Applied Mathematics and Computation 206 (2008) 810–817

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Multiplicity of positive solutions to second order Neumann boundaryvalue problems with impulse actions q

Qiuyue Li a,b, Fuzhong Cong b,*, Daqing Jiang c

a School of Mathematics, Jilin University, Changchun 130012, People’s Republic of Chinab Fundamental Department, Aviation University of Air Force, Changchun 130022, People’s Republic of Chinac School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, People’s Republic of China

a r t i c l e i n f o

Keywords:Neumann boundary value problemPositive solutionsAlternative principleFixed point theorem in cones

0096-3003/$ - see front matter Crown Copyright �doi:10.1016/j.amc.2008.09.045

q Partially supported by NSFC Grants (10571179,* Corresponding author.

E-mail address: congfz@263.net (F. Cong).

a b s t r a c t

This paper deals with existence theorems for single and multiple positive solutions to akind of Neumann boundary value problems with impulse actions. The proof of the mainresults relies on a nonlinear alternative principle of Leray–Schauder type and a well-knownfixed point theorem in cones. The impulses are not restricted to be monotonous in theresults obtained by us.

Crown Copyright � 2008 Published by Elsevier Inc. All rights reserved.

1. Introduction

We will be concerned with the existence of positive solutions of the Neumann boundary value problems with impulseactions

�ðpðxÞu0Þ0ðxÞ þ qðxÞuðxÞ ¼ gðx; uðxÞÞ; x 2 J; x – xk; k ¼ 1;2; . . . ; p;

�Dðpu0Þjx¼xk¼ IkðuðxkÞÞ; k ¼ 1;2; . . . ;p;

u0ð0Þ ¼ u0ð1Þ ¼ 0;

8><>: ð1:1Þ

where J ¼ ½0;1� and 0 < x1 < x2 < � � � < xp < 1 are given. Ik 2 CðJ;RþÞ. �Dðpu0Þjx¼xk¼ �pðxkÞðu0ðxk þ 0Þ � u0ðxk � 0ÞÞ. Here

u0ðxk þ 0Þ (respectively u0ðxk � 0Þ) denotes the right limit (respectively left limit) of u0ðxÞ at x ¼ xk.The type of perturbations gðx;uÞ we are mainly interested in is that gðx; uÞ has an attractive singularity near u ¼ 0 and

gðx;uÞ is superlinear near u ¼ þ1.From the physical explanation, (1.1) has a attractive singularity at u ¼ 0 if

limu!0þ

gðx; uÞ ¼ þ1 uniformly in x

and the superlinearity of gðx; uÞ near u ¼ þ1 means that

limu!þ1

gðx;uÞu¼ þ1 uniformly in x:

It is well known that there are abundant results about the existence of positive solutions of boundary value problems forsecond order ordinary differential equations. Some works can be found in [1–10,14–16] and references therein. They, mainly,investigated the case without impulse actions. Recently, Dirichlet boundary problems of second order impulsive differential

2008 Published by Elsevier Inc. All rights reserved.

10571021, 10871203) and NCET-07-0386 of China.

Q. Li et al. / Applied Mathematics and Computation 206 (2008) 810–817 811

equations have been studied in [11–13]. Motivated by the work above, this paper attempts to study the multiplicity of po-sitive solutions for the Neumann boundary value problems. The techniques we employ here involve an application of thefixed point theorem in cones to second order boundary value problem with impulse action, and Leray–Schauder alternativeprinciple. Moreover, we do not restrict impulses be monotonous.

Throughout this paper, we assume that the unperturbed part of (1.1), i.e.,

�ðpðxÞu0Þ0ðxÞ þ qðxÞuðxÞ ¼ 0 ð1:2Þ

satisfies the following standing hypothesis:

(A) The Green’s function, Gðx; yÞ, associated with the following problem:

�ðpðxÞu0Þ0ðxÞ þ qðxÞuðxÞ ¼ hðxÞ; x 2 J;

u0ð0Þ ¼ u0ð1Þ ¼ 0

(ð1:3Þ

is positive for all ðx; yÞ 2 ½0;1� � ½0;1�.

