Skip to main content

Triple solutions of complementary Lidstone boundary value problems via fixed point theorems

Abstract

We consider the following complementary Lidstone boundary value problem:

( − 1 ) m y ( 2 m + 1 ) ( t ) = F ( t , y ( t ) , y ′ ( t ) ) , t ∈ [ 0 , 1 ] , y ( 0 ) = 0 , y ( 2 k − 1 ) ( 0 ) = y ( 2 k − 1 ) ( 1 ) = 0 , 1 ≤ k ≤ m .

By using fixed point theorems of Leggett-Williams and Avery, we offer several criteria for the existence of three positive solutions of the boundary value problem. Examples are also included to illustrate the results obtained. We note that the nonlinear term F depends on y ′ and this derivative dependence is seldom investigated in the literature and a new technique is required to tackle the problem.

MSC:34B15, 34B18.

1 Introduction

In this paper we shall consider the complementary Lidstone boundary value problem

( − 1 ) m y ( 2 m + 1 ) ( t ) = F ( t , y ( t ) , y ′ ( t ) ) , t ∈ [ 0 , 1 ] , y ( 0 ) = 0 , y ( 2 k − 1 ) ( 0 ) = y ( 2 k − 1 ) ( 1 ) = 0 , 1 ≤ k ≤ m ,
(1.1)

where m≥1 and F is continuous at least in the interior of the domain of interest. It is noted that the nonlinear term F involves y ′ , a derivative of the dependent variable. Most research papers on boundary value problems consider nonlinear terms that involve y only, and derivative-dependent nonlinearities are seldom tackled as special techniques are required.

The complementary Lidstone interpolation and boundary value problems have been very recently introduced in [1], and drawn on by Agarwal et al. in [2, 3] where they consider an (2m+1)th order differential equation together with boundary data at the odd order derivatives

y(0)= a 0 , y ( 2 k − 1 ) (0)= a k , y ( 2 k − 1 ) (1)= b k ,1≤k≤m.
(1.2)

The boundary conditions (1.2) are known as complementary Lidstone boundary conditions, they naturally complement the Lidstone boundary conditions [4–7] which involve even order derivatives. To be precise, the Lidstone boundary value problem comprises an 2m th order differential equation and the Lidstone boundary conditions

y ( 2 k ) (0)= a k , y ( 2 k ) (1)= b k ,0≤k≤m−1.
(1.3)

There is a vast literature on Lidstone interpolation and boundary value problems. In fact, the Lidstone interpolation was first introduced by Lidstone [8] in 1929 and further characterized in the work of [9–16]. More recent research on Lidstone interpolation as well as Lidstone spline can be found in [1, 17–23]. Meanwhile, the Lidstone boundary value problems and several of its particular cases have been the subject matter of numerous investigations, see [4, 18, 24–37] and the references cited therein. In most of these works the nonlinear terms considered do not involve derivatives of the dependent variable, only a handful of papers [30, 31, 34, 35] tackle nonlinear terms that involve even order derivatives. In the present work, our study of the complementary Lidstone boundary value problem (1.1) where F depends on a derivative certainly extends and complements the rich literature on boundary value problems and notably on Lidstone boundary value problems. The literature on complementary Lidstone boundary value problems pales in comparison with that on Lidstone boundary value problems - after the first work [2] on complementary Lidstone boundary value problems, the recent paper [38] discusses the eigenvalue problem, while in [39] the existence of at least one or two positive solutions of the complementary Lidstone boundary value problem is derived by Leray-Schauder alternative and Krasnosel’skii’s fixed point theorem in a cone.

In the present work, we shall establish the existence of at least three positive solutions using fixed point theorems of Leggett and Williams [40] as well as of Avery [41]. Estimates on the norms of these solutions will also be provided. Besides achieving new results, we also compare the results in terms of generality and illustrate the importance of the results through some examples. As remarked earlier, the presence of the derivative y ′ in the nonlinear term F requires a special technique to tackle the problem.

The paper is organized as follows. Section 2 contains the necessary definitions and fixed point theorems. The existence criteria are developed and discussed in Section 3. Finally, examples are presented in Section 4 to illustrate the importance of the results obtained.

2 Preliminaries

In this section we shall state some necessary definitions, the relevant fixed point theorems and properties of certain Green’s function. Let B be a Banach space equipped with the norm ∥⋅∥.

Definition 2.1 Let C (⊂B) be a nonempty closed convex set. We say that C is a cone provided the following conditions are satisfied:

  1. (a)

    If x∈C and α≥0, then αx∈C;

  2. (b)

    If x∈C and −x∈C, then x=0.

Definition 2.2 Let C (⊂B) be a cone. A map ψ is a nonnegative continuous concave functional on C if the following conditions are satisfied:

  1. (a)

    ψ:C→[0,∞) is continuous;

  2. (b)

    ψ(ty+(1−t)z)≥tψ(y)+(1−t)ψ(z) for all y,z∈C and 0≤t≤1.

Definition 2.3 Let C (⊂B) be a cone. A map β is a nonnegative continuous convex functional on C if the following conditions are satisfied:

  1. (a)

    β:C→[0,∞) is continuous;

  2. (b)

    β(ty+(1−t)z)≤tβ(y)+(1−t)β(z) for all y,z∈C and 0≤t≤1.

Let γ, β, Θ be nonnegative continuous convex functionals on C and α, ψ be nonnegative continuous concave functionals on C. For nonnegative numbers w i , 1≤i≤3, we shall introduce the following notations:

C ( w 1 ) = { x ∈ C ∣ ∥ x ∥ < w 1 } , C ( ψ , w 1 , w 2 ) = { x ∈ C ∣ ψ ( x ) ≥ w 1  and  ∥ x ∥ ≤ w 2 } , P ( γ , w 1 ) = { x ∈ C ∣ γ ( x ) < w 1 } , P ( γ , α , w 1 , w 2 ) = { x ∈ C ∣ α ( x ) ≥ w 1  and  γ ( x ) ≤ w 2 } , Q ( γ , β , w 1 , w 2 ) = { x ∈ C ∣ β ( x ) ≤ w 1  and  γ ( x ) ≤ w 2 } , P ( γ , Θ , α , w 1 , w 2 , w 3 ) = { x ∈ C ∣ α ( x ) ≥ w 1 , Θ ( x ) ≤ w 2  and  γ ( x ) ≤ w 3 } , Q ( γ , β , ψ , w 1 , w 2 , w 3 ) = { x ∈ C ∣ ψ ( x ) ≥ w 1 , β ( x ) ≤ w 2  and  γ ( x ) ≤ w 3 } .

The following fixed point theorems are our main tools, the first is usually called Leggett-Williams’ fixed point theorem, and the second is known as the five-functional fixed point theorem.

