Skip to main content

Multiple positive solutions of nonlinear BVPs for differential systems involving integral conditions

Abstract

In this paper, we consider the following system of nonlinear third-order nonlocal boundary value problems (BVPs for short):

{ u ( t ) = f ( t , v ( t ) , v ( t ) ) , t ( 0 , 1 ) , v ( t ) = g ( t , u ( t ) , u ( t ) ) , t ( 0 , 1 ) , u ( 0 ) = 0 , a u ( 0 ) b u ( 0 ) = α [ u ] , c u ( 1 ) + d u ( 1 ) = β [ u ] , v ( 0 ) = 0 , a v ( 0 ) b v ( 0 ) = α [ v ] , c v ( 1 ) + d v ( 1 ) = β [ v ] ,

where f,gC([0,1]× R + × R + , R + ), α[u]= 0 1 u(t)dA(t) and β[u]= 0 1 u(t)dB(t) are linear functionals on C[0,1] given by Riemann-Stieltjes integrals and are not necessarily positive functionals; a, b, c, d are nonnegative constants with ρ:=ac+ad+bc>0. By using the Guo-Krasnoselskii fixed point theorem, some sufficient conditions are obtained for the existence of at least one or two positive solutions and nonexistence of positive solutions to the above problem. Two examples are also included to illustrate the main results.

MSC:34B15.

1 Introduction

The theory of BVPs with integral boundary conditions for ordinary differential equations arises in different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics can be reduced to nonlocal problems with integral boundary conditions. Moreover, BVPs with Riemann-Stieltjes integral boundary condition (BC for short) have been considered recently as both multipoint and Riemann integral type BCs are treated in a single framework. For more comments on Stieltjes integral BC and its importance, we refer the reader to the papers by Webb and Infante [13] and their other related works.

In recent years, third-order nonlocal BVPs have received much attention from many authors; see, for example [414]. It is worth mentioning that Sun and Li [12] studied the third-order BVP with integral boundary conditions

{ u ( t ) + f ( t , u ( t ) , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 0 ) = 0 , u ( 1 ) = 0 1 g ( t ) u ( t ) d t .
(1.1)

Their main tool was the Guo-Krasnoselskii fixed point theorem. Recently, we [14] were concerned with the existence of a monotone positive solution for the third-order BVP

{ u ( t ) + f ( t , u ( t ) , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = 0 , a u ( 0 ) b u ( 0 ) = α [ u ] , c u ( 1 ) + d u ( 1 ) = β [ u ] ,
(1.2)

by applying monotone iterative techniques, where fC([0,1]× R + × R + , R + ), α[u]= 0 1 u(t)dA(t) and β[u]= 0 1 u(t)dB(t) are linear functionals on C[0,1] given by Riemann-Stieltjes integrals.

Furthermore, motivated by the wide applications of systems of differential equations in biomathematics, the study of systems of BVPs has received increased interest; see [1528] and the references therein. In particular, Henderson and Luca [17] established the existence of positive solutions for the system of BVPs with multi-point boundary conditions

{ u ( t ) + λ c ( t ) f ( u ( t ) , v ( t ) ) = 0 , t ( 0 , T ) , v ( t ) + μ d ( t ) g ( u ( t ) , v ( t ) ) = 0 , t ( 0 , T ) , α u ( 0 ) β u ( 0 ) = 0 , u ( T ) = i = 1 m 2 a i u ( ξ i ) , m 3 , γ v ( 0 ) δ v ( 0 ) = 0 , v ( T ) = i = 1 n 2 b i v ( η i ) , m 3
(1.3)

by applying the fixed point index theory.

Yang [27] studied the existence of positive solutions for the system of second-order nonlocal BVPs

{ u ( t ) = f ( t , u , v ) , v ( t ) = g ( t , u , v ) , u ( 0 ) = v ( 0 ) = 0 , u ( 1 ) = H 1 ( 0 1 u ( τ ) d α ( τ ) ) , v ( 1 ) = H 2 ( 0 1 v ( τ ) d β ( τ ) )
(1.4)

by using fixed point index theory in a cone.

Infante and Pietramala [19] studied the existence of positive solutions for a system of perturbed Hammerstein integral equations by fixed point index theory for compact maps and illustrated their theory by studying the following system of BVPs:

{ u ( t ) + g 1 ( t ) f 1 ( t , u ( t ) , v ( t ) ) = 0 , 0 < t < 1 , v ( t ) + g 2 ( t ) f 2 ( t , u ( t ) , v ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = H 11 ( β 11 [ u ] ) , u ( 1 ) = H 12 ( β 12 [ u ] ) , v ( 0 ) = H 21 ( β 21 [ v ] ) , v ( 1 ) = H 22 ( β 22 [ v ] ) .
(1.5)

The result was quite general and covered a wide class of systems of BVPs. Here β i j [w] was of the form β i j [w]= 0 1 w(t)d B i j (t) involving positive Riemann-Stieltjes measures.

