ_{0}||x||}  (2.8)

Where σ_{0} is given by (2.5), θ ∈ (0, \frac{1}{2}). It is easy to see that Q is a closed convex cone of E and Q ⊂ P. Lemma 2.2. Let (H_{1}) − (H_{3}) be satisfied. Then T_{λ}^{0}(Q) ⊂ Q and T_{λ}^{0} : Q → Q is completely continuous and nondecreasing. Proof. For any x ∈ P, we have by (2.2) and (2.7)

T_{λ}^{0}x(t) = ∫_{0}^{1} G(t, s)λp(s)g(s)f(x(s))ds
≤ ∫_{0}^{1} λG(s, s)p(s)g(s)f(x(s))ds.

therefore ||T_{λ}^{0}x|| ≤ ∫_{0}^{1} λG(s, s)p(s)g(s)f(x(s))ds.

On the other hand, for any t ∈ J_{θ}, we have by (2.4) and (2.7)

min_{t∈J_{θ}} T_{λ}^{0}x(t) = min_{t∈J_{θ}} ∫_{0}^{1} G(t, s)λp(s)g(s)f(x(s))ds

≥ λσ_{0} ∫_{0}^{1} G(s, s)p(s)g(s)f(x(s))ds
≥ σ_{0}||T_{λ}^{0}x||.

Hence T_{λ}^{0}x ∈ Q. Therefore T_{λ}^{0}P ⊂ Q and therefore T_{λ}^{0}Q ⊂ Q by Q ⊂ P. By similar arguments in [1,3], T_{λ}^{0} : Q → Q is completely continuous. It is clear that T_{λ}^{0} is nondecreasing on [0, +∞) by (H_{3}). Remark 2.1. Similar to proving Lemma 2.1-Lemma 2.2, we have T_{λ}^{ρ} : Q → Q is completely continuous; x(t) is a solution of (2.6)_{λ}^{ρ} if and only if x(t) is a fixed point of T_{λ}^{ρ}. Lemma 2.3. Suppose λ ∈ S, S_{1} = (λ, +∞) ∩ S ≢ ∅. Then there exists R(λ) > 0, such that ||x_{λ′}|| ≤ R(λ), where λ′ ∈ S_{1}, x_{λ′} ∈ Q is a solution of (1.1)_{λ′}. Proof. For any λ′ ∈ S, let x_{λ′} is a solution of the BVP (1.1)_{λ′}. Then we have

\begin{matrix}x_{λ′}(t) = T_{λ′}^{0}x_{λ′}(t) \\ = ∫_{0}^{1} G(t, s)λ′p(s)g(s)f(x_{λ′}(s))ds. \end{matrix}

Let R(λ) = max{[λ′σ_{0}^{m+1}δ̄∫_{θ}^{1−θ} G(s, s)p(s)g(s)ds]^{−1}, 1}, next we prove ||x_{λ′}|| ≤ R(λ). Indeed, if ||x_{λ′}|| < 1, the result is easily obtained. On the other hand, if ||x_{λ′}|| ≥ 1, then we have by (H_{3})

\frac{1}{||x_{λ′} ||} ≥ \frac{min_{t∈J_{θ}} x_{λ′} (t)}{||x_{λ′} ||^{2}}

= \frac{1}{||x_{λ′} ||^{2}} min_{t∈J_{θ}} ∫_{0}^{1} G(t, s)λ′p(s)g(s)f(x_{λ′}(s))ds

≥ \frac{1}{||x_{λ′} ||^{2}} σ_{0} ∫_{θ}^{1−θ} G(s, s)λ′p(s)g(s)δ̄(x_{λ′}(s))^{m}ds

≥ \frac{1}{||x_{λ′} ||^{2}} σ_{0}^{m+1} ∫_{θ}^{1−θ} G(s, s)λ′p(s)g(s)δ̄||x_{λ′}||^{m}ds

≥ λ′σ_{0}^{m+1}δ̄∫_{θ}^{1−θ} G(s, s)p(s)g(s)ds.