Theorem 2.1 [40]

Let C (⊂B) be a cone, and w 4 >0 be given. Assume that ψ is a nonnegative continuous concave functional on C such that ψ(x)≤∥x∥ for all x∈ C ¯ ( w 4 ), and let S: C ¯ ( w 4 )→ C ¯ ( w 4 ) be a continuous and completely continuous operator. Suppose that there exist numbers w 1 , w 2 , w 3 , where 0< w 1 < w 2 < w 3 ≤ w 4 , such that

  1. (a)

    {x∈C(ψ, w 2 , w 3 )∣ψ(x)> w 2 }≠∅, and ψ(Sx)> w 2 for all x∈C(ψ, w 2 , w 3 );

  2. (b)

    ∥Sx∥< w 1 for all x∈ C ¯ ( w 1 );

  3. (c)

    ψ(Sx)> w 2 for all x∈C(ψ, w 2 , w 4 ) with ∥Sx∥> w 3 .

Then S has (at least) three fixed points x 1 , x 2 and x 3 in C ¯ ( w 4 ). Furthermore, we have

x 1 ∈ C ( w 1 ) , x 2 ∈ { x ∈ C ( ψ , w 2 , w 4 ) ∣ ψ ( x ) > w 2 } and x 3 ∈ C ¯ ( w 4 ) ∖ ( C ( ψ , w 2 , w 4 ) ∪ C ¯ ( w 1 ) ) .
(2.1)

Theorem 2.2 [41]

Let C (⊂B) be a cone. Assume that there exist positive numbers w 5 , M, nonnegative continuous convex functionals γ, β, Θ on C, and nonnegative continuous concave functionals α, ψ on C, with

α(x)≤β(x)and∥x∥≤Mγ(x)

for all x∈ P ¯ (γ, w 5 ). Let S: P ¯ (γ, w 5 )→ P ¯ (γ, w 5 ) be a continuous and completely continuous operator. Suppose that there exist nonnegative numbers w i , 1≤i≤4, with 0< w 2 < w 3 such that

  1. (a)

    {x∈P(γ,Θ,α, w 3 , w 4 , w 5 )∣α(x)> w 3 }≠∅, and α(Sx)> w 3 for all x∈P(γ,Θ,α, w 3 , w 4 , w 5 );

  2. (b)

    {x∈Q(γ,β,ψ, w 1 , w 2 , w 5 )∣β(x)< w 2 }≠∅, and β(Sx)< w 2 for all x∈Q(γ,β,ψ, w 1 , w 2 , w 5 );

  3. (c)

    α(Sx)> w 3 for all x∈P(γ,α, w 3 , w 5 ) with Θ(Sx)> w 4 ;

  4. (d)

    β(Sx)< w 2 for all x∈Q(γ,β, w 2 , w 5 ) with ψ(Sx)< w 1 .

Then S has (at least) three fixed points x 1 , x 2 and x 3 in P ¯ (γ, w 5 ). Furthermore, we have

β( x 1 )< w 2 ,α( x 2 )> w 3 andβ( x 3 )> w 2 with Î±( x 3 )< w 3 .
(2.2)

We also require the definition of an L 1 -Carathéodory function.

Definition 2.4 [42]

A function P:[0,1]× R 2 →R is an L 1 -Carathéodory function if the following conditions hold:

  1. (a)

    The map t→P(t,u) is measurable for all u∈ R 2 .

  2. (b)

    The map u→P(t,u) is continuous for almost all t∈[0,1].

  3. (c)

    For any r>0, there exists μ r ∈ L 1 [0,1] such that |u|≤r implies that |P(t,u)|≤ μ r (t) for almost all t∈[0,1].

To tackle the complementary Lidstone boundary value problem (1.1), let us review certain attributes of the Lidstone boundary value problem. Let g m (t,s) be the Green’s function of the Lidstone boundary value problem

x ( 2 m ) ( t ) = 0 , t ∈ [ 0 , 1 ] , x ( 2 k ) ( 0 ) = x ( 2 k ) ( 1 ) = 0 , 0 ≤ k ≤ m − 1 .
(2.3)

The Green’s function g m (t,s) can be expressed as [4, 5]

g m (t,s)= ∫ 0 1 g 1 (t,u) g m − 1 (u,s)du,m≥2,
(2.4)

where

g 1 (t,s)= { t ( s − 1 ) , 0 ≤ t ≤ s ≤ 1 , s ( t − 1 ) , 0 ≤ s ≤ t ≤ 1 .

Further, it is known that

| g m (t,s)|= ( − 1 ) m g m (t,s)and g m (t,s)= g m (s,t),(t,s)∈(0,1)×(0,1).
(2.5)

The following two lemmas give the upper and lower bounds of | g m (t,s)|, they play an important role in subsequent development. We remark that the bounds in the two lemmas are sharper than those given in the literature [4, 5, 35, 37].

Lemma 2.1 [38]

For (t,s)∈[0,1]×[0,1], we have

| g m (t,s)|≤ 1 π 2 m − 1 sinπs.

Lemma 2.2 [38]

Let δ∈(0, 1 2 ) be given. For (t,s)∈[δ,1−δ]×[0,1], we have

| g m (t,s)|≥ 2 δ π 2 m sinπs.

3 Triple positive solutions

In this section, we shall use the fixed point theorems stated in Section 2 to obtain the existence of at least three positive solutions of the complementary Lidstone boundary value problem (1.1). By a positive solution y of (1.1), we mean a nontrivial y∈C[0,1] satisfying (1.1) and y(t)≥0 for t∈[0,1].

To tackle (1.1), we first consider the initial value problem

y ′ ( t ) = x ( t ) , t ∈ [ 0 , 1 ] , y ( 0 ) = 0
(3.1)

whose solution is simply

y(t)= ∫ 0 t x(s)ds.
(3.2)

Taking into account (3.1) and (3.2), the complementary Lidstone boundary value problem (1.1) reduces to the Lidstone boundary value problem

( − 1 ) m x ( 2 m ) ( t ) = F ( t , ∫ 0 t x ( s ) d s , x ( t ) ) , t ∈ [ 0 , 1 ] , x ( 2 k − 2 ) ( 0 ) = x ( 2 k − 2 ) ( 1 ) = 0 , 1 ≤ k ≤ m .
(3.3)

If (3.3) has a solution x ∗ , then by virtue of (3.2), the boundary value problem (1.1) has a solution given by

y ∗ (t)= ∫ 0 t x ∗ (s)ds.
(3.4)

So the existence of a solution of the complementary Lidstone boundary value problem (1.1) follows from the existence of a solution of the Lidstone boundary value problem (3.3). It is clear from (3.4) that ∥ y ∗ ∥≤∥ x ∗ ∥; moreover if x ∗ is positive, so is y ∗ . With the tools in Section 2 and a technique to handle the nonlinear term F, we shall study the boundary value problem (1.1) via (3.3).

