Degree for Impulsive Boundary Value Problems under Non-Well-Ordered Upper and Lower

Boundary Value Problems

Volume 2008, Article ID 197205,21pages doi:10.1155/2008/197205

Research Article

Multiplicity Results via Topological

Degree for Impulsive Boundary Value Problems under Non-Well-Ordered Upper and Lower

Solution Conditions

Xu Xian,¹ Donal O’Regan,² and R. P. Agarwal³

1Department of Mathematics, Xuzhou Normal University, Xuzhou, Jiangsu 221116, China

2Department of Mathematics, National University of Ireland, Galway, Ireland

3Department of Mathematical Science, Florida Institute of Technology, Melbourne, FL 32901, USA

Correspondence should be addressed to R. P. Agarwal,agarwal@fit.edu Received 14 April 2008; Accepted 26 August 2008

Recommended by R. P. Gilbert

Some multiplicity results for solutions of an impulsive boundary value problem are obtained under the condition of non-well-ordered upper and lower solutions. The main ideas of this paper are to associate a Leray-Schauder degree with the lower or upper solution.

Copyrightq2008 Xu Xian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In this paper, we study multiplicity of solutions of the impulsive boundary value problem yf

t, yt, yt

0, t /tk, Δy|_ttk I_k

y t_k

, k1,2, . . . , m, Δy|_ttk I_k

y t_k

, k1,2, . . . , m,

y0 0y1−αyη,

1.1

wheref ∈ CJ ×R²,R,J 0,1,I_k, I_k ∈ CR,R,k 1,2, . . . , m,Δy|_ttk yt_k−yt⁻_k, Δy|ttk yt_k−yt⁻_k, 0t₀< t₁ <· · ·< t_m< t_m11,α∈0,1.

Impulsive differential equations arise naturally in a wide variety of applications, such as spacecraft control, inspection processes in operations research, drug administration, and threshold theory in biology. In the past twenty years, a significant development in the

theory of impulsive differential equations was seen. Many authors have studied impulsive differential equations using a variety of methodssee1–5and the references therein.

The purpose of this paper is to study the multiplicity of solutions of the impulsive boundary value problems 1.1 by the method of upper and lower solutions. The method of lower and upper solutions has a very long history. Some of the ideas can be traced back to Picard6. This method deals mainly with existence results for various boundary value problems. For an overview of this method for ordinary differential equations, the reader is referred to7. Usually, when one uses the method of upper and lower solutions to study the existence and multiplicity of solutions of impulsive differential equations, one assumes that the upper solution is larger than the lower solution, that is, the condition that upper and lower solutions are well ordered. For example, Guo1studied the PBVP for second-order integrodifferential equations of mixed type in real Banach spaceE:

−uft, u, Tu, Su ∀t∈0,2π, t /ti, Δu|_tti Liu

ti

, Δu|_tti L^∗_iu

t_i

, i1,2, . . . , m,

u0 u2π, u0 u2π,

1.2

wheref ∈C0,2π×E×E×E, E,TandS:E→Eare two linear operators, 0< t₁< t₂<· · ·<

tm<1,Li, L^∗_i i1,2, . . . , mare constants. In1Guo first obtained a comparison result, and then, by establishing two increasing and decreasing sequences, he proved an existence result for maximal and minimal solutions of the PBVP1.2in the ordered interval defined by the lower and upper solutions.

