RAVI P. AGARWAL , DONAL O’REGAN - Aust MS · PDF fileRAVI P. AGARWAL 1, DONAL...

ANZIAM J.47(2006), 309–332

ON CONSTANT-SIGN PERIODIC SOLUTIONS IN MODELLINGTHE SPREAD OF INTERDEPENDENT EPIDEMICS

RAVI P. AGARWAL 1, DONAL O’REGAN2 and PATRICIA J. Y. WONG3

(Received 4 April, 2005; revised 18 November, 2005)

Abstract

We consider the following model that describes thespread ofn types of epidemics whichare interdependent on each other:

ui .t/ =∫ t

t−−gi .t; s/ fi .s;u1.s/;u2.s/; : : : ;un.s// ds; t ∈ R; 1 ≤ i ≤ n:

Our aim is to establish criteria such that the above system has one or multipleconstant-signperiodicsolutions.u1;u2; : : : ;un/, that is, for each 1≤ i ≤ n; ui is periodic and�i ui ≥ 0where�i ∈ {1;−1} is fixed. Examples are also included to illustrate the results obtained.

2000Mathematics subject classification: 45G10, 47H10.Keywords and phrases: periodic solutions, integral equations, epidemics, fixed point theo-rems.

1. Introduction

In this paper we shall consider the following system of integral equations that describesthespread ofn types of epidemics which are interdependent on each other:

ui .t/ =∫ t

t−−gi .t; s/ fi .s; u1.s/; u2.s/; : : : ; un.s// ds; t ∈ R; 1 ≤ i ≤ n: (M1)

Here, for 1≤ i ≤ n, we letui .t/ represent the proportion of the population infectedwith type i disease at timet , fi .t; u1.t/; u2.t/; : : : ; un.t// denotes the proportion ofthe population newly infected with typei disease per unit time,gi .t; s/ is a certain per

1Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, Florida32901-6975, USA; e-mail:[email protected] of Mathematics, National University of Ireland, Galway, Ireland.3School of Electrical and Electronic Engineering, Nanyang Technological University, 50 NanyangAvenue, Singapore 639798, Singapore; e-mail:[email protected]© Australian Mathematical Society 2006, Serial-fee code 1446-8735/06

309

http://www.austms.org.au/Publ/ANZIAM/V47P3/2304.html

mailto:[email protected]

mailto:[email protected]

310 Ravi P. Agarwal, Donal O’Regan and Patricia J. Y. Wong [2]

unit time environmental factor associated with typei disease at timet , and− ∈ R+ isthe length of time an individual remains infectious with typei disease.

Throughout we shall denoteu = .u1; u2; : : : ; un/. We say thatu is a solutionof constant signof (M1) if for each 1≤ i ≤ n; we have�i ui .t/ ≥ 0 for t ∈ R;where�i ∈ {1;−1} is fixed. Let 0< ! < ∞: A solution u of (M1) is said tobe !-periodic if ui is !-periodic for each 1≤ i ≤ n. More precisely, we meanu ∈ .A!.R//n = A!.R/× A!.R/× · · · × A!.R/ (n times) where

A!.R/ = {y ∈ BC.R/ | y.t/ = y.t + !/ for all t ∈ R}

andBC.R/ is the space of bounded and continuous functions onR with values inR.A simplified model of (M1) whenn = 1 andgi .t; s/ = 1, namely

y.t/ =∫ t

t−−f .s; y.s// ds; t ∈ [−;∞/ (M2)

was first introduced in 1976 by Cooke and Kaplan [9] and Smith [19]. Using Kras-nosel’skii’s fixed point theorem, it is shown in [9] that (M2) has a nontrivial periodicsolution provided the effective contact ratea.t/ = lim y→0 f .t; y/=y exceeds a certainthreshold level. On the other hand, Nussbaum [17] and Smith [19] have verified theexistence of a nontrivial periodic solution when− exceeds some threshold value. Inthe eighties, Leggett and Williams [15, 20] employed their own fixed point theoremsto obtain existence criteria for (M2). Their criteria complement the threshold-typeresults in [9, 17, 19].

In the present work, we have generalised the well-known (M2) to the system (M1)which not only models the spread ofinterdependentepidemics but also incorporatesenvironmental factorsin the modelling, and therefore is more robust for real-worldapplications. Further, by employing a variety of fixed point theorems we shall establishnewexistence results forconstant-sign periodic solutions, which, when reduced to aspecial case, improve and generalise those in [9, 12, 15]. As a side remark, muchwork has been carried out on the related system of the form

ui .t/ =∫ 1

0

gi .t; s/ fi .s; u1.s/; u2.s/; : : : ; un.s// ds; t ∈ [0; 1]; 1 ≤ i ≤ n:

The reader may refer to [3, 4, 5, 2, 6, 7] which are motivated by the vast amount ofliterature on the existence of positive solutions [1, 8, 10, 11, 16, 18].

The paper is organised as follows. In Section2, we state some well-known resultsin the literature which will be used to obtain the main theorems. Existence results for(M1) are developed in Section3. We also include examples to illustrate the importanceof the results obtained.

[3] On constant-sign periodic solutions in modelling the spread of interdependent epidemics 311

2. Preliminaries

We require the definition of aCarath́eodory function.

DEFINITION 2.1. A function f : I × Rn → R is a Carath́eodory functionif thefollowing conditions hold:

(a) the mapt 7→ f .t; u/ is measurable for allu ∈ Rn;(b) the mapu 7→ f .t; u/ is continuous for almost allt ∈ I :

The next three theorems are fixed point theorems; they are the main tools used inlater sections. Theorem2.2is usually calledKrasnosel’skii’s fixed point theorem in acone, Theorem2.3 is known as theLeray-Schauder alternative, and Theorem2.4 isfrom Leggett and Williams [14].

THEOREM 2.2 ([13]). Let B = .B; ‖ · ‖/ be a Banach space, and letC ⊂ B be acone inB: Assume�1; �2 are open bounded subsets ofB with 0 ∈ �1, S�1 ⊂ �2, andlet S : C ∩ .S�2\�1/ → C be a continuous and completely continuous operator suchthat either

(a) ‖Su‖ ≤ ‖u‖, u ∈ C ∩ @�1, and‖Su‖ ≥ ‖u‖, u ∈ C ∩ @�2, or(b) ‖Su‖ ≥ ‖u‖, u ∈ C ∩ @�1, and‖Su‖ ≤ ‖u‖, u ∈ C ∩ @�2.

ThenShas a fixed point inC ∩ .S�2\�1/.

THEOREM 2.3 ([18]). Let B be a Banach space withE ⊆ B closed and convex.AssumeU is a relatively open subset ofE with 0 ∈ U andS : SU → E is a continuousand compact map. Then either

(a) Shas a fixed point inSU , or(b) there existsu ∈ @U and½ ∈ .0; 1/ such thatu = ½Su:

THEOREM 2.4 ([14]). Let B = .B; ‖ · ‖/ be a Banach space,C ⊂ B a conein B; r1 > 0; r2 > 0, r1 6= r2 with R = max{r1; r2} and r = min{r1; r2}. DefineSC� = {u ∈ C | ‖u‖ ≤ �} and

C.u0/ = {u ∈ C | there exists½ > 0 such thatu ≥ ½u0}= {u ∈ C | there exists½ > 0 such thatu − ½u0 ∈ C}:

Let S : SCR → C be a continuous and compact map such that

(a) there existsu0 ∈ C\{0} with Su 6≤ u (equivalentlyu − Su 6∈ C) for u ∈@SCr2 ∩ C.u0/, and(b) ‖Su‖ ≤ ‖u‖ for u ∈ @SCr1.

ThenShas at least one fixed pointu ∈ C with r ≤ ‖u‖ ≤ R.