Let the Banach space B=C[0,1] be equipped with the norm ∥x∥= sup t ∈ [ 0 , 1 ] |x(t)| for x∈B. Define the operator S:C[0,1]→C[0,1] by

S x ( t ) = ∫ 0 1 ( − 1 ) m g m ( t , s ) F ( s , ∫ 0 s x ( τ ) d τ , x ( s ) ) d s = ∫ 0 1 | g m ( t , s ) | F ( s , ∫ 0 s x ( τ ) d τ , x ( s ) ) d s , t ∈ [ 0 , 1 ] ,
(3.5)

where g m (t,s) is the Green’s function given in (2.4). A fixed point x ∗ of the operator S is clearly a solution of the boundary value problem (3.3), and as seen earlier y ∗ (t)= ∫ 0 t x ∗ (s)ds is a solution of (1.1).

For easy reference, we shall list the conditions that are needed later. In these conditions the sets K and K ˜ are defined by

K Ëœ = { x ∈ B ∣ x ( t ) ≥ 0 , t ∈ [ 0 , 1 ] } , K = { x ∈ K Ëœ ∣ x ( t ) > 0  on some subset of  [ 0 , 1 ]  of positive measure } .
(3.6)

(C1) F:[0,1]× R 2 →R is an L 1 -Carathéodory function.

(C2) We have

F ( t , u , v ) ≥ 0 , u , v ∈ K Ëœ , a.e.  t ∈ ( 0 , 1 ) and F ( t , u , v ) > 0 , u , v ∈ K , a.e.  t ∈ ( 0 , 1 ) .

(C3) There exist continuous functions f, ν, μ with f:[0,∞)×[0,∞)→[0,∞) and ν,μ:(0,1)→[0,∞) such that

μ(t)f(u,v)≤F(t,u,v)≤ν(t)f(u,v),u,v∈ K Ëœ ,a.e. t∈(0,1).

(C4) There exists a number 0<θ≤1 such that

μ(t)≥θν(t),a.e. t∈(0,1).

If (C2) and (C3) hold, then it follows from (3.5) that for x∈ K ˜ and t∈[0,1],

0 ≤ ∫ 0 1 | g m ( t , s ) | μ ( s ) f ( ∫ 0 s x ( τ ) d τ , x ( s ) ) d s ≤ S x ( t ) ≤ ∫ 0 1 | g m ( t , s ) | ν ( s ) f ( ∫ 0 s x ( τ ) d τ , x ( s ) ) d s .
(3.7)

Let δ∈(0, 1 2 ) be fixed. We define a cone C in B as

C= { x ∈ B | x ( t ) ≥ 0  for  t ∈ [ 0 , 1 ] , and  min t ∈ [ δ , 1 − δ ] x ( t ) ≥ 2 δ θ Ï€ ∥ x ∥ } ,
(3.8)

where θ is given in (C4). Clearly, we have C⊆ K ˜ .

Lemma 3.1 Let (C1)-(C4) hold. Then the operator S defined in (3.5) is continuous and completely continuous, and S maps C into C.

Proof From (2.4) we have g m (t,s)∈C[0,1]⊆ L ∞ [0,1], t∈[0,1] and the map t→ g m (t,s) is continuous from [0,1] to C[0,1]. This together with F:[0,1]× R 2 →R is an L 1 -Carathéodory function ensures (as in [[42], Theorem 4.2.2]) that S is continuous and completely continuous.

Let x∈C. From (3.7) we have Sx(t)≥0 for t∈[0,1]. Next, using (3.7) and Lemma 2.1 gives for t∈[0,1],

S x ( t ) ≤ ∫ 0 1 | g m ( t , s ) | ν ( s ) f ( ∫ 0 s x ( τ ) d τ , x ( s ) ) d s ≤ 1 π 2 m − 1 ∫ 0 1 ν ( s ) f ( ∫ 0 s x ( τ ) d τ , x ( s ) ) sin π s d s .
(3.9)

Hence, we have

∥Sx∥≤ 1 π 2 m − 1 ∫ 0 1 ν(s)f ( ∫ 0 s x ( τ ) d τ , x ( s ) ) sinπsds.
(3.10)

Now, employing (3.7), Lemma 2.2, (C4) and (3.10), we find for t∈[δ,1−δ],

S x ( t ) ≥ ∫ 0 1 | g m ( t , s ) | μ ( s ) f ( ∫ 0 s x ( τ ) d τ , x ( s ) ) d s ≥ 2 δ π 2 m ∫ 0 1 μ ( s ) f ( ∫ 0 s x ( τ ) d τ , x ( s ) ) sin π s d s ≥ 2 δ π 2 m ∫ 0 1 θ ν ( s ) f ( ∫ 0 s x ( τ ) d τ , x ( s ) ) sin π s d s ≥ 2 δ θ π ∥ S x ∥ .

This leads to

min t ∈ [ δ , 1 − δ ] Sx(t)≥ 2 δ θ π ∥Sx∥.

We have shown that Sx∈C. □

For subsequent results, we define the following constants for fixed δ∈(0, 1 2 ) and τ 1 , τ 2 , τ 3 , τ 4 ∈[0,1]:

{ q = 1 π 2 m − 1 ∫ 0 1 ν ( s ) sin π s d s , r = min t ∈ [ δ , 1 − δ ] ∫ 1 2 1 − δ | g m ( t , s ) | μ ( s ) d s , p 1 = min t ∈ [ τ 2 , τ 3 ] ∫ 1 2 τ 3 | g m ( t , s ) | μ ( s ) d s , p 2 = 1 π 2 m − 1 ∫ τ 1 τ 4 ν ( s ) sin π s d s , p 3 = 1 π 2 m − 1 ∫ 0 τ 1 ν ( s ) sin π s d s + 1 π 2 m − 1 ∫ τ 4 1 ν ( s ) sin π s d s .
(3.11)

Lemma 3.2 Let (C1)-(C4) hold, and assume

(C5) the function ν(s)sinπs>0 on a subset of [0,1] of positive measure.

Suppose that there exists a number d>0 such that for u,v∈[0,d],

f(u,v)< d q .
(3.12)

Then

S ( C ¯ ( d ) ) ⊆C(d)⊂ C ¯ (d).
(3.13)

Proof Let x∈ C ¯ (d). So ∥x∥≤d, which implies immediately that

∫ 0 s x(τ)dτ≤dandx(s)≤d,s∈[0,1].

Then, using (3.9), (C5) and (3.12), we find for t∈[0,1],

| S x ( t ) | ≤ 1 π 2 m − 1 ∫ 0 1 ν ( s ) f ( ∫ 0 s x ( τ ) d τ , x ( s ) ) sin π s d s < 1 π 2 m − 1 ∫ 0 1 ν ( s ) sin π s d s ⋅ d q = q ⋅ d q = d .

This implies ∥Sx∥<d. Together with the fact that Sx∈C (Lemma 3.1), we have shown that Sx∈C(d). Conclusion (3.13) is now immediate. □

Using a similar argument as Lemma 3.2, we have the following lemma.

Lemma 3.3 Let (C1)-(C4) hold. Suppose that there exists a number d>0 such that for u,v∈[0,d],

f(u,v)≤ d q .

Then

S ( C ¯ ( d ) ) ⊆ C ¯ (d).