Remark. In Section 2 we will give a sufficient condition to ensure the Green function G be positive, which is pðxÞ > 0 andqðxÞ > 0.

Under Assumption (A), we always denote

A ¼ min06x;y61

Gðx; yÞ; B ¼ max06x;y61

Gðx; yÞ; r ¼ AB: ð1:4Þ

Thus B > A > 0;0 < r < 1. We also use xðxÞ to denote the unique periodic solution of (1.3) with hðxÞ ¼ 1, i.e.,xðxÞ ¼

R 10 Gðx; yÞdy.

2. Preliminaries

For the reader’s convenience we introduce some preliminary results of Green’s functions. For the details, see [9]. Denote

J0 ¼ J n fx1; x2; . . . ; xpg;U ¼ fu : J ! Rju 2 CðJ;RÞ; u0jðxk ;xkþ1Þ 2 Cððxk; xkþ1Þ;RÞ;u0ðxk � 0Þ ¼ u0ðxkÞ; 9u0ðxk þ 0Þg:

Let kjujkU ¼maxfkukC ; ku0kUg, then, ðU; kj � jkÞ is a Banach space.Let Q ¼ J � J and Q 1 ¼ fðx; yÞ 2 Q j0 6 x 6 y 6 1g;Q2 ¼ fðx; yÞ 2 Q j0 6 y 6 x 6 1g. Consider the following equation:

�ðpðxÞu0Þ0 þ qðxÞu ¼ 0; u0ð0Þ ¼ u0ð1Þ ¼ 0; ð2:1Þ

where

pðxÞ > 0; qðxÞ > 0; x 2 J: ð2:2Þ

Let Lu � �ðpðxÞu0Þ0 þ qðxÞu. We would prove that the Green’s function of (2.1) can be written as

Gðx; yÞ :¼mðxÞnðyÞ

c�; ðx; yÞ 2 Q1;

mðyÞnðxÞc�

; ðx; yÞ 2 Q2;

(ð2:3Þ

where m and n are linearly independent, and m;n and c� satisfy the following lemma.

Lemma 2.1. Suppose that (2.2) holds. Then the Green’s function Gðx; yÞ, defined by (2.3), possesses the following properties:

(i) mðxÞ 2 C2ðJ;RÞ is increasing and mðxÞ > 0;(ii) nðxÞ 2 C2ðJ;RÞ is decreasing and nðxÞ > 0;

(iii) ðLmÞðxÞ � 0;mð0Þ ¼ 1;m0ð0Þ ¼ 0;(iv) ðLnÞðxÞ � 0;nð1Þ ¼ 1;n0ð1Þ ¼ 0;(v) pðxÞðm0ðxÞnðxÞ �mðxÞn0ðxÞÞ � c� with some positive constant c�;

(vi) Gðx; yÞ is continuous and symmetrical on Q;(vii) Gðx; yÞ has continuously partial derivative on Q1 and Q2, respectively;

(viii) for each fixed y 2 J;Gðx; yÞ satisfies LGðx; yÞ ¼ 0 for x – y; x 2 J. Moreover, G0xð0; yÞ ¼ G0xð1; yÞ ¼ 0 for y 2 ð0;1Þ;(ix) G0x has discontinuous point of the first kind at x ¼ y and G0xðyþ 0; yÞ � G0xðy� 0; yÞ ¼ � 1

pðyÞ ; y 2 ð0;1Þ.

Proof. Let m1ðxÞ and m2ðxÞ be the base solutions of the equation Lm ¼ 0. Since the Wronsky determinant

DðxÞ ¼m1ðxÞ m2ðxÞm01ðxÞ m02ðxÞ

��������– 0;

812 Q. Li et al. / Applied Mathematics and Computation 206 (2008) 810–817

we can take constants l1; l2; q1 and q2 such that

l1m1ð0Þ þ l2m2ð0Þ ¼ 1;l1m01ð0Þ þ l2m02ð0Þ ¼ 0

�

and

q1m1ð1Þ þ q2m2ð1Þ ¼ 1;q1m01ð1Þ þ q2m02ð1Þ ¼ 0:

�

Let mðxÞ ¼ l1m1ðxÞ þ l2m2ðxÞ;nðxÞ ¼ q1m1ðxÞ þ q2m2ðxÞ. It is easy to check that (iii) and (iv) are satisfied, and mðxÞ and nðxÞ arecontinuous.