3. Existence Results for (M1)

Criteria for the existence of one or moreconstant-sign periodic solutionsof model(M1) are presented in this section. These results will be developed via various fixedpoint theorems.

To begin, let 0< ! < ∞ and let the Banach spaceB = .A!.R//n be equippedwith the norm|u|! = max1≤i ≤n supt∈[0;!] |ui .t/| = max1≤i ≤n |ui |!, where we let|ui |! =supt∈[0;!] |ui .t/|, 1 ≤ i ≤ n.

Define the operatorS : B → .C.R//n by

Su.t/ = .S1u.t/; S2u.t/; : : : ; Snu.t//; t ∈ R; (3.1)

where

Si u.t/ =∫ t

t−−gi .t; s/ fi .s; u.s// ds; t ∈ R; 1 ≤ i ≤ n: (3.2)

Clearly, a fixed point of the operatorS is a solution of the system (M1).With �i ∈ {1;−1}; 1 ≤ i ≤ n fixed, define

[0;∞/i ={

[0;∞/; �i = 1;

.−∞; 0]; �i = −1;

K̃ = {u ∈ B | �i ui .t/ ≥ 0; t ∈ R; 1 ≤ i ≤ n}and

K = {u ∈ K̃ | � j u j .t/ > 0 for somej ∈ {1; : : : ; n} and somet ∈ R}= K̃\{0}:

Our first result uses Krasnosel’skii’s fixed point theorem (Theorem2.2).

THEOREM 3.1. Let 1 ≤ p ≤ ∞; q be such that1=p + 1=q = 1, 0 < ! < ∞ andlet �i ∈ {1;−1}, 1 ≤ i ≤ n be fixed. Assume the following hold for each1 ≤ i ≤ n:

gti .s/ ≡ gi .t; s/ ∈ Lq[0; !] for eacht ∈ [0; !]; (3.3)

the mapt 7→ gti is continuous from[0; !] to Lq[0; !]; (3.4)

gi .t; s/ ≥ 0 for all t ∈ [0; !] anda:e: s ∈ [0; !]; (3.5)

gi .t; s + !/ = gi .t; s/ for all t ∈ R anda:e: s ∈ R; (3.6)

gi .t + !; s/ = gi .t; s/ for all t ∈ R anda:e: s ∈ R; (3.7)

fi : [0; !] ×Rn → R is a Carath́eodory function; (3.8)

�i fi .t; u/ ≥ 0 for a.e.t ∈ [0; !] and allu ∈ K̃ ; (3.9)

fi .t + !; u/ = fi .t; u/ for a.e.t ∈ R and all u ∈ K̃ ; (3.10)


for eachr > 0, there exists¼ir ∈ L p[0; !] such that|u j | ≤ r ,

1 ≤ j ≤ n, implies | fi .t; u1; u2; : : : ; un/| ≤ ¼ir .t/ for a.e.

t ∈ [0; !];(3.11)

there exists a function i : ∏nj =1[0;∞/ j → [0;∞/ continuous and

‘nondecreasing’ in the sense that for each1 ≤ j ≤ n, if |u j | ≤ |v j |,then i .u1; : : : ; uj −1; u j ; u j +1; : : : ; un/ ≤ i .u1; : : : ; uj −1; v j ; u j +1;

: : : ; un/, a constant²i , 0 < ²i ≤ 1, and a functionbi : R → [0;∞/,bi ∈ L p[0; !], bi .t + !/ = bi .t/ for a.e.t ∈ R, with ²i bi .t/ i .u/ ≤�i fi .t; u/ ≤ bi .t/ i .u/ for a.e.t ∈ [0; !] and all u ∈ K̃ ;

(3.12)

{there exists a continuous function�i : .0; 1/ → .0;∞/ such that forany0< m< 1 andu ∈ [0;∞/n, we have i .mu/ ≥ �i .m/ i .u/;

(3.13)

K2;i = inft∈[0;!]

∫ t

t−−gi .t; s/bi .s/ ds> 0; (3.14)

{there exists0 < Mi < 1 with Mi ≤ ²i .K2;i =K1;i /�i .M0/, whereK1;i = supt∈[0;!]

∫ t

t−− gi .t; s/bi .s/ ds and M0 = min1≤ j ≤n M j ∈ .0; 1/. (3.15)

Moreover, suppose

there existsÞ > 0 with max1≤ j ≤n

j .Þ; Þ; : : : ; Þ/K1; j < Þ (3.16)

and

there existsþ > 0, þ 6= Þ, so that for eachz ∈ {1; 2; : : : ; n}, thereexists jz ∈ {1; 2; : : : ; n} such that for anyu ∈ [0;∞/n, we have jz.u/ ≥ ajzk.uk/ for each1 ≤ k ≤ n, whereajzk : [0;∞/ → [0;∞/

is continuous,ajzk.x/ > 0 if x > 0, andþ < ² jzajzz.M0þ/K2; jz.

(3.17)

Then(M1) has at least one constant-sign solutionu ∈ .A!.R//n satisfying

(a) 0< Þ < |u|! < þ and�i ui .t/ > MiÞ for all t ∈ R and somei ∈ {1; : : : ; n}, ifÞ < þ;(b) 0< þ < |u|! < Þ and�i ui .t/ > Miþ for all t ∈ R and somei ∈ {1; : : : ; n}, ifþ < Þ.

PROOF. We shall employ Theorem2.2. Define a coneC in B as

C = {u ∈ B | �i ui .t/ ≥ 0 and�i ui .t/ ≥ Mi |ui |! for 1 ≤ i ≤ n; t ∈ [0; !]}= {u ∈ B | �i ui .t/ ≥ 0 and�i ui .t/ ≥ Mi |ui |! for 1 ≤ i ≤ n; t ∈ R}; (3.18)

whereMi is defined in (3.15). Note thatC ⊆ K̃ .


First, we shall show thatSmapsC into .A!.R//n, that is, that

Si : C → A!.R/; 1 ≤ i ≤ n: (3.19)

Let u ∈ C. Then, using (3.6), (3.7) and (3.10), we get fort ∈ R and 1≤ i ≤ n,

Si u.t + !/ =∫ t+!

t+!−−gi .t + !; s/ fi .s; u.s// ds

=∫ t

t−−gi .t + !; x + !/ fi .x + !; u.x + !// dx

=∫ t

t−−gi .t + !; x + !/ fi .x + !; u.x// dx

=∫ t

t−−gi .t; x/ fi .x; u.x// dx = Si u.t/: (3.20)

We shall next show thatSi u ∈ C[0; !], 1 ≤ i ≤ n. Sinceu ∈ B ⊂ .BC.R//n; thereexistsr > 0 with |u|! ≤ r . Therefore (3.11) guarantees, for each 1≤ i ≤ n, theexistence of¼i

r ∈ L p[0; !]; such that| fi .s; u.s//| ≤ ¼ir .s/, a.e.s ∈ [0; !]. This,

together with (3.10), implies that there exists�ir ∈ L p[−−; !], 1 ≤ i ≤ n, with

| fi .s; u.s//| ≤ �ir .s/; a.e.s ∈ [−−; !]: (3.21)

(Alternatively we could assume without loss of generality that¼ir .t + !/ = ¼i

r .t/ fora.e.t ∈ R and take�i

r = ¼ir .) Now, for t1; t2 ∈ [0; !] with t2 > t1 andt2 − − < t1, we

have for each 1≤ i ≤ n;

Si u.t1/− Si u.t2/

=∫ t1

t1−−gi .t1; s/ fi .s; u.s// ds−

∫ t2

t2−−gi .t2; s/ fi .s; u.s// ds

=∫ t2−−

t1−−gi .t1; s/ fi .s; u.s// ds+

∫ t1

t2−−[gi .t1; s/− gi .t2; s/] fi .s; u.s// ds

−∫ t2

t1

gi .t2; s/ fi .s; u.s// ds

and so

|Si u.t1/− Si u.t2/|

≤(∫ t2−−

t1−−|gt1

i .s/|q ds

)1=q (∫ t2−−

t1−−| fi .s; u.s//|p ds

)1=p

+(∫ t1

t2−−|gt1

i .s/− gt2i .s/|q ds

)1=q (∫ t1

t2−−| fi .s; u.s//|p ds

)1=p


+(∫ t2

t1

|gt2i .s/|q ds

)1=q (∫ t2

t1

| fi .s; u.s//|p ds

)1=p

:

Using (3.21), it then follows that

|Si u.t1/− Si u.t2/|

≤ supt∈[0;!]