We are now ready to establish the existence of three positive solutions for the complementary Lidstone boundary value problem (1.1). The first result below uses Leggett-Williams’ fixed point theorem (Theorem 2.1).

Theorem 3.1 Let δ∈(0, 1 2 ) be fixed. Let (C1)-(C5) hold, and assume

(C6) for each t∈[δ,1−δ], the function | g m (t,s)|μ(s)>0 on a subset of [ 1 2 ,1−δ] of positive measure.

Suppose that there exist numbers w 1 , w 2 , w 3 with

0< w 1 < w 2 ( 1 2 − δ ) < w 2 < π w 2 2 δ θ ≤ w 3

such that the following hold:

  1. (P)

    f(u,v)< w 1 q for u,v∈[0, w 1 ];

  2. (Q)

    one of the following holds:

    1. (Q1)

      lim sup u → ∞ , v → ∞ f ( u , v ) u < 1 q or lim sup u → ∞ , v → ∞ f ( u , v ) v < 1 q ;

    2. (Q2)

      there exists a number d (≥ w 3 ) such that f(u,v)≤ d q for u,v∈[0,d];

  3. (R)

    f(u,v)> w 2 r for u∈[ w 2 ( 1 2 −δ), w 3 ( 1 2 −δ)] and v∈[ w 2 , w 3 ].

Then we have the following conclusions:

  1. (a)

    The Lidstone boundary value problem (3.3) has (at least) three positive solutions x 1 , x 2 , x 3 ∈C (where C is defined in (3.8)) such that

    { ∥ x 1 ∥ < w 1 ; x 2 ( t ) > w 2 , t ∈ [ δ , 1 − δ ] ; ∥ x 3 ∥ > w 1 and min t ∈ [ δ , 1 − δ ] x 3 ( t ) < w 2 .
    (3.14)
  2. (b)

    The complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions y 1 , y 2 , y 3 such that for i=1,2,3,

    { y i ( t ) = ∫ 0 t x i ( s ) d s , t ∈ [ 0 , 1 ] ; ∥ y i ∥ ≤ ∥ x i ∥ ; y i ( t ) ≥ 2 δ θ π ∥ x i ∥ ( t − δ ) , t ∈ [ δ , 1 − δ ]
    (3.15)

(where x i ’s are those in conclusion (a)). We further have

{ ∥ y 1 ∥ < w 1 ; y 2 ( t ) > w 2 ( t − δ ) , t ∈ [ δ , 1 − δ ] ; y 3 ( t ) > 2 δ θ π w 1 ( t − δ ) , t ∈ [ δ , 1 − δ ] .
(3.16)

Proof We shall employ Theorem 2.1 with the cone C defined in (3.8). First, we shall prove that condition (Q) implies the existence of a number w 4 , where w 4 ≥ w 3 , such that

S ( C ¯ ( w 4 ) ) ⊆ C ¯ ( w 4 ).
(3.17)

Suppose that (Q2) holds. Then by Lemma 3.3 we immediately have (3.17) where we pick w 4 =d. Suppose now that lim sup u → ∞ , v → ∞ f ( u , v ) u < 1 q of (Q1) is satisfied. Then there exist N>0 and ϵ< 1 q such that

f ( u , v ) u <ϵ,u,v>N.
(3.18)

Let

L= max u , v ∈ [ 0 , N ] f(u,v).

Noting (3.18), it is then clear that for u,v≥0,

f(u,v)≤L+ϵu.
(3.19)

Now, pick the number w 4 so that

w 4 >max { w 3 , L ( 1 q − ϵ ) − 1 } .
(3.20)

Let x∈ C ¯ ( w 4 ). Using (3.9), (3.19) and (3.20) yields for t∈[0,1],

| S x ( t ) | ≤ 1 π 2 m − 1 ∫ 0 1 ν ( s ) f ( ∫ 0 s x ( τ ) d τ , x ( s ) ) sin π s d s ≤ 1 π 2 m − 1 ∫ 0 1 ν ( s ) ( L + ϵ ∫ 0 s x ( τ ) d τ ) sin π s d s ≤ 1 π 2 m − 1 ∫ 0 1 ν ( s ) ( L + ϵ w 4 ) sin π s d s = q ( L + ϵ w 4 ) < q [ w 4 ( 1 q − ϵ ) + ϵ w 4 ] = w 4 .

Hence, ∥Sx∥< w 4 and so Sx∈C( w 4 )⊂ C ¯ ( w 4 ). Thus, (3.17) follows immediately. Note that the argument is similar if we assume that lim sup u → ∞ , v → ∞ f ( u , v ) v < 1 q of (Q1) is satisfied.

Let ψ:C→[0,∞) be defined by

ψ(x)= min t ∈ [ δ , 1 − δ ] x(t).

Clearly, ψ is a nonnegative continuous concave functional on C and ψ(x)≤∥x∥ for all x∈C.

We shall verify that condition (a) of Theorem 2.1 is satisfied. It is obvious that

x(t)= w 2 + w 3 2 ∈ { x ∈ C ( ψ , w 2 , w 3 ) ∣ ψ ( x ) > w 2 }

and so {x∈C(ψ, w 2 , w 3 )∣ψ(x)> w 2 }≠∅. Next, let x∈C(ψ, w 2 , w 3 ). Then ψ(x)≥ w 2 and ∥x∥≤ w 3 which imply

x(s)∈[ w 2 , w 3 ],s∈[δ,1−δ]and ∫ δ 1 2 x(τ)dτ∈ [ w 2 ( 1 2 − δ ) , w 3 ( 1 2 − δ ) ] .
(3.21)

Using (3.7), (3.21), (C6) and (R), it follows that

ψ ( S x ) = min t ∈ [ δ , 1 − δ ] S x ( t ) ≥ min t ∈ [ δ , 1 − δ ] ∫ 0 1 | g m ( t , s ) | μ ( s ) f ( ∫ 0 s x ( τ ) d τ , x ( s ) ) d s ≥ min t ∈ [ δ , 1 − δ ] ∫ 1 2 1 − δ | g m ( t , s ) | μ ( s ) f ( ∫ δ 1 2 x ( τ ) d τ , x ( s ) ) d s > min t ∈ [ δ , 1 − δ ] ∫ 1 2 1 − δ | g m ( t , s ) | μ ( s ) d s ⋅ w 2 r = r ⋅ w 2 r = w 2 .

Therefore, we have shown that ψ(Sx)> w 2 for all x∈C(ψ, w 2 , w 3 ).

Next, by condition (P) and Lemma 3.2 (with d= w 1 ), we have S( C ¯ ( w 1 ))⊆C( w 1 ). Hence, condition (b) of Theorem 2.1 is fulfilled.

