Existence of positive solutions for fourth-order m -point boundary value problems with a one-dimensional p -Laplacian operator

Nonlinear Analysis 71 (2009) 2985–2996

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Existence of positive solutions for fourth-order m-point boundary value problems with a one-dimensional p-Laplacian operator Jingbao Yang a,c,∗ , Zhongli Wei b,c a

Department of Sciences, Bozhou Teachers College, Mengcheng, Anhui 233500, China

b

School of Sciences, Shandong Jianzhu University, Jinan, Shandong 250101, China

c

School of Mathematics, Shandong University, Jinan, Shandong 250100, China

article

abstract

info

Article history: Received 18 June 2007 Accepted 23 January 2009

This paper discusses the existence of positive solutions for fourth-order m-point boundary value problems with a one-dimensional p-Laplacian operator

MSC: 34B15 34B16 34B18 Keywords: Multi-point boundary value problems Krasnoselskii’s fixed point theorem One-dimensional p-Laplacian operator Lower semi-continuous functions

 (φp (u00 (t )))00 − g (t )f (u(t )) = 0,    u00 (0) = u00(1) = 0,    m −2  X  au(0) − bu0 (0) = ai u(ξi ),  i=1   m −2  X   0  bi u(ξi ), cu(1) + du (1) =

t ∈ (0, 1),

i=1

where φp (s) = |s| s, p > 1, f is a lower semi-continuous function. By using Krasnoselskii’s fixed point theorems in a cone, the existence of one positive solution and multiple positive solutions for nonlinear singular boundary value problems is obtained. © 2009 Elsevier Ltd. All rights reserved. p−2

1. Introduction In [1], Il’in and Moiseev studied the existence of solutions for a linear multi-point boundary value problem. Motivated by the study of Il’in and Moiseev [1], Gupta [2] studied certain three-point boundary value problems for nonlinear ordinary differential equations. Since then, more general nonlinear multi-point boundary value problems have been studied by several authors because multi-point boundary value problems describe many phenomena of applied mathematics and physics. For example, the vibrations of a guy wine of a uniform cross-section and composed of N parts of different densities can be set up as a multi-point boundary value problem (see [3]) and many problems in the theory of elastic stability can be handled by the method of multi-point problems (see [4]). Motivated by [5,7–10], in this paper, we investigate the existence of positive solutions for fourth-order singular m-point boundary value problems (BVP for short) with a one-dimensional p-Laplacian operator

 (φp (u00 (t )))00 − g (t )f (u(t )) = 0,   u00 (0) = u00 (1) = 0,    m −2  X  au(0) − bu0 (0) = ai u(ξi ),  i =1   m −2  X   0  bi u(ξi ), cu(1) + du (1) =

t ∈ (0, 1),

i=1

∗

Corresponding author at: Department of Sciences, Bozhou Teachers College, Mengcheng, Anhui 233500, China. E-mail addresses: [email protected], [email protected] (J. Yang), [email protected] (Z. Wei).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.191

(1.1)

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J. Yang, Z. Wei / Nonlinear Analysis 71 (2009) 2985–2996

where φp (s) is a p-Laplacian operator, i.e. φp (s) = |s|p−2 s, p > 1, (φp )−1 = φq ( 1p + 1q = 1); m > 2 (m ∈ N); a ≥ 0, b ≥ 0, c ≥ 0, d ≥ 0, ρ = ac + bc + ad > 0; ξi ∈ (0, 1), ai , bi ∈ (0, +∞) (i = 1, 2, . . . , m − 2); g (t ) ∈ C ((0, 1), (0, ∞)) R1 and 0 < 0 g (t )dt < +∞; f is a nonnegative, lower semi-continuous function defined on [0, +∞). We show that the BVP (1.1) has positive solutions by using Krasnoselskii’s fixed point theorems in a cone. This paper is organized as follows. In Section 2, we present the expression and properties of Green’s function associated with the corresponding differential equation and the boundary value conditions. Then we give some preliminaries about the operator and Krasnoselakii’s fixed point theorem. In Section 3, we give the existence of one positive solution for the BVP (1.1) by using Krasnoselakii’s fixed point theorem. In Section 4, we give the existence of multiple positive solutions for the BVP (1.1). In Section 5 we give two examples as applications. 2. Preliminaries The basic space used in this paper is a real Banach space C [0, 1] with the norm k · k defined by kuk = maxt ∈[0,1] |u(t )|. For convenience, we make the following assumptions: (H1 ): f ∈ C ([0, +∞), [0, +∞)); (H∗1 ): f is a nonnegative, lower semi-continuous function defined on [0, +∞), i.e. ∃I ⊂ [0, +∞), ∀xn ∈ I , xn → x0 (n → ∞), one has f (x0 ) ≤ limn→∞ f (xn ). Moreover, f has only a finite number of discontinuity points in each compact subinterval of [0, +∞); R1 (H2 ): g (t ) ∈ C ((0, 1), [0, +∞)) and 0 < 0 g (t )dt < +∞. Moreover, g (t ) does not vanish identically on any subinterval of [0, 1]; c ≥ 0, d ≥ 0, ρ = ac + bc + ad > 0; ξi ∈ (0, 1), ai , bi ∈ (0, +∞) (i = 1, 2, . . . , m − 2); ρ − Pm(−H23 ): a ≥ 0, b ≥ 0, P m−2 ϕ(ξ ) > 0 , ρ − a i i i=1 i=1 bi ψ(ξi ) > 0, ∆ < 0, where

m−2 X − ai ψ(ξi ) i=1 ∆= m −2 X ρ − bi ψ(ξi )

m−2

ρ−

X

i=1 m−2

−

i=1

X i=1



ai ϕ(ξi )

bi ϕ(ξi )

and

ψ(t ) = b + at ,

ϕ(t ) = c + d − ct ,

t ∈ [0, 1]

