Non-negative solutions of systems of ODEs with coupled boundary conditions

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Non-negative solutions of systems of ODEs with coupled

boundary conditions

Gennaro Infante

a,⇑

, Feliz M. Minhós

b

, Paolamaria Pietramala

a

a_{Dipartimento di Matematica, Università della Calabria, 87036 Arcavacata di Rende, Cosenza, Italy} b_{Department of Mathematics, University of É vora, Rua Romão Ramalho 59, 7000-671 Évora, Portugal}

a r t i c l e i n f o

Article history:

Received 29 January 2012

Received in revised form 29 May 2012 Accepted 31 May 2012

Available online 15 June 2012

Keywords: Fixed point index Cone

System

Non-negative solution

a b s t r a c t

We provide a new existence theory of multiple positive solutions valid for a wide class of systems of boundary value problems that possess a coupling in the boundary conditions. Our conditions are fairly general and cover a large number of situations. The theory is illus-trated in details in an example. The approach relies on classical fixed point index.

Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction

The problem of the existence of positive solutions for systems of local and nonlocal boundary value problems (BVPs) has received an increased attention by researchers, see for example the papers of Agarwal et al.[1–3], Ahmad and Graef[4], Ahmad and Nieto[5], Henderson et al.[14], Lan and Lin[23], Precup[28,29], Yang and Kong[36], Yang and Zhang[37]and references therein. Between systems of BVPs of particular interest are those where the boundary conditions (BCs) are coupled. Systems with coupled BCs can be applied to Lotka–Volterra models, reaction-diffusion phenomena and interaction problems, see for example the works of Amann[7], Leung[24]and Mehmeti and Nicaise[27]. A recent paper in this line of research is the one by Asif and Khan[8], who study the four-point coupled system

u00_{ðtÞ þ f} 1ðt; uðtÞ;

v

ðtÞÞ ¼ 0; t 2 ð0; 1Þ;

v

00_{ðtÞ þ f} 2ðt; uðtÞ;

v

ðtÞÞ ¼ 0; t 2 ð0; 1Þ; uð0Þ ¼ 0; uð1Þ ¼ d12

v

ð

g

12Þ;

v

ð0Þ ¼ 0;

v

ð1Þ ¼ d22uð

g

22Þ: ð1Þ

The authors prove, via the well-known Guo–Krasnosel’skiı˘ theorem on cone compression-expansion, the existence of one positive solution of the system(1)by means of an associated auxiliary system of Hammerstein integral equations, namely

1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.cnsns.2012.05.025 ⇑Corresponding author.

E-mail addresses:[email protected](G. Infante),[email protected](F.M. Minhós),[email protected](P. Pietramala). Contents lists available atSciVerse ScienceDirect

Commun Nonlinear Sci Numer Simulat

uðtÞ ¼ Z 1 0 F1ðt; sÞf1ðs; uðsÞ;

v

ðsÞÞ ds þ Z 1 0 G1ðt; sÞf2ðs; uðsÞ;

v

ðsÞÞ ds;

v

ðtÞ ¼ Z 1 0 F2ðt; sÞf2ðs; uðsÞ;

v

ðsÞÞ ds þ Z 1 0 G2ðt; sÞf1ðs; uðsÞ;

v

ðsÞÞ ds: ð2Þ

An integral representation of the type(2)is also used in the papers[9,38]. Yuan et al. in[38]study, by means of a nonlinear alternative of Leray–Schauder type and the Guo–Krasnosel’skiı˘ fixed-point theorem, the existence of one or two positive solutions for a semi-positone system of fractional differential equations subject to four-point BCs. In[9]Cui and Sun study, via fixed point index theory, the existence of one positive solution for the system

u00_{ðtÞ þ f} 1ðt; uðtÞ;

v

ðtÞÞ ¼ 0; t 2 ð0; 1Þ;

v

00_{ðtÞ þ f} 2ðt; uðtÞ;

v

ðtÞÞ ¼ 0; t 2 ð0; 1Þ; uð0Þ ¼ 0; uð1Þ ¼ b12½

v

;

v

ð0Þ ¼ 0;

v

ð1Þ ¼ b22½u; ð3Þ

where bij½ are linear functionals defined via (positive) Stieltjes measures.