However, to the best of our knowledge, only in the last few years, it was shown that existence and multiplicity for impulsive differential equation under the condition that the upper solution is not larger than the lower solution, that is, the condition of non-well-ordered upper and lower solutions. In8, Rach ˚unkov´a and Tvrd ´y studied the existence of solutions of the nonlinear impulsive periodic boundary value problem

uf t, u, u

, t /ti

u t_i

J u

t_i

, i1,2, . . . , m, u

t_i Mi

u t_i

, i1,2, . . . , m,

u0 uT, u0 uT,

1.3

where f ∈ C0, T× R², Ji,Mi ∈ CR. Using Leray-Schauder degree, the authors of 8 showed some existence results for 1.3 under the non-well-ordered upper and lower solutions condition. For other results related to non-well-ordered upper and lower solutions, the reader is referred to7,9–14. Also, here we mention the main results of a very recent paper15. In that paper, we studied the second-order three-point boundary value problem

yt ft, y 0, 0≤t≤1,

y0 0, y1−αyη 0, 1.4

where 0< η <1, 0< α <1,f ∈C0,1×R,R. In15, we made the following assumption.

x₆

x3

u2

u1

x4

x1

v2

v1

x₅

x2

Figure 1: The positions ofu1, u2, v1, v2and six solutionsx1, x2, . . . , x6inTheorem 1.1.

A0There existsM >0 such that

f t, x₂

−f t, x₁

≥ −M

x₂−x₁

, t∈0,1, x2≥x₁. 1.5

Let the functionebeeet tfort∈0,1. In15, we proved the following theorem see,15, Theorem 3.4.

Theorem 1.1. Suppose thatA0holds,u1andu2are two strict lower solutions of 1.4,v1andv2

are two strict upper solutions of 1.4, andu₁< v₁,u₂< v₂,u₂/≤v₁,u₁/≤v₂. Moreover, assume

−ς0e≤u2−u1≤ς0e, −ς0e≤v2−v1≤ς0e 1.6

for some ς0 > 0. Then the three-point boundary value problem 1.4 has at least six solutions x1, x2, . . . , x6.

Theorem 1.1 establishes the existence of at least six solutions of the three-point boundary value problem1.4only under the condition of two pairs of strict lower and upper solutions. The positions ofu1, u2, v1, v2and six solutionsx1, x2, . . . , x6inTheorem 1.1can be illustrated roughly byFigure 1.

In some sense, we can say that these two pairs of lower and upper solutions are parallel to each other. The position of these two pairs of lower and upper solutions is sharply different from that of the lower and upper solutions of the main results in14,16,17. The technique to prove our main results of15is to use the fixed-point index of some increasing operator with respect to some closed convex sets, which are translations of some special conesseeQc, Q_cof15.

This paper is a continuation of the paper15. The aim of this paper is to study the multiplicity of solutions of the impulsive boundary value problem1.1under the conditions of non-well-ordered upper and lower solutions. In this paper, we will permit the presence of impulses and the first derivative. The main ideas of this paper are to associate a Leray- Schauder degree with the lower or upper solution. We will give some multiplicity results for at least eight solutions. To obtain this multiplicity result, an additional pair of lower and upper solutions is needed, that is, we will employ a condition of three pairs of lower and upper solutions. The position of these three pairs of lower and upper solutions will be illustrated inRemark 2.16.

2. Results for at least eight solutions

LetJJ\{t1, t₂, . . . , t_m}, PCJ,R {x|xis a map fromJ intoRsuch thatxtis continuous att /tk, left continuous at ttkand its right-hand limit xt_katttkexits}, and PC¹J,R {x|xis a map fromJ intoRsuch thatxtandxtare continuous att /t_k, left continuous attt_kand their right-hand limitsxt_kandxt_kattt_kexits}. For eachx∈PC¹J,R, let

x_PC¹max

xPC,x

PC

, 2.1

wherexPC sup_t∈J|xt|andxPC sup_t∈J|xt|. Then, PC¹J,Ris a real Banach space with the norm·_PC¹. The functionx∈PC¹J,R∩C²J,Ris called a solution of the boundary value problem1.1if it satisfies all the equalities of1.1.

Now, for convenience, we make the following assumptions.

H00t0< t1<· · ·< tm< η < t_m11, α∈0,1.

H1Ik k1,2, . . . , mis increasing onR.