(2.1)

are linearly independent solutions of the equation x (t ) = 0, t ∈ [0, 1]. Obviously, ψ is non-decreasing on [0, 1] and ϕ is non-increasing on [0, 1]. Let y(t ) = φp (u00 (t )), then the BVP 00



(φp (u00 (t )))00 − g (t )f (u(t )) = 0, x00 (0) = u00 (1) = 0,

t ∈ (0, 1),

is turned into the BVP



y00 (t ) − g (t )f (u(t )) = 0, y(0) = y(1) = 0.

t ∈ (0, 1)

(2.2)

Lemma 2.1. If (H1 ) and (H2 ) hold, then the BVP (2.2) has a unique solution y(t ) = −

1

Z

h(t , s)g (s)f (u(s))ds,

(2.3)

0

where h(t , s) =



t (1 − s), s(1 − t ),

0 ≤ t ≤ s ≤ 1, 0 ≤ s ≤ t ≤ 1.

(2.4)

By calculation, it is easy to prove that Lemma 2.1 holds. So we omit its proof here. It is evident that the Green’s function h(t , s) has the following properties. Proposition 2.1. For t , s ∈ [0, 1], we have 0 ≤ h(t , s) ≤ h(s, s) ≤

1 4

.

(2.5)

J. Yang, Z. Wei / Nonlinear Analysis 71 (2009) 2985–2996

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Proposition 2.2. Let θ ∈ (0, 21 ), then for all t ∈ [θ , 1 − θ ], s ∈ [0, 1], we have h(t , s) ≥ θ h(s, s).

(2.6)

Proof. For θ ≤ t ≤ s < 1, by (2.4), we have h(t , s) h(s, s)

=

t ( 1 − s) s(1 − s)

=

t s

≥ t ≥ θ.

On the other hand, for 0 < s ≤ t ≤ 1 − θ , by (2.4), we also have h(t , s) h(s, s)

=

s(1 − t ) s(1 − s)

=

1−t 1−s

≥ 1 − t ≥ θ.

If s = 0 or s = 1, it is obvious that (2.6) holds, too. The proof of Proposition 2.3 is complete.  Lemma 2.2. If (H3 ) holds, then for y ∈ C [0, 1], the BVP

 00 u (t ) = φq (y(t )), t ∈ [0, 1],   m −2  X   au(0) − bu0 (0) = ai u(ξi ),

(2.7)

i =1  m −2  X   0  bi u(ξi ), cu(1) + du (1) = i =1

has a unique solution u( t ) = −

1

Z



G(t , s)φq (y(s))ds + A(φq (y))ψ(t ) + B(φq (y))ϕ(t ) ,

(2.8)

0

where

ψ(s)ϕ(t ), 0 ≤ s ≤ t ≤ 1, ρ ψ(t )ϕ(s), 0 ≤ t ≤ s ≤ 1, m−2 Z 1 X ai G(ξi , s)φq (y(s))ds 1 i=1 0 A(φq (y)) = X −2 Z 1 ∆ m bi G(ξi , s)φq (y(s))ds G(t , s) =

1



0

i =1

m−2 X − ai ψ(ξi ) 1 i=1 B(φq (y)) = m −2 X ∆ ρ − bi ψ(ξi ) i=1

m −2

X

0

i=1

1

Z bi 0

m−2

ρ−

X

, bi ϕ(ξi )

i=1 m−2

−

X i=1



ai ϕ(ξi )

(2.10)



G(ξi , s)φq (y(s))ds

ai

i =1 m−2

X

1

Z

(2.9)

. G(ξi , s)φq (y(s))ds

(2.11)

The proof of Lemma 2.2 is similar to that of Lemma 5.5.1 in [5], so we omit it here. Remark 2.1. For fixed integrable function y, it is obvious that A(φq (y)) and B(φq (y)) are constants. For convenience, let

  ϕ(1 − θ ) ψ(θ ) α = max{1, kψk, kϕk}, β = min , , ψ(1)   ϕ(0) n γo γ = min min ϕ(t ), min ψ(t ), 1 , Γ = min β, . t ∈[θ,1−θ] t ∈[θ,1−θ] α Proposition 2.3. For t , s ∈ [0, 1], we have 0 ≤ G(t , s) ≤ G(s, s) ≤

α2 . ρ

(2.12)

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J. Yang, Z. Wei / Nonlinear Analysis 71 (2009) 2985–2996

Proof. By the monotonicity of ϕ and ψ , it is evident that (2.12) holds.



Proposition 2.4. For t ∈ [θ , 1 − θ ], we have G(t , s) ≥ β G(s, s),

s ∈ [0, 1].