Finally, we shall show that condition (c) of Theorem 2.1 holds. Let x∈C(ψ, w 2 , w 4 ) with ∥Sx∥> w 3 . Using (3.7), Lemma 2.2, (C4), (3.10) and the inequality Ï€ w 2 2 δ θ ≤ w 3 , we find

ψ ( S x ) ≥ min t ∈ [ δ , 1 − δ ] ∫ 0 1 | g m ( t , s ) | μ ( s ) f ( ∫ 0 s x ( τ ) d τ , x ( s ) ) d s ≥ 2 δ π 2 m ∫ 0 1 μ ( s ) f ( ∫ 0 s x ( τ ) d τ , x ( s ) ) sin π s d s ≥ 2 δ π 2 m ∫ 0 1 θ ν ( s ) f ( ∫ 0 s x ( τ ) d τ , x ( s ) ) sin π s d s ≥ 2 δ θ π ∥ S x ∥ > 2 δ θ π w 3 ≥ w 2 .

Hence, we have proved that ψ(Sx)> w 2 for all x∈C(ψ, w 2 , w 4 ) with ∥Sx∥> w 3 .

It now follows from Theorem 2.1 that the Lidstone boundary value problem (3.3) has (at least) three positive solutions x 1 , x 2 , x 3 ∈ C ¯ ( w 4 ) satisfying (2.1). It is easy to see that here (2.1) reduces to (3.14). This completes the proof of conclusion (a).

Finally, it is observed from (3.4) that the complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions y 1 , y 2 , y 3 such that for i=1,2,3,

y i (t)= ∫ 0 t x i (s)ds,t∈[0,1]and∥ y i ∥≤∥ x i ∥.
(3.22)

Moreover, since x i ∈C, we get for t∈[δ,1−δ],

y i (t)= ∫ 0 t x i (s)ds≥ ∫ δ t x i (s)ds≥ ∫ δ t 2 δ θ π ∥ x i ∥ds= 2 δ θ π ∥ x i ∥(t−δ).
(3.23)

Combining (3.22) and (3.23) gives (3.15) immediately.

Further, since x 2 (t)> w 2 for t∈[δ,1−δ], we have for t∈[δ,1−δ],

y 2 (t)= ∫ 0 t x 2 (s)ds> ∫ δ t x 2 (s)ds> ∫ δ t w 2 ds= w 2 (t−δ).
(3.24)

Hence, noting (3.14), (3.15) and (3.24), we get (3.16). This completes the proof of conclusion (b). □

We shall now employ the five-functional fixed point theorem (Theorem 2.2) to give other existence criteria. In applying Theorem 2.2 it is possible to choose the functionals and constants in different ways, indeed we shall do so and derive two results. Our first result below turns out to be a generalization of Theorem 3.1.

Theorem 3.2 Let δ∈(0, 1 2 ) be fixed. Let (C1)-(C4) hold. Assume that there exist numbers τ j , 1≤j≤4, with

0≤ τ 1 ≤δ≤ τ 2 < 1 2 < τ 3 ≤1−δ≤ τ 4 ≤1

such that

(C7) for each t∈[ τ 2 , τ 3 ], the function | g m (t,s)|μ(s)>0 on a subset of [ 1 2 , τ 3 ] of positive measure;

(C8) the function ν(s)sinπs>0 on a subset of [ τ 1 , τ 4 ] of positive measure.

Suppose that there exist numbers w i , 2≤i≤5, with

{ 0 < w 2 < w 3 < π w 3 2 δ θ ≤ w 4 ≤ w 5 < q w 2 p 3 , τ 1 w 5 + ( τ 4 − τ 1 ) w 2 < w 3 ( 1 2 − τ 2 )

such that the following hold:

  1. (P)

    f(u,v)< 1 p 2 ( w 2 − w 5 p 3 q ) for u∈[0, τ 1 w 5 +( τ 4 − τ 1 ) w 2 ] and v∈[0, w 2 ];

  2. (Q)

    f(u,v)≤ w 5 q for u,v∈[0, w 5 ];

  3. (R)

    f(u,v)> w 3 p 1 for u∈[ w 3 ( 1 2 − τ 2 ), w 4 ( 1 2 − τ 2 )] and v∈[ w 3 , w 4 ].

Then we have the following conclusions:

  1. (a)

    The Lidstone boundary value problem (3.3) has (at least) three positive solutions x 1 , x 2 , x 3 ∈ C ¯ ( w 5 ) (where C is defined in (3.8)) such that

    { x 1 ( t ) < w 2 , t ∈ [ τ 1 , τ 4 ] ; x 2 ( t ) > w 3 , t ∈ [ τ 2 , τ 3 ] ; max t ∈ [ τ 1 , τ 4 ] x 3 ( t ) > w 2 and min t ∈ [ τ 2 , τ 3 ] x 3 ( t ) < w 3 .
    (3.25)
  2. (b)

    The complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions y 1 , y 2 , y 3 such that (3.15) holds for i=1,2,3. We further have

    { y 1 ( t ) < τ 1 max s ∈ [ 0 , τ 1 ] x 1 ( s ) + ( τ 4 − τ 1 ) w 2 , t ∈ [ τ 1 , τ 4 ] ; y 2 ( t ) > w 3 ( t − τ 2 ) , t ∈ [ τ 2 , τ 3 ] ; y 3 ( t ) > 2 δ θ π w 2 ( t − δ ) , t ∈ [ δ , 1 − δ ] .
    (3.26)

Proof We shall apply Theorem 2.2 with the cone C defined in (3.8). We define the following five functionals on the cone C:

{ γ ( x ) = ∥ x ∥ , ψ ( x ) = min t ∈ [ δ , 1 − δ ] x ( t ) , β ( x ) = Θ ( x ) = max t ∈ [ τ 1 , τ 4 ] x ( t ) , α ( x ) = min t ∈ [ τ 2 , τ 3 ] x ( t ) .
(3.27)

First, we shall show that the operator S maps P ¯ (γ, w 5 ) into P ¯ (γ, w 5 ). Note that P ¯ (γ, w 5 )= C ¯ ( w 5 ). By (Q) and Lemma 3.3 (with d= w 5 ), we immediately have S( C ¯ ( w 5 ))⊆ C ¯ ( w 5 ).

Next, to see that condition (a) of Theorem 2.2 is fulfilled, we note that

{ x ∈ P ( γ , Θ , α , w 3 , w 4 , w 5 ) ∣ α ( x ) > w 3 } ≠∅

since it has an element x(t)= 1 2 ( w 3 + w 4 ). Let x∈P(γ,Θ,α, w 3 , w 4 , w 5 ). Then by definition we have α(x)≥ w 3 and Θ(x)≤ w 4 , which imply

x ( s ) ∈ [ w 3 , w 4 ] , s ∈ [ τ 2 , τ 3 ] and ∫ τ 2 1 2 x ( τ ) d τ ∈ [ w 3 ( 1 2 − τ 2 ) , w 4 ( 1 2 − τ 2 ) ] .
(3.28)

Noting (3.7), (3.28), (C7) and (R), we find

α ( S x ) = min t ∈ [ τ 2 , τ 3 ] S x ( t ) ≥ min t ∈ [ τ 2 , τ 3 ] ∫ 0 1 | g m ( t , s ) | μ ( s ) f ( ∫ 0 s x ( τ ) d τ , x ( s ) ) d s ≥ min t ∈ [ τ 2 , τ 3 ] ∫ 1 2 τ 3 | g m ( t , s ) | μ ( s ) f ( ∫ τ 2 1 2 x ( τ ) d τ , x ( s ) ) d s > min t ∈ [ τ 2 , τ 3 ] ∫ 1 2 τ 3 | g m ( t , s ) | μ ( s ) d s ⋅ w 3 p 1 = p 1 ⋅ w 3 p 1 = w 3 .