Letx, y∈PCJ,R. Now, we define the ordering≺by

x≺y iffxt< yt∀t∈J, xt_k< yt_k for eachk1,2, . . . , m. 2.2

Definition 2.1. The functionu∈PC¹J,R∩C²J,Ris called a strict lower solution of1.1if ut f

t, ut, ut

>0, t /t_k,

u0<0, u1−αuη<0, 2.3

wheneverIi0x/0 orIj0x/0 for somei0, j0∈ {1,2, . . . , m}and somex∈R Δu|_ttk I_k

u t_k

, k1,2, . . . , m, Δu|_ttk > I_k

u t_k

, k1,2, . . . , m,

2.4

wheneverI_kx I_kx 0 for eachx∈Randk∈ {1,2, . . . , m},Δu|_ttk Δu|_ttk 0 for each k∈ {1,2, . . . , m}.

The functionv∈PC¹J,R∩C²J, Ris called a strict upper solution of1.1if vt f

t, vt, vt

<0, t /t_k,

v0>0, v1−αvη>0, 2.5

wheneverIi0x/0 orIj0x/0 for somei0, j0∈ {1,2, . . . , m}and somex∈R Δv|_ttk I_k

v t_k

, k1,2, . . . , m, Δv|ttk < Ik

v tk

, k1,2, . . . , m, 2.6

and wheneverIkx Ikx 0 for eachx∈Randk∈ {1,2, . . . , m},Δv|ttk Δv|ttk 0 for eachk∈ {1,2, . . . , m}.

Definition 2.2. Letut, vt ∈ PC¹J,R∩C²J,R,ut ≤ vtfor allt ∈ J. We say thatf satisfies Nagumo condition with respect tou, vif there exists functionφ∈C0,∞,0,∞ such that

ft, x, y≤φ

|y|

, ∀t, x, y∈J×

ut, vt ×R, _∞

0

s

φsds∞. 2.7

Definition 2.3. Let r1t, r2t· · ·rnt be strict upper solutions of 1.1 and rt min{r1t, r2t, . . . rnt}for eacht ∈J. Then, we say the upper solutionsr1t, r2t, . . . rnt are well ordered if for eachk∈ {1,2, . . . , m}, there existi₀, j₀ ∈ {1,2, . . . , n}andδ₀ >0 small enough such that

rt

⎧⎨

⎩

ri0t, t∈

tk−δ0, tk , r_j0t, t∈

t_k, t_kδ₀ .

2.8

Definition 2.4. Let l1t, l2t, . . . lnt be strict lower solutions of 1.1 and lt max{l1t, l2t, . . . lnt} for eacht ∈ J. Then, we say the lower solutionsl1t, l2t, . . . lnt are well ordered if for eachk∈ {1,2, . . . , m}, there existi₀, j₀ ∈ {1,2, . . . , n}andδ₀ >0 small enough such that

lt

⎧⎨

⎩

li0t, t∈

tk−δ0, tk , l_j0t, t∈

t_k, t_kδ₀ .

2.9

From18, Lemma 5.4.1, we have the following lemma.

Lemma 2.5. H⊂PC¹J,Ris a relative compact set if and only if for allx∈H,xtandxtare uniformly bounded onJ and equicontinuous on eachJ_k k 1,2, . . . , m, whereJ₁ 0, t1, Ji ti−1, t_i, i2,3, . . . , m1.

The following lemma can be easily proved.

Lemma 2.6. Suppose thatx∈PC¹J,R∩C²J,Rsatisfies

−xt f

t, xt, xt

, t /t_k k1,2, . . . , m. 2.10 Then

xt x0− _t

0

f

s, xs, xs

ds

0<tk<t

x t_k

−x

t_k ∀t∈J,

xt x0 x0t− _t

0

t−sf

s, xs, xs

ds

0<tk<t

x t_k

−x t_k

0<tk<t

x t_k

−x

t_k t−t_k

∀t∈J.