(∫ w−−

−−|gt

i .s/|q ds

)1=q (∫ t2−−

t1−−[�i

r .s/]p ds

)1=p

+(∫ w

−−|gt1

i .s/− gt2i .s/|q ds

)1=q (∫ w

−−[�i

r .s/]p ds

)1=p

+ supt∈[0;!]

(∫ w

0

|gti .s/|q ds

)1=q (∫ t2

t1

[¼ir .s/]p ds

)1=p

: (3.22)

We also note that (3.3) and (3.6) guarantee that

gti ∈ Lq[−−; !] for t ∈ [0; !]: (3.23)

In view of (3.4) and (3.23), it is clear from (3.22) that |Si u.t1/ − Si u.t2/| → 0 ast1 → t2. Hence

Si u ∈ C[0; !]; 1 ≤ i ≤ n: (3.24)

Further, noting (3.23) and (3.21), we have fort ∈ [0; !] and 1≤ i ≤ n,

|Si u.t/| ≤∫ t

t−−|gi .t; s/|�i

r .s/ ds ≤(∫ !

−−|gt

i .s/|q ds

)1=q (∫ !

−−[�i

r .s/]p ds

)1=p

< ∞:

Thus

Si u is bounded, 1≤ i ≤ n: (3.25)

Having established (3.20), (3.24) and (3.25), we have completed the proof of (3.19).Next, we shall verify that

S : C → .A!.R//n is continuous: (3.26)

Let {um}, u ∈ C ({um} is a sequence) withum → u. Sinceum; u ∈ B ⊂ .BC.R//n,there existsr > 0 with |um|!; |u|! ≤ r . Therefore, as in a previous argument, (3.10)and (3.11) lead to the existence of�i

r ∈ L p[−−; !], 1 ≤ i ≤ n, such that (3.21) and(3.21)|u=um hold. Hence

| fi .s; um.s//− fi .s; u.s//| ≤ 2�ir .s/; a.e. s ∈ [−−; !] (3.27)


and also from (3.8) and (3.10) we have

| fi .s; um.s//− fi .s; u.s//| → 0 pointwise for a.e.s ∈ [−−; !]: (3.28)

Now, for t ∈ [0; !] and 1≤ i ≤ n; we find

|Si um.t/− Si u.t/|≤∫ t

t−−|gi .t; s/[ fi .s; um.s//− fi .s; u.s//]| ds

≤(∫ t

t−−|gt

i .s/|q ds

)1=q (∫ t

t−−| fi .s; um.s//− fi .s; u.s//|p ds

)1=p

≤ supx∈[0;!]

(∫ !

−−|gx

i .s/|q ds

)1=q (∫ !

−−| fi .s; um.s//− fi .s; u.s//|p ds

)1=p

m→∞−−−→ 0;

where we have used (3.23), (3.27) and (3.28). It follows that

|Sum − Su|! = max1≤i ≤n

|Si um − Si u|! = max

1≤i ≤nsup

t∈[0;!]|Si u

m.t/− Si u.t/| m→∞−−−→ 0:

Hence we have proved (3.26).Next, we shall show that

S : C → .A!.R//n is completely continuous. (3.29)

Let � be a bounded set inC, that is, there existsr > 0 such that|u|! ≤ r for allu ∈ �. We shall prove thatSi�, 1 ≤ i ≤ n, is relatively compact inA!.R/. Let{um} be a sequence in�. Then, for each 1≤ i ≤ n, {Si um} is a sequence inSi�.Now, using once again a previous argument, (3.10) and (3.11) lead to the existence of�i

r ∈ L p[−−; !], 1 ≤ i ≤ n, such that (3.21)|u=um holds. Hence, we get fort ∈ [0; !]and 1≤ i ≤ n,

|Si um.t/| ≤

(∫ t

t−−|gt

i .s/|q ds

)1=q (∫ t

t−−| fi .s; um.s//|p ds

)1=p

≤ supx∈[0;!]

(∫ !

−−|gx

i .s/|q ds

)1=q (∫ !

−−[�i

r .s/]p ds

)1=p

< ∞;

where we have also used (3.23). Thus{Si um} is a uniformly bounded sequence inC[0; !]. An argument similar to that in (3.22) guarantees that{Si um} is equicontinuouson[0; !]. The Arzela-Ascoli theorem guarantees anSi u ∈ C[0; !] and a subsequence{Si umk} of {Si um} which converges uniformly on[0; !] to Si u. SinceSi umk.t + !/ =


Si umk.t/, by letting k → ∞ we haveSi u.t + !/ = Si u.t/. ThusSi u ∈ A!.R/ andSi umk → Si u in A!.R/. So Si : C → A!.R/ is completely continuous, 1≤ i ≤ n.This completes the proof of (3.29).

We also need to show that

S : C → C: (3.30)

Let u ∈ C. We note that for a fixedt ∈ [0; !], anys ∈ [t − −; t] can be written ass + z! = s!, wherez is some integer ands! ∈ [0; !]. Therefore, in view of (3.5),(3.6), (3.9), (3.10) and (3.12), we get fort ∈ [0; !], a.e.s ∈ [t − −; t] and 1≤ i ≤ n,

gi .t; s/ = gi .t; s!/ ≥ 0; (3.31)

�i fi .s; u.s// = �i fi .s!; u.s// ≥ 0; (3.32)

�i fi .s; u.s// = �i fi .s!; u.s// ≤ bi .s!/ i .u.s// = bi .s/ i .u.s// (3.33)

and

�i fi .s; u.s// = �i fi .s!; u.s// ≥ ²i bi .s!/ i .u.s// = ²i bi .s/ i .u.s//; (3.34)

where we have also used the fact thatbi .x + !/ = bi .x/ for a.e.x ∈ R. Then, noting(3.31) and (3.32), it is clear that

�i Si u.t/ =∫ t

t−−gi .t; s/�i fi .s; u.s// ds ≥ 0; t ∈ [0; !]; 1 ≤ i ≤ n: (3.35)

Now, using (3.12), (3.31), (3.33) and (3.35), we find fort ∈ [0; !] and 1≤ i ≤ n,

|Si u.t/| = �i Si u.t/ ≤∫ t

t−−gi .t; s/bi .s/ i .u.s// ds

≤ i

(|u1|!; |u2|!; : : : ; |un|!) ∫ t

t−−gi .t; s/bi .s/ ds

≤ i

(|u1|!; |u2|!; : : : ; |un|!)K1;i :

This yields

|Si u|! ≤ K1;i i .|u1|!; |u2|!; : : : ; |un|!/ ; 1 ≤ i ≤ n: (3.36)

On the other hand, sinceu ∈ C, we have

|ui .t/| = �i ui .t/ ≥ Mi |ui |! ≥ M0|ui |!; t ∈ R; 1 ≤ i ≤ n; (3.37)

whereM0 = min1≤ j ≤n M j . Therefore (3.13)–(3.15), (3.31) and (3.34)–(3.37) give fort ∈ [0; !] and 1≤ i ≤ n,

�i Si u.t/ ≥ ²i

∫ t



≥ ²i i

(M0|u1|!;M0|u2|!; : : : ;M0|un|!