We mention that Stieltjes integrals are also used in the framework of nonlinear coupled BCs in the paper of Kang and Wei

[18](where the Leggett and Williams fixed point theorem is used) and in two recent papers of Goodrich[11,12](who uses the Guo–Krasnosel’skiı˘ fixed-point theorem); one interesting feature of[11,12]is the possibility to use signed measures, in the line of the paper by Webb and Infante[32].

Here we provide a new approach for a wide class of systems of BVPs that possess, along the coupling in the nonlinearities of the differential equations, also a coupling in the BCs and we prove, under suitable conditions, existence of multiple non-negative solutions. Our idea is to give an existence theory valid for systems of perturbed Hammerstein integral equations of the type uðtÞ ¼

c

11ðtÞb11½u þ

c

12ðtÞb12½

v

 þ Z 1 0 k1ðt; sÞg1ðsÞf1ðs; uðsÞ;

v

ðsÞÞ ds;

v

ðtÞ ¼

c

21ðtÞb21½

v

 þ

c

22ðtÞb22½u þ Z 1 0 k2ðt; sÞg2ðsÞf2ðs; uðsÞ;

v

ðsÞÞ ds; ð4Þ

where

c

ijare continuous functions and bij½ are linear functionals defined via Stieltjes measures. A system of perturbed

Ham-merstein integral equations similar to(4)is investigated by Infante and Pietramala in[16], with the intent of dealing with BVPs with nonlinear BCs, allowing a coupling in the nonlinearities f1and f2but not in the BCs. The methodology of[16]relies

on an extensions of the results of[32]to the context of systems.

We illustrate our theory with an example of a system of second and fourth order ordinary differential equations u00_{ðtÞ þ g}

1ðtÞf1ðt; uðtÞ;

v

ðtÞÞ ¼ 0; t 2 ð0; 1Þ;

v

ð4Þ_{ðtÞ ¼ g}

2ðtÞf2ðt; uðtÞ;

v

ðtÞÞ; t 2 ð0; 1Þ;

ð5Þ subject to the nonlocal boundary conditions

uð0Þ ¼ b11½u; uð1Þ ¼ b12½

v

;

v

ð0Þ ¼ b21½

v

;

v

ð1Þ ¼ 0;

v

00ð0Þ ¼ 0;

v

00ð1Þ þ b22½u ¼ 0:

ð6Þ The system of ordinary differential equations(5), with local BCs, can be used as a model for the stationary states of a one-dimensional bridge, with a coupling between the cable and the roadbed. Here the cable is seen as vibrating string and the roadbed as a vibrating beam, see for example the papers of Lazer and McKenna[20], Lü et al.[25], Sun[30]and the doctoral thesis of Matas[26]. The boundary conditions(6)involve functionals of the form

bij½w ¼

Z 1 0

wðsÞ dBijðsÞ;

and include, as special cases, m-point and integral conditions, when bij½w ¼ Xm j¼1 dijwð

g

ijÞ and bij½w ¼ Z 1 0 dijðsÞwðsÞ ds:

For previous work on Riemann–Stieltjes integral BCs we refer the reader, for example, to the papers of Karakostas and Tsamatos[19]and Webb[31,33]. We point out that nonlocal conditions have a physical interpretation; for example the cou-pled condition

uð0Þ ¼ uð1Þ ¼

v

ð1Þ ¼

v

00_{ð0Þ ¼}

_v

_{ð0Þ ¼ 0;}

_v

00_{ð1Þ þ duð}

_g

_{Þ ¼ 0;}

models a feedback control mechanism, where the bending moment in the right end of the beam is related to the displace-ment registered in a point

g

of the string.

We prove our results by means of the classical fixed point index theory (see for example the review of Amann[6]and the book of Guo and Lakshmikantham[13]) and also make use of ideas from the papers[16,17,32].

2. Positive solutions for systems of integral equations

In order to utilize the classical fixed point index theory to find positive solutions of the system of integral equations uðtÞ ¼

c

₁₁ðtÞb11½u þ

c

12ðtÞb12½

v

 þ Z 1 0 k1ðt; sÞg1ðsÞf1ðs; uðsÞ;

v

ðsÞÞ ds;

v

ðtÞ ¼

c

21ðtÞb21½

v

 þ

c

22ðtÞb22½u þ Z 1 0 k2ðt; sÞg2ðsÞf2ðs; uðsÞ;

v

ðsÞÞ ds; ð7Þ

we make the following hypotheses on the terms that occur in(7):