2.11

Lemma 2.7. Letg ∈PCJ,Randak, bk∈Rk0,1,2, . . . , m. Then,x∈PC¹J,R∩C²J,R is a solution of

−xt gt, t /t_k, k1,2, . . . , m, Δx|ttk a_k, k1,2, . . . , m, Δx|_ttk bk, k1,2, . . . , m,

x0 a₀, x1−αxη b₀

2.12

if and only ifx∈PCJ,Rsatisfies

xt a0

1− 1−α 1−αηt

b0t

1−αη t 1−αη

₁

0

1−sgsds− αt 1−αη

_η

0

η−sgsds

− t 1−αη

m k1

1−αak

1−tk−α

η−tk bk

− _t

0

t−sgsds

0<tk<t

akbk

t−tk , t∈J.

2.13

Proof. Letx∈PC¹J,R∩C²J,Rbe a solution of2.12. FromLemma 2.6, we have

xt x0 x0t− _t

0

t−sgsds

0<tk<t

Δx|ttk

0<tk<t

Δx|ttk

t−t_k

a0x0t− _t

0

t−sgsds

0<tk<t

ak

0<tk<t

bk

t−tk

.

2.14

Thus,

x1 a0x0− ₁

0

1−sgsds^m

k1

ak^m

k1

bk

1−tk

,

xη a0x0η− _η

0

η−sgsds^m

k1

ak^m

k1

bk

η−tk

.

2.15

Using the boundary value conditionx1−αxη b0, we have

x0 1

1−αηb0− 1−α

1−αηa0 1 1−αη

₁

0

1−sgsds− α

1−αη _η

0

η−sgsds

− 1−α 1−αη

m k1

a_k− 1 1−αη

m k1

b_k 1−t_k

α 1−αη

m k1

b_k η−t_k

.

2.16

The equality2.13now follows from2.14and2.16.

On the other hand, ifx∈PCJ,Rsatisfies2.13, by direct computation, we can easily show thatxsatisfies2.12. The proof is complete.

Let us define the operatorA: PC¹J,R→PC¹J,Rby

Axt t 1−αη

₁

0

1−sf

s, xs, xs

ds− αt 1−αη

_η

0

η−sf

s, xs, xs ds

− _t

0

t−sf

s, xs, xs

ds

0<tk<t

I_k x

t_k I_k

x t_k

t−t_k

− t 1−αη

m k1

1−αIk

x t_k

1−t_k−α

η−t_k I_k x

t_k .

2.17

FromLemma 2.5,A: PC¹J,R→PC¹J,Ris a completely continuous operator.

Theorem 2.8. Suppose thatH0andH1hold. Letα_i, β_i i1,2, . . . , nbenpairs of strict lower and upper solution, and

αt max

α₁t, α2t, . . . , αnt

, t∈J,

βt min

β1t, β2t, . . . , βnt

, t∈J.

2.18

Suppose thatα_i ≺β_i i1,2, . . . , n,α≺β,fsatisfies Nagumo condition with respect toα1, β₁. Moreover, the strict lower solutionsα1, α2, . . . , αnand the strict upper solutionsβ1, β2, . . . , βnare well ordered wheneverI_i0x/0 orI_j0x/0 for somei₀, j₀ ∈ {1,2, . . . , m}and somex∈R. Then, there existR0 >0 andL0 >0 sufficiently large such that for eachR≥R0andL > L0

degI−A,Ω, θ 1, 2.19

where

Ω

x∈Bθ, R|α≺x≺β, −L≺x≺L ,

Bθ, R

x∈PC¹J, R| x_PC¹ < R .

2.20

Proof. We only prove the case whenIi0x/0 orIj0x/0 for somei0, j0 ∈ {1,2, . . . , m}and somex∈R. The conclusion is achieved in four steps.

Step 1. Since f satisfies Nagumo condition with respect to α1, β₁, then there exists φ ∈ C0,∞,0,∞such that

ft, x, y≤φ

|y|

, t, x, y∈J×

αt, βt ×R, _∞

0

s

φsds∞. 2.21

Letμ0min_1≤k≤m1tk−t_k−1. Takeλ >0 such that

λ >max_1≤i≤nsup_t∈Jβ_it−min_1≤i≤ninf_t∈Jα_it

μ0 , 2.22

andN >0 such that

_N

λ

s

φsds >2λ. 2.23

LetL₀max{N,2λ,max_1≤i≤nα_iPC,max_1≤i≤nβ_iPC}. Define the functionsg, h:J×R² →R by

gt, x, y

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

ft, x, L, y > L, ft, x, y, −L≤y≤L, ft, x,−L, y <−L,

ht, x, y

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ g

t, βt, y

, x > βt, gt, x, y, αt≤x≤βt, g

t, αt, y

, x < αt.

2.24

For eachk∈ {1,2, . . . , m}, let us define the functionsJk, J_k:R→Rby

Jkx

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ I_k

β t_k

, x > β t_k

,

Ikx, α

tk

≤x≤β tk

,

Ik

α tk

, x < α tk

,

J_kx

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ Ik

β tk

, x > β tk

,

I_kx, α

t_k

≤x≤β t_k

, Ik

α tk

, x < α tk

.

2.25

It is easy to see that there existsM₁ >0 such that

ht, x, y≤M1, t, x, y∈J×R²,

Jkx≤M1, J_kx≤M1, x∈R, k1,2, . . . , m. 2.26

Let us define the operatorA^∗: PC¹J,R→PC¹J,Rby A^∗x

t t 1−αη

₁

0

1−sh

s, xs, xs

ds− αt 1−αη

_η

0

η−sh

s, xs, xs ds

− _t

0

t−sh

s, xs, xs

ds

0<tk<t

J_k x

t_k

t−t_k J_k

x t_k

− t 1−αη

m k1

1−αJk

x t_k

1−t_k

−α

η−t_k J_k x

t_k .

2.27

By2.26, we have A^∗x

t≤ 1 1−αη

₁

0

1−sh

s, xs, xsds α 1−αη

_η

0

η−sh

s, xs, xsds

₁

0

1−sh

s, xs, xsds^m

k1

J_k x

t_k

1−t_kJ_k x

t_k

2 1−αη

m k1

J_k x

t_kJ_k x

t_k

≤ M1

1−αη 1

2 1 2αη² 1

22m4m

≤ M1

1−αη36m, t∈J, A^∗x

t≤ 1 1−αη

₁

0

1−sh

s, xs, xsds α 1−αη

_η

0

η−sh

s, xs, xsds

₁

0

h

s, xs, xsds^m

k1

J_k x

tk 2

1−αη m k1

Jk

x

tkJ_k x

tk

≤ M₁ 1−αη

1 2 1

2αη²15m

≤ M₁

1−αη5m3, t∈J.

2.28 From2.28, we haveA^∗x_PC¹ ≤ M1/1−αη11m6for eachx ∈ PC¹J,R. LetR₀ M1/1−αη11m6 1. Then,A^∗PC¹J,R⊂Bθ, R0. By the properties of the Leray- Schauder degree, we have

deg

I−A^∗, Bθ, R, θ

1. 2.29

Thus,A^∗has at least one fixed pointx0. FromLemma 2.7,x0satisfies

x₀t h

t, x0t, x₀t

0, t /tk, Δx0|_ttk J_k

x₀ t_k

, k1,2, . . . , m Δx₀

ttk J_k x₀

t_k

, k1,2, . . . , m, x₀0 0x₀1−αx₀η,

2.30

Step 2. Next, we will show that

α≺x₀≺β, 2.31

−L≺x₀≺L. 2.32

We first show that

αt≤x0t≤βt ∀t∈J. 2.33

To begin, we show thatx₀t≤βtfor allt∈J. Suppose not, then there existst∈Jsuch that x₀t> βt. Setwt x₀t−βtfort∈J. There are a number of cases to consider.

1w0 sup_t∈Jwt>0, then, we have

0< w0 x₀0−β0 −β0<0, 2.34

which is a contradiction.

2w1 sup_t∈Jwt > 0; assume without loss of generality thatα > 0 andβ1 βi01for somei0 ∈ {1,2, . . . , n}, then, we have

0< w1 x₀1−β_i01≤αx₀η−αβ_i0η≤αx₀η−αβη αwη≤αw1, 2.35

which is a contradiction.

3 There exist k₀ ∈ {1,2, . . . , m, m 1} and τ₀ ∈ tk0−1, t_k0 such that wτ0

sup_t∈Jwt > 0. Assume without loss of generality that βτ0 βi0τ0 for some i0 ∈ {1,2, . . . , n}. We have the following two cases:

3Aβ_jτ0> β_i0τ0for eachj∈ {1,2, . . . , n}andj /i₀;

3Bthere existsj0∈ {1,2, . . . , n}, j0/i0such thatβj0τ0 βi0τ0.

For case3A, there existsδ₀ >0 small enough such thatτ0−δ₀, τ₀δ₀⊂tk0−1, t_k0 and

wt x0t−βi0t, t∈

τ0−δ0, τ0δ0 . 2.36

Then,w∈C²τ0−δ0, τ0δ0,wτ0is the maximum ofwonτ0−δ0, τ0δ0. Thus,wτ0

0, wτ0≤0. By2.30, we have

0≥w τ₀

x₀ τ₀

−β_i

0

τ₀ −h

τ₀, x₀ τ₀

, x₀ τ₀

−β_i

0

τ₀

−f τ₀, β_i0

τ₀ , β_i

0

τ₀

−β_i

0

τ₀ ,

2.37

which is a contradiction.

For case3B, setw1t x0t−βj0tfort∈tk0−1, tk0. For anyt∈tk0−1, tk0, we have

w₁ τ₀

x₀ τ₀

−β_j0 τ₀

x₀ τ₀

−β_i0 τ₀

w τ₀

≥w t

x₀ t

−β t

≥x₀ t

−β_j0 t

w₁ t

.

2.38

This implies that w₁τ0 is a local maximum. Since w₁ ∈ C²tk0−1, t_k0, then w₁τ0 0, w₁τ0≤0. Therefore,

0≥w₁ τ₀

x₀ τ₀

−β_j

0

τ₀ −f

τ₀, β_j0 τ₀

, β_j

0

τ₀

−β_j

0

τ₀

>0, 2.39

which is a contradiction.

4There existsk₀ ∈ {1,2, . . . , m}such thatwtk0 sup_t∈Jwt > 0. Without loss of generality, we may assumewτ<sup_t∈Jwtfor eachτ ∈tk−1, tkandk ∈ {1,2, . . . , m, m 1}.Otherwise, if there exists τ₀ ∈ tk0−1, t_k0 for somek₀ ∈ {1,2, . . . , m, m1} such that wτ0 sup_t∈Jwt, then we can get a contradiction as in case3. In this case, we have the following two subcases:

4Athere existsi₀∈ {1,2, . . . , n}such thatβ_i0tk0< β_jtk0forj 1,2, . . . , nandj /i₀; 4Bthere exists a subset{n1, n₂, . . . , n_s} ⊂ {1,2, . . . , n}such that

β tk0

βn1

tk0

βn2

tk0

· · ·βns

tk0

, 2.40

whileβ_ltk0> βtk0for eachl∈ {1,2, . . . , n} \ {n1, n₂, . . . , n_s}, s≥2.

First, we consider case4A. SinceIk0is increasing onR, then

β_i0 t_k

0

β_i0 t_k0

I_k0 β_i0

t_k0

< β_j t_k0

I_k0 β_j

t_k0 β_j

t_k

0

, j /i₀. 2.41

Then, there existsδ0 > 0 small enough such thatβt βi0tfort ∈ tk0−δ0, tk0δ0and so wt x₀t−β_i0tfort ∈ tk0−δ₀, t_k0 δ₀. Sinceβ_i0tis a strict upper solution, we

have

w t_k0

x0

t_k0

−βi0

t_k0

x₀

t_k0 J_k0

x₀

t_k0 − β_i0

t_k0 I_k0

β_i0 t_k0

x₀ t_k0

−β_i0

t_k0 J_k0

x₀ t_k0

−I_k0 β_i0

t_k0 w

t_k0

I_k0 β_i0

t_k0

−I_k0 β_i0

t_k0 w

t_k0 .

2.42

Sincewτ < wtk0 for eachτ ∈ tk0−1, t_k0, then we havewtk0 ≥ 0. Similarly, we have wt_k0≤0. Therefore,

0≥w t_k

0

x t_k

0

−β_i

0

t_k

0

>

x₀ tk0

J_k0 x0

tk0 − β_i0

tk0

Ik0

βi0

tk0

w t_k0

I_k0

β_i0 t_k0

−I_k0 β_i0

t_k0 w t_k0

≥0,

2.43

which is contradiction.

Now we consider case4B. SinceI_k0is increasing, then we have

β t_k

0

β_n1 t_k

0

β_n2 t_k

0

· · ·β_ns t_k

0

, 2.44

whileβ_lt_k

0 > βt_k

0for eachl ∈ {1,2, . . . , n} \ {n1, n₂, . . . , n_s}. For case4B, we have two subcases:

4Bathere existsδ₀ > 0 small enough andi₀ ∈ {n1, n₂, . . . , n_s}such thatβ_i0t βt fort∈tk0−δ₀, t_k0δ₀;

4Bbthere existsδ0>0 small enough andi0/j0,i0, j0∈ {n1, n2, . . . , ns}such that

βt

⎧⎨

⎩

β_i0t, t∈

t_k0−δ₀, t_k0 , βj0t, t∈

tk0, tk0δ0 .

2.45

For case4Baas in case4A, we can easily obtain a contradiction. For case4Bb, we have

wt

⎧⎨

⎩

x₀t−β_i0t, t∈

t_k0−δ₀, t_k0 , x0t−βj0t, t∈

tk0, tk0δ0 .

2.46

In the same way as in the proof of case4A, we see thatwt_k0 wtk0,wtk0≥0 and we havewt_k

0≤0. Note thatβ_j

0tk0≤β_i

0tk0, and we have 0≥w

t_k

0

x₀ t_k

0

−β_j

0

t_k

0

x₀ tk0

Δx₀|ttk0 − β_j0

tk0

Δβ_j0|ttk0

>

x₀ tk0

−β_j0

tk0 J_k0

x0

tk0

−Ik0

βj0

tk0

≥x₀ tk0

−β_i0 tk0

Ik0

βj0

tk0

−Ik0

βj0

tk0

w tk0

≥0,

2.47

which is a contradiction.

5There exists ak₀ ∈ {1,2, . . . , m}such thatwt_k

0 sup_t∈Jwt > 0. Without loss of generality, we may assume thatwτ < wt_k

0for eachk ∈ {1,2, . . . , m, m1} andτ ∈ tk−1, t_k. We have two subcases:

5Athere existsi0∈ {1,2, . . . , n}such thatβi0t_k0< βjt_k0for eachj /i0; 5Bthere exists a subset{n1, n2, . . . , ns} ⊂ {1,2, . . . , n}such that

β t_k

0

βn1

t_k

0

βn2

t_k

0

· · ·βns

t_k

0

, 2.48

whileβlt_k0> βt_k0for eachl∈ {1,2, . . . , n} \ {n1, n2, . . . , ns}, s≥2.

SinceI_k0is increasing, then for case5A, we have βi0

tk0

< βj

tk0 j /i0

, x0

tk0

> βi0

tk0

, 2.49

and for case5B, we havex₀tk0> β_i0tk0and β

tk0

βn1

tk0

βn2

tk0

· · ·βns

tk0

, 2.50

whileβltk0> βtk0for eachl∈ {1,2, . . . , n} \ {n1, n2, . . . , ns}. Therefore, we can use the same method as in case4to obtain a contradiction.

From the discussions of1–5, we see thatx₀t ≤ βtfort ∈ J. Similarly, we can prove thatαt≤x0tfort∈J. Thus,2.33holds.

Next, we prove thatα ≺ x₀ ≺ β. If the inequalityx₀ ≺ βdoes not hold, then either there existsτ₀∈Jsuch thatx₀τ0 βτ0or there existsk₀∈ {1,2, . . . , m}such thatx₀t_k

0

βt_k0. Setwt x0t−βtfort∈J. Then, we have eitherwτ0 sup_t∈Jwtorwt_k0 sup_t∈Jwtfor some k₀ ∈ {1,2, . . . , m}. Essentially the same reasoning as in 1–5above yields a contradiction. Thus,x0≺β. Similarly,α≺x0. Consequently,2.31holds.

Step 3. Now, we show2.32. Suppose not, then we have the following two subcases:

Ithere existss₁∈Jsuch that|x₀s1| ≥L;

IIthere existsk₀∈ {1,2, . . . , m}such that|x₀t_k

0| ≥L.

We only consider case II. A similar argument works for caseI. We may assume without loss of generality that x₀t_k

0 ≥ L. By the mean-value theorem, there exists s₂ ∈ tk0, t_k01such that

x₀ s2

x₀ t_k01

−x₀ t_k0

t_k01−t_k0 ≤ max_1≤i≤nsup_t∈Jβ_it−min_1≤i≤ninf_t∈Jα_it

μ₀ < λ < L. 2.51

LetL₁be such thatL₀< L₁< L, then, there exists₃, s₄∈tk0, s₂such thats₃< s₄,x₀s3 L₁, x₀s4 λ, andλ≤x₀s≤L₁fors∈s3, s₄. Therefore,

x₀sh

s, x₀s, x₀sf

s, x₀s, x₀s≤φ x₀s

, s∈

s₃, s₄ . 2.52

Consequently, _s4

s3

x₀sx₀s φ

x₀s ds ≤

_s4

s3

x₀sds _s4

s3

x₀sdsx0

s4

−x0

s3

< λ. 2.53

On the other hand, _s4

s3

x₀sx₀s φ

x₀s ds

_L1

λ

s φsds

≥ _N

λ

s

φsds > λ, 2.54

which is a contradiction. Thus,2.32holds.

Step 4. From the excision property of Leray-Schauder degree and2.29, we have

deg

I−A^∗,Ω, θ

deg

I−A^∗, Bθ, R, θ

1. 2.55

From2.31and2.32, we see thatAxA^∗xfor eachx∈Ω, and so

degI−A,Ω, θ 1. 2.56 The proof is complete.

Remark 2.9. From the proof of Theorem 2.8, we see thatAhas no fixed point on∂Ω.

Theorem 2.10. Suppose thatH0,H1hold,u1t, u2tare strict lower solutions,v1t, v0tare strict upper solutions,u1≺v1≺v0,u2≺v0,u2t> v1tfor somet∈J, andfsatisfies Nagumo condition with respect tou1, v₀. Moreover, the strict lower solutionsu₁t, u2tare well ordered wheneverIi0x/0 orIj0x/0 for somei0, j0 ∈ {1,2, . . . , m}and somex∈ R. Then,1.1has at least three solutionsx₁, x₂, andx₃, such that

u1≺x1≺v1, u1≺x2≺v0, u2≺x2≺v0, u1≺x3≺v0, 2.57 andv₁s1< x₃s1, x3s2< u₂s2for somes₁, s₂∈J.