Suppose that mðxÞ > 0 is not true. Since mð0Þ ¼ 1, then there exists a constant a 2 ð0;1� such that mðaÞ ¼ 0 andmðxÞ > 0; x 2 ½0; aÞ. Since �ðpðxÞm0Þ0 þ qðxÞm ¼ 0; x 2 J and qðxÞ > 0, we have

ðpðxÞm0ðxÞÞ0 P 0; x 2 ½0; a�;

which leads to

pðxÞm0ðxÞP pð0Þm0ð0Þ ¼ 0

and so

m0ðxÞP 0:

This is a contradiction with mð0Þ ¼ 1 and mðaÞ ¼ 0. Therefore, mðxÞ > 0.Next we prove that mðxÞ is increasing. Since �ðpðxÞm0Þ0 þ qðxÞm ¼ 0; x 2 J, and mðxÞ > 0, then

ðpðxÞm0ðxÞÞ0 > 0; x 2 J

and

pðxÞm0ðxÞ > pð0Þm0ð0Þ ¼ 0:

Therefore,

m0ðxÞ > 0; x 2 J;

i.e., mðxÞ is increasing in J. So conclusion (i) is satisfied.Similarly, we can prove (ii). By Lm ¼ 0 and Ln ¼ 0, we have

ðpðxÞðm0n� n0mÞÞ0 ¼ nLm�mLn ¼ 0;

which implies pðxÞðm0ðxÞnðxÞ �mðxÞn0ðxÞÞ � c� for some constant c�. According to (i) and (ii), we claim that c� > 0. In fact, bypðxÞ > 0;m0ðxÞnðxÞ � n0ðxÞmðxÞ – 0, we have c� – 0. Hence, (i), (ii) and the fact c� ¼ pðxÞðm0ðxÞnðxÞ � n0ðxÞmðxÞÞ ¼pð1Þm0ð1Þnð1Þ lead to the conclusion (v) of Lemma 2.1. It is easy to prove that Gðx; yÞ has the properties of (vi)–(ix). See[9], for the details. This completes the proof of Lemma 2.1. h

We consider

�ðpðxÞu0Þ0 þ qðxÞu ¼ hðxÞ; 0 6 x 6 1;u0ð0Þ ¼ u0ð1Þ ¼ 0

(ð2:4Þ

for the case gðx; yÞ ¼ hðxÞ;8x; y 2 ½0;1�.

Lemma 2.2 [9]. Under the condition of Lemma 2.1, the following conclusions hold:

(i) BVP (2.4) has a unique solution;(ii) if function vðxÞ is defined by vðxÞ ¼

R 10 Gðx; yÞhðyÞdy, then v 2 C2ðJÞ and vðxÞ is the solution of BVP (2.4);

(iii) if v 2 C2ðJÞ is the solution of BVP (2.4), then vðxÞ satisfies

vðxÞ ¼Z 1

0Gðx; yÞhðyÞdy

Lemma 2.3 [9]. If u is the solution of the following equation:

uðxÞ ¼Z 1

0Gðx; yÞhðyÞdyþ

Xp

k¼1

Gðx; xkÞIkðuðxkÞÞ; ð2:5Þ

where Gðx; yÞ is the Green’s function of the following equation:

�ðpðxÞu0Þ0ðxÞ þ qðxÞuðxÞ ¼ 0; x 2 J;

u0ð0Þ ¼ u0ð1Þ ¼ 0;

(ð2:6Þ

Q. Li et al. / Applied Mathematics and Computation 206 (2008) 810–817 813

then u is the solution of the following impulse actions problem:

�ðpðxÞu0Þ0ðxÞ þ qðxÞuðxÞ ¼ hðxÞ; x 2 J0;

�Dðpu0Þjx¼xk¼ IkðuðxkÞÞ; k ¼ 1;2; . . . ;p;

u0ð0Þ ¼ u0ð1Þ ¼ 0:

8><>: ð2:7Þ

Define

K ¼ fu 2 U : uðxÞP 0 for all x and min06x61

uðxÞP rkukg; ð2:8Þ

where r is as in (1.4). One may readily verify that K is a cone in U. If h 2 L1ð0;1Þ with hðxÞP 0 a.e. x, it is easy to see thatðLhÞ :¼

R 10 Gðx; yÞhðyÞdy 2 K .