(2.13)

Proof. For t ∈ [θ , 1 − θ ], s ∈ [0, 1], we have G(t , s)

n ψ(θ ) ϕ(1 − θ ) o n ψ(θ ) ϕ(1 − θ ) o ≥ min , ≥ min , = β, G(s, s) ψ(s) ϕ(s) ψ(1) ϕ(0)

i.e. G(t , s) ≥ β G(s, s).



From Lemmas 2.1 and 2.2, we know that u(t ) is a solution of the BVP (1.1) if and only if u(t ) =

1

Z

G(t , s)φq (W (s))ds + A(φq (W ))ψ(t ) + B(φq (W ))ϕ(t ),

(2.14)

0

where W (s) =

R1

h(s, τ )g (τ )f (u(τ ))dτ .

0

Lemma 2.3. Suppose that (H1 ), (H2 ) and (H3 ) hold, then the solution of the BVP (1.1) satisfies (i) u(t ) ≥ 0 for t ∈ [0, 1] and (ii) mint ∈[θ,1−θ ] u(t ) ≥ Γ kuk. Proof. (i) For (t , s) ∈ [0, 1] × [0, 1], by (2.3)–(2.4), (2.8)–(2.11) and the property of function φq , it is obvious that we have G(t , s) ≥ 0,

φq (W (s)) ≥ 0,

A(φq (W )) ≥ 0,

B(φq (W )) ≥ 0,

so we get u(t ) ≥ 0. (ii) From (2.12)–(2.14), for t ∈ [θ , 1 − θ], we have u(t ) =

1

Z

G(t , s)φq (W (s))ds + A(φq (W ))ψ(t ) + B(φq (W ))ϕ(t ) 0

≥β

1

Z

G(s, s)φq (W (s))ds + A(φq (W ))ψ(t ) + B(φq (W ))ϕ(t ) 0

γ · α · [A(φq (W )) + B(φq (W ))] α 0 Z 1  ≥Γ G(s, s)φq (W (s))ds + α[A(φq (W )) + B(φq (W ))] ≥β

1

Z

G(s, s)φq (W (s))ds +

0

≥ Γ kuk. Therefore, we get mint ∈[θ,1−θ] u(t ) ≥ Γ kuk.



Let K = {u(t ) ∈ C [0, 1] | u(t ) ≥ 0, mint ∈[θ,1−θ] u(t ) ≥ Γ kuk}, then it is clear that K is a cone in C [0, 1]. Define an operator T in K by

(Tu)(t ) =

1

Z

G(t , s)φq (W (s))ds + A(φq (W ))ψ(t ) + B(φq (W ))ϕ(t ),

(2.15)

0

R1

where W (s) = 0 h(s, τ )g (τ )f (u(τ ))dτ . Evidently, u(t ) is a solution of the BVP (1.1) if and only if u(t ) is a fixed point of operator T . Lemma 2.4. Suppose that the conditions (H1 ), (H2 ) and (H3 ) hold, then T (K ) ⊂ K and T : K → K is completely continuous. Proof. For any u ∈ K , by (2.15), we obtain (Tu)(t ) ≥ 0 and, for t ∈ [0, 1],

(Tu)(t ) =

1

Z

G(t , s)φq (W (s))ds + A(φq (W ))ψ(t ) + B(φq (W ))ϕ(t ) 0 1

Z

G(s, s)φq (W (s))ds + α[A(φq (W )) + B(φq (W ))].

≤ 0

Thus kTuk ≤

R1 0

G(s, s)φq (W (s))ds + α[A(φq (W )) + B(φq (W ))].

J. Yang, Z. Wei / Nonlinear Analysis 71 (2009) 2985–2996

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On the other hand, for t ∈ [θ , 1 − θ ], we have

(Tu)(t ) =

1

Z

G(t , s)φq (W (s))ds + A(φq (W ))ψ(t ) + B(φq (W ))ϕ(t ) 0

≥β

1

Z

G(s, s)φq (W (s))ds + A(φq (W ))ψ(t ) + B(φq (W ))ϕ(t ) 0

γ · α · [A(φq (W )) + B(φq (W ))] α 0  Z 1 G(s, s)φq (W (s))ds + α[A(φq (W )) + B(φq (W ))] ≥Γ ≥β

Z

1

G(s, s)φq (W (s))ds +

0

≥ Γ kTuk. Therefore we get T (K ) ⊂ K . By conventional arguments and Ascoli–Arzela theorem, one can prove T : K → K is completely continuous, so we omit it here.  To obtain positive solutions of the BVP (1.1), the following well-known Krasnoselskii’s fixed point theorem in a cone is very useful. Theorem 2.1 (Krasnoselskii’s Fixed Point Theorem (See [6])). Let E be a real Banach space, and K ⊂ E be a cone. Assume that Ω1 and Ω2 are two bounded open sets in E such that 0 ∈ Ω1 , Ω 1 ⊂ Ω2 , and let T : K ∩ ( Ω 2 \ Ω1 ) → K be completely continuous. Suppose that one of two conditions (i) kTuk ≤ kuk, u ∈ K ∩ ∂ Ω1

and kTuk ≥ kuk, u ∈ K ∩ ∂ Ω2

and (ii)

kTuk ≥ kuk, u ∈ K ∩ ∂ Ω1 and kTuk ≤ kuk, u ∈ K ∩ ∂ Ω2

is satisfied. Then T has at least one fixed point in K ∩ (Ω 2 \ Ω1 ). 3. The existence of one positive solution We define Ωl = {u ∈ K | kuk < l}, ∂ Ωl = {u ∈ K | kuk = l}, where l > 0. If u ∈ ∂ Ωl , for t ∈ [θ , 1 − θ ], we have Γ l ≤ u ≤ l. For convenience, we introduce the following notations. Let