) ∫ t


≥ ²i�i .M0/ i

(|u1|!; |u2|!; : : : ; |un|!) ∫ t


≥ ²i�i .M0/ i

(|u1|!; |u2|!; : : : ; |un|!)K2;i

≥ ²i�i .M0/|Si u|! K2;i

K1;i≥ Mi |Si u|!:

Together with (3.35), we have shown thatSu∈ C and hence (3.30) is proved.Let

�Þ = {u ∈ B | |u|! < Þ} and �þ = {u ∈ B | |u|! < þ}: (3.38)

We shall now show that

|Su|! < |u|! for u ∈ C ∩ @�Þ (3.39)

and

|Su|! > |u|! for u ∈ C ∩ @�þ: (3.40)

To verify (3.39), let u ∈ C ∩ @�Þ. Then|u|! = Þ. For t ∈ [0; !], using (3.12),(3.31), (3.33) and (3.35) we find, for 1≤ i ≤ n,

|Si u.t/| ≤ i

(|u|!; |u|!; : : : ; |u|!) ∫ t

t−−gi .t; s/bi .s/ ds ≤ i .Þ; Þ; : : : ; Þ/K1;i ;

which, together with (3.16), leads to

|Su|! = max1≤i ≤n

|Si u|! ≤ max1≤i ≤n

i .Þ; Þ; : : : ; Þ/K1;i < Þ = |u|!: (3.41)

Thus (3.39) is proved.Next, let u ∈ C ∩ @�þ . Then |u|! = þ = |uz|! for somez ∈ {1; 2; : : : ; n}.

For t ∈ [0; !]; using (3.17), (3.31), (3.34), (3.35) and (3.37), we get for somejz ∈ {1; 2; : : : ; n},

|Sjzu.t/| ≥ ² jz

∫ t

t−−gjz.t; s/bjz.s/ jz.u.s// ds

≥ ² jz jz

(M0|u1|!;M0|u2|!; : : : ;M0|un|!

) ∫ t

t−−gjz.t; s/bjz.s/ ds

≥ ² jzajzz.M0|uz|!/K2; jz = ² jzajzz.M0þ/K2; jz > þ = |u|!:Thus

|Su|! ≥ |Sjzu|! > |u|! (3.42)


and (3.40) is true.Now Theorem2.2 guarantees thatS has a fixed pointu with u ∈ C ∩ .S�Þ\�þ/

if þ < Þ, whereasu ∈ C ∩ .S�þ\�Þ/ if Þ < þ. Hence, equivalently (M1) has aconstant-sign solutionu ∈ .A!.R//n with min{Þ; þ} ≤ |u|! ≤ max{Þ; þ}. Note that|u|! 6= Þ and |u|! 6= þ. To see this, suppose|u|! = Þ or |u|! = þ. Then, sinceu = Suwe have, noting (3.41) and (3.42),

Þ = |u|! = |Su|! ≤ max1≤i ≤n

i .Þ; Þ; : : : ; Þ/K1;i < Þ = |u|!or þ = |u|! = |Su|! ≥ |Sjzu|! > |u|! = þ, which are contradictions. Hence

min{Þ; þ} < |u|! < max{Þ; þ}:Finally, |u|! = |ui |! for somei ∈ {1; 2; : : : ; n}. Sinceu ∈ C, we have, fort ∈ R,�i ui .t/ ≥ Mi |ui |! = Mi |u|! > Mi min{Þ; þ}.

REMARK 3.2. In Theorem3.1, it is possible to replace (3.3), (3.4), (3.6)–(3.8) and(3.11) with the following:

gti .s/ ≡ gi .t; s/ ∈ L1[0; !] for eacht ∈ [0; !]; (3.43)

the mapt 7→ gti is continuous from[0; !] to L1[0; !]; (3.44){

gi .t; s/ ≥ 0 for all t ∈ [0; !] and a.e.s ∈ [t − −; t];gi .t + !; s + !/ = gi .t; s/ for all t ∈ R and a.e.s ∈ R;

(3.45)

fi : [0; !] ×Rn → R is continuous. (3.46)

As in [7, Section 3], we need only notice that (3.43)–(3.46) imply that S : C → C iscontinuous and completely continuous.

Our next result gives the existence ofmultipleconstant-sign periodic solutions.

THEOREM3.3. Let1 ≤ p ≤ ∞, q be such that1=p+1=q = 1, 0< ! < ∞ and let�i ∈ {1;−1}, 1 ≤ i ≤ n, be fixed. Assume that(3.3)–(3.15) hold for each1 ≤ i ≤ n.Let (3.16) be satisfied forÞ = Þ`, ` = 1; : : : ; k, and (3.17) be satisfied forþ = þ`,` = 1; : : : ;m.

(a) If m = k + 1 and0 < þ1 < Þ1 < · · · < þk < Þk < þk+1, then(M1) has(atleast) 2k constant-sign solutionsu1; : : : ; u2k ∈ .A!.R//n such that

0< þ1 < |u1|! < Þ1 < |u2|! < þ2 < · · · < Þk < |u2k|! < þk+1:

(b) If m = k and0 < þ1 < Þ1 < · · · < þk < Þk; then(M1) has(at least) 2k − 1constant-sign solutionsu1; : : : ; u2k−1 ∈ .A!.R//n such that

0< þ1 < |u1|! < Þ1 < |u2|! < þ2 < · · · < þk < |u2k−1|! < Þk:


(c) If k = m + 1 and0 < Þ1 < þ1 < · · · < Þm < þm < Þm+1; then(M1) has(atleast) 2m constant-sign solutionsu1; : : : ; u2m ∈ .A!.R//n such that

0< Þ1 < |u1|! < þ1 < |u2|! < Þ2 < · · · < þm < |u2m|! < Þm+1:

(d) If k = m and0 < Þ1 < þ1 < · · · < Þk < þk; then(M1) has(at least) 2k − 1constant-sign solutionsu1; : : : ; u2k−1 ∈ .A!.R//n such that

0< Þ1 < |u1|! < þ1 < |u2|! < Þ2 < · · · < Þk < |u2k−1|! < þk:

EXAMPLE 1. Consider the nonlinear system of integral equations

ui .t/ =∫ t

t−−gi .t; s/�i hi .s/

[|u1.s/| + |u2.s/| + · · · + |un.s/| ]

ds; (3.47)

t ∈ R, 1 ≤ i ≤ n, where 0< < 1 and�i ∈ {−1; 1}, 1 ≤ i ≤ n, are fixed. For each1 ≤ i ≤ n, assume (3.3)–(3.7) hold,{

there exists! > 0 with hi .t + !/ = hi .t/ for a.e.t ∈ R;hi is nonnegative andhi ∈ L p[0; !] (3.48)

and

inft∈[0;!]

∫ t

t−−gi .t; s/hi .s/ ds> 0: (3.49)

Then (3.47) has at least one constant-sign solutionu ∈ .A!.R//n such that{0< min{Þ; þ} < |u|! < max{Þ; þ} and

�kuk.t/ > Mk min{Þ; þ} for all t ∈ R and somek ∈ {1; : : : ; n}; (3.50)

where

Mi = K2;i

K1;i

(min

1≤ j ≤n

K2; j

K1; j

) =.1− /∈ .0; 1/; 1 ≤ i ≤ n (3.51)

with

K2;i = inft∈[0;!]

∫ t

t−−gi .t; s/hi .s/ ds; K1;i = sup

t∈[0;!]