Suppose now that F : ½0;1� � R! ½0;1Þ is a continuous function. Define an operator T : U ! U by

ðTuÞðxÞ ¼Z 1

0Gðx; yÞFðy; uðyÞÞdyþ

Xp

k¼1

Gðx; xkÞIkðuðxkÞÞ ð2:9Þ

for u 2 U and x 2 ½0;1�.

Lemma 2.4 [9]. Let Ik 2 CðJ;RþÞ; k ¼ 1; . . . ; p. Then T is well defined and maps U into K. Moreover, T is continuous and completelycontinuous.

Next we give the theorem of fixed points in cones, one may refer to Guo and Lakshmikantham [8].

Theorem 2.5 [8]. Let X be a Banach space and K ð� XÞ be a cone. Assume that X1;X2 are open subsets of X with 0 2 X1;X1 � X2,and let

T : K \ ðX2 nX1Þ ! K

be a continuous and compact operator such that either

(i) kTukP kuk;u 2 K \ @X1 and kTuk 6 kuk;u 2 K \ @X2; or(ii) kTuk 6 kuk;u 2 K \ @X1 and kTukP kuk;u 2 K \ @X2.

Then T has a fixed point in K \ ðX2 nX1Þ.

Theorem 2.6 [9]. Assume X is a relatively open subset of a convex set K in a Banach space U. Let T: X! K be a compact map with0 2 X. Then either

(i) T has a fixed point in X, or(ii) there is a u 2 @X and a k < 1 such that u ¼ kTðuÞ.

3. Main results and proofs

In this section we establish the existence and multiplicity of positive solutions to (1.1). Since we are mainly interested inthe attractive-superlinear nonlinearities, we assume that gðx;uÞ satisfies

(H1) for each constant L > 0, there exists a function /L > 0 such that gðx;uÞP /LðxÞ for all ðx;uÞ 2 ½0;1� � ð0; L�;(H2) there exist continuous, non-negative functions f ðuÞ;hðuÞ and cðuÞ on ð0;1Þ such that

gðx;uÞ 6 f ðuÞ þ hðuÞ for all ðx;uÞ 2 ½0;1� � ð0;1Þ;max16k6p

IkðuÞ 6 cðuÞ for all ðx;uÞ 2 ½0;1� � ð0;1Þ

and f ðuÞ > 0 is non-increasing, and hðuÞ=f ðuÞ and cðuÞ are non-decreasing in u 2 ð0;1Þ;

(H3) there exists a positive number r such that

kxkf ðrrÞ 1þ hðrÞf ðrÞ

� �þ pBcðrÞ < r;

where r;B and xðxÞ are as in Section 1.

Theorem 3.1. Under Assumptions (A), (H1), (H2) and (H3), Eq. (1.1) has at least one positive periodic solution with 0 < kuk < r.

Proof. The existence is proved by using the Leray–Schauder alternative principle. Let N0 ¼ fn0;n0 þ 1; . . .g, wheren0 2 f1;2; . . .g is chosen such that

814 Q. Li et al. / Applied Mathematics and Computation 206 (2008) 810–817

kxkf ðrrÞ 1þ hðrÞf ðrÞ

� �þ pBcðrÞ þ 1

n0< r;

which is obtained from (H3). Fix n 2 N0. Consider the family of equations

�ðpðxÞu0Þ0ðxÞ þ qðxÞuðxÞ ¼ kgnðx; uðxÞÞ þ qðxÞn ; x 2 J0;

�DpðxÞu0jx¼xk¼ IkðuðxkÞÞ; k ¼ 1;2; . . . ;p;

u0ð0Þ ¼ u0ð1Þ ¼ 0;