Inspired greatly by the above-mentioned excellent works, in this paper, we are concerned with the following system of third-order BVPs:

{ u ( t ) = f ( t , v ( t ) , v ( t ) ) , t ( 0 , 1 ) , v ( t ) = g ( t , u ( t ) , u ( t ) ) , t ( 0 , 1 ) , u ( 0 ) = 0 , a u ( 0 ) b u ( 0 ) = α [ u ] , c u ( 1 ) + d u ( 1 ) = β [ u ] , v ( 0 ) = 0 , a v ( 0 ) b v ( 0 ) = α [ v ] , c v ( 1 ) + d v ( 1 ) = β [ v ] ,
(1.6)

where f,gC([0,1]× R + × R + , R + ), α[u]= 0 1 u(t)dA(t) and β[u]= 0 1 u(t)dB(t) are linear functionals on C[0,1] given by Riemann-Stieltjes integrals with signed measures; a, b, c, d are nonnegative constants with ρ:=ac+ad+bc>0. To the best of our knowledge, the study of existence of positive solutions of third-order differential systems (1.6) has not been done.

A vector (u,v) C 2 (0,1)× C 2 (0,1) is said to be a positive solution of BVP (1.6) if and only if u, v satisfy BVP (1.6) and u, v are positive on (0,1). The proof of our main results is based on the well-known Guo-Krasnoselskii fixed point theorem, which we present now.

Theorem 1.1 Let E be a Banach space, KE be a cone, and Ω 1 and Ω 2 be bounded open subsets of E with 0 Ω 1 , Ω ¯ 1 Ω 2 . Assume that A:K( Ω ¯ 2 Ω 1 )K is a completely continuous operator such that either

  1. (i)

    Auu for uK Ω 1 and Auu for uK Ω 2 ; or

  2. (ii)

    Auu for uK Ω 1 and Auu for uK Ω 2 .

Then A has a fixed point in K( Ω ¯ 2 Ω 1 ).

2 Preliminary lemmas

In this section, we adopt the ideas and the method which have been widely used and which are due to Webb and Infante in [1, 2].

In our case, the existence of positive solutions of nonlocal BVP

{ u ( t ) + f ( t , u ( t ) , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = 0 , a u ( 0 ) b u ( 0 ) = α [ u ] , c u ( 1 ) + d u ( 1 ) = β [ u ]
(2.1)

with two nonlocal boundary terms α[u], β[u] can be studied via a perturbed Hammerstein integral equation of the type

u(t)=γ(t)α[u]+δ(t)β[u]+ 0 1 G(t,s)f ( s , u ( s ) , u ( s ) ) ds=: T 1 u(t).
(2.2)

Here γ(t), δ(t) are linearly independent and given by

γ ( t ) = 0 , γ ( 0 ) = 0 , a γ ( 0 ) b γ ( 0 ) = 1 , c γ ( 1 ) + d γ ( 1 ) = 0 , δ ( t ) = 0 , δ ( 0 ) = 0 , a δ ( 0 ) b δ ( 0 ) = 0 , c δ ( 1 ) + d δ ( 1 ) = 1 ,

which imply γ(t)= 2 c t + 2 d t c t 2 2 ρ and δ(t)= a t 2 + 2 b t 2 ρ , t[0,1]. Let be the usual supremum norm in C[0,1]. A direct calculation shows that for t[θ,1θ], 0<θ< 1 2 ,

γ ( t ) c 1 γ , δ ( t ) c 2 δ , γ ( t ) d 1 γ and δ ( t ) d 2 δ ,
(2.3)

where c 1 = 2 c θ + 2 d θ c θ 2 c + 2 d , c 2 = a θ 2 + 2 b θ a + 2 b , d 1 = c θ + d c + d and d 2 = a θ + b a + b ; G(t,s) is Green’s function for the corresponding problem with local terms when α[u] and β[u] are identically 0, i.e.,

G(t,s)={ ( a t 2 + 2 b t ) ( c ( 1 s ) + d ) 2 ρ ( t s ) 2 2 , 0 s t 1 , ( a t 2 + 2 b t ) ( c ( 1 s ) + d ) 2 ρ , 0 t s 1 .

In the remainder of this paper, we always assume that

(H1) 0α[γ],β[δ]<1, α[δ],β[γ]0 and D:=(1α[γ])(1β[δ])α[δ]β[γ]>0;

(H2) A, B are functions of bounded variation, and K A (s), K B (s)0 for s[0,1], where

K A (s):= 0 1 G(t,s)dA(t)and K B (s):= 0 1 G(t,s)dB(t).

As shown in Theorem 2.3 in [1], if u is a fixed point of T 1 in (2.2), then u is a fixed point of S, which is now given by

S u ( t ) : = γ ( t ) D ( ( 1 β [ δ ] ) 0 1 K A ( s ) f ( s , u ( s ) , u ( s ) ) d s + α [ δ ] 0 1 K B ( s ) f ( s , u ( s ) , u ( s ) ) d s ) + δ ( t ) D ( β [ γ ] 0 1 K A ( s ) f ( s , u ( s ) , u ( s ) ) d s + ( 1 α [ γ ] ) 0 1 K B ( s ) f ( s , u ( s ) , u ( s ) ) d s ) + 0 1 G ( t , s ) f ( s , u ( s ) , u ( s ) ) d s = : 0 1 G S ( t , s ) f ( s , u ( s ) , u ( s ) ) d s

in our case. The kernel G S is Green’s function corresponding to BVP (2.1). By Lemma 2.1 and Lemma 2.2 in [14], we can get the following properties of Green’s function.

Lemma 2.1 Let ρ:=ac+ad+bc>0, c 3 = ρ 0 θ Φ ( τ ) d τ ( a + b ) ( c + d ) , 0<θ<1. Then G(t,s) satisfies

G ( t , s ) Φ ( s ) for  t [ 0 , 1 ] , s [ 0 , 1 ] , G ( t , s ) c 3 Φ ( s ) for  t [ θ , 1 θ ] , s [ 0 , 1 ] ,

where Φ(s)= 1 ρ (b+as)(d+c(1s)), s[0,1].

Lemma 2.2 Let c 0 =min{ c 1 , c 2 , c 3 }. Then G S (t,s) satisfies

G S ( t , s ) Φ 1 ( s ) for  t [ 0 , 1 ] , s [ 0 , 1 ] , G S ( t , s ) c 0 Φ 1 ( s ) for  t [ θ , 1 θ ] , s [ 0 , 1 ] ,

where Φ 1 (s):= γ D ((1β[δ]) K A (s)+α[δ] K B (s))+ δ D (β[γ] K A (s)+(1α[γ]) K B (s))+Φ(s), s[0,1].

Lemma 2.3 Let d 0 =min{ d 1 , d 2 , d 3 }, d 3 = ρ min t [ θ , 1 θ ] Φ ( t ) ( a + b ) ( c + d ) , 0<θ<1, and

Φ 2 ( s ) : = γ D [ ( 1 β [ δ ] ) K A ( s ) + α [ δ ] K B ( s ) ] + δ D [ β [ γ ] K A ( s ) + ( 1 α [ γ ] ) K B ( s ) ] + Φ ( s ) .

Then G S ( t , s ) t satisfies

G S ( t , s ) t Φ 2 ( s ) for  t [ 0 , 1 ] , s [ 0 , 1 ] , G S ( t , s ) t d 0 Φ 2 ( s ) for  t [ θ , 1 θ ] , s [ 0 , 1 ] .

Proof For s[0,1], by the fact

G ( t , s ) t Φ ( s ) ={ ( b + a s ) ( d + c ( 1 t ) ) ( b + a s ) ( d + c ( 1 s ) ) = ( b + a t ) ( d + c ( 1 t ) ) ( b + a t ) ( d + c ( 1 s ) ) ρ Φ ( t ) ( a + b ) ( c + d ) , 0 s t 1 , ( b + a t ) ( d + c ( 1 s ) ) ( b + a s ) ( d + c ( 1 s ) ) = ( b + a t ) ( d + c ( 1 t ) ) ( b + a s ) ( d + c ( 1 t ) ) ρ Φ ( t ) ( a + b ) ( c + d ) , 0 t s 1 ,

so G ( t , s ) t ρ min t [ θ , 1 θ ] Φ ( t ) ( a + b ) ( c + d ) Φ(s), t[θ,1θ], s[0,1], which together with (2.3) shows that G S ( t , s ) t d 0 Φ 2 (s) holds. □

Let E={ C 1 [0,1]:u(0)=0} equipped with the norm u=max{ u , u }, where u is the usual supremum norm in C[0,1]. Similar to Lemma 2.1 in [29], we can get the following lemma.

Lemma 2.4 If uE, then u u . And so, E is a Banach space when it is endowed with the norm u= u .

Define

K= { u E : u ( t ) 0 , u ( t ) 0 , t [ 0 , 1 ] , min t [ θ , 1 θ ] u ( t ) d 0 u } .

Then it is easy to verify that K is a cone in E.

For uK, we define

( T u ) ( t ) = 0 1 G S ( t , s ) f ( s , 0 1 G S ( s , τ ) g ( τ , u ( τ ) , u ( τ ) ) d τ , 0 1 G S ( s , τ ) s g ( τ , u ( τ ) , u ( τ ) ) d τ ) d s , t [ 0 , 1 ] .

It is easy to see that if x is a fixed point of T in K, then BVP (1.6) has one solution (u,v), where

{ u ( t ) = x ( t ) , v ( t ) = 0 1 G S ( t , s ) g ( s , x ( s ) , x ( s ) ) d s .

Lemma 2.5 T:KK.

Proof It is obvious that (Tu)(t)0 and ( T u ) (t)0. Moreover, for t[0,1], by Lemma 2.3, we have

( T u ) ( t ) = 0 1 G S ( t , s ) t f ( s , 0 1 G S ( s , τ ) g ( τ , u ( τ ) , u ( τ ) ) d τ , 0 1 G S ( s , τ ) s g ( τ , u ( τ ) , u ( τ ) ) d τ ) d s 0 1 Φ 2 ( s ) f ( s , 0 1 G S ( s , τ ) g ( τ , u ( τ ) , u ( τ ) ) d τ , 0 1 G S ( s , τ ) s g ( τ , u ( τ ) , u ( τ ) ) d τ ) d s

and hence

T u 0 1 Φ 2 ( s ) f ( s , 0 1 G S ( s , τ ) g ( τ , u ( τ ) , u ( τ ) ) d τ , 0 1 G S ( s , τ ) s g ( τ , u ( τ ) , u ( τ ) ) d τ ) d s .

Moreover, it follows from Lemma 2.2 that for t[θ,1θ],

( T u ) ( t ) = 0 1 G S ( t , s ) t f ( s , 0 1 G S ( s , τ ) g ( τ , u ( τ ) , u ( τ ) ) d τ , 0 1 G S ( s , τ ) s g ( τ , u ( τ ) , u ( τ ) ) d τ ) d s d 0 0 1 Φ 2 ( s ) f ( s , 0 1 G S ( s , τ ) g ( τ , u ( τ ) , u ( τ ) ) d τ , 0 1 G S ( s , τ ) s g ( τ , u ( τ ) , u ( τ ) ) d τ ) d s d 0 T u .

Then we get

min t [ θ , 1 θ ] ( T u ) (t) d 0 Tu,

which shows that TKK. □

Similar to the proof of Lemma 2.4 in [28], we can get the following lemma.

Lemma 2.6 T:KK is completely continuous.

3 Main results

Denote

f 0 = lim sup x + y 0 + max t [ 0 , 1 ] f ( t , x , y ) x + y and g 0 = lim sup x + y 0 + max t [ 0 , 1 ] g ( t , x , y ) x + y , f 0 = lim inf x + y 0 + min t [ θ , 1 θ ] f ( t , x , y ) x + y and g 0 = lim inf x + y 0 + min t [ θ , 1 θ ] g ( t , x , y ) x + y , f = lim sup x + y + max t [ 0 , 1 ] f ( t , x , y ) x + y and g = lim sup x + y + max t [ 0 , 1 ] g ( t , x , y ) x + y , f = lim inf x + y + min t [ θ , 1 θ ] f ( t , x , y ) x + y and g = lim inf x + y + min t [ θ , 1 θ ] g ( t , x , y ) x + y , A 1 = 0 1 Φ 2 ( s ) d s , B 1 = 2 0 1 ( Φ 1 ( s ) + Φ 2 ( s ) ) d s , A 2 = d 0 θ 1 θ Φ 2 ( s ) d s , B 2 = d 0 θ 1 θ ( c 0 Φ 1 ( s ) + d 0 Φ 2 ( s ) ) d s .

Theorem 3.1 Assume that A 1 f 0 <1< A 2 f and B 1 g 0 <1< B 2 g . Then BVP (1.6) has at least one positive solution.

Proof In view of A 1 f 0 <1 and B 1 g 0 <1, there exists ε 1 >0 such that

A 1 ( f 0 + ε 1 ) 1, B 1 ( g 0 + ε 1 ) 1.
(3.1)

By the definition of f 0 , g 0 , we may choose σ 1 >0 so that

f ( t , x , y ) ( f 0 + ε 1 ) ( x + y ) , g ( t , x , y ) ( g 0 + ε 1 ) ( x + y ) , t [ 0 , 1 ] , ( x + y ) [ 0 , σ 1 ] .
(3.2)

Set Ω 1 ={uE|u< σ 1 /2}. It follows from (3.1), (3.2), Lemmas 2.2 and 2.3 that for any uK Ω 1 , s[0,1],

0 1 ( G S ( s , τ ) + G S ( s , τ ) s ) g ( τ , u ( τ ) , u ( τ ) ) d τ 0 1 ( Φ 1 ( τ ) + Φ 2 ( τ ) ) ( g 0 + ε 1 ) ( u ( τ ) + u ( τ ) ) d τ 2 u ( g 0 + ε 1 ) 0 1 ( Φ 1 ( τ ) + Φ 2 ( τ ) ) d τ σ 1 .
(3.3)

Then, by (3.1), (3.2) and (3.3), we have

( T u ) ( t ) = 0 1 G S ( t , s ) t f ( s , 0 1 G S ( s , τ ) g ( τ , u ( τ ) , u ( τ ) ) d τ , 0 1 G S ( s , τ ) s g ( τ , u ( τ ) , u ( τ ) ) d τ ) d s 0 1 Φ 2 ( s ) ( f 0 + ε 1 ) ( 0 1 G S ( s , τ ) g ( τ , u ( τ ) , u ( τ ) ) d τ + 0 1 G S ( s , τ ) s g ( τ , u ( τ ) , u ( τ ) ) d τ ) d s 0 1 Φ 2 ( s ) ( f 0 + ε 1 ) ( 0 1 ( Φ 1 ( τ ) + Φ 2 ( τ ) ) ( g 0 + ε 1 ) ( u ( τ ) + u ( τ ) ) d τ ) d s 0 1 Φ 2 ( s ) d s ( f 0 + ε 1 ) ( 0 1 ( Φ 1 ( s ) + Φ 2 ( s ) ) d s ) ( g 0 + ε 1 ) 2 u u , t [ 0 , 1 ] .

Therefore,

Tuu,uK Ω 1 .
(3.4)

On the other hand, since 1< A 2 f and 1< B 2 g , there exists ε 2 >0 such that

A 2 ( f ε 2 )1, B 2 ( g ε 2 )1.
(3.5)

By the definition of f , g , we may choose σ 2 > σ 1 so that

f ( t , x , y ) ( f ε 2 ) ( x + y ) , g ( t , x , y ) ( g ε 2 ) ( x + y ) , t [ θ , 1 θ ] , ( x + y ) [ σ 2 , + ) .
(3.6)

Let σ 2 =max{2 σ 1 , σ 2 / d 0 } and set Ω 2 ={uE|u< σ 2 }. Then uK Ω 2 implies that σ 2 d 0 u u (t), t[θ,1θ]. So, for s[θ,1θ], in view of Lemmas 2.2 and 2.3, we have

0 1 ( G S ( s , τ ) + G S ( s , τ ) s ) g ( τ , u ( τ ) , u ( τ ) ) d τ θ 1 θ ( c 0 Φ 1 ( τ ) + d 0 Φ 2 ( τ ) ) ( g ε 2 ) ( u ( τ ) + u ( τ ) ) d τ u ( g ε 2 ) d 0 θ 1 θ ( c 0 Φ 1 ( τ ) + d 0 Φ 2 ( τ ) ) d τ σ 2 .
(3.7)

Then, for t[θ,1θ], by (3.5), (3.6), (3.7), Lemmas 2.2 and 2.3, we have

( T u ) ( t ) = 0 1 G S ( t , s ) t f ( s , 0 1 G S ( s , τ ) g ( τ , u ( τ ) , u ( τ ) ) d τ , 0 1 G S ( s , τ ) s g ( τ , u ( τ ) , u ( τ ) ) d τ ) d s d 0 θ 1 θ Φ 2 ( s ) ( f ε 2 ) ( 0 1 ( G S ( s , τ ) + G S ( s , τ ) s ) g ( τ , u ( τ ) , u ( τ ) ) d τ ) d s d 0 θ 1 θ Φ 2 ( s ) d s ( f ε 2 ) σ 2 u , t [ 0 , 1 ] .

Therefore,

Tuu,uK Ω 2 .
(3.8)

Therefore, it follows from the first part of Theorem 1.1 that T has a fixed point u 1 K( Ω ¯ 2 Ω 1 ). Consequently, BVP (1.6) has a positive solution (u,v)K×K, here

{ u ( t ) = u 1 ( t ) , v ( t ) = 0 1 G S ( t , s ) g ( s , u 1 ( s ) , u 1 ( s ) ) d s .

 □

Theorem 3.2 Assume that A 1 f <1< A 2 f 0 and B 1 g <1< B 2 g 0 . Then BVP (1.6) has at least one positive solution.

Proof The proof is similar to Theorem 3.1 and therefore omitted. □

Theorem 3.3 Assume that A 2 f 0 >1, A 2 f >1, B 2 g 0 >1, B 2 g >1, B 2 g 0 <2 and there is a μ>0 such that

max { g ( t , x , y ) , t [ 0 , 1 ] , ( x + y ) [ 0 , μ ] } < 2 μ B 1 ;
(3.9)
max { f ( t , x , y ) , t [ 0 , 1 ] , ( x + y ) [ 0 , μ ] } < μ 2 A 1 .
(3.10)

Then BVP (1.6) has at least two positive solutions.

Proof Firstly, in view of A 2 f 0 >1 and B 2 g 0 >1, there exists ε>0 such that

A 2 ( f 0 ε)1, B 2 ( g 0 ε)1.
(3.11)

By the definition of f 0 , g 0 , we may choose σ ˆ 1 >0 so that

f ( t , x , y ) ( f 0 ε ) ( x + y ) , g ( t , x , y ) ( g 0 ε ) ( x + y ) , t [ 0 , 1 ] , ( x + y ) [ 0 , σ ˆ 1 ] .
(3.12)

Moreover, from B 2 g 0 <2, take ρ 1 satisfying 0< ρ 1 < B 2 B 1 σ 1 ˆ <μ such that

g(t,x,y) 2 ρ 1 B 2 ,t[0,1],x+y[0, ρ 1 ].

Set Ω 1 ={uE|u< ρ 1 /2}. It follows from (3.11), (3.12), Lemmas 2.2 and 2.3 that for any uK Ω 1 ,

0 1 ( G S ( s , τ ) + G S ( s , τ ) s ) g ( τ , u ( τ ) , u ( τ ) ) d τ 2 ρ 1 B 2 0 1 ( Φ 1 ( τ ) + Φ 2 ( τ ) ) d τ = B 1 B 2 ρ 1 < σ ˆ 1 , s [ 0 , 1 ] .
(3.13)

Then, for t[θ,1θ], by (3.11), (3.12), (3.13), Lemmas 2.2 and 2.3, we have

( T u ) ( t ) = 0 1 G S ( t , s ) t f ( s , 0 1 G S ( s , τ ) g ( τ , u ( τ ) , u ( τ ) ) d τ , 0 1 G S ( s , τ ) s g ( τ , u ( τ ) , u ( τ ) ) d τ ) d s d 0 θ 1 θ Φ 2 ( s ) ( f 0 ε ) ( 0 1 ( G S ( s , τ ) + G S ( s , τ ) s ) g ( τ , u ( τ ) , u ( τ ) ) d τ ) d s d 0 θ 1 θ Φ 2 ( s ) ( f 0 ε ) ( θ 1 θ ( G S ( s , τ ) + G S ( s , τ ) s ) g ( τ , u ( τ ) , u ( τ ) ) d τ ) d s d 0 θ 1 θ Φ 2 ( s ) ( f 0 ε ) × ( θ 1 θ ( c 0 Φ 1 ( τ ) + d 0 Φ 2 ( τ ) ) ( g 0 ε ) ( u ( τ ) + u ( τ ) ) d τ ) d s d 0 θ 1 θ Φ 2 ( s ) d s ( f 0 ε ) ( θ 1 θ ( c 0 Φ 1 ( τ ) + d 0 Φ 2 ( τ ) ) d τ ) ( g 0 ε ) d 0 u u , t [ 0 , 1 ] .

Thus,

Tuu,uK Ω 1 .
(3.14)

Secondly, similar to the proof of (3.8), we may choose σ 2 >μ and set Ω 2 ={uE|u< σ 2 }, and easily get

Tuu,uK Ω 2 .
(3.15)

Let Ω 3 ={uE|u<μ/2}. Then, for any uK Ω 3 , it follows by (3.9) and (3.10) that

0 1 ( G S ( s , τ ) + G S ( s , τ ) s ) g ( τ , u ( τ ) , u ( τ ) ) d τ 2 μ B 1 0 1 ( Φ 1 ( τ ) + Φ 2 ( τ ) ) d τ μ , s [ 0 , 1 ]
(3.16)

and

( T u ) ( t ) = 0 1 G S ( t , s ) t f ( s , 0 1 G S ( s , τ ) g ( τ , u ( τ ) , u ( τ ) ) d τ , 0 1 G S ( s , τ ) s g ( τ , u ( τ ) , u ( τ ) ) d τ ) d s < 0 1 Φ 2 ( s ) d s μ 2 A 1 = μ 2 = u , t [ 0 , 1 ] .

Thus,

Tu<u,uK Ω 3 ,
(3.17)

which together with (3.13), (3.14) shows that T has at least two fixed points in u 1 K( Ω ¯ 2 Ω 3 ) and u 2 K( Ω ¯ 3 Ω 1 ). □

Similarly, we can get the following theorem.

Theorem 3.4 Assume that A 1 f 0 <1, A 1 f < 1 2 , B 1 g 0 <1, B 2 g > d 0 and there exists η>0 such that

max { g ( t , x , y ) , t [ θ , 1 θ ] , ( x + y ) [ d 0 η , + ) } > d 0 2 η B 2 ;
(3.18)
max { f ( t , x , y ) , t [ θ , 1 θ ] , ( x + y ) [ d 0 η , + ) } > η A 2 .
(3.19)

Then BVP (1.6) has at least two positive solutions.

Theorem 3.5 If A 1 f(t,x,y)<(x+y) and B 1 g(t,x,y)<(x+y) for t[0,1] and (x+y)[0,+), then BVP (1.6) has no monotone positive solution.

Proof Suppose on the contrary that u is a monotone positive solution of BVP (1.6). Then u(t)0 and u (t)0 for t[0,1], and

u ( t ) = 0 1 G S ( t , s ) t f ( s , 0 1 G S ( s , τ ) g ( τ , u ( τ ) , u ( τ ) ) d τ , 0 1 G S ( s , τ ) s g ( τ , u ( τ ) , u ( τ ) ) d τ ) d s < 1 A 1 0 1 Φ 2 ( s ) ( 0 1 G S ( s , τ ) g ( τ , u ( τ ) , u ( τ ) ) d τ + 0 1 G S ( s , τ ) s g ( τ , u ( τ ) , u ( τ ) ) d τ ) d s < 1 A 1 1 B 1 0 1 Φ 2 ( s ) d s 0 1 ( Φ 1 ( s ) + Φ 2 ( s ) ) d s 0 1 ( u ( τ ) + u ( τ ) ) d τ < u ,

which shows that u<u. This is a contradiction. □

Similarly, we can prove the following theorem.

Theorem 3.6 If A 2 f(t,x,y)>(x+y) and B 2 g(t,x,y)>(x+y) for t[θ,1θ] and (x+y)[0,+), then BVP (1.6) has no monotone positive solution.

4 Example

In this section, we give an example to illustrate our main results.

Consider the BVP:

{ u ( t ) = f ( t , v ( t ) , v ( t ) ) , t ( 0 , 1 ) , v ( t ) = g ( t , u ( t ) , u ( t ) ) , t ( 0 , 1 ) , u ( 0 ) = 0 , u ( 0 ) = α [ u ] = 0 1 ( 1 s ) u ( s ) d s , u ( 1 ) = β [ u ] = 0 1 s u ( s ) d s , v ( 0 ) = 0 , v ( 0 ) = α [ v ] = 0 1 ( 1 s ) v ( s ) d s , v ( 1 ) = β [ v ] = 0 1 s v ( s ) d s .
(4.1)

For this BCs, the corresponding γ(t)= 2 t t 2 2 and δ(t)= t 2 2 . A simple calculation shows that

α [ γ ] = 1 8 , α [ δ ] = 1 24 , β [ γ ] = 5 24 , β [ δ ] = 1 8 , D = ( 1 α [ γ ] ) ( 1 β [ δ ] ) α [ δ ] β [ γ ] = 109 144 , K A ( s ) : = 0 1 G ( t , s ) ( 1 t ) d t = s 8 s 2 4 + s 3 6 s 4 24 , K B ( s ) : = 0 1 G ( t , s ) t d t = 5 s 24 s 2 4 + s 4 24 , Φ 1 ( s ) = 265 s 218 145 s 2 109 + 13 s 3 109 s 4 218 , Φ 2 ( s ) = 156 s 109 181 s 2 109 + 26 s 3 109 s 4 109 .

Let θ=1/4, then A 1 =719/3,2700.2199, A 2 =126,517/3,348,4800.0378, B 1 =2,767/3,2700.8462, B 2 =4,438,339/428,605,4400.0104.

Example 4.1 Let

f ( t , v ( t ) , v ( t ) ) = 1 1 + t [ v ( t ) + v ( t ) e v ( t ) + v ( t ) + 1 , 000 ( v ( t ) + v ( t ) ) 2 1 + v ( t ) + v ( t ) ] , g ( t , u ( t ) , u ( t ) ) = 1 10 ( 1 + t ) [ u ( t ) + u ( t ) e u ( t ) + u ( t ) + 2 , 000 ( u ( t ) + u ( t ) ) 2 1 + u ( t ) + u ( t ) ] .

It is easy to compute that f 0 =1, f = 4 , 000 7 , g 0 = 1 10 , g = 800 7 , which show that A 1 f 0 <1< A 2 f and B 1 g 0 <1< B 2 g . So, it follows from Theorem 3.1 that BVP (4.1) has at least one positive solution.

Example 4.2 Let

f ( t , v ( t ) , v ( t ) ) = 1 1 + t [ 50 ( v ( t ) + v ( t ) ) e v ( t ) + v ( t ) + 50 ( v ( t ) + v ( t ) ) 2 1 , 000 + v ( t ) + v ( t ) ] , g ( t , u ( t ) , u ( t ) ) = [ 1 + ( t 3 4 ) 2 ] [ 100 ( u ( t ) + u ( t ) ) e 200 ( u ( t ) + u ( t ) ) + 100 ( u ( t ) + u ( t ) ) 2 1 , 000 + u ( t ) + u ( t ) ] .

It is easy to compute that f 0 = 200 7 , f = 200 7 , g 0 =100, g =100 and g 0 = 625 4 , which show that A 2 f 0 >1, A 2 f >1, B 2 g 0 >1, B 2 g >1 and B 2 g 0 <2.

Choose μ=14,

max { g ( t , x , y ) , t [ 0 , 1 ] , ( x + y ) [ 0 , 14 ] } = 50 e + 4 , 900 507 < 29 < 2 μ B 1 ; max { f ( t , x , y ) , t [ 0 , 1 ] , ( x + y ) [ 0 , 14 ] } = 25 16 ( 1 2 e + 9 , 800 507 ) < 21 < μ 2 A 1 .

So, it follows from Theorem 3.3 that BVP (4.1) has at least two positive solutions.

References

  1. Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems: a unified approach. J. Lond. Math. Soc. 2006, 74(2):673-693.

    Article  MathSciNet  Google Scholar 

  2. Webb JRL, Infante G: Nonlocal boundary value problems of arbitrary order. J. Lond. Math. Soc. 2009, 79(2):238-258.

    MathSciNet  Google Scholar 

  3. Webb JRL: Positive solutions of some higher order nonlocal boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2009, 29: 1-15.

    Google Scholar 

  4. Boucherif A, Bouguima SM, Al-Malki N, Benbouziane Z: Third order differential equations with integral boundary conditions. Nonlinear Anal. 2009, 71: e1736-e1743. 10.1016/j.na.2009.02.055

    Article  MathSciNet  Google Scholar 

  5. Du Z, Ge W, Zhou M: Singular perturbations for third-order nonlinear multi-point boundary value problem. J. Differ. Equ. 2005, 218: 69-90. 10.1016/j.jde.2005.01.005

    Article  MathSciNet  Google Scholar 

  6. El-Shahed M: Positive solutions for nonlinear singular third order boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 2009, 14: 424-429. 10.1016/j.cnsns.2007.10.008

    Article  MathSciNet  Google Scholar 

  7. Graef JR, Webb JRL: Third order boundary value problems with nonlocal boundary conditions. Nonlinear Anal. 2009, 71: 1542-1551. 10.1016/j.na.2008.12.047

    Article  MathSciNet  Google Scholar 

  8. Graef JR, Yang B: Positive solutions of a third order nonlocal boundary value problem. Discrete Contin. Dyn. Syst., Ser. S 2008, 1: 89-97.

    MathSciNet  Google Scholar 

  9. Henderson J, Tisdell CC: Five-point boundary value problems for third-order differential equations by solution matching. Math. Comput. Model. 2005, 42: 133-137. 10.1016/j.mcm.2004.04.007

    Article  MathSciNet  Google Scholar 

  10. Hopkins B, Kosmatov N: Third-order boundary value problems with sign-changing solutions. Nonlinear Anal. 2007, 67: 126-137. 10.1016/j.na.2006.05.003

    Article  MathSciNet  Google Scholar 

  11. Ma R: Multiplicity results for a third order boundary value problem at resonance. Nonlinear Anal. 1998, 32: 493-499. 10.1016/S0362-546X(97)00494-X

    Article  MathSciNet  Google Scholar 

  12. Sun JP, Li HB: Monotone positive solution of nonlinear third-order BVP with integral boundary conditions. Bound. Value Probl. 2010., 2010: Article ID 874959 10.1155/2010/874959

    Google Scholar 

  13. Zhao JF, Wang PG, Ge WG: Existence and nonexistence of positive solutions for a class of third order BVP with integral boundary conditions in Banach spaces. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 402-413. 10.1016/j.cnsns.2009.10.011

    Article  MathSciNet  Google Scholar 

  14. Zhang HE, Sun JP: Existence and iteration of monotone positive solutions for third-order nonlocal BVPs involving integral conditions. Electron. J. Qual. Theory Differ. Equ. 2012, 18: 1-9.

    Google Scholar 

  15. Du ZJ: Singularly perturbed third-order boundary value problem for nonlinear systems. Appl. Math. Comput. 2007, 189(1):869-877. 10.1016/j.amc.2006.11.167

    Article  MathSciNet  Google Scholar 

  16. Henderson J, Luca R: On a system of higher-order multi-point boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2012, 49: 1-14.

    Article  MathSciNet  Google Scholar 

  17. Henderson J, Luca R: Positive solutions for a system of second-order multi-point boundary value problems. Appl. Math. Comput. 2012, 218: 6083-6094. 10.1016/j.amc.2011.11.092

    Article  MathSciNet  Google Scholar 

  18. Infante G, Pietramala P: Eigenvalues and non-negative solutions of a system with nonlocal BCs. Nonlinear Stud. 2009, 16: 187-196.

    MathSciNet  Google Scholar 

  19. Infante G, Pietramala P: Existence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations. Nonlinear Anal. 2009, 71: 1301-1310. 10.1016/j.na.2008.11.095

    Article  MathSciNet  Google Scholar 

  20. Infante G, Minhós FM, Pietramala P: Non-negative solutions of systems of ODEs with coupled boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 4952-4960. 10.1016/j.cnsns.2012.05.025

    Article  MathSciNet  Google Scholar 

  21. Infante, G, Pietramala, P: Multiple positive solutions of systems with coupled nonlinear BCs. arXiv:1306.5556

  22. Li YH, Guo YP, Li GG: Existence of positive solutions for systems of nonlinear third-order differential equations. Commun. Nonlinear Sci. Numer. Simul. 2009, 14: 3792-3797. 10.1016/j.cnsns.2009.02.019

    Article  MathSciNet  Google Scholar 

  23. Li SJ, Zhang XG, Wu YH, Caccetta L: Extremal solutions for P -Laplacian differential systems via iterative computation. Appl. Math. Lett. 2013, 26(12):1151-1158. 10.1016/j.aml.2013.06.014

    Article  MathSciNet  Google Scholar 

  24. Li WT, Sun JP: Multiple positive solutions of BVPs for third-order discrete difference systems. Appl. Math. Comput. 2004, 149: 389-398. 10.1016/S0096-3003(03)00147-4

    Article  MathSciNet  Google Scholar 

  25. Jankowski T: Nonnegative solutions to nonlocal boundary value problems for systems of second-order differential equations dependent on the first-order derivatives. Nonlinear Anal. 2013, 87: 83-101.

    Article  MathSciNet  Google Scholar 

  26. Wang GW, Zhou MR, Sun L: Existence of solutions of boundary value problem for 3rd order nonlinear system. Appl. Math. Comput. 2007, 189: 1131-1138. 10.1016/j.amc.2006.12.003

    Article  MathSciNet  Google Scholar 

  27. Yang Z: Positive solutions to a system of second-order nonlocal boundary value problems. Nonlinear Anal. 2005, 62: 1251-1265. 10.1016/j.na.2005.04.030

    Article  MathSciNet  Google Scholar 

  28. Zhang HE, Sun JP: Existence of positive solution to singular systems of second-order four-point BVPs. J. Appl. Math. Comput. 2009, 29: 325-339. 10.1007/s12190-008-0133-5

    Article  MathSciNet  Google Scholar 

  29. Zhong Y, Chen SH, Wang CP: Existence results for a fourth-order ordinary differential equation with a four-point boundary condition. Appl. Math. Lett. 2008, 21: 465-470. 10.1016/j.aml.2007.03.029

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The work is supported by the Science and Technology Foundation of Hebei Province (Z2013016) and the Science and Technology Plan Foundation of Tangshan (12110233b) and the Scientific Research Foundation of Tangshan College (13011B). The author would like to thank the anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of the work.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hai-E Zhang.

Additional information

Competing interests

The author declares that she has no competing interests.

Author’s contributions

The author read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Zhang, HE. Multiple positive solutions of nonlinear BVPs for differential systems involving integral conditions. Bound Value Probl 2014, 61 (2014). https://doi.org/10.1186/1687-2770-2014-61

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-2770-2014-61

Keywords