Hence ||x_{λ′}|| ≤ R(λ). It follows the result of Lemma 2.3. Lemma 2.4. (see [3]) Suppose f : [0, +∞) → (0, +∞) is continuous and increasing. For given s, s_{0} and M such that 0 < s < s_{0}, M > 0, then there exist s̄ ∈ (s, s_{0}), ρ_{0} ∈ (0, 1) such that

sf(x + ρ) < s̄f(x), x ∈ [0, M], ρ ∈ (0, ρ_{0}).  (2.9)

3 Main results

In this section, we give our main results and proofs. Our approach depends on the upper and lower solutions method and the fixed point index theory. In addition, we let c ≥ 1 when we prove the conclusion (3) of the Theorem 3.1. Theorem 3.1. Let (H_{1}) − (H_{4}) be satisfied. Then there exists 0 < λ^{∗} < +∞ such that:

(1) the BVP (1.1)_{λ} has no solution for λ > λ^{∗};

(2) the BVP (1.1)_{λ} has at least one positive solution for λ = λ^{∗};

(3) the BVP (1.1)_{λ} has at least two positive solutions for 0 < λ < λ^{∗}. Proof. Firstly, we prove the conclusion (1) of Theorem 3.1 is held. Let β(t) is a solution of the following BVP

\{ \begin{matrix}\frac{1}{p(t)}(p(t)x′(t))′ + g(t) = 0,  0 < t < 1, \\ ax(0) − b lim_{t→0^{+}} p(t)x′(t) = 0, \\ cx(1) + d lim_{t→1^{−}} p(t)x′(t) = 0. \end{matrix}  (3.1)

therefore we have by Lemma 2.1 β(t) = ∫_{0}^{1} G(t, s)p(s)g(s)ds.

Let β_{0} = max_{t∈[0,1]} β(t). Therefore by (H_{3}) and (2.4) we have

T_{λ}^{0}β(t) ≤ T_{λ}^{0}β_{0} = ∫_{0}^{1} G(t, s)λp(s)g(s)f(β_{0})ds < β(t), ∀0 < λ < \frac{1}{f(β_{0})}.

This implies that β(t) is an upper solution of T_{λ}^{0}. On the other hand, let α(t) ≡ 0, t ∈ [0, 1]. Then it is clear that α(t) is a lower solution of T_{λ}^{0}, and α(t) < β(t), t ∈ [0, 1]. By Lemma 2.2, T_{λ}^{0} is completely continuous on [α, β]. Therefore, T_{λ}^{0} has a fixed point x_{λ} ∈ [α, β], and x_{λ} is a solution of the BVP (1.1)_{λ} by Lemma 2.1. Hence, for any 0 < λ < \frac{1}{f(β_{0})}, we have λ ∈ S, which implies S ≠ ∅.

On the other hand, if λ_{1} ∈ S, then we must have (0, λ_{1}) ⊂ S. In fact, let x_{λ_{1}} be a solution of the BVP (1.1)_{λ_{1}}. Then we have by Lemma 2.1

x_{λ_{1}}(t) = T_{λ_{1}}^{0}x_{λ_{1}}(t), t ∈ [0, 1].

Therefore, for any λ ∈ (0, λ_{1}), we have by (2.7)

T_{λ}^{0}x_{λ_{1}}(t) = ∫_{0}^{1} G(t, s)λp(s)g(s)f(x_{λ_{1}}(s)ds
≤ ∫_{0}^{1} G(t, s)λ_{1}p(s)g(s)f(x_{λ_{1}}(s)ds
= T_{λ_{1}}^{0}x_{λ_{1}}(t)
= x_{λ_{1}}(t),

which implies that x_{λ_{1}} is an upper solution of T_{λ}^{0}. Combining this with the fact that α(t) ≡ 0 (t ∈ [0, 1]) is a lower solution of T_{λ}^{0}, then by Lemma 2.1, the BVP (1.1)_{λ} has a solution. Thus λ ∈ S and we have (0, λ_{1}) ⊂ S.

Let λ^{∗} = supS. Now we prove λ^{∗} < +∞. If not, then we must have N ⊂ S, where N denotes natural number set. Therefore, for any n ∈ N, by Lemma 2.1, there exists x_{n} ∈ Q satisfying

\begin{matrix}x_{n} = T_{n}^{0}x_{n} \\ = ∫_{0}^{1} G(t, s)np(s)g(s)f(x_{n}(s)ds. \end{matrix}

Let K = [δ̄σ_{0}^{m+1} ∫_{θ}^{1−θ} G(s, s)p(s)g(s)ds]^{−1} and ∥x_{n}∥ ≥ 1. Then, by Lemma 2.1 and (H_{3}), we have

1 ≥ \frac{1}{||x_{n}||}

≥ \frac{min_{θ≤t≤1−θ} x_{n}(t)}{||x_{n}||^{2}}

= \frac{1}{||x_{n}||^{2}} min_{θ≤t≤1−θ} ∫_{0}^{1} G(t, s)np(s)g(s)f(x_{n}(s)ds

≥ \frac{1}{||x_{n}||^{2}} σ_{0} ∫_{θ}^{1−θ} G(s, s)np(s)g(s)δ̄(x_{n}(s))^{m}ds ≥ \frac{1}{||x_{n}||^{2}} σ_{0}^{m+1} ∫_{θ}^{1−θ} G(s, s)np(s)g(s)δ̄||x_{n}||^{m}ds ≥ nσ_{0}^{m+1}δ̄∫_{θ}^{1−θ} G(s, s)p(s)g(s)ds.

If ∥x_{n}∥ ≤ 1, then we have

1 ≥ ∥x_{n}∥ ≥ min_{θ≤t≤1−θ} ∫_{0}^{1} G(t, s)np(s)g(s)f(x_{n}(s)ds ≥ σ_{0} ∫_{θ}^{1−θ} G(s, s)np(s)g(s)f(0)ds.

Hence n ≤ {K, [σ_{0} ∫_{θ}^{1−θ} G(s, s)p(s)g(s)f(0)ds]^{−1}}, this contradicts the fact that N is unbounded, therefore λ^{∗} < +∞, and therefore the proof of the conclusion (1) is complete.

Secondly, we verify the conclusion (2) of Theorem 3.1. Let\{λ_{n}\} ⊂ [\frac{λ^{∗}}{2} , λ^{∗}), λ_{n} → λ^{∗}(n → ∞), {λ_{n}} be an increasing sequence. Suppose x_{n} is solution of (1.1)_{λ_{n}}, by Lemma 2.3, there exists R(\frac{λ^{∗}}{2} ) > 0 such that ||x_{n}|| ≤ R(\frac{λ^{∗}}{2} ), n = 1, 2, · · ·. Hence x_{n} is a bounded set. It is clear that {x_{n}} is equicontinuous set of C[0,1]. Therefore we have by Ascoli-Arzela theorem {x_{n}} is compact set, and therefore {x_{n}} has convergent subsequence. No loss of generality, we suppose x_{n} is convergent: x_{n} → x^{∗}(n → +∞). Since x_{n} = T_{λ_{n}}^{0}x_{n}, by control convergence theorem, we have x^{∗} = T_{λ^{∗}}^{0}x^{∗}. Therefore, by Lemma 2.1, x^{∗} is a solution of the BVP (1.1)_{λ^{∗}}. Hence the conclusion (2) of Theorem 3.1 is held.

Finally, we prove the conclusion (3) of Theorem 3.1.

Let α(t) ≡ 0(t ∈ [0, 1]). Then for any λ ∈ (0, λ^{∗}), α(t) is a lower solution of the BVP (2.6)_{λ}^{ρ}.

On the other hand, By Lemma 2.3, there exists R(λ) > 0 such that ||x_{λ′}|| ≤ R(λ), λ′ ∈ [λ, λ^{∗}], where x_{λ′} is a solution of the BVP (1.1)_{λ′}. And by Lemma 2.4, there exist λ̄ ∈ [λ, λ^{∗}], ρ_{0} ∈ (0, 1) satisfying

λf(x + ρ) < λ̄f(x), x ∈ [0, R(λ)], ρ ∈ (0, ρ_{0}).

Let xλ̄ be a solution of the BVP (1.1)λ̄. Suppose x̄_{λ}(t) = xλ̄ + ρ, ρ ∈ (0, ρ_{0}). Then

x̄_{λ}(t) = xλ̄ + ρ
= ∫_{0}^{1} G(t, s)λ̄g(s)f(xλ̄(s))ds + ρ
≥ ρ + ∫_{0}^{1} G(t, s)λg(s)f(xλ̄(s) + ρ)ds
≥ ρh(t) + ∫_{0}^{1} G(t, s)λg(s)f((xλ̄(s) + ρ))ds
= T_{λ}^{ρ}x̄_{λ}(t)

Combining this with ax̄(0) − b lim_{t→0^{+}} p(t)x̄′(t) ≥ 0, cx̄(1) + d lim_{t→1^{−}} p(t)x̄′(t) ≥ ρ, we

have that x̄_{λ}(t) is an upper solution of the BVP (2.6)_{λ}^{ρ}. Therefore the BVP (2.6)_{λ}^{ρ} has solution and let v_{λ}(t) be a solution of the BVP (2.6)_{λ}^{ρ}. Let Ω = {y ∈ Q|y(t) < v_{λ}(t), t ∈ [0, 1]}. It is clear that Ω ⊂ Q is a bounded open set. If y ∈ ∂Ω, then there exists t_{0} ∈ [0, 1], such that y(t_{0}) = v_{λ}(t_{0}). Therefore, for any μ ≥ 1, ρ ∈ (0, ρ_{0}), y ∈ ∂Ω, we have

T_{λ}^{0}y(t_{0}) < ρh(t) + T_{λ}^{0}y(t_{0})
= ρh(t) + T_{λ}^{0}v_{λ}(t_{0})
= T_{λ}^{ρ}v_{λ}(t_{0})
= v_{λ}(t_{0})
= y(t_{0})
≤ μy(t_{0}).

Hence for any μ ≥ 1, we have T_{λ}^{0}y ≠ μy, y ∈ ∂Ω. Therefore by Lemma 1.1 we have

i(T_{λ}^{0}, Ω, Q) = 1.  (3.2)

It remains to prove that the conditions of Lemma 1.2 are held.

Firstly, we check the condition (1) of Lemma 1.2 is satisfied. In fact, for any x ∈ Q, by (H_{4}) and (2.5) we have

\begin{matrix}T_{λ}^{0}x(\frac{1}{2}) = ∫_{0}^{1} G(\frac{1}{2}, s)λp(s)g(s)f(x(s))ds \\ ≥ ∫_{θ}^{1−θ} G(\frac{1}{2}, s)λp(s)g(s)δ̄σ_{0}^{m}||x||^{m}ds \\ = ||x||^{m} ∫_{θ}^{1−θ} G(\frac{1}{2}, s)λp(s)g(s)δ̄σ_{0}^{m}ds \\ = ||x||^{m−1} ∫_{θ}^{1−θ} G(\frac{1}{2}, s)λp(s)g(s)δ̄σ_{0}^{m}ds||x|| \end{matrix}  (3.3)

Taking R̄ > 0, such that R̄^{m−1} ∫_{θ}^{1−θ} G(\frac{1}{2}, s)λp(s)g(s)δ̄σ_{0}^{m}ds > 1. Therefore, for any R > R̄ and B_{R} ⊂ Q, we have by (3.3)

||T_{λ}^{0}x|| > ||x|| > 0, x ∈ ∂B_{R},  (3.4)

where B_{R} = {x ∈ Q|||x|| < R}. Hence the condition (1) of Lemma 1.2 is held.

Now we prove the condition (2) of Lemma 1.2 is satisfied. In fact, if the condition (2) of Lemma 1.2 is not held, then there exist x_{1} ∈ Q ⋂ ∂B_{R}, 0 < μ_{1} ≤ 1, such that T_{λ}^{0}x_{1} = μ_{1}x_{1}. Therefore ||T_{λ}^{0}x_{1}|| ≤ ||x_{1}||. This conflicts with (3.4). Hence the condition (2) of Lemma 1.2 is satisfied. By Lemma 1.2 we have

i(T_{λ}^{0}, B_{R}, Q) = 0.  (3.5)

Consequently, by the additivity of the fixed point index,

0 = i(T_{λ}^{0}, B_{R}, Q) = i(T_{λ}^{0}, Ω, Q) + i(T_{λ}^{0}, B_{R} \ Ω̄, Q).

Since i(T_{λ}^{0}, Ω, Q) = 1, i(T_{λ}^{0}, B_{R}Ω̄, Q) = −1. Therefore, by the solution property of the fixed point index, there is a fixed point of T_{λ}^{0} in Ω and a fixed point of T_{λ}^{0} in B_{R} \ Ω̄, respectively. Therefore the BVP (1.1)_{λ} by Lemma 2.1 has at least two solutions. Furthermore, the BVP (1.1)_{λ} has at least two positive solutions by (H_{1}) − (H_{3}). The proof of Theorem 3.1 is complete. Example. consider the following BVP

\{ \begin{matrix}\sqrt{t}(1 − t)(\frac{1}{\sqrt{t}(1−t)}x′(t))′ + λ\frac{1−t}{\sqrt[3]{t}}2^{2x} = 0, 0 < t < 1, \\ x(0) = x(1) = 0. \end{matrix}  (3.6)_{λ}

where λ > 0. It is clear that the BVP (3.6)_{λ} is not resolved by the results of [1-15].

Let p(t) = \frac{1}{\sqrt{t}(1−t)}, g(t) = \frac{1−t}{\sqrt[3]{t}}, f(x) = 2^{2x}, 0 < t < 1, a = c = 1, b = d = 0. It is clearly that p(t), g(t) are singular at t = 0 and/or at t = 1 respectively. It is easy to prove that (H_{1}) and (H_{2}) hold. By calculation, we obtain the Green^{,}s function of the BVP (3.6)_{λ}:

G(t, s) = \{ \begin{matrix}\frac{1}{15}s^{\frac{3}{2}} (5 − 3s)(2 − 5t^{\frac{3}{2}} + 3t^{\frac{5}{2}} ), 0 ≤ s ≤ t ≤ 1, \\ \frac{1}{15}t^{\frac{3}{2}} (5 − 3t)(2 − 5t^{\frac{3}{2}} + 3s^{\frac{5}{2}} ), 0 ≤ t ≤ s ≤ 1. \end{matrix}

It is easy to see that 0 ≤ G(s, s) ≤ 1. In addition, for δ̄ = 1 > 0, m = 2, f(x) = 2^{2x} = δ̄2^{2x} > x^{2} = x^{m} > 0. Hence (H_{3}) and (H_{4}) are held.

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(Received June 22, 2006)

Meiqiang Feng Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, PR China Department of Fundamental Sciences, Beijing Information Technology Institute, Beijing 100101, PR China E-mail address: meiqiangfeng@sina.com Xuemei Zhang Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, PR China E-mail address:zxm74@sina.com Weigao Ge Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, PR China E-mail address:gew@bit.edu.cn