∫ t

t−−gi .t; s/hi .s/ ds (3.52)

andÞ; þ are positive numbers satisfying

Þ >

(n max

1≤ j ≤nK1; j

)1=.1− /and þ < M =.1− /

0

(min

1≤ j ≤nK2; j

)1=.1− /; (3.53)


whereM0 = min1≤ j ≤n M j ∈ .0; 1/.To see that the above is true, we shall apply Theorem3.1with

fi .t; u/ = �i hi .t/

[|u1| + |u2| + · · · + |un| ];

²i = 1; bi = hi ; i .u/ = |u1| + |u2| + · · · + |un| ;�i .m/ = m and ajzk.x/ = x ; 1 ≤ i; jz; k ≤ n:

(3.54)

Note that (3.8)–(3.14) are clearly satisfied. Next, the inequality in condition (3.15) isreduced toMi ≤ .K2;i =K1;i /M

0 , 1 ≤ i ≤ n, which will be satisfied if we set

Mi = K2;i

K1;iM

0 ; 1 ≤ i ≤ n: (3.55)

It follows immediately thatM0 = min1≤ j ≤n M j = min1≤ j ≤n.K2; j=K1; j /M

0 or

M0 =(

min1≤ j ≤n

K2; j

K1; j

)1=.1− /: (3.56)

Substituting (3.56) in (3.55) yields (3.51).Further, (3.16) holds since noting (3.54) we find

max1≤ j ≤n

j .Þ; Þ; : : : ; Þ/ K1; j = nÞ max1≤ j ≤n

K1; j < Þ Þ1− = Þ;

where we have also used (3.53).Finally, (3.17) is fulfilled since in view of (3.53) and (3.54) we have

² jzajzz.M0þ/K2; jz = .M0þ/ K2; jz ≥ .M0þ/

min1≤ j ≤n

K2; j

> .M0þ/ þ1− M−

0 = þ:

We now conclude from Theorem3.1that the system (3.47) has at least one constant-sign solutionu ∈ .A!.R//n satisfying (3.50)–(3.53).

The next result uses the nonlinear alternative (Theorem2.3) to show the existenceof a periodic solution (which neednot be of constant sign).

THEOREM 3.4. Let 1 ≤ p ≤ ∞, q be such that1=p + 1=q = 1 and0 < ! < ∞.For each1 ≤ i ≤ n, assume that(3.3), (3.4), (3.6)–(3.8) and (3.11) hold and

fi .t + !; u/ = fi .t; u/ for a.e.t ∈ R and u ∈ Rn: (3.57)

Suppose there exists a constantc, independent of½, such that

|u|! 6= c (3.58)


for any solutionu ∈ .A!.R//n of the system

ui .t/ = ½

∫ t

t−−gi .t; s/ fi .s; u.s// ds; t ∈ R; 1 ≤ i ≤ n; .3:59/½

where½ ∈ .0; 1/. Then,(M1) has at least one solutionu ∈ .A!.R//n with |u|! ≤ c.

PROOF. We shall apply Theorem2.3. Let E = B = ..A!.R//n; | · |!/ andU = {u ∈ B | |u|! < c}. Clearly, a solution of.3:59/½ is a fixed point of theequationu = ½Su. As seen in the proof of Theorem3.1, (3.3), (3.4), (3.6)–(3.8),(3.11) and (3.57) guarantee thatS : B → B is continuous and completely continuous.In view of (3.58), we cannot have conclusion (b) of Theorem2.3, hence conclusion (a)of Theorem2.3must hold, that is,Shas a fixed point inSU , or equivalently the system(M1) has a solutionu ∈ SU with |u|! ≤ c.

Using Theorem3.4, we shall obtain the existence of aconstant-signperiodicsolution in the next result.

THEOREM 3.5. Let 1 ≤ p ≤ ∞, q be such that1=p + 1=q = 1, 0 < ! < ∞ andlet �i ∈ {1;−1}, 1 ≤ i ≤ n, be fixed. For each1 ≤ i ≤ n, assume that(3.3)–(3.9),(3.11) and (3.57) hold and

there exists a function i : ∏nj =1[0;∞/ j → [0;∞/ continu-

ous and ‘nondecreasing’ in the sense that for each1 ≤ j ≤n, if |u j | ≤ |v j |, then i .u1; : : : ; uj −1; u j ; u j +1; : : : ; un/ ≤ i .u1; : : : ; uj −1; v j ; u j +1; : : : ; un/, and a functionbi : R →[0;∞/, bi ∈ L p[0; !], bi .t + !/ = bi .t/ for a.e. t ∈ R, with�i fi .t; u/ ≤ bi .t/ i .u/ for a.e.t ∈ [0; !] and allu ∈ K̃ .

(3.60)

Moreover, suppose{there existsÞ > 0 with max1≤ j ≤n j .Þ; Þ; : : : ; Þ/K1; j < Þ, whereK1; j = supt∈[0;!]

∫ t

t−− gj .t; s/bj .s/ ds. (3.61)

Then,(M1) has at least one constant-sign solutionu ∈ .A!.R//n with |u|! < Þ.

PROOF. We shall employ Theorem3.4. To begin, we consider the system

ui .t/ =∫ t

t−−gi .t; s/ f̂i .s; u.s// ds; t ∈ R; 1 ≤ i ≤ n; (3.62)

where f̂i : R×Rn → R is defined by

f̂i .t; u1; u2; : : : ; un/ = fi .t; �1|u1|; �2|u2|; : : : ; �n|un|/; 1 ≤ i ≤ n: (3.63)


We shall prove that (3.62) has a solution. For this, we consider the system

ui .t/ = ½

∫ t

t−−gi .t; s/ f̂i .s; u.s// ds; t ∈ R; 1 ≤ i ≤ n; .3:64/½

where½ ∈ .0; 1/. Let u ∈ .A!.R//n be any solution of.3:64/½. If we can show that

|u|! 6= Þ; (3.65)

then by Theorem3.4 it follows that (3.62) has a solution.Now, using (3.5), (3.9), the fact that.�1|u1|; �2|u2|; : : : ; �n|un|/ ∈ K̃ , (3.31), (3.32)

and (3.63) we get fort ∈ [0; !] and 1≤ i ≤ n,

�i ui .t/ = ½

∫ t

t−−gi .t; s/�i f̂i .s; u1.s/; u2.s/; : : : ; un.s// ds

= ½

∫ t

t−−gi .t; s/�i fi .s; �1|u1.s/|; �2|u2.s/|; : : : ; �n|un.s/|/ ds ≥ 0;

which means that

|ui .t/| = �i ui .t/; t ∈ [0; !]; 1 ≤ i ≤ n: (3.66)

An application of (3.33),(3.60) and (3.66) yields for t ∈ [0; !] and 1≤ i ≤ n,

|ui .t/| = �i ui .t/ ≤∫ t

t−−gi .t; s/�i fi .s; �1|u1.s/|; : : : ; �n|un.s/|/ ds

≤∫ t

t−−gi .t; s/bi .s/ i .�1|u1.s/|; : : : ; �n|un.s/|/ ds

≤ i .|u|!; : : : ; |u|!/∫ t


≤ i .|u|!; : : : ; |u|!/K1;i :

This immediately leads to

|ui |! ≤ i .|u|!; : : : ; |u|!/K1;i ; 1 ≤ i ≤ n: (3.67)

Now |u|! = |uz|! for somez ∈ {1; 2; : : : ; n}. Then it follows from (3.67) that

|u|! ≤ z.|u|!; : : : ; |u|!/K1;z: (3.68)

Noting (3.61) and (3.68), we conclude that|u|! 6= Þ. Hence (3.65) is proved.It now follows from Theorem3.4 that the system (3.62) has a solutionu∗ =

.u∗1; u∗

2; : : : ; u∗n/ ∈ .A!.R//n with ‖u∗‖ ≤ Þ, and

u∗i .t/ =

∫ t

t−−gi .t; s/ f̂i .s; u∗.s// ds; t ∈ R; 1 ≤ i ≤ n:


Using a similar argument as above, together withu∗ ∈ .A!.R//n, it can be easily seenthat

|u∗i .t/| = �i u

∗i .t/; t ∈ R; 1 ≤ i ≤ n; (3.69)

and

|u∗|! 6= Þ: (3.70)

Thereforeu∗ is of constant sign and|u∗|! < Þ: Further, using (3.63) and (3.69), wehave fort ∈ R and 1≤ i ≤ n,

u∗i .t/ =

∫ t

t−−gi .t; s/ f̂i .s; u∗.s// ds =

∫ t

t−−gi .t; s/ fi .s; �1|u∗

1.s/|; : : : ; �n|u∗n.s/|/ ds

=∫ t

t−−gi .t; s/ fi .s; �

21u∗

1.s/; : : : ; �2nu∗

n.s// ds

=∫ t

t−−gi .t; s/ fi .s; u∗

1.s/; : : : ; u∗n.s// ds:

Henceu∗ is in fact a solution of (M1). The proof is now complete.