Hence, α(Sx)> w 3 for all x∈P(γ,Θ,α, w 3 , w 4 , w 5 ).

We shall now verify that condition (b) of Theorem 2.2 is satisfied. Let w 1 be such that 0< w 1 < w 2 . Note that

{ x ∈ Q ( γ , β , ψ , w 1 , w 2 , w 5 ) ∣ β ( u ) < w 2 } ≠∅

because it has an element x(t)= 1 2 ( w 1 + w 2 ). Let x∈Q(γ,β,ψ, w 1 , w 2 , w 5 ). Then we have β(x)≤ w 2 and γ(x)≤ w 5 , i.e.,

x(s)∈[0, w 2 ],s∈[ τ 1 , τ 4 ]andx(s)∈[0, w 5 ],s∈[0,1],
(3.29)

which lead to the following:

∫ 0 s x ( τ ) d τ ≤ w 5 , s ∈ [ 0 , τ 1 ] ∪ [ τ 4 , 1 ] ; ∫ 0 s x ( τ ) d τ ≤ ∫ 0 τ 4 x ( τ ) d τ = ∫ 0 τ 1 x ( τ ) d τ + ∫ τ 1 τ 4 x ( τ ) d τ ∫ 0 s x ( τ ) d τ ≤ τ 1 w 5 + ( τ 4 − τ 1 ) w 2 , s ∈ [ τ 1 , τ 4 ] .
(3.30)

Using (3.9), (3.29), (3.30), (C8), (P) and (Q) successively, we find

β ( S x ) = max t ∈ [ τ 1 , τ 4 ] S x ( t ) ≤ 1 π 2 m − 1 ∫ 0 1 ν ( s ) f ( ∫ 0 s x ( τ ) d τ , x ( s ) ) sin π s d s = 1 π 2 m − 1 ( ∫ 0 τ 1 + ∫ τ 1 τ 4 + ∫ τ 4 1 ) ν ( s ) f ( ∫ 0 s x ( τ ) d τ , x ( s ) ) sin π s d s < [ 1 π 2 m − 1 ∫ 0 τ 1 ν ( s ) sin π s d s + 1 π 2 m − 1 ∫ τ 4 1 ν ( s ) sin π s d s ] w 5 q + [ 1 π 2 m − 1 ∫ τ 1 τ 4 ν ( s ) sin π s d s ] 1 p 2 ( w 2 − w 5 p 3 q ) = p 3 ⋅ w 5 q + p 2 ⋅ 1 p 2 ( w 2 − w 5 p 3 q ) = w 2 .

Therefore, β(Sx)< w 2 for all x∈Q(γ,β,ψ, w 1 , w 2 , w 5 ).

Next, we shall show that condition (c) of Theorem 2.2 is met. Let x∈C. Clearly, we have

Θ(Sx)= max t ∈ [ τ 1 , τ 4 ] Sx(t)≤∥Sx∥.
(3.31)

Moreover, using the fact that S maps C into C, we find

α(Sx)= min t ∈ [ τ 2 , τ 3 ] Sx(t)≥ min t ∈ [ δ , 1 − δ ] Sx(t)≥ 2 δ θ π ∥Sx∥.
(3.32)

Combining (3.31) and (3.32) yields

α(Sx)≥ 2 δ θ π Θ(Sx),x∈C.
(3.33)

Now, let x∈P(γ,α, w 3 , w 5 ) with Θ(Sx)> w 4 . Then it follows from (3.33) and the inequality π w 3 2 δ θ ≤ w 4 that

α(Sx)≥ 2 δ θ π Θ(Sx)> 2 δ θ π w 4 ≥ 2 δ θ π π w 3 2 δ θ = w 3 .
(3.34)

Thus, α(Sx)> w 3 for all x∈P(γ,α, w 3 , w 5 ) with Θ(Sx)> w 4 .

Finally, we shall prove that condition (d) of Theorem 2.2 is fulfilled. Let x∈Q(γ,β, w 2 , w 5 ) with ψ(Sx)< w 1 . Then we have β(x)≤ w 2 and γ(x)≤ w 5 which give (3.29) and (3.30). As in proving condition (b), we get β(Sx)< w 2 . Hence, condition (d) of Theorem 2.2 is satisfied.

It now follows from Theorem 2.2 that the Lidstone boundary value problem (3.3) has (at least) three positive solutions x 1 , x 2 , x 3 ∈ P ¯ (γ, w 5 )= C ¯ ( w 5 ) satisfying (2.2). Furthermore, (2.2) reduces to (3.25) immediately. This completes the proof of conclusion (a).

Finally, as in the proof of Theorem 3.1, we see that (3.15) holds for the positive solutions y i , i=1,2,3, of the complementary Lidstone boundary value problem (1.1). Moreover, noting that x 1 (t)< w 2 for t∈[ Ï„ 1 , Ï„ 4 ], we find for t∈[ Ï„ 1 , Ï„ 4 ],

y 1 ( t ) = ∫ 0 t x 1 ( s ) d s ≤ ∫ 0 τ 4 x 1 ( s ) d s = ∫ 0 τ 1 x 1 ( s ) d s + ∫ τ 1 τ 4 x 1 ( s ) d s < τ 1 max s ∈ [ 0 , τ 1 ] x 1 ( s ) + ( τ 4 − τ 1 ) w 2 .

Next, noting x 2 (t)> w 3 for t∈[ τ 2 , τ 3 ], we get for t∈[ τ 2 , τ 3 ],

y 2 (t)= ∫ 0 t x 2 (s)ds> ∫ τ 2 t x 2 (s)ds> w 3 (t− τ 2 ).

Lastly, using (3.15) and ∥ x 3 ∥≥ max t ∈ [ τ 1 , τ 4 ] x 3 (t)> w 2 , we find for t∈[δ,1−δ],

y 3 (t)≥ 2 δ θ π ∥ x 3 ∥(t−δ)> 2 δ θ π w 2 (t−δ).

The proof of conclusion (b) is complete. □

We shall now consider the special case of Theorem 3.2 when

τ 1 =0, τ 2 =δ, τ 3 =1−δand τ 4 =1.

Then, from definitions (3.11), we see that

p 1 =r, p 2 =qand p 3 =0.

In this case Theorem 3.2 yields the following corollary.

Corollary 3.1 Let δ∈(0, 1 2 ) be fixed. Let (C1)-(C4) hold, and assume

(C7)′ for each t∈[δ,1−δ], the function | g m (t,s)|μ(s)>0 on a subset of [ 1 2 ,1−δ] of positive measure;

(C8)′ the function ν(s)sinπs>0 on a subset of [0,1] of positive measure.

Suppose that there exist numbers w i , 2≤i≤5, with

0< w 2 < w 3 ( 1 2 − δ ) < w 3 < π w 3 2 δ θ ≤ w 4 ≤ w 5

such that the following hold:

  1. (P)

    f(u,v)< w 2 q for u,v∈[0, w 2 ];

  2. (Q)

    f(u,v)≤ w 5 q for u,v∈[0, w 5 ];

  3. (R)

    f(u,v)> w 3 r for u∈[ w 3 ( 1 2 −δ), w 4 ( 1 2 −δ)] and v∈[ w 3 , w 4 ].

Then we have the following conclusions:

  1. (a)

    The Lidstone boundary value problem (3.3) has (at least) three positive solutions x 1 , x 2 , x 3 ∈ C ¯ ( w 5 ) (where C is defined in (3.8)) such that

    { ∥ x 1 ∥ < w 2 ; x 2 ( t ) > w 3 , t ∈ [ δ , 1 − δ ] ; ∥ x 3 ∥ > w 2 and min t ∈ [ δ , 1 − δ ] x 3 ( t ) < w 3 .
    (3.35)
  2. (b)

    The complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions y 1 , y 2 , y 3 such that (3.15) holds for i=1,2,3. We further have

    { ∥ y 1 ∥ < w 2 ; y 2 ( t ) > w 3 ( t − δ ) , t ∈ [ δ , 1 − δ ] ; y 3 ( t ) > 2 δ θ π w 2 ( t − δ ) , t ∈ [ δ , 1 − δ ] .
    (3.36)

Remark 3.1 Corollary 3.1 is actually Theorem 3.1. Since Corollary 3.1 is a special case of Theorem 3.2, this shows that Theorem 3.2 is more general than Theorem 3.1.

The next theorem illustrates another application of Theorem 2.2. Compared to the conditions in Theorem 3.2, here the numbers w 1 , Ï„ 1 and Ï„ 4 have different ranges and condition (P) is also different. Note that in the proof of Theorem 3.3 the functionals ψ and Θ are chosen differently from those in Theorem 3.2.

Theorem 3.3 Let δ∈(0, 1 2 ) be fixed. Let (C1)-(C4) hold. Assume that there exist numbers τ j , 1≤j≤4, with

δ≤ τ 1 ≤ τ 2 < 1 2 < τ 3 ≤ τ 4 ≤1−δ

such that (C7) and (C8) hold. Suppose that there exist numbers w i , 1≤i≤5, with

{ 0 < w 1 ≤ 2 δ θ w 2 π < w 2 < w 3 < π w 3 2 δ θ ≤ w 4 ≤ w 5 < q w 2 p 3 , τ 1 w 5 + ( τ 4 − τ 1 ) w 2 < w 3 ( 1 2 − τ 2 )

such that the following hold:

  1. (P)

    f(u,v)< 1 p 2 ( w 2 − w 5 p 3 q ) for u∈[0, τ 1 w 5 +( τ 4 − τ 1 ) w 2 ] and v∈[ w 1 , w 2 ];

  2. (Q)

    f(u,v)≤ w 5 q for u,v∈[0, w 5 ];

  3. (R)

    f(u,v)> w 3 p 1 for u∈[ w 3 ( 1 2 − τ 2 ), w 4 ( 1 2 − τ 2 )] and v∈[ w 3 , w 4 ].

Then we have conclusions (a) and (b) of Theorem  3.2.

Proof To apply Theorem 2.2, we shall define the following functionals on the cone C (see (3.8)):

{ γ ( x ) = ∥ x ∥ , ψ ( x ) = min t ∈ [ τ 1 , τ 4 ] x ( t ) , β ( x ) = max t ∈ [ τ 1 , τ 4 ] x ( t ) , α ( u ) = min t ∈ [ τ 2 , τ 3 ] x ( t ) , Θ ( u ) = max t ∈ [ τ 2 , τ 3 ] x ( t ) .
(3.37)

As in the proof of Theorem 3.2, using (Q) and Lemma 3.3 we can show that S: P ¯ (γ, w 5 )→ P ¯ (γ, w 5 ).

Next, to see that condition (a) of Theorem 2.2 is fulfilled, we use (R) and a similar argument as in the proof of Theorem 3.2.

We shall now prove that condition (b) of Theorem 2.2 is satisfied. Note that

x(t)= 1 2 ( w 1 + w 2 )∈ { x ∈ Q ( γ , β , ψ , w 1 , w 2 , w 5 ) ∣ β ( u ) < w 2 } ≠∅.

Let x∈Q(γ,β,ψ, w 1 , w 2 , w 5 ). Then we have ψ(x)≥ w 1 , β(x)≤ w 2 and γ(x)≤ w 5 which imply

x(s)∈[ w 1 , w 2 ],s∈[ τ 1 , τ 4 ]andx(s)∈[0, w 5 ],s∈[0,1]
(3.38)

and also (3.30). In view of (3.9), (3.38), (3.30), (C8), (P) and (Q), we obtain, as in the proof of Theorem 3.2, that β(Sx)< w 2 . Therefore, condition (b) of Theorem 2.2 is fulfilled.

Next, using a similar argument as in the proof of Theorem 3.2, we see that condition (c) of Theorem 2.2 is met.

Finally, we shall verify that condition (d) of Theorem 2.2 is fulfilled. Let x∈C. It is clear that

β(Sx)= max t ∈ [ τ 1 , τ 4 ] Sx(t)≤∥Sx∥.
(3.39)

Noting that S maps C into C, we find

ψ(Sx)= min t ∈ [ τ 1 , τ 4 ] Sx(t)≥ min t ∈ [ δ , 1 − δ ] Sx(t)≥ 2 δ θ π ∥Sx∥.
(3.40)

A combination of (3.39) and (3.40) gives

ψ(Sx)≥ 2 δ θ π β(Sx),x∈C.
(3.41)

Let x∈Q(γ,β, w 2 , w 5 ) with ψ(Sx)< w 1 . Then (3.41) and the inequality w 1 ≤ 2 δ θ w 2 π lead to

β(Sx)≤ π 2 δ θ ψ(Sx)< π 2 δ θ w 1 ≤ π 2 δ θ 2 δ θ π w 2 = w 2 .

Thus, β(Sx)< w 2 for all x∈Q(γ,β, w 2 , w 5 ) with ψ(Sx)< w 1 .

Conclusion (a) now follows from Theorem 2.2 immediately, while conclusion (b) is similarly obtained as in Theorem 3.2. □

4 Examples

In this section, we shall present examples to illustrate the usefulness as well as to compare the generality of the results obtained in Section 3.

Example 4.1 Consider the complementary Lidstone boundary value problem (1.1) with m=3 and the nonlinear term F given by

F ( t , u , v ) = f ( u , v ) = { w 1 2 q , ( u , v ) ∈ [ 0 , w 1 ] × [ 0 , w 1 ] ≡ A 1 , 1 2 ( d q + w 2 r ) , ( u , v ) ∈ [ w 2 ( 1 2 − δ ) , ∞ ) × [ w 2 , ∞ ) ≡ A 2 , ϕ ( u , v ) , ( u , v ) ∈ R 2 ∖ ( A 1 ∪ A 2 ) ,
(4.1)

where Ï•(u,v) is continuous in each argument and satisfies

{ ϕ ( 0 , v ) = ϕ ( w 1 , v ) = ϕ ( u , 0 ) = ϕ ( u , w 1 ) = w 1 2 q , u , v ∈ [ 0 , w 1 ] ; ϕ ( w 2 ( 1 2 − δ ) , v ) = ϕ ( u , w 2 ) = 1 2 ( d q + w 2 r ) , u ∈ [ w 2 ( 1 2 − δ ) , ∞ ) , v ∈ [ w 2 , ∞ ) ; 0 ≤ ϕ ( u , v ) ≤ 1 2 ( d q + w 2 r ) , ( u , v ) ∈ R 2 ∖ ( A 1 ∪ A 2 ) .
(4.2)

Here, δ= 1 4 is fixed and the w i ’s and d are in the context of Theorem 3.1 satisfying

0< w 1 < w 2 ( 1 2 − δ ) < w 2 < π w 2 2 δ θ ≤ w 3 ≤dandd> q w 2 r .
(4.3)

Let the functions μ=ν≡1 (which implies θ=1). Then it is clear that (C1)-(C6) are fulfilled. Moreover, by direct computation we get q=2.080× 10 − 3 , and on using Lemma 2.2 we find

r= min t ∈ [ δ , 1 − δ ] ∫ 1 2 1 − δ | g m (t,s)|μ(s)ds≥ 2 δ π 2 m ∫ 1 2 1 − δ sinπsds=1.171× 10 − 4 .

For convenience, we take r=1.171× 10 − 4 although this will lead to more stringent conditions.

Hence, (4.3) reduces to

0< w 1 <0.25 w 2 < w 2 <6.283 w 2 ≤ w 3 ≤dandd>17.77 w 2
(4.4)

and clearly we can easily find numbers w i ’s and d that satisfy (4.4).

We shall check the conditions of Theorem 3.1. First, condition (P) is obviously satisfied. Next, from (4.3) we have w 2 r < d q , therefore for (u,v)∈[0,d]×[0,d] it follows that

f(u,v)≤ 1 2 ( d q + w 2 r ) < 1 2 ( d q + d q ) = d q .

Hence, condition (Q2) is met. Finally, (R) is satisfied since for (u,v)∈[ w 2 ( 1 2 −δ), w 3 ( 1 2 −δ)]×[ w 2 , w 3 ], we have

f(u,v)= 1 2 ( d q + w 2 r ) > 1 2 ( w 2 r + w 2 r ) = w 2 r .

By Theorem 3.1 (conclusion (b)), the boundary value problem (1.1) with m=3, δ= 1 4 , (4.1) and (4.2) has (at least) three positive solutions y 1 , y 2 , y 3 such that (from (3.16))

{ ∥ y 1 ∥ < w 1 ; y 2 ( t ) > w 2 ( t − 1 4 ) , t ∈ [ 1 4 , 3 4 ] ; y 3 ( t ) > 1 2 π w 1 ( t − 1 4 ) , t ∈ [ 1 4 , 3 4 ] ,
(4.5)

where w i ’s satisfy (4.4).

Example 4.2 Consider the complementary Lidstone boundary value problem (1.1) with m=3 and the nonlinear term F given by

F ( t , u , v ) = f ( u , v ) = { 1 2 p 2 ( w 2 − w 5 p 3 q ) , ( u , v ) ∈ [ 0 , τ 1 w 5 + ( τ 4 − τ 1 ) w 2 ] × [ 0 , w 2 ] ≡ A 3 , 1 2 ( w 3 r + w 3 p 1 ) , ( u , v ) ∈ [ w 3 ( 1 2 − τ 2 ) , ∞ ) × [ w 3 , ∞ ) ≡ A 4 , κ ( u , v ) , ( u , v ) ∈ R 2 ∖ ( A 3 ∪ A 4 ) ,
(4.6)

where κ(u,v) is continuous in each argument and satisfies

{ κ ( 0 , v ) = κ ( τ 1 w 5 + ( τ 4 − τ 1 ) w 2 , v ) = κ ( u , 0 ) = κ ( u , w 2 ) = 1 2 p 2 ( w 2 − w 5 p 3 q ) , κ ( 0 , v ) = κ ( τ 1 w 5 + ( τ 4 − τ 1 ) w 2 , v ) = κ ( u ∈ [ 0 , τ 1 w 5 + ( τ 4 − τ 1 ) w 2 ] , v ∈ [ 0 , w 2 ] ; κ ( w 3 ( 1 2 − τ 2 ) , v ) = κ ( u , w 3 ) = 1 2 ( w 3 r + w 3 p 1 ) , u ∈ [ w 3 ( 1 2 − τ 2 ) , ∞ ) , v ∈ [ w 3 , ∞ ) ; 0 ≤ κ ( u , v ) ≤ 1 2 ( w 3 r + w 3 p 1 ) , ( u , v ) ∈ R 2 ∖ ( A 3 ∪ A 4 ) .
(4.7)

Here, we fix

δ=0.05, τ 1 =0, τ 2 =0.1, τ 3 = τ 4 =1−δ=0.95
(4.8)

and the w i ’s are in the context of Theorem 3.2 satisfying

{ 0 < w 2 < w 3 < π w 3 2 δ θ ≤ w 4 ≤ w 5 < q w 2 p 3 , τ 1 w 5 + ( τ 4 − τ 1 ) w 2 < w 3 ( 1 2 − τ 2 ) and w 5 > q w 3 r .
(4.9)

Let the functions μ=ν≡1 (which implies θ=1). Then it is clear that (C1)-(C4), (C7) and (C8) are fulfilled. Moreover, by direct computation we have

q=2.080× 10 − 3 , p 2 =2.068× 10 − 3 , p 3 =1.281× 10 − 5 .

By using Lemma 2.2, we get

p 1 = min t ∈ [ 0.1 , 0.95 ] ∫ 1 2 0.95 | g m ( t , s ) | μ ( s ) d s > r = min t ∈ [ 0.05 , 0.95 ] ∫ 1 2 0.95 | g m ( t , s ) | μ ( s ) d s ≥ 2 δ π 2 m ∫ 1 2 0.95 sin π s d s = 3.270 × 10 − 5 .

For convenience, we take r=3.270× 10 − 5 although this will lead to more stringent conditions. