Open Access

Solvability for system of nonlinear singular differential equations with integral boundary conditions

Boundary Value Problems20142014:158

https://doi.org/10.1186/s13661-014-0158-7

Received: 4 April 2014

Accepted: 13 June 2014

Published: 23 September 2014

Abstract

By using the Banach contraction principle and the Leggett-Williams fixed point theorem, this paper investigates the uniqueness and existence of at least three positive solutions for a system of mixed higher-order nonlinear singular differential equations with integral boundary conditions:

{ u ( n 1 ) ( t ) + a 1 ( t ) f 1 ( t , u ( t ) , v ( t ) ) = 0 , 0 < t < 1 , v ( n 2 ) ( t ) + a 2 ( t ) f 2 ( t , u ( t ) , v ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 0 ) = = u ( n 1 2 ) ( 0 ) = 0 , u ( 1 ) = g 1 ( β 1 [ u ] , β 1 [ v ] ) , v ( 0 ) = v ( 0 ) = = v ( n 2 2 ) ( 0 ) = 0 , v ( 1 ) = g 2 ( β 2 [ u ] , β 2 [ v ] ) ,

where the nonlinear terms f i , g i satisfy some growth conditions, β i [ ] are linear functionals given by β i [ w ] = 0 1 w ( s ) d ϕ i ( s ) , involving Stieltjes integrals with positive measures, and i = 1 , 2 . We give an example to illustrate our result.

MSC: 34B16, 34B18.

Keywords

positive solutionsintegral boundary conditionshigher-order differential equationsfixed point theorem

1 Introduction

The purpose of this paper is to establish the uniqueness and existence of at least three positive solutions for a system of mixed higher-order nonlinear singular differential equations with integral boundary conditions,
{ u ( n 1 ) ( t ) + a 1 ( t ) f 1 ( t , u ( t ) , v ( t ) ) = 0 , 0 < t < 1 , v ( n 2 ) ( t ) + a 2 ( t ) f 2 ( t , u ( t ) , v ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 0 ) = = u ( n 1 2 ) ( 0 ) = 0 , u ( 1 ) = g 1 ( β 1 [ u ] , β 1 [ v ] ) , v ( 0 ) = v ( 0 ) = = v ( n 2 2 ) ( 0 ) = 0 , v ( 1 ) = g 2 ( β 2 [ u ] , β 2 [ v ] ) ,
(1.1)

where n i 3 , a i ( t ) C ( ( 0 , 1 ) , [ 0 , + ) ) , a i ( t ) are allowed to be singular at t = 0 and/or t = 1 , f i C ( [ 0 , 1 ] × [ 0 , + ) × [ 0 , + ) , [ 0 , + ) ) , g i C ( [ 0 , + ) × [ 0 , + ) , [ 0 , + ) ) , a i ( t ) f i ( t , 0 , 0 ) do not vanish identically on any subinterval of ( 0 , 1 ) , the functionals β i [ ] are linear functionals given by β i [ w ] = 0 1 w ( s ) d ϕ i ( s ) , involving Stieltjes integrals with positive measures, and i = 1 , 2 .

The theory of boundary value problems with integral conditions for ordinary differential equations arises in different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, and plasma physics can be reduced to boundary value problems with integral conditions, which included, as special cases, two-point, three-point and multi-point boundary value problems considered by many authors (see [1]–[5]).

In recent years, to the best of our knowledge, although there are many papers concerning the existence of positive solutions for n th order boundary value problems with different kinds of boundary conditions for system (see [6]–[10] and the references therein), results for the system (1.1) are rarely seen. Moreover, the methods mainly depend on the Krasonsel’skii fixed point theorem, fixed point index theory, the upper and lower solution technique, some new fixed point theorem for cones, etc. For example, in [7], by applying the Krasonsel’skii fixed point theorem, Henderson and Ntouyas studied the existence of at least one positive solution for the following system:
{ u ( n ) ( t ) + λ h 1 ( t ) f 1 ( v ( t ) ) = 0 , 0 < t < 1 , v ( n ) ( t ) + λ h 2 ( t ) f 2 ( u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 0 ) = = u ( n 2 ) ( 0 ) = 0 , u ( 1 ) = α u ( η ) , v ( 0 ) = v ( 0 ) = = v ( n 2 ) ( 0 ) = 0 , v ( 1 ) = α u ( η ) .
(1.2)
In [9], by using fixed point index theory, Xu and Yang extended the results of [7], [8] and established the existence of at least one and two positive solutions for the following system:
{ u ( n ) ( t ) + a 1 ( t ) f 1 ( t , u ( t ) , v ( t ) ) = 0 , 0 < t < 1 , v ( n ) ( t ) + a 2 ( t ) f 2 ( t , u ( t ) , v ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 0 ) = = u ( n 2 ) ( 0 ) = 0 , u ( 1 ) = 0 , v ( 0 ) = v ( 0 ) = = v ( n 2 ) ( 0 ) = 0 , v ( 1 ) = 0 ,
(1.3)

where h i ( t ) and a i ( t ) are nonsingular. In [10], u ( 1 ) = 0 , v ( 1 ) = 0 of the system (1.3) are replaced by u ( 1 ) = α u ( η ) , v ( 1 ) = β v ( η ) , and in [6], u ( 1 ) = 0 , v ( 1 ) = 0 of the system (1.3) are replaced by u ( 1 ) = i = 1 m 2 α i u ( ξ i ) , v ( 1 ) = i = 1 m 2 β i v ( η i ) , where a i ( t ) is singular. By using fixed point index theory and the Krasonsel’skii fixed point theorem, the existence of one and/or two positive solutions is established.

On the other hand, Webb [11] gave a unified method of tackling many nonlocal boundary value problems, which have been applied to the study of the problem with Stieltjes integrals,
{ u ( n ) ( t ) + g 1 ( t ) f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 0 ) = = u ( n 2 ) ( 0 ) = 0 , u ( 1 ) = α [ u ] .
(1.4)
We mention that Stieltjes integrals are also used in the framework of nonlinear boundary conditions in several papers (see [12]–[17] and the references therein). In particular, Yang [12] studied the existence of positive solutions for the following system by using fixed point index theory in a cone:
{ u ( t ) + f 1 ( t , u ( t ) , v ( t ) ) = 0 , 0 < t < 1 , v ( t ) + f 2 ( t , u ( t ) , v ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = 0 , u ( 1 ) = H 1 ( 0 1 u ( s ) d B 1 ( s ) ) , v ( 0 ) = 0 , v ( 1 ) = H 2 ( 0 1 v ( s ) d B 2 ( s ) ) .
(1.5)
Infante and Pietramala [14] studied the following system as a special case to illustrate the obtained theory:
{ u ( t ) + f 1 ( t , u ( t ) , v ( t ) ) = 0 , 0 < t < 1 , v ( t ) + f 2 ( t , u ( t ) , v ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = H 11 ( β 12 [ u ] ) , u ( 1 ) = H 12 ( β 12 [ u ] ) , v ( 0 ) = H 21 ( β 21 [ v ] ) , v ( 1 ) = H 22 ( β 21 [ v ] ) .
(1.6)
By constructing a special cone and using fixed point index theory, Cui and Sun [15] studied the existence of at least one positive solution for the system with Stieltjes integrals,
{ u ( t ) + f 1 ( t , u ( t ) , v ( t ) ) = 0 , 0 < t < 1 , v ( t ) + f 2 ( t , u ( t ) , v ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = 0 , u ( 1 ) = β 12 [ v ] , v ( 0 ) = 0 , v ( 1 ) = β 22 [ u ] .
(1.7)

By using fixed point index theory and a priori estimates achieved by utilizing some properties of concave functions, Xu and Yang [16] showed the existence and multiplicity positive solutions for the system of the generalized Lidstone problems, where the system are mixed higher-order differential equations.

Motivated by the work of the above papers, we aim to investigate the solvability for the system (1.1). The main features are as follows: Firstly, the method we adopt, which has been widely used, is different from [5]–[12], [14]–[17]. Secondly, the nonlinear terms f i we considered here satisfy some growth conditions. In [6]–[8], [10], [11], [15], [17], the sublinear or superlinear conditions are used for f i . Moreover, the form of the Stieltjes integrals we consider here is quite general, which involves that of the Stieltjes integrals in [11]–[13], [15], [17] and is different from [14]. This implies that the case of boundary conditions (1.1) covers the multi-point boundary conditions and also the integral boundary conditions in a single framework.

The rest of the paper is organized as follows. In Section 2, we present some preliminaries and several lemmas. In Section 3, by applying the fixed-point theorem, we obtain the uniqueness and existence of at least three positive solutions for the system (1.1). In Section 4, we give an example to illustrate our result.

2 Preliminaries and lemmas

Definition 2.1

Let E be a real Banach space. A nonempty, closed, convex set P E is said to be a cone, which satisfies the following conditions:
  1. (1)

    x P , λ > 0 λ x P ;

     
  2. (2)

    x , x P x = 0 .

     

Definition 2.2

Let E be a real Banach space with cone P . A map β : P [ 0 , + ) is said to be a non-negative continuous concave functional on P if β is continuous and
β ( t x + ( 1 t ) y ) t β ( x ) + ( 1 t ) β ( y ) ,

for all x , y P and t [ 0 , 1 ] .

Let a , b be two numbers such that 0 < a < b and β be a non-negative continuous concave functional on P . We define the following convex sets:
P a = { x P : x < a } , P ( β , a , b ) = { x P : a β ( x ) , x b } .

Lemma 2.3

(see [18])

Let A : P ¯ c P ¯ c be completely continuous operator and β be a non-negative continuous concave functional on P such that β ( x ) x for x P ¯ c . Suppose there exist 0 < a < b < d c such that

(A1): { x P ( β , b , d ) : β ( x ) > b } ϕ and β ( A x ) > b for x P ( β , b , d ) ,

(A2): A x < a for x a ,

(A3): β ( A x ) > b for x P ( β , b , c ) with A x > d .

Then A has at least three fixed points x 1 , x 2 , x 3 in P ¯ c such that
x 1 < a , b < β ( x 2 ) and x 3 > a with  β ( x 3 ) < b .

Definition 2.4

( u , v ) C n 1 ( [ 0 , 1 ] , [ 0 , + ) ) × C n 2 ( [ 0 , 1 ] , [ 0 , + ) ) is said to be a positive solution of the system (1.1) if and only if ( u , v ) satisfies the system (1.1) and u ( t ) 0 , v ( t ) 0 , for any t [ 0 , 1 ] .

Lemma 2.5

Let x ( t ) , y ( t ) C [ 0 , 1 ] , then the boundary value problem
{ u ( n 1 ) ( t ) + x ( t ) = 0 , v ( n 2 ) ( t ) + y ( t ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 0 ) = = u ( n 1 2 ) ( 0 ) = 0 , u ( 1 ) = g 1 ( β 1 [ u ] , β 1 [ v ] ) , v ( 0 ) = v ( 0 ) = = v ( n 2 2 ) ( 0 ) = 0 , v ( 1 ) = g 2 ( β 2 [ u ] , β 2 [ v ] ) ,
(2.1)
has the integral representation
{ u ( t ) = 0 1 K 1 ( t , s ) x ( s ) d s + t n 1 1 g 1 ( β 1 [ u ] , β 1 [ v ] ) , v ( t ) = 0 1 K 2 ( t , s ) y ( s ) d s + t n 2 1 g 2 ( β 2 [ u ] , β 2 [ v ] ) ,
(2.2)
where
K i ( t , s ) = 1 ( n i 1 ) ! { t n i 1 ( 1 s ) n i 1 ( t s ) n i 1 , 0 s t 1 , t n i 1 ( 1 s ) n i 1 , 0 t s 1 , i = 1 , 2 .
(2.3)

Proof

By Taylor’s formula, we have
u ( t ) = u ( 0 ) + t u ( 0 ) + + t n 1 1 ( n 1 1 ) ! u ( n 1 1 ) ( 0 ) u ( t ) = + 1 ( n 1 1 ) ! 0 t ( t s ) n 1 1 u ( n 1 ) ( s ) d s , v ( t ) = v ( 0 ) + t v ( 0 ) + + t n 2 1 ( n 2 1 ) ! v ( n 2 1 ) ( 0 ) v ( t ) = + 1 ( n 2 1 ) ! 0 t ( t s ) n 2 1 v ( n 2 ) ( s ) d s ,
so, we reduce the equation of problem (2.1) to an equivalent integral equation,
u ( t ) = 1 ( n 1 1 ) ! 0 t ( t s ) n 1 1 x ( s ) d s + t n 1 1 ( n 1 1 ) ! u ( n 1 1 ) ( 0 ) ,
(2.4)
v ( t ) = 1 ( n 2 1 ) ! 0 t ( t s ) n 2 1 y ( s ) d s + t n 2 1 ( n 2 1 ) ! v ( n 2 1 ) ( 0 ) .
(2.5)
By (2.4) and (2.5), combining with the conditions u ( 1 ) = g 1 ( β 1 [ u ] , β 1 [ v ] ) , v ( 1 ) = g 2 ( β 2 [ u ] , β 2 [ v ] ) , and letting t = 1 , we have
u ( n 1 1 ) ( 0 ) = 0 1 ( 1 s ) n 1 1 x ( s ) d s + ( n 1 1 ) ! g 1 ( β 1 [ u ] , β 1 [ v ] ) , v ( n 2 1 ) ( 0 ) = 0 1 ( 1 s ) n 2 1 y ( s ) d s + ( n 2 1 ) ! g 2 ( β 2 [ u ] , β 2 [ v ] ) .
Substituting u ( n 1 1 ) ( 0 ) and v ( n 2 1 ) ( 0 ) into (2.4) and (2.5), we have
u ( t ) = 1 ( n 1 1 ) ! 0 t ( t s ) n 1 1 x ( s ) d s + 1 ( n 1 1 ) ! 0 1 t n 1 1 ( 1 s ) n 1 1 x ( s ) d s u ( t ) = + t n 1 1 g 1 ( β 1 [ u ] , β 1 [ v ] ) u ( t ) = 1 ( n 1 1 ) ! 0 t [ t n 1 1 ( 1 s ) n 1 1 ( t s ) n 1 1 ] x ( s ) d s u ( t ) = + 1 ( n 1 1 ) ! t 1 t n 1 1 ( 1 s ) n 1 1 x ( s ) d s + t n 1 1 g 1 ( β 1 [ u ] , β 1 [ v ] ) u ( t ) = 0 1 K 1 ( t , s ) x ( s ) d s + t n 1 1 g 1 ( β 1 [ u ] , β 1 [ v ] ) , v ( t ) = 0 1 K 2 ( t , s ) y ( s ) d s + t n 2 1 g 2 ( β 2 [ u ] , β 2 [ v ] ) ,

which is equivalent to the boundary value problem (2.1). □

Lemma 2.6

(see [11])

The function K i ( t , s ) , i = 1 , 2 defined by (2.3) has the following properties:
  1. (1)

    K i ( t , s ) 0 , for t , s [ 0 , 1 ] ;

     
  2. (2)

    c i ( t ) G i ( s ) K i ( t , s ) G i ( s ) , for t , s [ 0 , 1 ] ,

     
where
G i ( s ) : = τ i ( s ) n i 2 s ( 1 s ) n i 1 ( n i 1 ) ! , τ i ( s ) : = s ( 1 ( 1 s ) n i 1 n i 2 )
and
c i ( t ) : = min { ( n i 1 ) n i 1 t n i 2 ( 1 t ) ( n i 2 ) n i 2 , t n i 1 } .

Throughout this paper, we assume that the following condition is satisfied.

(H1): a i ( t ) does not vanish identically on any subinterval of ( 0 , 1 ) , 0 < 0 1 G i ( s ) a i ( s ) d s < + , i = 1 , 2 , where G i ( s ) is defined by Lemma 2.6 and there exists t 0 ( 0 , 1 ) such that a i ( t 0 ) > 0 .

Remark 2.7

By (H1), we can choose a subinterval [ ξ , η ] ( 0 , 1 ) such that t 0 [ ξ , η ] . Let γ : = min { c i ( t ) : t [ ξ , η ] , i = 1 , 2 } ; it is easy to see that 0 < γ < 1 . By Lemma 2.6, we have min t [ ξ , η ] K i ( t , s ) γ G i ( s ) , s [ 0 , 1 ] .

By Lemma 2.5, it is easy to prove that ( u , v ) C n 1 [ 0 , 1 ] × C n 2 [ 0 , 1 ] is a positive solution of the system (1.1) if and only if ( u , v ) C [ 0 , 1 ] × C [ 0 , 1 ] is a positive solution of the following integral system:
{ u ( t ) = 0 1 K 1 ( t , s ) a 1 ( s ) f 1 ( s , u ( s ) , v ( s ) ) d s + t n 1 1 g 1 ( β 1 [ u ] , β 1 [ v ] ) , v ( t ) = 0 1 K 2 ( t , s ) a 2 ( s ) f 2 ( s , u ( s ) , v ( s ) ) d s + t n 2 1 g 2 ( β 2 [ u ] , β 2 [ v ] ) .
Let E = C ( [ 0 , 1 ] , R ) × C ( [ 0 , 1 ] , R ) be a Banach space endowed with the norm ( u , v ) : = u + v , where u = max 0 t 1 | u ( t ) | , v = max 0 t 1 | v ( t ) | and define the cone K E by
K : = { ( u , v ) E : u ( t ) 0 , v ( t ) 0 , t [ 0 , 1 ] , min t [ ξ , η ] ( u ( t ) + v ( t ) ) γ ( u , v ) } .

It is easy to prove that E is a Banach space and K is a cone in E .

Define the operator T : K E by
T ( u , v ) ( t ) = ( T 1 ( u , v ) ( t ) , T 2 ( u , v ) ( t ) ) , t [ 0 , 1 ] ,
where
T 1 ( u , v ) ( t ) = 0 1 K 1 ( t , s ) a 1 ( s ) f 1 ( s , u ( s ) , v ( s ) ) d s + t n 1 1 g 1 ( β 1 [ u ] , β 1 [ v ] ) ,
(2.6)
T 2 ( u , v ) ( t ) = 0 1 K 2 ( t , s ) a 2 ( s ) f 2 ( s , u ( s ) , v ( s ) ) d s + t n 2 1 g 2 ( β 2 [ u ] , β 2 [ v ] ) .
(2.7)

Lemma 2.8

The operator T : K K .

Proof

For any ( u , v ) K , considering K i ( t , s ) 0 , i = 1 , 2 , we have T 1 ( u , v ) ( t ) 0 , T 2 ( u , v ) ( t ) 0 , for t [ 0 , 1 ] . From (2.6) and Lemma 2.6, we have
T 1 ( u , v ) 0 1 G 1 ( s ) a 1 ( s ) f 1 ( s , u ( s ) , v ( s ) ) d s + g 1 ( β 1 [ u ] , β 1 [ v ] ) .
(2.8)
It follows from (2.8) and Lemma 2.6 that we have
min t [ ξ , η ] T 1 ( u , v ) ( t ) = min t [ ξ , η ] [ 0 1 K 1 ( t , s ) a 1 ( s ) f 1 ( s , u ( s ) , v ( s ) ) d s + t n 1 1 g 1 ( β 1 [ u ] , β 1 [ v ] ) ] γ 0 1 G 1 ( s ) a 1 ( s ) f 1 ( s , u ( s ) , v ( s ) ) d s + γ g 1 ( β 1 [ u ] , β 1 [ v ] ) γ T 1 ( u , v ) .
Similarly, it follows from (2.7) and Lemma 2.6 that we have
min t [ ξ , η ] T 2 ( u , v ) ( t ) γ T 2 ( u , v ) .
Therefore,
min t [ ξ , η ] ( T 1 ( u , v ) ( t ) + T 2 ( u , v ) ( t ) ) min t [ ξ , η ] T 1 ( u , v ) ( t ) + min t [ ξ , η ] T 2 ( u , v ) ( t ) γ T 1 ( u , v ) + γ T 2 ( u , v ) = γ ( T 1 ( u , v ) , T 2 ( u , v ) ) .

From the above, we conclude that T ( u , v ) = ( T 1 ( u , v ) , T 2 ( u , v ) ) K , that is, T : K K . □

3 Main result

For convenience, we use the following notation:
M i = max t [ 0 , 1 ] 0 1 K i ( t , s ) a i ( s ) d s , m i = min t [ ξ , η ] ξ η K i ( t , s ) a i ( s ) d s , L i = 1 β i [ 1 ] , i = 1 , 2 .

Then 0 m i M i , i = 1 , 2 .

Theorem 3.1

Suppose that the condition (H1) is satisfied and there exist non-negative numbers h i , k i , d i , e i , i = 1 , 2 such that for all t [ 0 , 1 ] and ( u , v ) , ( u ¯ , v ¯ ) K :
| f i ( t , u , v ) f i ( t , u ¯ , v ¯ ) | h i | u u ¯ | + k i | v v ¯ | , i = 1 , 2 ,
(3.1)
| g i ( u , v ) g i ( u ¯ , v ¯ ) | d i | u u ¯ | + e i | v v ¯ | , i = 1 , 2
(3.2)
and
A 1 + A 2 < 1 ,
(3.3)

where A i = ( h i + k i ) M i + ( d i + e i ) β i [ 1 ] , i = 1 , 2 . Then the system (1.1) has a unique positive solution in K .

Proof

By Lemma 2.5, the system (1.1) has a unique positive solution if and only if the operator T has a unique fixed point in K .

Define sup t [ 0 , 1 ] f i ( t , 0 , 0 ) = N i < , i = 1 , 2 and g i ( 0 , 0 ) = G i < , i = 1 , 2 such that
r M 1 N 1 + G 1 + M 2 N 2 + G 2 1 A 1 A 2 .
First we show that T B r B r , where B r = { ( u , v ) | ( u , v ) K , ( u , v ) r } . For ( u , v ) B r , we have
| T 1 ( u , v ) ( t ) | 0 1 K 1 ( t , s ) a 1 ( s ) [ | f 1 ( s , u ( s ) , v ( s ) ) f 1 ( s , 0 , 0 ) | + | f 1 ( s , 0 , 0 ) | ] d s + t n 1 1 [ | g 1 ( β 1 [ u ] , β 1 [ v ] ) g 1 ( 0 , 0 ) | + | g 1 ( 0 , 0 ) | ] M 1 ( h 1 u + k 1 v + N 1 ) + β 1 [ 1 ] ( d 1 u + e 1 v ) + G 1 A 1 r + M 1 N 1 + G 1 ,
hence
T 1 ( u , v ) A 1 r + M 1 N 1 + G 1 .
In the same way, we obtain
T 2 ( u , v ) A 2 r + M 2 N 2 + G 2 .

Consequently, T ( u , v ) = T 1 ( u , v ) + T 1 ( u , v ) r .

Now we shall prove that T is a contraction. Let ( u , v ) , ( u ¯ , v ¯ ) K ; applying (2.6) we get
T 1 ( u , v ) ( t ) T 1 ( u ¯ , v ¯ ) ( t ) = 0 1 K 1 ( t , s ) a 1 ( s ) [ f 1 ( s , u ( s ) , v ( s ) ) f 1 ( s , u ¯ ( s ) , v ¯ ( s ) ) ] d s + t n 1 1 [ g 1 ( β 1 [ u ] , β 1 [ v ] ) g 1 ( β 1 [ u ¯ ] , β 1 [ v ¯ ] ) ] .
With the help of (3.1) and (3.2) we obtain
| T 1 ( u , v ) ( t ) T 1 ( u ¯ , v ¯ ) ( t ) | h 1 M 1 max t [ 0 , 1 ] | u ( t ) u ¯ ( t ) | + k 1 M 1 max t [ 0 , 1 ] | v ( t ) v ¯ ( t ) | + d 1 β 1 [ 1 ] max t [ 0 , 1 ] | u ( t ) u ¯ ( t ) | + e 1 β 1 [ 1 ] max t [ 0 , 1 ] | v ( t ) v ¯ ( t ) | = ( h 1 M 1 + d 1 β 1 [ 1 ] ) u u ¯ + ( k 1 M 1 + e 1 β 1 [ 1 ] ) v v ¯ ,
this together with (3.3) implies
T 1 ( u , v ) T 1 ( u ¯ , v ¯ ) A 1 ( u u ¯ + v v ¯ ) .
(3.4)
Similarly, applying (2.7), with the help of (3.1) and (3.2) we have
T 2 ( u , v ) T 2 ( u ¯ , v ¯ ) A 2 ( u u ¯ + v v ¯ ) .
(3.5)
Taking (3.4) and (3.5) into account we have
T ( u , v ) T ( u ¯ , v ¯ ) = T 1 ( u , v ) T 1 ( u ¯ , v ¯ ) + T 2 ( u , v ) T 2 ( u ¯ , v ¯ ) ( A 1 + A 2 ) ( u u ¯ + v v ¯ ) ,

where A 1 + A 2 < 1 . So, T is a contraction, hence it has a unique point fixed in K which is the unique positive solution of the system (1.1). The proof is completed. □

Define the non-negative continuous concave functional on K by
β ( u , v ) = min t [ ξ , η ] ( u ( t ) + v ( t ) ) .

We observe here that β ( u , v ) ( u , v ) , for each ( u , v ) K .

Throughout this section, we assume that p i , q i , i = 1 , 2 are four positive numbers satisfying 1 p 1 + 1 p 2 + 1 q 1 + 1 q 2 1 .

Theorem 3.2

Suppose that the condition (H1) is satisfied and there exist non-negative numbers: a , b , c such that 0 < a < b min { γ , m 1 p 1 M 1 , m 2 p 2 M 2 } c and f i ( t , u , v ) , g i ( x , y ) satisfy the following growth conditions:

(H2): g i ( x , y ) 1 q i L i ( x + y ) , x + y [ 0 , c β i [ 1 ] ] , i = 1 , 2 ;

(H3): f i ( t , u , v ) 1 p i c M i , t [ 0 , 1 ] , u + v [ 0 , c ] , i = 1 , 2 ;

(H4): f i ( t , u , v ) > b m i , t [ ξ , η ] , u + v [ b , b γ ] , i = 1 , 2 ;

(H5): f i ( t , u , v ) < 1 p i a M i , t [ 0 , 1 ] , u + v [ 0 , a ] , i = 1 , 2 .

Then the system (1.1) has at least three positive solutions ( u 1 , v 1 ) , ( u 2 , v 2 ) , ( u 3 , v 3 ) such that ( u 1 , v 1 ) < a , b < min t [ ξ , η ] ( u 2 ( t ) + v 2 ( t ) ) and ( u 3 , v 3 ) > a with min t [ ξ , η ] ( u 3 ( t ) + v 3 ( t ) ) < b .

Proof

It is clear that the existence of positive solutions for the system (1.1) is equivalent to the existence of fixed points of T in K .

We first prove that T : K ¯ c K ¯ c is a completely continuous operator. In fact, if ( u , v ) K ¯ c , then ( u , v ) c and by condition (H2), we have
g i ( β i [ u ] , β i [ v ] ) 1 q i L i β i [ u + v ] 1 q i L i c β i [ 1 ] = 1 q i c , i = 1 , 2 .
Thus, by condition (H3), we have
T ( u , v ) = max t [ 0 , 1 ] | T 1 ( u , v ) ( t ) | + max t [ 0 , 1 ] | T 2 ( u , v ) ( t ) | = max t [ 0 , 1 ] [ 0 1 K 1 ( t , s ) a 1 ( s ) f 1 ( s , u ( s ) , v ( s ) ) d s + t n 1 1 g 1 ( β 1 [ u ] , β 1 [ v ] ) ] + max t [ 0 , 1 ] [ 0 1 K 2 ( t , s ) a 2 ( s ) f 2 ( s , u ( s ) , v ( s ) ) d s + t n 2 1 g 2 ( β 2 [ u ] , β 2 [ v ] ) ] max t [ 0 , 1 ] 0 1 K 1 ( t , s ) a 1 ( s ) f 1 ( s , u ( s ) , v ( s ) ) d s + 1 q 1 c + max t [ 0 , 1 ] 0 1 K 2 ( t , s ) a 2 ( s ) f 2 ( s , u ( s ) , v ( s ) ) d s + 1 q 2 c 1 p 1 c M 1 M 1 + 1 q 1 c + 1 p 2 c M 2 M 2 + 1 q 2 c c .

Therefore, T ( u , v ) c , that is, T : K ¯ c K ¯ c . Standard applications of the Arzelà-Ascoli theorem imply that T is a completely continuous operator.

Now, we show that conditions (A1)-(A3) of Lemma 2.3 are satisfied.

Firstly, let u ( t ) = b 2 , v ( t ) = b 2 γ , it follows that β ( u , v ) > b , ( u , v ) < b γ , which shows that { ( u , v ) P ( β , b , b γ ) : β ( u , v ) > b } , and, for ( u , v ) P ( β , b , b γ ) , we have b u ( s ) + v ( s ) b γ , s [ ξ , η ] . By condition (H4) of Theorem 3.2, we obtain
β ( T ( u , v ) ( t ) ) = min t [ ξ , η ] ( T 1 ( u , v ) ( t ) + T 2 ( u , v ) ( t ) ) min t [ ξ , η ] ξ η K 1 ( t , s ) a 1 ( s ) f 1 ( s , u ( s ) , v ( s ) ) d s + γ g 1 ( β 1 [ u ] , β 1 [ v ] ) + min t [ ξ , η ] ξ η K 2 ( t , s ) a 2 ( s ) f 2 ( s , u ( s ) , v ( s ) ) d s + γ g 2 ( β 2 [ u ] , β 2 [ v ] ) > b m 1 min t [ ξ , η ] ξ η K 1 ( t , s ) a 1 ( s ) d s = b m 1 m 1 = b .
Similarly, by condition (H4) of Theorem 3.2, we can obtain
β ( T ( u , v ) ( t ) ) = min t [ ξ , η ] ( T 1 ( u , v ) ( t ) + T 2 ( u , v ) ( t ) ) > b m 2 min t [ ξ , η ] ξ η K 2 ( t , s ) a 2 ( s ) d s = b m 2 m 2 = b .

Therefore, condition (A1) of Lemma 2.3 is satisfied.

Secondly, in a completely analogous argument to the proof of T : K ¯ c K ¯ c , by condition (H5) of Theorem 3.2, condition (A2) of Lemma 2.3 is satisfied.

Finally, we show that condition (A3) of Lemma 2.3 is satisfied. If ( u , v ) P ( β , b , b γ ) and T ( u , v ) ( t ) > b γ , then
β ( T ( u , v ) ( t ) ) = min t [ ξ , η ] ( T 1 ( u , v ) ( t ) + T 2 ( u , v ) ( t ) ) γ T ( u , v ) ( t ) > b .

Therefore, condition (A3) of Lemma 2.3 is satisfied.

Thus, all conditions of Lemma 2.3 are satisfied. By Lemma 2.3, the system (1.1) has at least three positive solutions ( u 1 , v 1 ) , ( u 2 , v 2 ) , ( u 3 , v 3 ) such that ( u 1 , v 1 ) < a , b < min t [ ξ , η ] ( u 2 ( t ) + v 2 ( t ) ) and ( u 3 , v 3 ) > a , with min t [ ξ , η ] ( u 3 ( t ) + v 3 ( t ) ) < b . The proof is completed. □

4 Example

Example 4.1

Consider the following system of nonlinear mixed-order ordinary differential equations:
{ u ( 3 ) ( t ) + a 1 ( t ) f 1 ( t , u ( t ) , v ( t ) ) = 0 , 0 < t < 1 , v ( 4 ) ( t ) + a 2 ( t ) f 2 ( t , u ( t ) , v ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 0 ) = 0 , u ( 1 ) = g 1 ( β 1 [ u ] , β 1 [ v ] ) , v ( 0 ) = v ( 0 ) = v ( 0 ) = 0 , v ( 1 ) = g 2 ( β 2 [ u ] , β 2 [ v ] ) .
(4.1)
Then the system (4.1) is equivalent to the following system of nonlinear integral equations:
{ u ( t ) = 0 1 K 1 ( t , s ) a 1 ( s ) f 1 ( s , u ( s ) , v ( s ) ) d s + t 2 g 1 ( β 1 [ u ] , β 1 [ v ] ) , v ( t ) = 0 1 K 2 ( t , s ) a 2 ( s ) f 2 ( s , u ( s ) , v ( s ) ) d s + t 3 g 2 ( β 2 [ u ] , β 2 [ v ] ) ,
where
K 1 ( t , s ) = 1 2 { t 2 ( 1 s ) 2 ( t s ) 2 , 0 s t 1 , t 2 ( 1 s ) 2 , 0 t s 1 , K 2 ( t , s ) = 1 6 { t 3 ( 1 s ) 3 ( t s ) 3 , 0 s t 1 , t 3 ( 1 s ) 3 , 0 t s 1 .
We choose a 1 ( t ) = 5 t , a 2 ( t ) = 50 , β 1 [ u ] = 0 1 u ( s ) d s , β 2 [ u ] = 2 0 1 u ( s ) d s , β 1 [ v ] = 0 1 v ( s ) d s , β 2 [ v ] = 2 0 1 v ( s ) d s , and
f 1 ( t , u , v ) = { 0.1 t + 0.01 ( u + v ) 2 , t [ 0 , 1 ] , 0 u + v 2 , 0.1 t + 15 [ ( u + v ) 2 2 ( u + v ) ] + 0.04 , t [ 0 , 1 ] , 2 < u + v < 4 , 0.1 t + 15 [ 3 log 2 ( u + v ) + ( u + v ) / 2 ] + 0.04 , t [ 0 , 1 ] , 4 u + v 16 , 0.1 t + 300.04 , t [ 0 , 1 ] , u + v > 16
and
f 2 ( t , u , v ) = { 0.01 t + 0.02 ( u + v ) 2 , t [ 0 , 1 ] , 0 u + v 2 , 0.01 t + 17 [ ( u + v ) 2 2 ( u + v ) ] + 0.08 , t [ 0 , 1 ] , 2 < u + v < 4 , 0.01 t + 17 [ 3 log 2 ( u + v ) + ( u + v ) / 2 ] + 0.08 , t [ 0 , 1 ] , 4 u + v 16 , 0.01 t + 340.08 , t [ 0 , 1 ] , 16 < u + v < + ,
and
g 1 ( β 1 [ u ] , β 1 [ v ] ) = { 0.2 ln ( β 1 [ u ] + β 1 [ v ] + 1 ) , 0 u + v 800 , 0.2 ln 801 , 800 < u + v < + , g 1 ( β 1 [ u ] , β 1 [ v ] ) = { 0.125 ln ( β 1 [ u ] + β 1 [ v ] + 1 ) , 0 u + v 1 , 600 , 0.125 ln 1 , 601 , 1 , 600 < u + v < + .
By Lemma 2.6, we have
c 1 ( t ) = { t 2 , 0 t 0.8 , 4 t ( 1 t ) , 0.8 t 1 , c 2 ( t ) = { t 3 , 0 t 0.87 , 27 t 2 ( 1 t ) / 4 , 0.87 t 1 .
Choose [ ξ , η ] = [ 0.6 , 0.8 ] ; by Remark 2.7, we obtain γ = 0.216 . Then by direct calculation we obtain
M 1 0.248 , m 1 0.054 , M 2 0.220 , m 2 0.048 , L 1 = 1 , L 2 = 0.5 , β 1 [ 1 ] = 1 , β 2 [ 1 ] = 2 .

It is easy to verify that the condition (H1) holds. Let q 1 = 5 , q 2 = 4 , p 1 = 5 , p 2 = 7 , a = 1 , b = 4 , c = 800 . Also, it is easy to verify that f 1 , f 2 , g 1 , g 2 satisfy conditions (H2)-(H5).

Thus, by Theorem 3.2, the system (4.1) has at least three positive solutions ( u 1 , v 1 ) , ( u 2 , v 2 ) , ( u 3 , v 3 ) such that ( u 1 , v 1 ) < 1 , 4 < min t [ 0.6 , 0.8 ] ( u 2 , v 2 ) and ( u 3 , v 3 ) > 1 with min t [ 0.6 , 0.8 ] ( u 3 , v 3 ) < 4 .

Declarations

Acknowledgements

The authors are grateful to the referees for their careful reading. This research is supported by the Nature Science Foundation of Anhui Provincial Education Department (Grant Nos. KJ2014A252 and KJ2013A248) and Professors (Doctors) Scientific Research Foundation of Suzhou University (Grant No. 2013jb04).

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Suzhou University, Suzhou, China

References

  1. Yang J, Wei Z:Positive solutions of n th order m -point boundary value problem. Appl. Math. Comput. 2008, 202: 715-720. 10.1016/j.amc.2008.03.009MathSciNetView ArticleGoogle Scholar
  2. Sun J, Xu X, O’Regan D:Nodal solutions for m -point boundary value problems using bifurcation. Nonlinear Anal. 2008, 68: 3034-3046. 10.1016/j.na.2007.02.043MathSciNetView ArticleGoogle Scholar
  3. Li Y, Wei Z:Multiple positive solutions for n th order multi-point boundary value problem. Bound. Value Probl. 2010., 2010:Google Scholar
  4. Graef JR, Kong L: Necessary and sufficient conditions for the existence of symmetric positive solutions of multi-point boundary value problems. Nonlinear Anal. 2008, 68: 1529-1552. 10.1016/j.na.2006.12.037MathSciNetView ArticleGoogle Scholar
  5. 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.2011.11.092MathSciNetView ArticleGoogle Scholar
  6. Su H, Wei Z, Zhang X:Positive solutions of n -order m -order multi-point boundary value system. Appl. Math. Comput. 2007, 188: 1234-1243. 10.1016/j.amc.2006.10.077MathSciNetView ArticleGoogle Scholar
  7. Henderson J, Ntouyas SK:Existence of positive solutions for systems of n th-order three-point nonlocal boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2007., 2007:Google Scholar
  8. Xu J, Yang Z:Positive solutions of boundary value problem for system of nonlinear n th-order ordinary differential equations. J. Syst. Sci. Math. Sci. 2010, 30: 633-641. (in Chinese)Google Scholar
  9. Xu J, Yang Z:Positive solutions for a systems of n th-order nonlinear boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2011., 2011: 10.1186/1687-1847-2011-4Google Scholar
  10. Xie S, Zhu J:Positive solutions for the systems of n th-order singular nonlocal boundary value problems. J. Appl. Math. Comput. 2011, 37: 119-132. 10.1007/s12190-010-0424-5MathSciNetView ArticleGoogle Scholar
  11. Webb J: Nonlocal conjugate type boundary value problems of higher order. Nonlinear Anal. TMA 2009, 71: 1933-1940. 10.1016/j.na.2009.01.033View ArticleGoogle Scholar
  12. Yang Z: Positive solutions to a system of second-order nonlocal boundary value problems. Nonlinear Anal. 2005, 62: 1251-1265. 10.1016/j.na.2005.04.030MathSciNetView ArticleGoogle Scholar
  13. Xu J, Yang Z: Three positive solutions for a system of singular generalized Lidstone problems. Electron. J. Differ. Equ. 2009., 2009: 10.1155/2009/169321Google Scholar
  14. 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.2008.11.095MathSciNetView ArticleGoogle Scholar
  15. Cui Y, Sun J: On existence of positive solutions of coupled integral boundary value problems for a nonlinear singular superlinear differential system. Electron. J. Qual. Theory Differ. Equ. 2012., 2012: 10.1186/1687-1847-2012-41Google Scholar
  16. Xu J, Yang Z: Positive solutions for a system of generalized Lidstone problems. J. Appl. Math. Comput. 2011, 37: 13-35. 10.1007/s12190-010-0418-3MathSciNetView ArticleGoogle Scholar
  17. Jiang J, Liu L, Wu Y: Multiple positive solutions of singular fractional differential system involving Stieltjes integral conditions. Electron. J. Qual. Theory Differ. Equ. 2012., 2012: 10.1186/1687-1847-2012-43Google Scholar
  18. Leggett RW, Williams LR: Multiple positive fixed points of nonlinear operator on ordered Banach spaces. Indiana Univ. Math. J. 1979, 28: 673-688. 10.1512/iumj.1979.28.28046MathSciNetView ArticleGoogle Scholar

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© Li and Zhang; licensee Springer. 2014

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