In Theorem3.5, it is possible for|u|! to be zero. However, we can combineTheorem3.5with Theorem3.1to obtain the existence of multiplenontrivial constant-sign periodic solutions, stated as the next result.

THEOREM 3.6. Let 1 ≤ p ≤ ∞, q be such that1=p + 1=q = 1, 0 < ! < ∞ andlet �i ∈ {1;−1}, 1 ≤ i ≤ n, be fixed. Assume that(3.3)–(3.15) and (3.57) hold foreach1 ≤ i ≤ n. Let (3.16) be satisfied forÞ = Þ`, ` = 1; 2; : : : ; k, and (3.17) besatisfied forþ = þ`, ` = 1; 2; : : : ;m.

(a) If m = k + 1 and0 < þ1 < Þ1 < · · · < þk < Þk < þk+1, then(M1) has(atleast) 2k constant-sign solutionsu1; : : : ; u2k ∈ .A!.R//n such that

0< þ1 < |u1|! < Þ1 < |u2|! < þ2 < · · · < Þk < |u2k|! < þk+1:

(b) If m = k and0 < þ1 < Þ1 < · · · < þk < Þk; then(M1) has(at least) 2k − 1constant-sign solutionsu1; : : : ; u2k−1 ∈ .A!.R//n such that

0< þ1 < |u1|! < Þ1 < |u2|! < þ2 < · · · < þk < |u2k−1|! < Þk:

(c) If k = m + 1 and0 < Þ1 < þ1 < · · · < Þm < þm < Þm+1, then(M1) has(atleast) 2m + 1 constant-sign solutionsu0; : : : ; u2m ∈ .A!.R//n such that

0 ≤ |u0|! < Þ1 < |u1|! < þ1 < |u2|! < Þ2 < · · · < þm < |u2m|! < Þm+1:


(d) If k = m and 0 < Þ1 < þ1 < · · · < Þk < þk, then (M1) has (at least) 2kconstant-sign solutionsu0; : : : ; u2k−1 ∈ .A!.R//n such that

0 ≤ |u0|! < Þ1 < |u1|! < þ1 < |u2|! < Þ2 < · · · < Þk < |u2k−1|! < þk:

PROOF. In (a) and (b), we just apply Theorem3.1 repeatedly. In (c) and (d),Theorem3.5 is used to obtain the existence ofu0 ∈ .A!.R//n with 0 ≤ ‖u0‖ < Þ1;the results then follow by repeated use of Theorem3.1.

Our next result makes use of Leggett and Williams’ fixed point theorem (Theo-rem2.4).

THEOREM 3.7. Let 1 ≤ p ≤ ∞, q be such that1=p + 1=q = 1, 0 < ! < ∞ andlet �i ∈ {1;−1}, 1 ≤ i ≤ n, be fixed. Assume that(3.3)–(3.11) and (3.60) hold foreach1 ≤ i ≤ n. Moreover, suppose

∃ j ∈ {1; : : : ; n} such that K2; j = inft∈[0;!]

∫ t

t−−gj .t; s/bj .s/ ds> 0; (3.71)

for the same j as in (3.71), there existsr > 0 with r <� j .r; : : : ; r /K2; j , where � j : ∏n

i =1[0;∞/i → [0;∞/ is continu-ous,� j .u1; : : : ; un/=|u j | is ‘nonincreasing’ in the sense that for each1 ≤ k ≤ n, if 0 < |uk| ≤ |vk| ≤ r , then� j .u1; : : : ; un/=|u j | ≥� j .v1; : : : ; vn/=|v j |, and bj .t/� j .u/ ≤ � j f j .t; u/ for a.e. t ∈ [0; !]and all u ∈ K

(3.72)

and{there existsR .6= r / with R > max1≤i ≤n i .R; : : : ; R/K1;i , whereK1;i = supt∈[0;!]

∫ t

t−− gi .t; s/bi .s/ ds. (3.73)

Then(M1) has at least one constant-sign solutionu ∈ .A!.R//n with

min{r; R} ≤ |u|! ≤ max{r; R} and |u|! 6= R: (3.74)

PROOF. Let B = ..A!.R//n; | · |!/ and

C = {u ∈ B | for each 1≤ i ≤ n; �i ui .t/ ≥ 0 for t ∈ [0; !]}= {u ∈ B | for each 1≤ i ≤ n; �i ui .t/ ≥ 0 for t ∈ R}:

Also, letu0 ≡ .�1; : : : ; �n/. Then

C.u0/ = {u ∈ C | there exists½ > 0 with u.t/− ½u0 ∈ C for t ∈ [0; !]}= {u ∈ C | for each 1≤ i ≤ n; �i ui .t/ > 0 for t ∈ [0; !]}:


As seen in the proof of Theorem3.1, conditions (3.3)–(3.11) guarantee thatS : C → Cis continuous and completely continuous. To apply Theorem2.4, we shall first showthat

|Su|! ≤ |u|! for u ∈ @SCR: (3.75)

Let u ∈ @SCR. Then|u|! = R. Using (3.33) and (3.60), we find for t ∈ [0; !] and1 ≤ i ≤ n,

|Si u.t/| = �i Si u.t/ ≤∫ t


≤ i

(|u1|!; : : : ; |un|!) ∫ t

t−−gi .t; s/bi .s/ ds ≤ i .R; : : : ; R/K1;i :

This yields, together with (3.73),

|Su|! = max1≤i ≤n

|Si u|! ≤ max1≤i ≤n

i .R; : : : ; R/K1;i < R = |u|! (3.76)

and hence (3.75) is proved.Next, we shall verify that

Su 6≤ u; that is; u − Su 6∈ C; for u ∈ @SCr ∩ C.u0/: (3.77)

Let u ∈ @SCr ∩ C.u0/. Then

|u|! = r and r ≥ �i ui .t/ > 0; t ∈ [0; !]; 1 ≤ i ≤ n: (3.78)

By a similar argument as in getting (3.31)–(3.34) from (3.72) we have the followingfor u ∈ C, t ∈ [0; !], a.e.s ∈ [t − −; t] and somej ∈ {1; 2; : : : ; n} (thesamej as in(3.71)):

� j f j .s; u.s// = � j f j .s!; u.s// ≥ bj .s!/� j .u.s// = bj .s/� j .u.s//: (3.79)

Thus, fort ∈ [0; !] and thesamej as in (3.71), using (3.78) and (3.79) we get

� j Sj u.t/ ≥∫ t

t−−gj .t; s/bj .s/� j .u.s//; ds

=∫ t

t−−gj .t; s/bj .s/

� j .u.s//

� j u j .s/� j u j .s/ ds

≥ � j .r; : : : ; r /

r

∫ t

t−−gj .t; s/bj .s/� j u j .s/ ds: (3.80)


Let t0; j ∈ [0; !] be such that inft∈[0;!] � j u j .t/ = � j u j .t0; j / > 0. Then it follows from(3.72) and (3.80) that fort ∈ [0; !],

� j Sj u.t/ ≥ � j .r; : : : ; r /

r� j u j .t0; j /

∫ t

t−−gj .t; s/bj .s/ds

≥[� j .r; : : : ; r /

rK2; j

]� j u j .t0; j / > � j u j .t0; j /:

Thus in particular we have� j Sj u.t0; j / > � j u j .t0; j /, and so (3.77) is proved.It now follows from Theorem2.4 that system (M1) has a constant-sign solution

u ∈ .A!.R//n with min{r; R} ≤ |u|! ≤ max{r; R}. Note that|u|! 6= R. In fact, if|u|! = R, then from (3.76) we have|u|! = |Su|! < R = |u|! which is a contradiction.This completes the proof.

REMARK 3.8. If the inequality in (3.73) is changed to

R ≥ max1≤i ≤n

i .R; : : : ; R/K1;i ;

then (3.74) is correspondingly changed to min{r; R} ≤ |u|! ≤ max{r; R}.

REMARK 3.9. Theorem3.7 improves the results obtained in [9].

A repeated application of Theorem3.7yields the existence ofmultiplesolutions asfollows.

THEOREM3.10. Let 1 ≤ p ≤ ∞, q be such that1=p + 1=q = 1, 0< ! < ∞ andlet �i ∈ {1;−1}, 1 ≤ i ≤ n, be fixed. Assume that(3.3)–(3.11) and (3.60) hold foreach1 ≤ i ≤ n, and(3.71) holds. Let(3.72) be satisfied forr = r`, ` = 1; 2; : : : ; k,and (3.73) be satisfied forR = R`, ` = 1; 2; : : : ;m.

(a) If m = k + 1 and0 < R1 < r1 < · · · < Rk < rk < Rk+1; then(M1) has(atleast) 2k constant-sign solutionsu1; : : : ; u2k ∈ .A!.R//n such that

0< R1 < |u1|! ≤ r1 ≤ |u2|! < R2 < · · · ≤ rk ≤ |u2k|! < Rk+1:

(b) If m = k and0 < R1 < r1 < · · · < Rk < rk; then(M1) has(at least) 2k − 1constant-sign solutionsu1; : : : ; u2k−1 ∈ .A!.R//n such that

0< R1 < |u1|! ≤ r1 ≤ |u2|! < R2 < · · · < Rk < |u2k−1|! ≤ rk:

(c) If k = m + 1 and0 < r1 < R1 < · · · < rm < Rm < rm+1, then(M1) has(atleast) 2m constant-sign solutionsu1; : : : ; u2m ∈ .A!.R//n such that

0< r1 ≤ |u1|! < R1 < |u2|! ≤ r2 ≤ · · · < Rm < |u2m|! ≤ rm+1:


(d) If k = m and0 < r1 < R1 < · · · < rk < Rk; then(M1) has(at least) 2k − 1constant-sign solutionsu1; : : : ; u2k−1 ∈ .A!.R//n such that

0< r1 ≤ |u1|! < R1 < |u2|! ≤ r2 ≤ · · · ≤ rk ≤ |u2k−1|! < Rk:

EXAMPLE 2. Consider the nonlinear system of integral equations

ui .t/ =∫ t

t−−gi .t; s/�i hi .s/

[|u1.s/| + · · · + |un.s/| + |u j .s/|Ž]

ds; (3.81)

t ∈ R, 1 ≤ i ≤ n; where > 0, 0 < Ž < 1, j ∈ {1; 2; : : : ; n} and�i ∈ {−1; 1},1 ≤ i ≤ n, are fixed. For each 1≤ i ≤ n, assume (3.3)–(3.7) and (3.48) hold, andalso

inft∈[0;!]

∫ t

t−−gj .t; s/hj .s/ ds> 0: (3.82)

Then (3.81) has at least one constant-sign solutionu ∈ .A!.R//n such that

min{r; R} ≤ |u|! ≤ max{r; R} and |u|! 6= R; (3.83)

wherer andR are positive numbers satisfying

r <

{inf

t∈[0;!]

∫ t

t−−gj .t; s/hj .s/ds

}1=.1−Ž/(3.84)

and

n R −1 + RŽ−1 <

(max1≤i ≤n

K1;i

)−1

(3.85)

with

K1;i = supt∈[0;!]

∫ t

t−−gi .t; s/hi .s/ ds; 1 ≤ i ≤ n: (3.86)

To see that the above is true, we shall apply Theorem3.7with

fi .t; u/ = �i hi .t/[|u1| + · · · + |un| + |u j |Ž

];

bi = hi ; i .u/ = |u1| + · · · + |un| + |u j |Ž; 1 ≤ i ≤ n

� j .u/ = |u j |Ž:(3.87)

Note that (3.8)–(3.11) and (3.60) are clearly satisfied. Next, in view of (3.82) and(3.87), condition (3.71) is satisfied with the fixedj in (3.81). For this fixed j , itis obvious that� j .u/=|u j | = 1=|u j |1−Ž is nonincreasing. Moreover, the inequalityr < � j .r; : : : ; r /K2; j in (3.72) reduces to

r

� j .r; r; : : : ; r /= r 1−Ž < K2; j = inf

t∈[0;!]

∫ t

t−−gj .t; s/hj .s/ ds


which is equivalent to (3.84). Hence condition (3.72) is fulfilled. Finally, the inequalityin (3.73) is reduced toR> .n R + RŽ/

(max1≤i ≤n K1;i

)which leads to (3.85).

It now follows from Theorem3.7 that system (3.81) has at least one constant-signsolutionu ∈ .A!.R//n satisfying (3.83)–(3.86).

Our next result also employs Leggett and Williams’ fixed point theorem (Theo-rem2.4).

THEOREM3.11. Let 1 ≤ p ≤ ∞, q be such that1=p + 1=q = 1, 0< ! < ∞ andlet �i ∈ {1;−1}, 1 ≤ i ≤ n, be fixed. Assume that(3.3)–(3.11) and (3.60) hold foreach1 ≤ i ≤ n. Moreover, suppose

there exists somej ∈ {1; 2; : : : ; n} such thatgj .t; s/ ≥ aj .s/ for allt ∈ [0; !] and a.e.s ∈ [0; !], whereaj ∈ Lq[0; !], aj is nonnegative,andaj .t + !/ = aj .t/ for a.e.t ∈ R,

(3.88)

for the samej as in (3.88), there existsr > 0 and a continuousfunction � j : ∏n

i =1[0;∞/i → [0;∞/, where � j .u1; : : : ; un/=|u j |is ‘nonincreasing’ in the sense that for each1 ≤ k ≤ n, if0 < |uk| ≤ |vk| ≤ r , then� j .u1; : : : ; un/=|u j | ≥ � j .v1; : : : ; vn/=|v j |,with bj .t/� j .u/ ≤ � j f j .t; u/ for a.e.t ∈ [0; !] and all u ∈ K ,

(3.89)

for the samej andr as in(3.89), we have(� j .r; : : : ; r /

r

)N(

N∏i =1

∫Ii

aj .s/bj .s/ ds

)> 1,

where N is the smallest positive integer such that!=N ≤ −=2 andI i = [.i − 1/!=N; i!=N], i = 0; 1; : : : ; N,

(3.90)

and also(3.73) holds. Then(M1) has at least one constant-sign solutionu ∈ .A!.R//n

with min{r; R} ≤ |u|! ≤ max{r; R} and|u|! 6= R.

PROOF. Let B, C andu0 be defined as in the proof of Theorem3.7. Then the samearguments give (3.75).

Next, we shall show that (3.77) is true. Letu ∈ @SCr ∩ C.u0/. Then (3.78) follows.Further, from (3.89) we obtain (3.79) for u ∈ C, t ∈ [0; !], a.e.s ∈ [t − −; t] andsome j ∈ {1; : : : ; n} (thesamej as in (3.88)). Thus fort ∈ [0; !] and thesamej asin (3.88), using (3.78), (3.79), (3.88) and (3.89) we find

� j Sj u.t/ ≥∫ t

t−−gj .t; s/bj .s/� j .u.s// ds

≥∫ t

t−−aj .s/bj .s/

� j .u.s//

� j u j .s/� j u j .s/ ds


≥ � j .r; : : : ; r /

r

∫ t

t−−aj .s/bj .s/� j u j .s/ ds: (3.91)

We claim that there exists at0; j ∈ [0; !] with

� j .r; : : : ; r /

r

∫ t0; j

t0; j −−aj .s/bj .s/� j u j .s/ ds> � j u.t0; j /: (3.92)

If our claim is true, then it follows from (3.91) that

� j Sj u.t0; j / ≥ � j .r; : : : ; r /

r

∫ t0; j

t0; j −−aj .s/bj .s/� j u j .s/ ds> � j u.t0; j /

and therefore (3.77) holds. Using a similar argument as in the proof of Theorem3.7,we then apply Theorem2.4to obtain the result.

It remains to prove our claim. Suppose (3.92) is false. Then

� j u j .t/ ≥ � j .r; r; : : : ; r /

r

∫ t

t−−aj .s/bj .s/� j u j .s/ds for all t ∈ [0; !]: (3.93)

Note that ift ∈ Ii for somei ∈ {1; : : : ; N}, then Ii −1 ⊆ [t − −; t] since!=N ≤ −=2.This together with (3.93) gives fori ∈ {1; : : : ; N},∫

Ii

aj .t/bj .t/� j u j .t/ dt

≥ � j .r; : : : ; r /

r

∫Ii

aj .t/bj .t/∫ t

t−−aj .s/bj .s/� j u j .s/ ds dt

≥ � j .r; : : : ; r /

r

∫Ii

aj .t/bj .t/∫

Ii−1

aj .s/bj .s/� j u j .s/ ds dt

= � j .r; : : : ; r /

r

(∫Ii

aj .t/bj .t/ dt

)(∫Ii−1

aj .s/bj .s/� j u j .s/ ds

):

Applying the above repeatedly yields∫I N

aj .t/bj .t/� j u j .t/ dt

≥(� j .r; : : : ; r /

r

)N(

N∏i =1

∫Ii

aj .t/bj .t/ dt

)(∫I0


)

=(� j .r; : : : ; r /

r

)N(

N∏i =1

∫Ii

aj .t/bj .t/ dt

)(∫I N


): (3.94)


If∫

I Naj .t/bj .t/� j u j .t/ dt > 0, then (3.94) leads to

(� j .r; : : : ; r /

r

)N(

N∏i =1

∫Ii

aj .t/bj .t/ dt

)≤ 1;

which contradicts (3.90). On the other hand, if∫

I Naj .t/bj .t/� j u j .t/ dt = 0, then

since� j u j .t/ > 0 for t ∈ [0; !]; we must haveaj .t/bj .t/ = 0 for a.e.t ∈ I N. Thisagain contradicts (3.90). Hence we have shown that our claim (3.92) is true. Theproof is now complete.

REMARK 3.12. Remark3.8also holds for Theorem3.11.

REMARK 3.13. Theorem3.11extends the results obtained in [15].

Our final result generalises Theorem3.11to give the existence ofmultipleconstant-sign periodic solutions.

THEOREM 3.14. Let 1 ≤ p ≤ ∞, q be such that1=p + 1=q = 1, 0 < ! < ∞and let �i ∈ {1;−1}, 1 ≤ i ≤ n, be fixed. Assume that(3.3)–(3.11) and (3.60)hold for each1 ≤ i ≤ n, and (3.88) holds. Let(3.89) and (3.90) be satisfied forr = r`, ` = 1; 2; : : : ; k, and (3.73) be satisfied forR = R`, ` = 1; 2; : : : ;m. Thenconclusions(a)–(d)of Theorem3.10hold.

REMARK 3.15. Similar to Remark3.2, in Theorems3.3–3.14we can replace con-ditions (3.3), (3.4), (3.6)–(3.8) and (3.11) with (3.43)–(3.46).

References

[1] R. P. Agarwal, D. O’Regan and P. J. Y. Wong,Positive solutions of differential, difference andintegral equations(Kluwer, Dordrecht, 1999).

[2] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, “Constant-signL p solutions for a system of integralequations”,Results Math.46 (2004) 195–219.

[3] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, “Constant-sign solutions of a system of Fredholmintegral equations”,Acta Appl. Math.80 (2004) 57–94.

[4] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, “Eigenvalues of a system of Fredholm integralequations”,Math. Comput. Modelling39 (2004) 1113–1150.

[5] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, “Triple solutions of constant sign for a system ofFredholm integral equations”,Cubo6 (2004) 1–45.

[6] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, “Constant-sign solutions of a system of integralequations: The semipositone and singular case”,Asymptot. Anal.43 (2005) 47–74.

[7] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, “Constant-sign periodic and almost periodic solutionsfor a system of integral equations”,Acta Appl. Math.(to appear) .


[8] R. P. Agarwal and P. J. Y. Wong,Advanced topics in difference equations(Kluwer, Dordrecht,1997).

[9] K. L. Cooke and J. L. Kaplan, “A periodicity threshold theorem for epidemics and populationgrowth”, Mat. Biosc.31 (1976) 87–104.

[10] L. H. Erbe, S. Hu and H. Wang, “Multiple positive solutions of some boundary value problems”,J. Math. Anal. Appl.184(1994) 640–648.

[11] L. H. Erbe and H. Wang, “On the existence of positive solutions of ordinary differential equations”,Proc. Amer. Math. Soc.120(1994) 743–748.

[12] D. Guo and V. Lakshmikantham,Nonlinear problems in abstract cones(Academic Press, SanDiego, 1988).

[13] M. A. Krasnosel’skii,Positive solutions of operator equations(Noordhoff, Groningen, 1964).[14] R. W. Leggett and L. R. Williams, “Multiple positive fixed points of nonlinear operators on ordered

Banach spaces”,Indiana Univ. Math. J.28 (1979) 673–688.[15] R. W. Leggett and L. R. Williams, “A fixed point theorem with application to an infectious disease

model”,J. Math. Anal. Appl.76 (1980) 91–97.[16] W. Lian, F. Wong and C. Yeh, “On the existence of positive solutions of nonlinear second order

differential equations”,Proc. Amer. Math. Soc.124(1996) 1117–1126.[17] R. Nussbaum, “A periodicity threshold theorem for some nonlinear integral equations”,SIAM J.

Anal.9 (1978) 356–376.[18] D. O’Regan and M. Meehan,Existence theory for nonlinear integral and integrodifferential equa-

tions(Kluwer, Dordrecht, 1998).[19] H. Smith, “On periodic solutions of delay integral equations modelling epidemics and population

growth”, Ph. D. Thesis, University of Iowa City, 1976.[20] L. R. Williams and R. W. Leggett, “Nonzero solutions of nonlinear integral equations modeling

infectious disease”,SIAM J. Anal.13 (1982) 112–121.

RAVI P. AGARWAL , DONAL O’REGAN - Aust MS · PDF fileRAVI P. AGARWAL 1, DONAL...

Documents

Transcript of RAVI P. AGARWAL , DONAL O’REGAN - Aust MS · PDF fileRAVI P. AGARWAL 1, DONAL...