 For every i ¼ 1; 2; fi:½0; 1  ½0; 1Þ  ½0; 1Þ ! ½0; 1Þ satisfies Carathéodory conditions, that is, fið; u;

v

Þ is measurable for

each fixed ðu;

v

Þ and fiðt; ; Þ is continuous for almost every (a.e.) t 2 ½0; 1, and for each r > 0 there exists /i;r 2 L 1

½0; 1 such that

fiðt; u;

v

Þ 6 /i;rðtÞ for u;

v

2 ½0; r and a:e: t 2 ½0; 1:

 For every i ¼ 1; 2; ki:½0; 1  ½0; 1 ! ½0; 1Þ is measurable, and for every

s

2 ½0; 1 we have

lim

t!sjkiðt; sÞ  kið

s

;sÞj ¼ 0 for a:e: s 2 ½0; 1:

 For every i ¼ 1; 2, there exist a subinterval ½ai;bi # ½0; 1, a function Ui2 L1½0; 1, and a constant ci2 ð0; 1, such that

kiðt; sÞ 6UiðsÞ for t 2 ½0; 1 and a:e: s 2 ½0; 1;

kiðt; sÞ P ciUiðsÞ for t 2 ½ai;bi and a:e: s 2 ½0; 1:

 For every i ¼ 1; 2; giUi2 L 1

½0; 1; giP0 a.e., and

Rb_i

a_i UiðsÞgiðsÞ ds > 0.

 For every i; j ¼ 1; 2; bij½ is a linear functional given by

bij½w ¼

Z 1 0

wðsÞ dBijðsÞ;

involving Riemann–Stieltjes integrals; Bijis of bounded variation and dBij is a positive measure.

 For every i; j ¼ 1; 2;

c

_ij2 C½0; 1;

c

_ijðtÞ P 0 for every t 2 ½0; 1; bi1½

c

i1 < 1 and there exists cij2 ð0; 1 such that

c

ijðtÞ P cijk

c

ijk1 for every t 2 ½ai;bi;

where kwk1:¼ maxfjwðtÞj; t 2 ½0; 1g.

We work in the space C½0; 1  C½0; 1 endowed with the norm kðu;

v

Þk :¼ maxfkuk₁;k

v

k₁g:

Let e

Ki:¼ fw 2 C½0; 1 : wðtÞ P 0 for t 2 ½0; 1 and min t2½ai;bi

wðtÞ P ~cikwk1g;

where ~ci¼ minfci;ci1;ci2g, and consider the cone K in C½0; 1  C½0; 1 defined by

K :¼ fðu;

v

Þ 2 eK1 eK2g:

For a positive solution of the system(7)we mean a solution ðu;

v

Þ 2 K of(7)such that kðu;

v

Þk > 0. Under our assumptions, we show that the integral operator

Tðu;

v

ÞðtÞ ¼ T1ðu;

v

ÞðtÞ T2ðu;

v

ÞðtÞ   :¼

c

11ðtÞb11½u þ

c

12ðtÞb12½

v

 þ F1ðu;

v

ÞðtÞ

c

21ðtÞb21½

v

 þ

c

22ðtÞb22½u þ F2ðu;

v

ÞðtÞ   ; ð8Þ where Fiðu;

v

ÞðtÞ :¼ Z 1 0 kiðt; sÞgiðsÞfiðs; uðsÞ;

v

ðsÞÞ ds;

leaves the cone K invariant and is compact.

Proof. Take ðu;

v

Þ 2 K such that kðu;

v

Þk 6 r. Then we have, for t 2 ½0; 1, T1ðu;

v

ÞðtÞ ¼

c

11ðtÞb11½u þ

c

12ðtÞb12½

v

 þ Z 1 0 k1ðt; sÞg1ðsÞf1ðs; uðsÞ;

v

ðsÞÞ ds therefore kT1ðu;

v

Þk 6 k

c

11k1b11½u þ k

c

12k1b12½

v

 þ Z 1 0 U1ðsÞg1ðsÞf1ðs; uðsÞ;

v

ðsÞÞ ds: Then we obtain min t2½a1;b1 T1ðu;

v

ÞðtÞ P c11k

c

11k1b11½u þ c12k

c

12k1b12½

v

 þ c1 Z 1 0 U1ðsÞg1ðsÞf1ðs; uðsÞ;

v

ðsÞÞ ds P ~c1kT1ðu;

v

Þk:

Hence we have T1ðu;

v

Þ 2 eK1. In a similar manner we proceed for T2ðu;

v

Þ.

Moreover, the map T is compact since the components Tiare sum of two compact maps: the compactness of Fiis

well-known and, since

c

_i1 and

c

_i2 are continuous, the perturbation

c

_i1ðtÞbi1½u þ

c

i2ðtÞbi2½

v

 maps bounded sets into bounded

subsets of a finite dimensional space. h

We use the following (relative) open bounded sets in K: Kq¼ fðu;

v

Þ 2 K : kðu;

v

Þk <

q

g;

and

Vq¼ fðu;

v

Þ 2 K : min

t2½a1;b1

uðtÞ <

q

and min

t2½a2;b2

v

ðtÞ <

q

g:

The set Vq(in the context of systems) was introduced by Infante and Pietramala[15]and is equal to the set called Xq=cby Franco, Infante and O’Regan[10]. Xq=c_{is an extension to the case of systems of a set given by Lan[22]. The advantage of the}

notation Vqis that sheds light on the fact (see also the paper by Infante and Webb[17]) that choosing c as large as possible provides a weaker condition to be satisfied by the functions fi in Lemmas 3 and 4. Note that Kq Vq Kq=c, where

c ¼ minf ~c1; ~c2g. We denote by @Kqand @Vqthe boundary of Kqand Vqrelative to K. In the next Lemma we make use of the notation

KijðsÞ :¼

Z 1 0

kiðt; sÞ dBijðtÞ; i; j ¼ 1; 2;

and we prove that the index is 1 on Kq. Lemma 2. Assume that

ðI1qÞ there exists

q

>0 such that for every i ¼ 1; 2 k

c

i1k1bi1½

c

i2bi2½1 ð1  bi1½

c

i1Þ þ k

c

_i2k1bi2½1 þ f 0;q i 1 mi þ k

c

i1k1 ð1  bi1½

c

i1Þ Z 1 0 Ki1ðsÞgiðsÞ ds   <1; ð9Þ where f_i0;q¼ sup fiðt; u;

v

Þ

q

:ðt; u;

v

Þ 2 ½0; 1  ½0;

q

  ½0;

q

   and 1 mi¼ supt2½0;1 Z 1 0 kiðt; sÞgiðsÞ ds:

Then the fixed point index, iKðT; KqÞ, is equal to 1.

Proof. We show that

l

ðu;

v

Þ – Tðu;

v

Þ for every ðu;

v

Þ 2 @Kqand for every

l

P1; this ensures that the index is 1 on Kq. In fact, if this does not happen, there exist

l

P1 and ðu;

v

Þ 2 @Kqsuch that

l

ðu;

v

Þ ¼ Tðu;

v

Þ. Assume, without loss of generality, that kuk1¼

q

and k

v

k16

q

. Then

l

uðtÞ ¼

c

₁₁ðtÞb11½u þ

c

12ðtÞb12½

v

 þ F1ðu;

v

ÞðtÞ

and therefore, since

v

ðtÞ 6

q

, for all t 2 ½0; 1,

l

uðtÞ 6

c

11ðtÞb11½u þ

c

12ðtÞb12½

q

 þ F1ðu;

v

ÞðtÞ ¼

c

11ðtÞb11½u þ

qc

12ðtÞb12½1 þ F1ðu;

v

ÞðtÞ: ð10Þ

Applying b11to both sides of(10)gives

l

b11½u 6 b11½

c

11b11½u þ

q

b11½

c

12b12½1 þ b11½F1ðu;

v

Þ:

Thus we have

that is b11½u 6

q

b11½

c

12b12½1 ð

l

 b11½

c

11Þ þ b11½F1ðu;

v

Þ ð

l

 b11½

c

11Þ : Substituting into(10)gives

l

uðtÞ 6

c

11ðtÞ

q

b11½

c

12b12½1 ð

l

 b11½

c

11Þ þ b11½F1ðu;

v

Þ ð

l

 b11½

c

11Þ   þ

qc

12ðtÞb12½1 þ F1ðu;

v

ÞðtÞ ¼

q

c

11ðtÞb11½

c

12b12½1 ð

l

 b11½

c

11Þ þ

c

11ðtÞ ð

l

 b11½

c

11Þ Z 1 0 K11ðsÞg1ðsÞf1ðs; uðsÞ;

v

ðsÞÞ ds þ

qc

12ðtÞb12½1 þ F1ðu;

v

ÞðtÞ:

Since

l

P1, one has 1

lb11½c11 6 1 1b11½c11and therefore

l

uðtÞ 6

q

c

11ðtÞb11½

c

12b12½1 ð1  b11½

c

11Þ þ

c

11ðtÞ ð1  b11½

c

11Þ Z 1 0 K11ðsÞg1ðsÞf1ðs; uðsÞ;

v

ðsÞÞ ds þ

qc

12ðtÞb12½1 þ F1ðu;

v

ÞðtÞ:

Taking the supremum of t on ½0; 1 gives

l

q

6

q

k

c

11k1b11½

c

12b12½1 ð1  b11½

c

11Þ þ

q

f₁0;q k

c

11k1 ð1  b11½

c

11Þ Z 1 0 K11ðsÞg1ðsÞ ds þ

q

k

c

12k1b12½1 þ

q

f 0;q 1 1 m1 : Using the hypothesis(9)we obtain

l

q

<

q

. This contradicts the fact that

l

P1 and proves the result. h

We give a first Lemma that shows that the index is 0 on a set Vq, here we assume that the nonlinearities f1;f2have the

same growth.

Lemma 3. Assume that

ðI0_qÞ there exist

q

>0 such that for every i ¼ 1; 2 fi;ðq;q=cÞ ci1k

c

i1k1 ð1  bi1½

c

i1Þ Z bi ai Ki1ðsÞgiðsÞ ds þ 1 Mi ! >1; ð11Þ where f1;ðq;q=cÞ¼ inf f1ðt; u;

v

Þ

q

:ðt; u;

v

Þ 2 ½a1;b1  ½

q

;

q

=c  ½0;

q

=c   ; f2;ðq;q=cÞ¼ inf f2ðt; u;

v

Þ

q

:ðt; u;

v

Þ 2 ½a2;b2  ½0;

q

=c  ½

q

;

q

=c   and 1 Mi ¼ inf t2½ai;bi Z bi ai kiðt; sÞgiðsÞ ds: Then iKðT; VqÞ ¼ 0.

Proof. Let eðtÞ  1 for t 2 ½0; 1. Then ðe; eÞ 2 K. We prove that ðu;

v

Þ – Tðu;

v

Þ þ

l

ðe; eÞ for ðu;

v

Þ 2 @Vqand

l

P0:

In fact, if this does not happen, there exist ðu;

v

Þ 2 @Vqand

l

P0 such that ðu;

v

Þ ¼ Tðu;

v

Þ þ

l

ðe; eÞ. Without loss of gener-ality, we can assume that for all t 2 ½a1;b1 we have

q

6uðtÞ 6

q

=c; min uðtÞ ¼

q

and 0 6

v

_{ðtÞ 6}

q

=c: Then, for t 2 ½a1;b1, we obtain

uðtÞ ¼

c

₁₁ðtÞb11½u þ

c

12ðtÞb12½

v

 þ

Z 1 0

k1ðt; sÞg1ðsÞf1ðs; uðsÞ;

v

ðsÞÞds þ

l

eðtÞ

and therefore

uðtÞ P

c

11ðtÞb11½u þ F1ðu;

v

ÞðtÞ þ

l

eðtÞ: ð12Þ

Applying b11to both sides of(12)gives

b11½u P b11½

c

11b11½u þ b11½F1ðu;

v

Þ þ

l

b11½e:

This can be written in the form

that is b11½u P b11½F1ðu;

v

Þ ð1  b11½

c

11Þ þ

l

b11½e ð1  b11½

c

11Þ : Thus,(12)becomes uðtÞ P

c

11ðtÞb11½F1ðu;

v

Þ ð1  b11½

c

11Þ þ

l

c

11ðtÞb11½e ð1  b11½

c

11Þ þ F1ðu;

v

ÞðtÞ þ

l

eðtÞ ¼

c

11ðtÞ ð1  b11½

c

11Þ  Z 1 0 K11ðsÞg1ðsÞf1ðs; uðsÞ;

v

ðsÞÞ ds þ

l

c

₁₁ðtÞb11½e ð1  b11½

c

11Þ þ Z 1 0 k1ðt; sÞg1ðsÞf1ðs; uðsÞ;

v

ðsÞÞ ds þ

l

:

Then we have, for t 2 ½a1;b1,

uðtÞ P c11k

c

11k1 ð1  b11½

c

11Þ  Z b1 a1 K11ðsÞg1ðsÞf1ðs; uðsÞ;

v

ðsÞÞ ds þ

l

c11k

c

11k1b11½e ð1  b11½

c

11Þ þ Z b1 a1 k1ðt; sÞg1ðsÞf1ðs; uðsÞ;

v

ðsÞÞ ds þ

l

P c11k

c

11k1 ð1  b11½

c

11Þ  Z b1 a1 K11ðsÞg1ðsÞf1ðs; uðsÞ;

v

ðsÞÞ ds þ Z b1 a1 k1ðt; sÞg1ðsÞf1ðs; uðsÞ;

v

ðsÞÞ ds þ

l

:

Taking the minimum over ½a1;b1 gives

min t2½a1;b1 uðtÞ P

q

f1;ðq;q=cÞ c11k

c

11k1 ð1  b11½

c

11Þ  Z b1 a1 K11ðsÞg1ðsÞ ds þ

q

f1;ðq;q=cÞ 1 M1 þ

l

¼

q

f1;ðq;q=cÞ c11k

c

11k1 ð1  b11½

c

11Þ  Z b1 a1 K11ðsÞg1ðsÞ ds þ f1;ðq;q=cÞ 1 M1 ! þ

l

:

Using the hypothesis(11)we obtain

q

¼ mint2½a1;b1uðtÞ >

q

þ

l

, a contradiction. h

The following Lemma also shows that the index is 0 on Vq; the idea here is similar to the one in Lemma 4 of[16]: this time we have to control the growth of just one nonlinearity fi, at the cost of having to deal with a larger domain. For other results

on the existence of solutions with different growth on the nonlinearities see the works[28,29]and the paper by Yang[35].

Lemma 4. Assume that ðI0qÞ

H

there exist

q

>0 such that for some i ¼ 1; 2 f i;ð0;q=cÞ ci1k

c

i1k1 ð1  bi1½

c

i1Þ Z bi ai Ki1ðsÞgiðsÞ ds þ 1 Mi ! >1; ð13Þ where f i;ð0;q=cÞ¼ inf fiðt; u;

v

Þ

q

:ðt; u;

v

Þ 2 ½ai;bi  ½0;

q

=c  ½0;

q

=c   : Then iKðT; VqÞ ¼ 0.

Proof. Suppose that the condition(13)holds for i ¼ 1. Let eðtÞ  1 for t 2 ½0; 1. Then ðe; eÞ 2 K. We prove that ðu;

v

Þ – Tðu;

v

Þ þ

l

ðe; eÞ for ðu;

v

Þ 2 @Vqand

l

P0:

In fact, if this does not happen, there exist ðu;

v

Þ 2 @Vq and

l

P0 such that ðu;

v

Þ ¼ Tðu;

v

Þ þ

l

ðe; eÞ. So, for all t 2 ½a1;b1; min uðtÞ 6

q

and for t 2 ½a2;b2; min

v

ðtÞ 6

q

. We have, for t 2 ½0; 1,

uðtÞ ¼

c

₁₁ðtÞb11½u þ

c

12ðtÞb12½

v

 þ

Z 1 0

k1ðt; sÞg1ðsÞf1ðs; uðsÞ;

v

ðsÞÞds þ

l

eðtÞ

and, as in the proof ofLemma 3,

uðtÞ P

c

11ðtÞ ð1  b11½

c

11Þ  Z 1 0 K11ðsÞg1ðsÞf1ðs; uðsÞ;

v

ðsÞÞ ds þ

l

c

11ðtÞb11½e ð1  b11½

c

11Þ þ Z 1 0 k1ðt; sÞg1ðsÞf1ðs; uðsÞ;

v

ðsÞÞ ds þ

l

: Then we have min t2½a1;b1 uðtÞ P

q

f 1;ð0;q=cÞ c11k

c

11k1 ð1  b11½

c

11Þ  Z b1 a1 K11ðsÞg1ðsÞ ds þ

q

f1;ð0; q=cÞ 1 M1 þ

l

¼

q

f 1;ð0;q=cÞ c11k

c

11k1 ð1  b11½

c

11Þ  Z b1 a1 K11ðsÞg1ðsÞ ds þ

q

f1;ð0; q=cÞ 1 M1þ

l

:

Using the hypothesis(13)we obtain mint2½a1;b1uðtÞ >

q

þ

l

P

q

, a contradiction. h

The above Lemmas can be combined to prove the following Theorem, here we deal with the existence of at least one, two or three solutions. We stress that, by expanding the lists in conditions ðS5Þ; ðS6Þ below, it is possible to state results for four

or more positive solutions, see for example the paper by Lan[21]for the type of results that might be stated. We omit the proof which follows from the properties of fixed point index.

Theorem 5. The system(7)has at least one positive solution in K if either of the following conditions hold.

ðS1Þ There exist

q

1;

q

22 ð0; 1Þ with

q

1=c <

q

2 such that ðI 0 q₁Þ ½or ðI 0 q₁Þ H ; ðI1q₂Þ hold. ðS2Þ There exist

q

1;

q

22 ð0; 1Þ with

q

1<

q

2 such that ðI

1

q1Þ; ðI

0

q2Þ hold.

The system(7)has at least two positive solutions in K if one of the following conditions hold: ðS3Þ There exist

q

1;

q

2;

q

32 ð0; 1Þ with

q

1=c <

q

2<

q

3such that ðI

0

q₁Þ ½or ðI

0

q₁Þ

H

; ðI1q₂Þ and ðI

0

q₃Þ hold. ðS4Þ There exist

q

1;

q

2;

q

32 ð0; 1Þ with

q

1<

q

2and

q

2=c <

q

3 such that ðI

1 q1Þ; ðI 0 q2Þ and ðI 1 q3Þ hold.

The system(7)has at least three positive solutions in K if one of the following conditions hold: ðS5Þ There exist

q

1;

q

2;

q

3;

q

42 ð0; 1Þ with

q

1=c <

q

2<

q

3and

q

3=c <

q

4such that ðI

0 q₁Þ ½or ðI 0 q₁Þ H ; ðI1q₂Þ; ðI 0 q₃Þ and ðI 1 q₄Þ hold. ðS6Þ There exist

q

1;

q

2;

q

3;

q

42 ð0; 1Þ with

q

1<

q

2and

q

2=c <

q

3<

q

4such that ðI

1 q1Þ; ðI 0 q2Þ; ðI 1 q3Þ and ðI 0 q4Þ hold.

Remark 1. Note that, if the nonlinearities f1and f2have some extra positivity properties, a solution ðu;

v

Þ achieved by means

of the above Theorem has further positivity properties. For example, if the condition ðS1Þ holds and moreover we assume that

f1ðt; 0;

v

Þ > 0 in ½a1;b1  f0g  ½0;

q

2 and f2ðt; u; 0Þ > 0 in ½a2;b2  ½0;

q

2  f0g, the solution ðu;

v

Þ of the system(7)is such

that kuk1and k

v

k1are strictly positive. The Remark 2 of[16]should read accordingly.

3. An application to coupled systems of BVPs

We study the existence of positive solutions for the system of second order differential equations u00_{ðtÞ þ g}

1ðtÞf1ðt; uðtÞ;

v

ðtÞÞ ¼ 0; t 2 ð0; 1Þ;

v

ð4Þ_{ðtÞ ¼ g}

2ðtÞf2ðt; uðtÞ;

v

ðtÞÞ; t 2 ð0; 1Þ;

ð14Þ with the nonlocal boundary conditions

uð0Þ ¼ b11½u; uð1Þ ¼ b12½

v

;

v

ð0Þ ¼ b21½

v

;

v

ð1Þ ¼ 0;

v

00ð0Þ ¼ 0;

v

00ð1Þ þ b22½u ¼ 0:

ð15Þ We rewrite this differential system in the integral form

uðtÞ ¼ ð1  tÞb11½u þ tb12½u þ

Z 1 0 k1ðt; sÞg1ðsÞf1ðs; uðsÞ;

v

ðsÞÞ ds;

v

ðtÞ ¼ ð1  tÞb21½

v

 þ 1 6tð1  t 2 Þb22½u þ Z 1 0 k2ðt; sÞg2ðsÞf2ðs; uðsÞ;

v

ðsÞÞ ds; where k1ðt; sÞ ¼ sð1  tÞ; s 6 t; tð1  sÞ; s > t;  and k2ðt; sÞ ¼ 1 6sð1  tÞð2t  s 2_{ t}2_{Þ; s 6 t;} 1 6tð1  sÞð2s  t 2_{ s}2_{Þ; s > t;} (

are non-negative continuous functions on ½0; 1  ½0; 1.

The intervals ½a1;b1 and ½a2;b2 may be chosen arbitrarily in ð0; 1Þ. It is easy to check that

k1ðt; sÞ 6 sð1  sÞ :¼U1ðsÞ; min t2½a1;b1

k1ðt; sÞ P c1sð1  sÞ;

where c1¼ minf1  b1;a1g. Furthermore, see the paper by Webb et al.[34], we have that

k2ðt; sÞ 6U2ðsÞ :¼ ffiffi 3 p 27sð1  s 2_Þ32_{; for 0 6 s 6}1 2; ffiffi 3 p 27ð1  sÞs 3 2ð2  sÞ 3 2_{; for} 1 2<s 6 1; 8 < : and k2ðt; sÞ P c2ðtÞU2ðsÞ; where

c2ðtÞ ¼ 3pffiffi3 2 tð1  t 2 Þ; for t 2 ½0; 1=2; 3pffiffi3 2 tð1  tÞð2  tÞ; for t 2 ð1=2; 1; ( so that c2¼ min t2½a2;b2 c2ðtÞ > 0:

The existence of multiple solutions of the system(14)–(15)follows fromTheorem 5. In the next example we illustrate the constants that occur in our theory.

Example 1. Consider the system u00 þ ð1=8Þðu3

þ t3

v

3Þ þ 2 ¼ 0; t 2 ð0; 1Þ;

v

ð4Þ_¼pffiffiffiffiffi_{tuþ 13}

_v

2_{; t 2 ð0; 1Þ;}

uð0Þ ¼ ð1=2Þuð1=4Þ; uð1Þ ¼ ð1=3Þ

v

ð3=4Þ;

v

ð0Þ ¼ ð1=4Þ

v

ð1=3Þ;

v

ð1Þ ¼

v

00_{ð0Þ ¼ 0;}

_v

00_{ð1Þ þ ð1=5Þuð2=3Þ ¼ 0:}

ð16Þ

In this case the nonlocal conditions are given by the functionals bij½w ¼ dijwð

g

ijÞ.

The choice ½a1;b1 ¼ ½a2;b2 ¼ ½1=4; 3=4 gives

c1¼ 1=4; c2¼ 45 ffiffiffi 3 p =128; c11¼ c12¼ c21¼ 1=4; c22¼ 45 ffiffiffi 3 p =128; m1¼ 8; M1¼ 16; m2¼ 384=5; M2¼ 768=5:

We have that b11½

c

11 ¼ 3=8 and b21½

c

21 ¼ 1=6.

Since Ki1ðsÞ ¼ di1kið

g

i1;sÞ we obtain Z 1 0 K11ðsÞ ds ¼ 3=64 and Z 3=4 1=4 K11ðsÞ ds ¼ 1=32; Z 1 0 K21ðsÞ ds ¼ 11=3888 and Z 3=4 1=4 K21ðsÞ ds ¼ 3985=1990656:

Then, for

q

1¼ 1=8;

q

2¼ 1 and

q

3¼ 11, we have (the constants that follow have been rounded to 2 decimal places unless

exact) inf ff1ðt; u;

v

Þ : ðt; u;

v

Þ 2 ½1=4; 3=4  ½0; 1=2  ½0; 1=2g ¼ f1ð1=4; 0; 0Þ > 13:34

q

1; sup ff1ðt; u;

v

Þ : ðt; u;

v

Þ 2 ½0; 1  ½0; 1  ½0; 1g ¼ f1ð1; 1; 1Þ < 3

q

2; sup ff2ðt; u;

v

Þ : ðt; u;

v

Þ 2 ½0; 1  ½0; 1  ½0; 1g ¼ f2ð1; 1; 1Þ < 59:95

q

2; inf ff1ðt; u;

v

Þ : ðt; u;

v

Þ 2 ½1=4; 3=4  ½11; 44  ½0; 44g ¼ f1ð1=4; 11; 0Þ > 13:34

q

3; inf ff2ðt; u;

v

Þ : ðt; u;

v

Þ 2 ½1=4; 3=4  ½0; 44  ½11; 44g ¼ f2ð1=4; 0; 11Þ > 140:63

q

3;

that is the conditions ðI0q1Þ

H

;ðI1q2Þ and ðI

0

q3Þ are satisfied; therefore the system(16)has at least two positive solutions in K.

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