Multiple monotone positive solutions for higher order differential equations with integral boundary conditions

Abstract

This paper investigates the higher order differential equations with nonlocal boundary conditions

${ u ( n ) ( t ) + f ( t , u ( t ) , u ′ ( t ) , … , u ( n − 2 ) ( t ) ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = u ′ ( 0 ) = ⋯ = u ( n − 3 ) ( 0 ) = 0 , u ( n − 2 ) ( 0 ) = ∫ 0 1 u ( n − 2 ) ( s ) d A ( s ) , u ( n − 2 ) ( 1 ) = ∫ 0 1 u ( n − 2 ) ( s ) d B ( s ) .$

The existence results of multiple monotone positive solutions are obtained by means of fixed point index theory for operators in a cone.

MSC:34B10, 34B18.

1 Introduction

In this paper, we are concerned with the existence of multiple monotone positive solutions for the higher order differential equation

$u ( n ) (t)+f ( t , u ( t ) , u ′ ( t ) , … , u ( n − 2 ) ( t ) ) =0,t∈(0,1),$
(1.1)

subject to the following integral boundary conditions:

${ u ( 0 ) = u ′ ( 0 ) = ⋯ = u ( n − 3 ) ( 0 ) = 0 , u ( n − 2 ) ( 0 ) = ∫ 0 1 u ( n − 2 ) ( s ) d A ( s ) , u ( n − 2 ) ( 1 ) = ∫ 0 1 u ( n − 2 ) ( s ) d B ( s ) ,$
(1.2)

where $n≥3$, $f:[0,1]× ( R + ) n − 1 → R +$ is continuous in which $R + =[0,+∞)$, A and B are right continuous on $[0,1)$, left continuous at $t=1$, and nondecreasing on $[0,1]$, with $A(0)=B(0)=0$; $∫ 0 1 v(s)dA(s)$ and $∫ 0 1 v(s)dB(s)$ denote the Riemann-Stieltjes integrals of v with respect to A and B, respectively.

Boundary value problems (BVPs for short) for nonlinear differential equations arise in many areas of applied mathematics and physics. Many authors have discussed the existence of positive solutions for second order or higher order differential equations with boundary conditions defined at a finite number of points, for instance, . In , Graef and Yang considered the following n th-order multi-point BVP:

${ u ( n ) ( t ) + λ g ( t ) f ( u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = u ′ ( 0 ) = ⋯ = u ( n − 2 ) ( 0 ) = ∑ i = 1 m a i u ( n − 2 ) ( ξ i ) − u ( n − 2 ) ( 1 ) = 0 ,$

where $n≥3$, $λ>0$ is a parameter, g and f are continuous functions, $1 2 ≤ ξ 1 < ξ 2 <⋯< ξ m <1$, $a i >0$ for $1≤i≤m$ and $∑ i = 1 m a i =1$. The authors obtained the existence and nonexistence results of positive solutions by using Krasnosel’skii’s fixed point theorem in cones. In , we studied the following second order m-point nonhomogeneous BVP:

${ u ″ ( t ) + a ( t ) f ( t , u ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 , u ( 1 ) − ∑ i = 1 m − 2 k i u ( ξ i ) = b ,$

where $b>0$, $k i >0$ ($i=1,2,…,m−2$), $0< ξ 1 < ξ 2 <⋯< ξ m − 2 <1$, $∑ i = 1 m − 2 k i ξ i <1$. The authors obtained the existence, nonexistence and multiplicity of positive solutions by using the Krasnosel’skii-Guo fixed point theorem, the upper-lower solutions method and topological degree theory.

Boundary value problems with integral boundary conditions for ordinary differential equations represent a very interesting and important class of problems and arise in the study of various physical, biological and chemical processes , such as heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics. They include two, three, multi-point and nonlocal BVPs as special cases. The existence and multiplicity of positive solutions for such problems have received a great deal of attention, see  and the references therein. In , Feng, Ji and Ge considered the existence and multiplicity of positive solutions for a class of nonlinear boundary value problems of second order differential equations with integral boundary conditions in ordered Banach spaces

${ x ″ ( t ) + f ( t , x ( t ) ) = θ , t ∈ ( 0 , 1 ) , x ( 0 ) = ∫ 0 1 g ( t ) x ( t ) d t , x ( 1 ) = θ , or x ( 0 ) = θ , x ( 1 ) = ∫ 0 1 g ( t ) x ( t ) d t .$

The arguments are based upon a specially constructed cone and fixed point theory in a cone for strict set contraction operators.

Motivated by the works mentioned above, in this paper, we consider the existence of multiple monotone positive solutions for BVP (1.1) and (1.2). In comparison with previous works, our paper has several new features. Firstly, we consider higher order boundary value problems, and we allow the nonlinearity f to contain derivatives of the unknown function $u(t)$ up to $n−2$ order. Secondly, we discuss the boundary value problem with integral boundary conditions, i.e., BVP (1.1) and (1.2), which includes two-point, three-point, multi-point and nonlocal boundary value problems as special cases. Thirdly, we consider the existence of multiple monotone positive solutions. To our knowledge, few papers have considered the monotone positive solutions for a higher order differential equation with integral boundary conditions. We shall emphasize here that with these new features our work improves and generalizes the results of  and some other known results to some degree. In this work we shall also utilize the following fixed point theorem in cones.

Lemma 1.1 ([33, 34])

Let K be a cone in a Banach space E. Let D be an open bounded subset of E with $D K =D∩K≠∅$ and $D ¯ K ≠K$. Assume that $A: D ¯ K →K$ is a compact operator such that $u≠Au$ for $u∈∂ D K$. Then the following results hold.

1. (1)

If $∥Au∥≤∥u∥$, $u∈∂ D K$, then $i K (A, D K )=1$.

2. (2)

If there exists $e∈K∖{0}$ such that $u≠Au+λe$ for all $u∈∂ D K$ and $λ>0$, then $i K (A, D K )=0$.

3. (3)

Let U be open in E such that $U ¯ ⊂ D K$. If $i K (A, D K )=1$ and $i K (A, U K )=0$, then A has a fixed point in $D K ∖ U ¯ K$. The same result holds if $i K (A, D K )=0$ and $i K (A, U K )=1$.

2 Preliminary lemmas

Let $E={u∈ C n − 2 [0,1]: u ( i ) (0)=0,0≤i≤n−3}$, then E is a Banach space with the norm $∥u∥= sup t ∈ [ 0 , 1 ] | u ( n − 2 ) (t)|$ for each $u∈E$.

We make the following assumptions:

($H 1$) $f:[0,1]× ( R + ) n − 1 → R +$ is continuous.

($H 2$) $k 1 >0$, $k 4 >0$, $k>0$, where $k= k 1 k 4 − k 2 k 3$,

$k 1 = 1 − ∫ 0 1 ( 1 − s ) d A ( s ) , k 2 = ∫ 0 1 s d A ( s ) , k 3 = ∫ 0 1 ( 1 − s ) d B ( s ) , k 4 = 1 − ∫ 0 1 s d B ( s ) .$

Lemma 2.1 Assume that ($H 2$) holds. Then, for any $y∈C[0,1]$, the BVP

${ − u ″ ( t ) = y ( t ) , t ∈ ( 0 , 1 ) , u ( 0 ) = ∫ 0 1 u ( t ) d A ( t ) , u ( 1 ) = ∫ 0 1 u ( t ) d B ( t )$
(2.1)

has a unique solution u that can be expressed in the form

$u(t)= ∫ 0 1 H(t,s)y(s)ds,t∈[0,1],$
(2.2)

where

$H ( t , s ) = G ( t , s ) + t k 3 + ( 1 − t ) k 4 k G A ( s ) + t k 1 + ( 1 − t ) k 2 k G B ( s ) , G A ( s ) = ∫ 0 1 G ( τ , s ) d A ( τ ) , G B ( s ) = ∫ 0 1 G ( τ , s ) d B ( τ ) , G ( t , s ) = { t ( 1 − s ) , 0 ≤ t ≤ s ≤ 1 , s ( 1 − t ) , 0 ≤ s ≤ t ≤ 1 .$
(2.3)

Proof Firstly, we prove that if u is a solution of BVP (2.1), then it will take the form of (2.2). Now, integrating differential equation (2.1) from 0 to t twice, we have

$u(t)=u(0)+ u ′ (0)t− ∫ 0 t (t−s)y(s)ds.$
(2.4)

Letting $t=1$ in (2.4), we get

$u ′ (0)=u(1)−u(0)+ ∫ 0 1 (1−s)y(s)ds.$
(2.5)

Substituting the boundary conditions of (2.1) and (2.5) into (2.4) yields

$u ( t ) = u ( 0 ) + [ u ( 1 ) − u ( 0 ) + ∫ 0 1 ( 1 − s ) y ( s ) d s ] t − ∫ 0 t ( t − s ) y ( s ) d s = ( 1 − t ) ∫ 0 1 u ( t ) d A ( t ) + t ∫ 0 1 u ( t ) d B ( t ) + ∫ 0 1 G ( t , s ) y ( s ) d s ,$
(2.6)

and, consequently,

$∫ 0 1 u ( t ) d A ( t ) = ∫ 0 1 ( 1 − t ) d A ( t ) ∫ 0 1 u ( t ) d A ( t ) + ∫ 0 1 t d A ( t ) ∫ 0 1 u ( t ) d B ( t ) + ∫ 0 1 ∫ 0 1 G ( t , s ) y ( s ) d s d A ( t ) , ∫ 0 1 u ( t ) d B ( t ) = ∫ 0 1 ( 1 − t ) d B ( t ) ∫ 0 1 u ( t ) d A ( t ) + ∫ 0 1 t d B ( t ) ∫ 0 1 u ( t ) d B ( t ) + ∫ 0 1 ∫ 0 1 G ( t , s ) y ( s ) d s d B ( t ) .$

Solving the above two equations for $∫ 0 1 u(t)dA(t)$ and $∫ 0 1 u(t)dB(t)$, we have

$( k 1 − k 2 − k 3 k 4 ) ( ∫ 0 1 u ( t ) d A ( t ) ∫ 0 1 u ( t ) d B ( t ) ) = ( ∫ 0 1 ∫ 0 1 G ( t , s ) y ( s ) d s d A ( t ) ∫ 0 1 ∫ 0 1 G ( t , s ) y ( s ) d s d B ( t ) ) ,$

and so

$( ∫ 0 1 u ( t ) d A ( t ) ∫ 0 1 u ( t ) d B ( t ) ) = k − 1 ( k 4 k 2 k 3 k 1 ) ( ∫ 0 1 ∫ 0 1 G ( t , s ) y ( s ) d s d A ( t ) ∫ 0 1 ∫ 0 1 G ( t , s ) y ( s ) d s d B ( t ) ) .$
(2.7)

Hence, (2.2) follows from (2.6) and (2.7).

Next we prove that the u given by (2.2) satisfies the differential equation and boundary conditions of (2.1). From (2.2), we have

$u ( t ) = ∫ 0 t s ( 1 − t ) y ( s ) d s + ∫ t 1 t ( 1 − s ) y ( s ) d s + t k 3 + ( 1 − t ) k 4 k ∫ 0 1 [ ∫ 0 1 G ( τ , s ) d A ( τ ) ] y ( s ) d s + t k 1 + ( 1 − t ) k 2 k ∫ 0 1 [ ∫ 0 1 G ( τ , s ) d B ( τ ) ] y ( s ) d s .$
(2.8)

Direct differentiation of (2.8) gives $u ″ (t)=−y(t)$. Also, from (2.2) we have

$∫ 0 1 u ( t ) d A ( t ) = ∫ 0 1 ∫ 0 1 G ( t , s ) y ( s ) d s d A ( t ) + k − 1 [ k 3 k 2 + k 4 ( 1 − k 1 ) ] ∫ 0 1 [ ∫ 0 1 G ( τ , s ) d A ( τ ) ] y ( s ) d s + k − 1 [ k 1 k 2 + k 2 ( 1 − k 1 ) ] ∫ 0 1 [ ∫ 0 1 G ( τ , s ) d B ( τ ) ] y ( s ) d s = k − 1 k 4 ∫ 0 1 ∫ 0 1 G ( τ , s ) y ( s ) d A ( τ ) d s + k − 1 k 2 ∫ 0 1 ∫ 0 1 G ( τ , s ) y ( s ) d B ( τ ) d s ,$

and, similarly,

$∫ 0 1 u(t)dB(t)= k − 1 k 3 ∫ 0 1 ∫ 0 1 G(τ,s)y(s)dA(τ)ds+ k − 1 k 1 ∫ 0 1 ∫ 0 1 G(τ,s)y(s)dB(τ)ds.$

Therefore, by solving the above two equations with the double integrals as unknowns, we have

$∫ 0 1 ∫ 0 1 G(τ,s)y(s)dA(τ)ds= k 1 ∫ 0 1 u(t)dA(t)− k 2 ∫ 0 1 u(t)dB(t)$
(2.9)

and

$∫ 0 1 ∫ 0 1 G(τ,s)y(s)dB(τ)ds=− k 3 ∫ 0 1 u(t)dA(t)+ k 4 ∫ 0 1 u(t)dB(t).$
(2.10)

Hence (2.6) follows from (2.2), (2.9) and (2.10), and thus $u(0)= ∫ 0 1 u(t)dA(t)$, $u(1)= ∫ 0 1 u(t)dB(t)$. This completes the proof. □

Defining

$G 3 (t,s)= ∫ 0 t H(v,s)dv, G i (t,s)= ∫ 0 t G i − 1 (v,s)dv,i≥4,$

then $G n (t,s)$ is the Green function of BVP (1.1) and (1.2). Moreover, solving BVP (1.1) and (1.2) is equivalent to finding a solution of the following integral equation:

$u(t)= ∫ 0 1 G n (t,s)f ( s , u ( s ) , u ′ ( s ) , … , u ( n − 2 ) ( s ) ) ds,t∈[0,1].$

Remark 2.1 If ($H 2$) holds, then for any $t,s∈[0,1]$, it is easy to testify that

$G(t,s)≥0,H(t,s)≥0, G n (t,s)≥0,n≥3.$
(2.11)

Lemma 2.2 Let $δ∈(0, 1 2 )$, then for any $t∈[δ,1−δ]$, $η,s∈[0,1]$, we have

$H(t,s)≥δH(η,s).$

Proof It is easy to show that $G(t,s)≥δG(η,s)$, $∀t∈[δ,1−δ]$, $η,s∈[0,1]$. For $t∈[δ,1−δ]$, $η,s∈[0,1]$, we have

$H ( t , s ) = G ( t , s ) + t k 3 + ( 1 − t ) k 4 k ∫ 0 1 G ( τ , s ) d A ( τ ) + t k 1 + ( 1 − t ) k 2 k ∫ 0 1 G ( τ , s ) d B ( τ ) ≥ δ G ( η , s ) + δ ( k 3 + k 4 ) k ∫ 0 1 G ( τ , s ) d A ( τ ) + δ ( k 1 + k 2 ) k ∫ 0 1 G ( τ , s ) d B ( τ ) ≥ δ [ G ( η , s ) + η k 3 + ( 1 − η ) k 4 k ∫ 0 1 G ( τ , s ) d A ( τ ) + η k 1 + ( 1 − η ) k 2 k ∫ 0 1 G ( τ , s ) d B ( τ ) ] = δ H ( η , s ) .$

For any $s∈[0,1]$, we define $H(s)= max t ∈ [ 0 , 1 ] H(t,s)$. From Lemma 2.2, we know that

$δH(s)≤H(t,s)≤H(s),t∈[δ,1−δ],s∈[0,1].$
(2.12)

□

Lemma 2.3 Assume that ($H 2$) holds. If $u∈ C n [0,1]$ satisfies the boundary conditions (1.2) and

$u ( n ) (t)≤0,t∈[0,1],$

then

$u(t)≥0and u ′ (t)≥0fort∈[0,1].$
(2.13)

Proof Let $m(t)= u ( n − 2 ) (t)$, $t∈[0,1]$, then we have

${ m ″ ( t ) ≤ 0 , t ∈ [ 0 , 1 ] , m ( 0 ) = ∫ 0 1 m ( t ) d A ( t ) , m ( 1 ) = ∫ 0 1 m ( t ) d B ( t ) .$

For $t∈[0,1]$, $m ″ (t)≤0$ implies that

$m(t)=m [ ( 1 − t ) ⋅ 0 + t ⋅ 1 ] ≥(1−t)m(0)+tm(1).$

Thus

$m ( 0 ) = ∫ 0 1 m ( t ) d A ( t ) ≥ m ( 0 ) ∫ 0 1 ( 1 − t ) d A ( t ) + m ( 1 ) ∫ 0 1 t d A ( t ) = ( 1 − k 1 ) m ( 0 ) + k 2 m ( 1 ) ,$

i.e.,

$m(0)≥ k 2 k 1 m(1).$

On the other hand,

$m ( 1 ) = ∫ 0 1 m ( t ) d B ( t ) ≥ m ( 0 ) ∫ 0 1 ( 1 − t ) d B ( t ) + m ( 1 ) ∫ 0 1 t d B ( t ) = k 3 m ( 0 ) + ( 1 − k 4 ) m ( 1 ) ,$

and so

$m(1)≥ k 3 k 4 m(0)≥ k 2 k 3 k 1 k 4 m(1),$

i.e., $km(1)≥0$, therefore $m(1)≥0$, and so $m(0)≥0$.

Now, $m(0)≥0$, $m(1)≥0$ and $m(t)$ is concave downward, so we have

$m(t)= u ( n − 2 ) (t)≥0,t∈[0,1].$
(2.14)

From (2.14) and $u(0)= u ′ (0)=⋯= u ( n − 3 ) (0)=0$, we obtain (2.13). This completes the proof of Lemma 2.3. □

Remark 2.2 From Lemma 2.3, if u is a positive solution of BVP (1.1) and (1.2), then u is nondecreasing on $[0,1]$, i.e., u is a monotone positive solution of BVP (1.1) and (1.2).

Let

$K= { u ∈ E : u ( n − 2 ) ( t ) ≥ 0 , t ∈ [ 0 , 1 ] , u ( n − 2 ) ( t ) ≥ δ ∥ u ∥ , t ∈ [ δ , 1 − δ ] } .$

Obviously, K is a cone in E. For any $ρ>0$, let $K ρ ={u∈K:∥u∥<ρ}$, $∂ K ρ ={u∈K:∥u∥=ρ}$ and $K ¯ ρ ={u∈K:∥u∥≤ρ}$. Define an operator $T:K→E$ as follows:

$(Tu)(t)= ∫ 0 1 G n (t,s)f ( s , u ( s ) , u ′ ( s ) , … , u ( n − 2 ) ( s ) ) ds,t∈[0,1].$
(2.15)

Then u is a solution of BVP (1.1) and (1.2) if and only if u solves the operator equation $u=Tu$.

Lemma 2.4 Suppose that ($H 1$) and ($H 2$) hold, then $T:K→K$ is completely continuous.

Proof For all $u∈K$, $t∈[0,1]$, by ($H 1$), (2.11), (2.12) and (2.15), we have

$( T u ) ( n − 2 ) ( t ) = ∫ 0 1 H ( t , s ) f ( s , u ( s ) , u ′ ( s ) , … , u ( n − 2 ) ( s ) ) d s ≥ 0 , ( T u ) ( n − 2 ) ( t ) = ∫ 0 1 H ( t , s ) f ( s , u ( s ) , u ′ ( s ) , … , u ( n − 2 ) ( s ) ) d s ≤ ∫ 0 1 H ( s ) f ( s , u ( s ) , u ′ ( s ) , … , u ( n − 2 ) ( s ) ) d s ,$

and

$∥Tu∥≤ ∫ 0 1 H(s)f ( s , u ( s ) , u ′ ( s ) , … , u ( n − 2 ) ( s ) ) ds.$

Thus, further from the first inequality of (2.12), we have

$( T u ) ( n − 2 ) ( t ) = ∫ 0 1 H ( t , s ) f ( s , u ( s ) , u ′ ( s ) , … , u ( n − 2 ) ( s ) ) d s ≥ δ ∫ 0 1 H ( s ) f ( s , u ( s ) , u ′ ( s ) , … , u ( n − 2 ) ( s ) ) d s ≥ δ ∥ T u ∥ , t ∈ [ δ , 1 − δ ] .$

Hence, $Tu∈K$ and $T(K)⊂K$.

Next by standard methods and the Ascoli-Arzela theorem, one can prove that $T:K→K$ is completely continuous. So this is omitted. □

Let

$Ω ρ = { u ∈ K : min δ ≤ t ≤ 1 − δ u ( n − 2 ) ( t ) < δ ρ } = { u ∈ E : u ( n − 2 ) ( t ) ≥ 0 , t ∈ [ 0 , 1 ] , δ ∥ u ∥ ≤ min δ ≤ t ≤ 1 − δ u ( n − 2 ) ( t ) < δ ρ } .$

Proceeding as for the proof of Lemma 2.5 in , we have the following.

Lemma 2.5 $Ω ρ$ has the following properties:

1. (a)

$Ω ρ$ is open relative to K;

2. (b)

$K δ ρ ⊂ Ω ρ ⊂ K ρ$;

3. (c)

$u∈∂ Ω ρ$ if and only if $min δ ≤ t ≤ 1 − δ u ( n − 2 ) (t)=δρ$;

4. (d)

if $u∈∂ Ω ρ$, then $δρ≤ u ( n − 2 ) (t)≤ρ$ for $δ≤t≤1−δ$.

Now for convenience we introduce the following notations:

$f δ ρ , ρ = min { min δ ≤ t ≤ 1 − δ f ( t , u 0 , … , u n − 2 ) ρ : u 0 , u 1 , … , u n − 3 ∈ [ 0 , ρ ] , u n − 2 ∈ [ δ ρ , ρ ] } , f 0 , ρ = max { max 0 ≤ t ≤ 1 f ( t , u 0 , u 1 , … , u n − 2 ) ρ : ( u 0 , u 1 , … , u n − 2 ) ∈ [ 0 , ρ ] n − 1 } , f 0 = lim u 0 , u 1 , … , u n − 2 → 0 max 0 ≤ t ≤ 1 f ( t , u 0 , u 1 , … , u n − 2 ) u n − 2 , f ∞ = lim u 0 + u 1 + ⋯ + u n − 2 → + ∞ max 0 ≤ t ≤ 1 f ( t , u 0 , u 1 , … , u n − 2 ) u 0 + u 1 + ⋯ + u n − 2 , f 0 = lim u 0 , u 1 , … , u n − 2 → 0 min δ ≤ t ≤ 1 − δ f ( t , u 0 , u 1 , … , u n − 2 ) u n − 2 , f ∞ = lim u 0 + u 1 + ⋯ + u n − 2 → + ∞ min δ ≤ t ≤ 1 − δ f ( t , u 0 , u 1 , … , u n − 2 ) u 0 + u 1 + ⋯ + u n − 2 , m = ( max 0 ≤ t ≤ 1 ∫ 0 1 H ( t , s ) d s ) − 1 , M = ( min δ ≤ t ≤ 1 − δ ∫ δ 1 − δ H ( t , s ) d s ) − 1 .$

To prove our main results, we need the following lemmas.

Lemma 2.6 Assume that ($H 1$), ($H 2$) hold and f satisfies

$f 0 , ρ ≤mandu≠Tuforu∈∂ K ρ ,$
(2.16)

then $i K (T, K ρ )=1$.

Proof For $u∈∂ K ρ$, we have $0≤ u ( n − 2 ) (t)≤ρ$ and $0≤ u ( i ) (t)= ∫ 0 t u ( i + 1 ) (s)ds≤ max 0 ≤ t ≤ 1 u ( i + 1 ) (t)≤∥u∥=ρ$, $t∈[0,1]$, $i=0,1,…,n−3$. Then by (2.16) we have, for $t∈[0,1]$,

$( T u ) ( n − 2 ) ( t ) = ∫ 0 1 H ( t , s ) f ( s , u ( s ) , u ′ ( s ) , … , u ( n − 2 ) ( s ) ) d s ≤ m ρ ∫ 0 1 H ( t , s ) d s ≤ ρ = ∥ u ∥ .$

This implies that $∥Tu∥≤∥u∥$ for $u∈∂ K ρ$. By the point (1) in Lemma 1.1, we have $i K (T, K ρ )=1$. □

Lemma 2.7 Assume that ($H 1$), ($H 2$) hold and f satisfies

$f δ ρ , ρ ≥δMandu≠Tuforu∈∂ Ω ρ ,$
(2.17)

then $i K (T, K ρ )=0$.

Proof Let $e(t)= t n − 2 ( n − 2 ) !$, $t∈[0,1]$, then $e∈K$ with $∥e∥=1$. Next we prove that

$u≠Tu+λe,u∈∂ Ω ρ ,λ>0.$

In fact, if not, then there exist $u 0 ∈∂ Ω ρ$ and $λ 0 >0$ such that $u 0 =T u 0 + λ 0 e$. By (2.17) and the point (d) in Lemma 2.5, we have, for $t∈[δ,1−δ]$,

$u 0 ( n − 2 ) ( t ) = ( T u 0 ) ( n − 2 ) ( t ) + λ 0 e ( n − 2 ) ( t ) = ∫ 0 1 H ( t , s ) f ( s , u 0 ( s ) , u 0 ′ ( s ) , … , u 0 ( n − 2 ) ( s ) ) d s + λ 0 ≥ ∫ δ 1 − δ H ( t , s ) f ( s , u 0 ( s ) , u 0 ′ ( s ) , … , u 0 ( n − 2 ) ( s ) ) d s + λ 0 ≥ δ M ρ ∫ δ 1 − δ H ( t , s ) d s + λ 0 ≥ δ ρ + λ 0 > δ ρ .$

This implies that $min t ∈ [ δ , 1 − δ ] u 0 ( n − 2 ) (t)>δρ$, and so by the point (c) in Lemma 2.5, this is a contradiction. It follows from the point (2) of Lemma 1.1 that $i K (T, K ρ )=0$. □

3 Main results

In the following, we shall give the main results on the existence of multiple positive solutions of BVP (1.1) and (1.2).

Theorem 3.1 Suppose that ($H 1$) and ($H 2$) are satisfied. In addition, assume that one of the following conditions holds.

($H 3$) There exist $ρ 1 , ρ 2 , ρ 3 ∈(0,+∞)$ with $ρ 1 <δ ρ 2$ and $ρ 2 < ρ 3$ such that

$f 0 , ρ 1 ≤m, f δ ρ 2 , ρ 2 ≥δM,u≠Tuforu∈∂ Ω ρ 2 and f 0 , ρ 3 ≤m.$

($H 4$) There exist $ρ 1 , ρ 2 , ρ 3 ∈(0,+∞)$ with $ρ 1 < ρ 2 <δ ρ 3$ such that

$f δ ρ 1 , ρ 1 ≥δM, f 0 , ρ 2 ≤m,u≠Tuforu∈∂ K ρ 2 and f δ ρ 3 , ρ 3 ≥δM.$

Then BVP (1.1) and (1.2) has two nondecreasing positive solutions $u 1$, $u 2$ in K. Moreover, if in ($H 3$) $f 0 , ρ 1 ≤m$ is replaced by $f 0 , ρ 1 , then BVP (1.1) and (1.2) has a third nondecreasing positive solution $u 3 ∈ K ρ 1$.

Proof Assume that ($H 3$) holds. We show that either T has a fixed point $u 1 ∈∂ K ρ 1$ or in $Ω ρ 2 ∖ K ¯ ρ 1$. If $u≠Tu$ for $u∈∂ K ρ 1 ∪∂ K ρ 3$, by Lemmas 2.6 and 2.7, we have $i K (T, K ρ 1 )=1$, $i K (T, Ω ρ 2 )=0$, and $i K (T, K ρ 3 )=1$. By Lemma 2.5(b), we have $K ¯ ρ 1 ⊂ K δ ρ 2 ⊂ Ω ρ 2$ since $ρ 1 <δ ρ 2$. It follows from Lemma 1.1(3) that T has a fixed point $u 1 ∈ Ω ρ 2 ∖ K ¯ ρ 1$. Similarly, T has a fixed point $u 2 ∈ K ρ 3 ∖ Ω ¯ ρ 2$. The proof is similar when ($H 4$) holds. □

Corollary 3.1 If there exists $ρ>0$ such that one of the following conditions holds:

($H 5$) $0≤ f 0 , $f δ ρ , ρ ≥δM$, $u≠Tu$ for $u∈∂ Ω ρ$, $0≤ f ∞ < m n − 1$,

($H 6$) $M< f 0 ≤∞$, $f 0 , ρ ≤m$, $u≠Tu$ for $u∈∂ K ρ$, $M< f ∞ ≤∞$,

then BVP (1.1) and (1.2) has at least two nondecreasing positive solutions in K.

Proof We show that ($H 5$) implies ($H 3$). It is easy to verify that $0≤ f 0 implies that there exists $ρ 1 ∈(0,δρ)$ such that $f 0 , ρ 1 . Let $k∈( f ∞ , m n − 1 )$, by $f ∞ < m n − 1$, there exists $r>ρ$ such that $f(t, u 0 , u 1 ,…, u n − 2 )≤k( u 0 + u 1 +⋯+ u n − 2 )$ for $t∈[0,1]$, $u 0 + u 1 +⋯+ u n − 2 ∈[r,+∞)$. Let

$M ′ = max { max t ∈ [ 0 , 1 ] f ( t , u 0 , u 1 , … , u n − 2 ) : u 0 + u 1 + ⋯ + u n − 2 ∈ [ 0 , r ] } , ρ 3 > max { M ′ m − ( n − 1 ) k , r } ,$

then for $( u 0 , u 1 ,…, u n − 2 )∈ [ 0 , ρ 3 ] n − 1$, we have

$max t ∈ [ 0 , 1 ] f(t, u 0 , u 1 ,…, u n − 2 )≤ M ′ +k( u 0 + u 1 +⋯+ u n − 2 )≤ M ′ +k(n−1) ρ 3

This implies that $f 0 , ρ 3 ≤m$ and ($H 3$) holds. Similarly, ($H 6$) implies ($H 4$). This completes the proof. □

Remark 3.1 We establish the multiplicity of monotone positive solutions for a higher order differential equation with integral boundary conditions, and we allow the nonlinearity f to contain derivatives of the unknown function $u(t)$ up to $n−2$ order, so our work improves and generalizes the results of  to some degree.

References

1. 1.

Eloe PW, Ahmad B: Positive solutions of a nonlinear n th order boundary value problem with nonlocal conditions. Appl. Math. Lett. 2005, 18: 521-527. 10.1016/j.aml.2004.05.009

2. 2.

Graef JR, Yang B: Positive solutions to a multi-point higher order boundary value problem. J. Math. Anal. Appl. 2006, 316: 409-421. 10.1016/j.jmaa.2005.04.049

3. 3.

Graef JR, Henderson J, Wong PJY, Yang B: Three solutions of an n th order three-point focal type boundary value problem. Nonlinear Anal. 2008, 69: 3386-3404. 10.1016/j.na.2007.09.024

4. 4.

Hao X, Liu L, Wu Y: Positive solutions for second order differential systems with nonlocal conditions. Fixed Point Theory 2012, 13: 507-516.

5. 5.

Hao X, Liu L, Wu Y: On positive solutions of m -point nonhomogeneous singular boundary value problem. Nonlinear Anal. 2010, 73: 2532-2540. 10.1016/j.na.2010.06.028

6. 6.

Henderson J, Luca R: On a system of second-order multi-point boundary value problems. Appl. Math. Lett. 2012, 25: 2089-2094. 10.1016/j.aml.2012.05.005

7. 7.

Karaca IY: Positive solutions of an n th order multi-point boundary value problem. J. Comput. Anal. Appl. 2012, 14: 181-193.

8. 8.

Ma R: Existence of positive solutions for superlinear semipositone m -point boundary-value problems. Proc. Edinb. Math. Soc. 2003, 46: 279-292. 10.1017/S0013091502000391

9. 9.

Hao X, Xu N, Liu L: Existence and uniqueness of positive solutions for fourth-order m -point boundary value problems with two parameters. Rocky Mt. J. Math. 2013, 43: 1161-1180. 10.1216/RMJ-2013-43-4-1161

10. 10.

Zhang X: Eigenvalue of higher-order semipositone multi-point boundary value problems with derivatives. Appl. Math. Comput. 2008, 201: 361-370. 10.1016/j.amc.2007.12.031

11. 11.

Zhang X, Liu L: A necessary and sufficient condition of positive solutions for nonlinear singular differential systems with four-point boundary conditions. Appl. Math. Comput. 2010, 215: 3501-3508. 10.1016/j.amc.2009.10.044

12. 12.

Zhang X, Liu L: Positive solutions of four-order multi-point boundary value problems with bending term. Appl. Math. Comput. 2007, 194: 321-332. 10.1016/j.amc.2007.04.028

13. 13.

Gallardo JM: Second order differential operators with integral boundary conditions and generation of semigroups. Rocky Mt. J. Math. 2000, 30: 1265-1292. 10.1216/rmjm/1021477351

14. 14.

Karakostas GL, Tsamatos PC: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems. Electron. J. Differ. Equ. 2002, 2002: 1-17.

15. 15.

Lomtatidze A, Malaguti L: On an nonlocal boundary-value problems for second order nonlinear singular differential equations. Georgian Math. J. 2000, 7: 133-154.

16. 16.

Boucherif A: Second-order boundary value problems with integral boundary conditions. Nonlinear Anal. 2009, 70: 364-371. 10.1016/j.na.2007.12.007

17. 17.

Feng M, Ji D, Ge W: Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces. J. Comput. Appl. Math. 2008, 222: 351-363. 10.1016/j.cam.2007.11.003

18. 18.

Hao X, Liu L, Wu Y, Sun Q: Positive solutions for nonlinear n th-order singular eigenvalue problem with nonlocal conditions. Nonlinear Anal. 2010, 73: 1653-1662. 10.1016/j.na.2010.04.074

19. 19.

Hao X, Liu L, Wu Y, Xu N: Multiple positive solutions for singular n th-order nonlocal boundary value problem in Banach spaces. Comput. Math. Appl. 2011, 61: 1880-1890. 10.1016/j.camwa.2011.02.017

20. 20.

Infante G, Webb JRL: Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations. Proc. Edinb. Math. Soc. 2006, 49: 637-656. 10.1017/S0013091505000532

21. 21.

Jiang J, Liu L, Wu Y: Second-order nonlinear singular Sturm-Liouville problems with integral boundary conditions. Appl. Math. Comput. 2009, 215: 1573-1582. 10.1016/j.amc.2009.07.024

22. 22.

Kang P, Wei Z: Three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations. Nonlinear Anal. 2008, 70: 444-451.

23. 23.

Kong L: Second order singular boundary value problems with integral boundary conditions. Nonlinear Anal. 2010, 72: 2628-2638. 10.1016/j.na.2009.11.010

24. 24.

Ma H: Symmetric positive solutions for nonlocal boundary value problems of fourth order. Nonlinear Anal. 2008, 68: 645-651. 10.1016/j.na.2006.11.026

25. 25.

Webb JRL: Nonlocal conjugate type boundary value problems of higher order. Nonlinear Anal. 2009, 71: 1933-1940. 10.1016/j.na.2009.01.033

26. 26.

Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems: a unified approach. J. Lond. Math. Soc. 2006, 74: 673-693. 10.1112/S0024610706023179

27. 27.

Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems involving integral conditions. Nonlinear Differ. Equ. Appl. 2008, 15: 45-67. 10.1007/s00030-007-4067-7

28. 28.

Liu L, Hao X, Wu Y: Positive solutions for singular second order differential equations with integral boundary conditions. Math. Comput. Model. 2013, 57: 836-847. 10.1016/j.mcm.2012.09.012

29. 29.

Yang Z: Existence and nonexistence results for positive solutions of an integral boundary value problem. Nonlinear Anal. 2006, 65: 1489-1511. 10.1016/j.na.2005.10.025

30. 30.

Yang Z: Existence of nontrivial solutions for a nonlinear Sturm-Liouville problem with integral boundary conditions. Nonlinear Anal. 2008, 68: 216-225. 10.1016/j.na.2006.10.044

31. 31.

Zhang X, Feng M, Ge W: Symmetric positive solutions for p -Laplacian fourth-order differential equations with integral boundary conditions. J. Comput. Appl. Math. 2008, 222: 561-573. 10.1016/j.cam.2007.12.002

32. 32.

Zhang X, Han Y: Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations. Appl. Math. Lett. 2012, 25: 555-560. 10.1016/j.aml.2011.09.058

33. 33.

Lan KQ: Multiple positive solutions of semilinear differential equations with singularities. J. Lond. Math. Soc. 2001, 63: 690-704. 10.1112/S002461070100206X

34. 34.

Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, New York; 1988.

Acknowledgements

Research supported by the National Natural Science Foundation of China (11371221, 11201260), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20123705120004, 20123705110001), a Project of Shandong Province Higher Educational Science and Technology Program (J11LA06) and Foundation of Qufu Normal University (BSQD20100103).

Author information

Correspondence to Xinan Hao.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

XH wrote the first manuscript and LL corrected and improved the final version. Both authors 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 