   f (u) l u ∈ [Γ l, l] , u ∈ [0, l] , f = sup φp (l) φp (l) f (u) fµ = lim inf , (µ := 0+ or + ∞), u→µ φp (u) f (u) f µ = lim sup , (µ := 0+ or + ∞), φp (u) u→µ   q −1  2 Z 1   1 1 α e e = × φq g (τ )dτ + α A + α B , ω 4 ρ 0  Z 1−θ   1 2(q−1) q −1 β b b =θ (1 − θ ) × ψ(θ )ϕ(1 − θ )φq g (τ )dτ + γ A + γ B , M ρ θ 

fl = inf

f (u)

where e A and e B defined by (3.1) and (3.2), b A and b B defined by (3.3) and (3.4), respectively. From correlative definition, it is easy to get M > ω. We always assume that (H∗1 ), (H2 ) and (H3 ) hold in the following theorems. Theorem 3.1. Suppose that there exist constants r , R > 0 with r < Γ R for r < R (or MR < ωr for R < r), such that the following two conditions

(H4 ): f r ≤ φp (ω)

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J. Yang, Z. Wei / Nonlinear Analysis 71 (2009) 2985–2996

and

(H5 ): fR ≥ φp (M ) hold. Then the BVP (1.1) has at least one positive solution u ∈ K such that 0 < r ≤ kuk ≤ R

(or 0 < R ≤ kuk ≤ r ).

Proof. Case 1. We shall prove that the result holds when (H1 ) is satisfied. Without loss of generality, we suppose that r < Γ R for r < R. By (H4 ), (2.5), (2.10) and (2.11), for u ∈ ∂ Ωr , we have m−2

m−2 Z 1  Z 1 X g (τ )dτ ds G(ξi , s)φq ai ( 1 )q−1 ωr i=1 0 0 |A(φq (W ))| ≤ 4 m  Z Z −2 1 1 X ∆ g (τ )dτ ds G(ξi , s)φq bi 1

=

4

0

0

i =1

 q−1

ρ− ai ϕ(ξi ) i=1 m −2 X − bi ϕ(ξi ) X

i=1

ωre A

(3.1)

and

m−2 X − ai ψ(ξi ) 1 q −1 ( 4 ) ωr i=1 |B(φq (W ))| ≤ m −2 X ∆ ρ − bi ψ(ξi ) i=1

 q−1 1

=

4

m−2

1

1

ai G(ξi , s)φq g (τ )dτ ds 0 0 i =1 Z 1  m −2 Z 1 X bi G(ξi , s)φq g (τ )dτ ds X

i=1

Z

Z

0



0

ωre B.

(3.2)

Therefore, by (H4 ), (2.5), (2.12), (2.15), (3.1) and (3.2), for t ∈ [0, 1] and u ∈ ∂ Ωr , we have

(Tu)(t ) =

1

Z

G(t , s)φq (W (s))ds + A(φq (W ))ψ(t ) + B(φq (W ))ϕ(t ) 0

Z 1   q−1   q −1 α2 1 1 e ≤ rω × × φq g (τ )dτ + ωr Aψ(t ) + ωre Bϕ(t ) 4 ρ 4 4 0   q −1  2 Z 1   1 α ≤ rω × φq g (τ )dτ + αe A + αe B 4 ρ 0 = r = kuk.  q−1 1

This implies that kTuk ≤ kuk for u ∈ ∂ Ωr . On the other hand, by (H5 ), (2.6), (2.10) and (2.11), for u ∈ ∂ ΩR , we have

m−2 Z 1−θ Z 1−θ  X a G (ξ , s )φ g (τ ) d τ ds i i q θ 2(q−1) (1 − θ )q−1 MR i=1 θ θ |A(φq (W ))| ≥ m Z 1−θ  −2 Z 1−θ X ∆ b G (ξ , s )φ g (τ ) d τ ds i i q i=1

=θ

2(q−1)

(1 − θ )

q −1

θ

m−2

ρ−

X

i=1 m−2

−

X

θ

i =1



ai ϕ(ξi )

bi ϕ(ξi )

MRb A

(3.3)

and

m−2 X − ai ψ(ξi ) 2(q−1) q −1 θ (1 − θ ) MR i=1 |B(φq (W ))| ≥ m −2 X ∆ ρ − bi ψ(ξi ) i=1

=θ

2(q−1)

(1 − θ )

q−1

MRb B.

m−2

X i =1 m−2

X i=1

1−θ

Z ai

θ 1−θ

Z bi

G(ξi , s)φq

θ

G(ξi , s)φq

1−θ

Z

g (τ )dτ

θ 1−θ

Z θ





ds

g (τ )dτ ds 

(3.4)

J. Yang, Z. Wei / Nonlinear Analysis 71 (2009) 2985–2996

2991

Therefore, by (H5 ), (2.6), (2.13), (2.15), (3.3) and (3.4), for t ∈ [0, 1] and u ∈ ∂ ΩR , we have

(Tu)(t ) =

1

Z

G(t , s)φq (W (s))ds + A(φq (W ))ψ(t ) + B(φq (W ))ϕ(t ) 0

≥ θ 2(q−1) (1 − θ )q−1 MR



β × ψ(θ )ϕ(1 − θ )φq ρ

1−θ

Z

g (τ )dτ

θ



 + γb A + γb B

= R = kuk. This implies that kTuk ≥ kuk for u ∈ ∂ ΩR . Therefore, by Theorem 2.1, it follows that T has a fixed-point u in K ∩ (Ω R \ Ωr ). This means that the BVP (1.1) has at least one positive solution u ∈ K such that 0 < r ≤ kuk ≤ R. Case 2. When (H∗1 ) holds, by applying the linear approaching method on the domain of discontinuous points of f we can establish sequence{fj }∞ j=1 satisfying the following two conditions (i) fj ∈ C [0, ∞) and 0 ≤ fj ≤ fj+1 on [0, ∞) and (ii) limj→∞ fj = f , j = 1, 2, . . ., is pointwisely convergent on [0, ∞). By virtue of proof of situation 1, we know that when f = fj , the BVP (1.1) has a positive solution uj (t ), where uj (t ) =

1

Z

G(t , s)φq 0

1

Z

h(s, τ )g (τ )fj (uj (τ ))dτ

 ds

0

m−2 Z 1 Z 1  X a G (ξ , s )φ h ( s , τ ) g (τ ) f ( u (τ )) d τ ds i i q j j ψ(t ) i=1 0 0 + X Z 1  −2 Z 1 ∆ m b G (ξ , s )φ h ( s , τ ) g (τ ) f ( u (τ )) d τ ds i i q j j i=1

0

0

m−2

ρ−

X

i =1 m−2

−

X i=1



ai ϕ(ξi )

bi ϕ(ξi )

m−2 Z 1  m −2 Z 1 X X − a ψ(ξ ) a G (ξ , s )φ h ( s , τ ) g (τ ) f ( u (τ )) d τ ds i i i i q j j ϕ(t ) 0 0 i=1 i =1 + Z 1  m −2 Z 1 m −2 X X ∆ ρ − bi ψ(ξi ) bi G(ξi , s)φq h(s, τ )g (τ )fj (uj (τ ))dτ ds 0 0 i=1 i=1 Z 1  Z 1 = G(t , s)φq h(s, τ )g (τ )fj (uj (τ ))dτ ds + Aj · ψ(t ) + Bj · ϕ(t ) 0

0

for all t ∈ [0, 1] and r ≤ kuj k ≤ R, r , R are independent of j. By uniform continuity of G(t , s) on [0, 1] × [0, 1], ψ(t ) and ϕ(t ) on [0, 1], for any ε > 0 (adequate small), there exists δ > 0 such that for t1 , t2 ∈ [0, 1] and |t1 − t2 | < δ , one has |G(t1 , s) − G(t2 , s)| < ε, |ψ(t1 ) − ψ(t2 )| < ε and |ϕ(t1 ) − ϕ(t2 )| < ε. Thus, for t1 , t2 ∈ [0, 1] and |t1 − t2 | < δ , one has

|uj (t1 ) − uj (t2 )| Z Z 1 ≤ |G(t1 , s) − G(t2 , s)| · φq 0

1 4

h(s, τ )g (τ )fj (uj (τ ))dτ



ds + Aj |ψ(t1 ) − ψ(t2 )| + Bj |ϕ(t1 ) − ϕ(t2 )|

0

  q −1 ≤

1

1

Z

· max fj (uj ) · φq kuj k≤R

g (τ )dτ



· ε + Aj · ε + Bj · ε.

0

So we get that {uj }∞ j=1 are equicontinuous on [0, 1]. Thus, by Arzela–Asoli theorem, we know that there exists a convergent ∞ subsequence of {uj }∞ j=1 . For convenience, we denote this convergent subsequence with {uj }j=1 . Without loss of generality, we suppose limj→∞ uj (t ) = u(t ), ∀t ∈ [0, 1], and r ≤ kuk ≤ R. By Fatou Lemma and Lebesgue dominated convergence theorem, we have lim uj (t ) ≥

j→∞

1

Z

G(t , s)φq 0

1

Z

h(s, τ )g (τ ) lim fj (uj (τ ))dτ 0

j→∞

 ds

m−2 Z 1 Z 1  X a G (ξ , s )φ h ( s , τ ) g (τ ) lim f ( u (τ )) d τ ds i i q j j j→∞ ψ(t ) i=1 0 0 + X Z 1  −2 Z 1 ∆ m b G (ξ , s )φ h ( s , τ ) g (τ ) lim f ( u (τ )) d τ ds i i q j j j→∞ i=1

0

0

m−2

ρ−

X

i =1 m−2

−

X i=1



ai ϕ(ξi )

bi ϕ(ξi )

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J. Yang, Z. Wei / Nonlinear Analysis 71 (2009) 2985–2996

m−2 X − ai ψ(ξi ) ϕ(t ) i=1 + m −2 X ∆ ρ − bi ψ(ξi )

m−2

1

1

h(s, τ )g (τ ) lim fj (uj (τ ))dτ ds G(ξi , s)φq ai j →∞ 0 0 i=1  Z 1 m −2 Z 1 X h(s, τ )g (τ ) lim fj (uj (τ ))dτ ds G(ξi , s)φq bi j→∞ Z