8><>: ð3:1Þ

where k 2 ½0;1� and gnðx;uÞ ¼ gðx;maxfu;1=ngÞ; ðx;uÞ 2 ½0;1� � R. For every k and n, define an operator as follows:

ðTk;nuÞðxÞ ¼ kZ 1

0Gðx; yÞgnðy;uðyÞÞdyþ

Xp

k¼1

gðx; xkÞIkðuðxkÞÞ:

To find a solution of system (3.1) is equivalent to the following fixed point problem in U:

u ¼ Tk;nuþ 1n: ð3:2Þ

We claim that any fixed point u of (3.2) for any k 2 ½0;1� must satisfy kuk – r. Otherwise, assume that u is a solution of(3.2) for some k 2 ½0;1� such that kuk ¼ r. Note that gnðx;uÞP 0. By Lemma 2.4, for all x;uðxÞP 1=n andr P uðxÞP 1=nþ rku� 1=nk. By the choice of n0;1=n 6 1=n0 < r. Hence, for all x,

uðxÞP 1n

and r P uðxÞP 1nþ r u� 1

n

��������P

1nþ r r � 1

n

� �> rr: ð3:3Þ

By Assumption (H2), we have

IkðuðxkÞÞ 6max16j6p

IjðuðxkÞÞ 6 cðuðxkÞÞ 6 cðrÞ; ð3:4Þ

which leads to

Xp

k¼1

IkðuðxkÞÞ 6 pcðrÞ: ð3:5Þ

Using Assumption (H2), (3.3) and (3.5), we have, for all x,

uðxÞ ¼ kZ 1

0Gðx; yÞgnðy;uðyÞÞdyþ

Xp

k¼1

Gðx; xkÞIkðuðxkÞÞ þ1n6

Z 1

0Gðx; yÞgðy; uðyÞÞdyþ

Xp

k¼1

Gðx; xkÞIkðuðxkÞÞ þ1n

6

Z 1

0Gðx; yÞf ðuðyÞÞ 1þ hðuðyÞÞ

f ðuðyÞÞ

� �dyþ B

Xp

k¼1

IkðuðxkÞÞ þ1n6 xðxÞf ðrrÞ 1þ hðrÞ

f ðrÞ

� �þ pBcðrÞ þ 1

n0

6 kxkf ðrrÞ 1þ hðrÞf ðrÞ

� �þ pBcðrÞ þ 1

n0:

Therefore,

r ¼ kuk 6 kxkf ðrrÞ 1þ hðrÞf ðrÞ

� �þ pBcðrÞ þ 1

n0< r:

This is a contradiction to the choice of n0 and the claim is proved.From this claim the Leray–Schauder alternative principle implies that (3.2) (with k ¼ 1) has a fixed point, denoted by un,

in Br , i.e., Eq. (3.1) (with k ¼ 1) has a solution un with kunk < r. Since un satisfies (3.2), unðxÞP 1=n for all x and un is actually apositive solution of (3.1) (with k ¼ 1).

Next we claim that these solutions un have a uniform positive lower bound, i.e., there exists a constant d > 0, independentof n 2 N0, such that

minx

unðxÞP d ð3:6Þ

for all n 2 N0. In fact, we know from (H1) there exists a function /rðxÞ > 0 such that gðx;uÞP /rðxÞ for ðx;uÞ 2 ½0;1� � ð0; r�.Then

unðxÞ ¼Z 1

0Gðx; yÞgnðy; unðyÞÞdyþ

Xp

k¼1

Gðx; xkÞIkðunðxkÞÞ þ1n¼Z 1

0Gðx; yÞgðy; unðyÞÞdyþ

Xp

k¼1

Gðx; xkÞIkðunðxkÞÞ þ1n

PZ 1

0Gðx; yÞ/rðyÞdyþ

Xp

k¼1

Gðx; xkÞIkðunðxkÞÞ þ1n

PZ 1

0Gðx; yÞ/rðyÞdy P A

Z 1

0/rðyÞdy ¼: d:

Q. Li et al. / Applied Mathematics and Computation 206 (2008) 810–817 815

In order to pass the solutions un of the truncation equation (3.1) (with k ¼ 1) to that of the original equation (1.1), we needthe following fact:

ku0nk 6 H ð3:7Þ

for some constant H > 0 and for all n P n0.Since d 6 unðxÞ 6 r, let

M1 ¼ maxx2½0;1�

maxu2½d;r�

gðx; uÞ;

M2 ¼ maxx;y2½0;1�

jG0xðx; yÞj;

M3 ¼maxu2½d;r�

Xp

k¼1

IkðuÞ;

then

ku0nk ¼ sup06x61

ju0nðxÞj ¼ sup06x61

Z 1

0G0xðx; yÞgðy;unðyÞÞdyþ

Xp

k¼1

G0xðx; xkÞIkðunðxkÞÞ�����

����� 6 M1M2 þM3M2 ¼: H:

The fact kunk < r and (3.7) show that fungn2N0is a bounded and equi-continuous family on [0,1]. Now the Arzela–Ascoli the-

orem guarantees that fungn2N0has a subsequence, funj

gj2N , converging uniformly on ½0;1� to a function u 2 C½0;1�. From thefact kunk < r and (3.6), u satisfies d 6 uðxÞ 6 r for all x. Moreover, unj

satisfies the integral equation

unjðxÞ ¼

Z 1

0Gðx; yÞgðy; unj

ðyÞÞdyþXp

k¼1

Gðx; xkÞIkðunjðxkÞÞ þ

1nj:

Letting j!1, we arrive at

uðxÞ ¼Z 1

0Gðx; yÞgðy;uðyÞÞdyþ

Xp

k¼1

Gðx; xkÞIkðuðxkÞÞ;

where the uniform continuity of gðx;uÞ on ½0;1� � ½d; r� is used. Therefore, u is a positive solution of (1.1).Finally, it is not difficult to show that kuk < r, by noting that if kuk ¼ r, the argument similar to the proof of the first claim

will yield a contradiction. This completes the proof of Theorem 3.1. h

Corollary 3.2. Let the nonlinearity and IkðuÞ in (1.1) be the following form:

gðx; uÞ ¼ bðxÞu�a þ lcðxÞub þ eðxÞ; 0 6 x 6 1; x – xk;

IkðuÞ ¼ ckjuj; k ¼ 1;2; . . . ;p;

(ð3:8Þ

where a > 0; b P 0; ck P 0; bðxÞ; cðxÞ; eðxÞ 2 C½0;1� are non-negative functions and bðxÞ > 0 for all x, and l > 0 is a positiveparameter and pBmax16k6pck < 1. Then

(i) if b < 1, (1.1) has at least one positive solution for each l > 0; and(ii) there is a positive constant l� > 0 such that if b P 1, (1.1) has at least one positive solution for each 0 < l < l�.

Proof. We will apply Theorem 3.1. To this end, Assumption (H1) is fulfilled by /LðxÞ ¼ L�a �minxbðxÞ. Let c� ¼max16k6pck. Toverify (H2), one may simply take

f ðuÞ ¼ b0u�a; hðuÞ ¼ lc0ub þ e0; cðuÞ ¼ c�juj;

where

b0 ¼maxx

bðxÞ > 0; c0 ¼maxx

cðxÞP 0; e0 ¼ maxx

eðxÞP 0:

Now Assumption (H3) becomes

l <ð1� pBc�Þraraþ1 � ðb0 þ e0raÞkxk

c0raþbkxk

for some r > 0. So (1.1) has at least one positive solution for l with the following inequality:

0 < l < l� :¼ supr>0

ð1� pBc�Þraraþ1 � ðb0 þ e0raÞkxkc0raþbkxk :

Note that l� ¼ 1 if b < 1 and l� <1 if b P 1. We have the desired results. h

816 Q. Li et al. / Applied Mathematics and Computation 206 (2008) 810–817

Next we will find another positive periodic solution to Eq. (1.1) by using Theorem 2.5 for certain nonlinearities.

Theorem 3.3. Suppose that (A) and (H1)–(H3) are satisfied. Furthermore, assume that

(H4) there exist continuous, non-negative functions f1ðuÞ;h1ðuÞ and c1ðuÞ on ð0;1Þ such that

gðx;uÞP f1ðuÞ þ h1ðuÞ for all ðx;uÞ 2 ½0;1� � ð0;1Þ;min16k6p

IkðuÞP c1ðuÞ; Ik P 0; k ¼ 1; . . . ; p

and f1ðuÞ > 0 is non-increasing, and h1ðuÞ=f1ðuÞ and c1ðuÞ is non-decreasing in u 2 ð0;1Þ;

(H5) there exists a positive number R > r such that

R 6 kxkrf1ðRÞ 1þ h1ðrRÞf1ðrRÞ

� �þ pAc1ðrRÞ;

where r;A and xðxÞ are as in Section 1.

Then, besides the solution u constructed in Theorem 3.1, Eq. (1.1) has another positive solution ~u with r < k~uk 6 R.

Proof. Let U ¼ C½0;1� and K be the cone in U in Section 2. Let X1 ¼ Br and X2 ¼ BR be balls in U. The operatorT : K \ ðX2=X1Þ ! K is defined by (2.9). Note that any u 2 K \ ðX2=X1Þ satisfies 0 < rr 6 uðxÞ 6 R by the definition of K.

First we have kTuk < kuk for u 2 K \ @X1. In fact, if u 2 K \ @X1, then kuk ¼ r. Now the estimate kTuk < r can be obtainedalmost following the same ideas in proving Theorem 3.1. We omit the details.

Next we show that kTukP kuk for u 2 K \ @X2. To see this, let u 2 K \ @X2. Then kuk ¼ R and uðxÞP rR. As a result, itfollows from (H4) and (H5) that, for 0 6 x 6 1, similarly,

ðTuÞðxÞ ¼Z 1

0Gðx; yÞgðy; uðyÞÞdyþ

Xp

k¼1

Gðx; xkÞIkðuðxkÞÞPZ 1

0Gðx; yÞf1ðuðyÞÞ 1þ h1ðuðyÞÞ

f1ðuðyÞÞ

� �dyþ A

Xp

k¼1

IkðuðxkÞÞ

PZ 1

0Gðx; yÞf1ðRÞ 1þ h1ðrRÞ

f1ðrRÞ

� �dyþ pAc1ðrRÞ ¼ f1ðRÞ 1þ h1ðrRÞ

f1ðrRÞ

� �xðxÞ þ pAc1ðrRÞ

P rkxkf1ðRÞ 1þ h1ðrRÞf1ðrRÞ

� �þ pAc1ðrRÞP R ¼ kuk: ð3:9Þ

This implies kTukP kuk. The proof is ended. h

Now Theorem 2.5 guarantees that T has a fixed point ~u 2 K \ ðX2 nX1Þ. Thus r 6 k~uk 6 R. Clearly, ~u is a positive solution of(1.1) and actually satisfies k~uk > r.

Let us consider again system (3.8) in Corollary 3.2 for the superlinear case, i.e., b > 1. We assume also that cðxÞ > 0 for allx. Denote c# ¼min16k6pck. To verify (H4), one may simply take

f1ðuÞ ¼ b1u�a; h1ðuÞ ¼ lc1ub þ e1; c1ðuÞ ¼ c#juj;

where

b1 ¼minx

bðxÞ > 0; c1 ¼minx

cðxÞ > 0; e1 ¼ minx

eðxÞP 0:

Now Assumption (H5) becomes

l Pð1� pAc#rÞRaþ1 � rkxkðb1 þ e1RaÞ

c1Raþbraþbþ1kxk: ð3:10Þ

Since b > 1, the right-hand side goes to 0 as R! þ1. Thus, for any given 0 < l < l�, where l� is as in Corollary 3.2, it isalways possible to find such R r that (3.10) is satisfied. Thus, (1.1) has an additional solution ~u such that k~uk > r.

Corollary 3.4. Assume in (3.8) that b > 1; ck > 0; pBc� < 1 and bðxÞ > 0; cðxÞ > 0 for all x. Then, for each l with 0 < l < l�, thecorresponding Eq. (1.1) has at least two different positive solutions.

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