X

i=1

Z

0

0

i=1



i.e. u(t ) ≥

1

Z

G(t , s)φq (W (s))ds + A(φq (W ))ψ(t ) + B(φq (W ))ϕ(t ),

(3.5)

0

R1

where W (s) = uj (t ) ≤

h(s, τ )g (τ )f (u(τ ))dτ . On the other hand, by the conditions (i) and (ii), we have

0 1

Z

G(t , s)φq

1

Z

h(s, τ )g (τ )f (uj (τ ))dτ

 ds

0

0

m−2 Z 1  Z 1 X h ( s , τ ) g (τ ) f ( u (τ )) d τ ds G (ξ , s )φ a j i q i ψ(t ) i=1 0 0 + X Z 1  −2 Z 1 ∆ m b G (ξ , s )φ h ( s , τ ) g (τ ) f ( u (τ )) d τ ds i i q j 0

i=1

0

m−2 X − ai ψ(ξi ) ϕ(t ) i=1 + m −2 X ∆ ρ − bi ψ(ξi ) i=1

m−2

1

m−2

ρ− ai ϕ(ξi ) i=1 m −2 X − bi ϕ(ξi ) X

i=1

1

ai G(ξi , s)φq h(s, τ )g (τ )f (uj (τ ))dτ ds 0 0 i =1 Z 1  . m −2 Z 1 X bi G(ξi , s)φq h(s, τ )g (τ )f (uj (τ ))dτ ds Z

X

Z

0

i=1



0

By the lower semi-continuity of f , taking limits in above inequality as j → ∞, we have u(t ) ≤

1

Z

G(t , s)φq 0

1

Z

h(s, τ )g (τ )f (u(τ ))dτ

 ds

0

m−2 Z 1 Z 1  X a G (ξ , s )φ h ( s , τ ) g (τ ) f ( u (τ )) d τ ds i i q ψ(t ) i=1 0 0 + X Z 1  −2 Z 1 ∆ m b G (ξ , s )φ h ( s , τ ) g (τ ) f ( u (τ )) d τ ds i i q 0

i=1

m−2 X − ai ψ(ξi ) ϕ(t ) i=1 + m −2 X ∆ ρ − bi ψ(ξi ) i=1

0

m −2

X

i=1

G(ξi , s)φq

ai 0

i =1 m−2

X

1

Z

G(ξi , s)φq

bi 0

X

i=1 m−2

−

X i =1

bi ϕ(ξi )

h(s, τ )g (τ )f (u(τ ))dτ 1

Z 0



ai ϕ(ξi )

1

Z 0

1

Z

m−2

ρ−





ds

h(s, τ )g (τ )f (u(τ ))dτ ds 

i.e. u(t ) ≤

1

Z

G(t , s)φq (W (s))ds + A(φq (W ))ψ(t ) + B(φq (W ))ϕ(t ),

(3.6)

0

R1

where W (s) = 0 h(s, τ )g (τ )f (u(τ ))dτ . By (3.5) and (3.6), we have u(t ) =

1

Z

G(t , s)φq (W (s))ds + A(φq (W ))ψ(t ) + B(φq (W ))ϕ(t ), 0

R1

where W (s) = 0 h(s, τ )g (τ )f (u(τ ))dτ . Therefore u(t ) is a positive solution of the BVP (1.1). This completes the proof of Theorem 3.1.



Remark 3.1. It is obvious that the condition r < Γ R for r < R (or MR < ωr for R < r) is a necessary condition when (H4 ) and (H5 ) hold. Similarly, we can obtain the following conclusion.

J. Yang, Z. Wei / Nonlinear Analysis 71 (2009) 2985–2996

2993

Theorem 3.1∗ . Assume that there exist constants r , R > 0 with r < Γ R for r < R (or MR < ωr for R < r), such that the following conditions

(H4 )∗ : f r < φp (ω) and

(H5 )∗ : fR > φp (M ) hold. Then the BVP (1.1) has at least one positive solution u ∈ K such that 0 < r < kuk < R (or 0 < R < kuk < r ). Theorem 3.2. Assume that one of the following two conditions

(H6 ): f 0 ≤ φp (ω),

f∞ ≥ φp



M



Γ

and

(H7 ): f0 ≥ φp



M



Γ

,

f ∞ ≤ φp (ω)

is satisfied. Then the BVP (1.1) has at least one positive solution. Proof. All we need to do is to prove that the results of Theorem 3.2 hold when f is nonnegative and continuous on [0, ∞). And by the similar proof process of Theorem 3.1 we can prove the results of Theorem 3.2 when f is nonnegative and lower semi-continuous on [0, ∞). We show that (H6 ) implies (H4 ) and (H5 ). Suppose that (H6 ) holds, then there exist r and R with 0 < r < Γ R, such that f (u) ≤ φp (ω), φp (u)

0
and f (u)

φp (u)

≥ φp



M



Γ

,

u ≥ Γ R.

Hence, we obtain f (u) ≤ φp (ω)φp (u) ≤ φp (ω)φp (r ) = φp (r ω),

0
and f (u) ≥ φp



M



Γ

φp (u) ≥ φp



M



Γ

φp (Γ R) = φp (RM ),

u ≥ Γ R.

Hence, (H4 ) and (H5 ) holds. Therefore, by Theorem 3.1, the BVP (1.1) has at least one positive solution. Now suppose that (H7 ) holds, then there exist 0 < r < R with Mr < ωR such that f (u)

φp (u)

≥ φp



M

Γ



,

0 < u ≤ r,

(3.7)

and f (u)

φp (u)

≤ φp (ω),

u ≥ R.

(3.8)

By (3.7), it follows that f (u) ≥ φp



M

Γ



φp (u) ≥ φp



M

Γ



φp (Γ r ) = φp (rM ),

Γ r ≤ u ≤ r.

So, the condition (H5 ) holds for r. For (3.8), we consider two cases. (i) If f (u) is bounded, there exists a constant D >n0 such that fo(u) ≤ D, for 0 ≤ u < ∞. By (3.8), there exists a constant

λ ≥ R with Mr < ωR ≤ λω satisfying φp (λ) ≥ max φp (R), φpD(ω) such that f (u) ≤ D ≤ φp (λω) for 0 ≤ u ≤ λ. This means that the condition (H4 ) holds for λ.

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J. Yang, Z. Wei / Nonlinear Analysis 71 (2009) 2985–2996

(ii) If f (u) is unbounded, there exist λ1 ≥ R with Mr < ωR ≤ λ1 ω such that f (u) ≤ f (λ1 ) for 0 ≤ u ≤ λ1 . This yields f (u) ≤ f (λ1 ) ≤ φp (λ1 ω) for 0 ≤ u ≤ λ1 . Thus, the condition (H4 ) holds for λ1 . Therefore, by Theorem 3.1, the BVP (1.1) has at least one positive solution. Theorem 3.2 is proved.  Remark 3.2. It is evident that Theorem 3.2 holds if f satisfies conditions f 0 = 0, f∞ = +∞ or f0 = +∞, f ∞ = 0. 4. The existence of multiple positive solution In this section, we give some conclusions about the existence of multiple positive solutions. We always suppose that (H∗1 ), (H2 ) and (H3 ) hold in the following theorems. Theorem 4.1. Assume that the following two conditions

(H4 )∗ : f r < φp (ω) and

(H8 ): f0 ≥ φp



M



Γ

,

f∞ ≥ φp



M



Γ

are satisfied, then BVP (1.1) has at least two positive solutions such that 0 < ku1 k < r < ku2 k. Proof. By the proof of Theorem 3.2, we can take 0 < r1 < r < Γ r2 such that f (u) ≥ φp (r1 M ) for Γ r1 ≤ u ≤ r1 and f (u) ≥ φp (r2 M ) for Γ r2 ≤ u ≤ r2 . So, by Theorems 3.1 and 3.1∗ , it follows that BVP (1.1) has at least two positive solutions such that 0 < ku1 k < r < ku2 k.  Theorem 4.2. Assume that the following two conditions

(H5 )∗ : fR > φp (M ) and

(H9 ): f 0 ≤ φp (ω),

f ∞ ≤ φp (ω),

are satisfied, then BVP (1.1) has at least two positive solutions such that 0 < ku1 k < R < ku2 k. Theorem 4.3. Assume (H6 ) (or (H7 )) holds, and there exist constants r1 , r2 > 0 with r1 M < r2 ω (or r1 < Γ r2 ) such that (H4 )∗ holds for r = r2 (or r = r1 ) and (H5 )∗ holds for R = r1 (or R = r2 ). Then the BVP (1.1) has at least three positive solutions such that 0 < ku1 k < r1 < ku2 k < r2 < ku3 k. The proofs of Theorems 4.2 and 4.3 are similar to that of Theorem 4.1, so we omit it here. From Theorems 4.1–4.3, it is easy to see that if the conditions (H4 )∗ , (H5 )∗ , (H6 )–(H9 ) are combined properly, then the BVP (1.1) has relevantly multiple positive solutions. Theorem 4.4. Let n1 = 2k + 1, k ∈ N. Assume (H6 ) (or (H7 )) holds. If there exist constants r1 , r2 , . . . , rn1 −1 > 0 with r2i < Γ r2i+1 , for 1 ≤ i ≤ k − 1 and r2i−1 M < r2i ω, for 1 ≤ i ≤ k (or with r2i−1 < Γ r2i , for 1 ≤ i ≤ k and r2i M < r2i+1 ω, for 1 ≤ i ≤ k − 1) such that (H5 )∗ (or (H4 )∗ ) holds for r2i−1 , 1 ≤ i ≤ k and (H4 )∗ (or (H5 )∗ ) holds for r2i , 1 ≤ i ≤ k. Then the BVP (1.1) has at least n1 positive solutions u1 , u2 , . . . , un1 such that 0 < ku1 k < r1 < ku2 k < r2 < · · · < kun1 −1 k < rn1 −1 < kun1 k. Theorem 4.5. Let n2 = 2k, k ∈ N. Assume (H8 ) (or (H9 )) holds. If there exist constants r1 , r2 , . . . , rn2 −1 > 0 with r2i−1 < Γ r2i and r2i M < r2i+1 ω, for 1 ≤ i ≤ k − 1 (or with r2i < Γ r2i+1 and r2i−1 M < r2i ω, for 1 ≤ i ≤ k − 1) such that (H4 )∗ (or (H5 )∗ ) holds for r2i−1 , 1 ≤ i ≤ k and (H5 )∗ (or (H4 )∗ ) holds for r2i , 1 ≤ i ≤ k − 1. Then the BVP (1.1) has at least n2 positive solutions u1 , u2 , . . . , un2 such that 0 < ku1 k < r1 < ku2 k < r2 < · · · < kun2 −1 k < rn2 −1 < kun2 k.

J. Yang, Z. Wei / Nonlinear Analysis 71 (2009) 2985–2996

2995

5. Application Example 5.1. Consider the following singular boundary value problems (SBVP for short) with a p-Laplacian operator

 1  (φp (u00 (t )))00 − t − 2 f (u(t )) = 0,   00 00  u (0) = u (1) = 0,     1 1 u(0) − u0 (0) = u ,  2 2       1 1  u(1) + u0 (1) = u , 2

where p =

3 2

t ∈ (0, 1), (5.1)

2

,

 −u ue , 0 ≤ u ≤ 1, −u f (u) = (k√+ 1)e , k < u ≤ k + 1, k = 1, 2, . . . , 10,  u e , u ∈ (11, +∞). We note that m = 3,

q = 3,

a1 = b 1 = Let θ =

1 , 4

1 2

a = b = c = d = 1, 9

,

∆=− , 2

ρ = 3,

1 g (t ) = t − 2 ,

ξ1 =

f 0 = 0,

1 2

,

f∞ = +∞.

then

α = 2,

β=

5 8

,

γ = 1,

Γ =

1 2

,

ω=

24 13

,

M =d

262 144(2 +

√

3)

639

.

Thus, f 0 < φp (ω) and f∞ > φp ( M ). So the condition (H6 ) holds. Obviously, (H∗1 ), (H2 ) and (H3 ) hold. Therefore, by Γ Theorem 3.2, the SBVP (5.1) has at least one position solution. Example 5.2. Consider the following singular boundary value problems (SBVP for short) with a p-Laplacian operator

 1  (φp (u00 (t )))00 − t − 2 f (u(t )) = 0,    u00 (0) = u00 (1) = 0,     1 1 , u(0) − u0 (0) = u  2 2       1 1  u(1) + u0 (1) = u , 2

where p =

3 2

t ∈ (0, 1), (5.2)

2

,

 −u e , 0 ≤ u ≤ 1, −u f (u) = (k√+ 1)e , k < u ≤ k + 1, k = 1, 2, . . . , 10,  u e , u ∈ (11, +∞). We note that m = 3,

q = 3,

a1 = b 1 = Let θ =

1 , 4

1 2

a = b = c = d = 1, 9

,

∆=− , 2

ρ = 3,

1 g (t ) = t − 2 ,

f0 = +∞,

1 2

,

f∞ = +∞.

then

α = 2,

β=

5 8

,

γ = 1,

Γ =

1 2

,

ω=

Thus, f0 > φp ( M ) and f∞ > φp ( M ). We choose r = 1, then Γ Γ r

ξ1 =

f = sup



f ( u)

φp (r )



, u ∈ [0, r ] = 1 < φp (ω) =

r

24 13

.

24 13

,

M =

262 144(2 + 639

√

3)

.

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J. Yang, Z. Wei / Nonlinear Analysis 71 (2009) 2985–2996

Thus, (H∗4 ) and (H8 ) hold. Obviously, (H∗1 ), (H2 ) and (H3 ) hold. By Theorem 4.1, the SBVP (5.2) has at least two positive solutions u1 , u2 ∈ K such that 0 < ku1 k < 1 < ku2 k. Acknowledgments The authors thank the referees for their valuable suggestions and recommendations. Research supported by the Foundation for Bozhou Teachers College (BSKY0805), the NSF of Anhui Province (KJ2009B093), the NNSF-China (10471077) and the NSF of Shandong Province (Y2004A01). References [1] V.A. Il’in, E.I. Moiseev, Nonlocal boundary value problem of the first kind for a Sturm–Liouville operator in its differential and finite difference aspects, Differential Equations 23 (1987) 803–810. [2] C.P. Gupta, Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation, J. Math. Anal. Appl. 168 (1992) 540–551. [3] M. Moshinsky, Sobre los problemas de condiciones a la frontiera en una dimension de caracteristicas discontinuas, Bol. Soc. Mat. Mexicana 7 (1950) 1–25. [4] S. Timoshenko, Theory of Elastic Stability, McGraw-Hill, New York, 1961. [5] R.Y. Ma, Nonlocal Problems for the Nonlinear Ordinary Differential Equation, Science Press, Beijing, 2004 (in Chinese). [6] D.J. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cone, Academic Press, Boston, 1988. [7] E.R. Kaufmann, N. Kosmatov, A multiplicity result for boundary value problem with infinitely many singularities, J. Math. Anal. 269 (2002) 444–453. [8] H.Y. Feng, W.G. Ge, M. Jiang, Multiple positive solutions for m-point boundary value problems with a one-dimensional p-Laplacian, Nonlinear Anal. 68 (2008) 2269–2279. [9] H. Su, Z.L. Wei, B.H. Wang, The existence of positive solutions for a nonlinear four-point singular boundary value problem with a p-Laplacian operator, Nonlinear Anal. 66 (2007) 2204–2217. [10] Y.Y. Wang, W.G. Ge, Existence of multiple positive solutions for multi-point boundary value problems with a one-dimensional p-Laplacian, Nonlinear Anal. 67 (2007) 476–485.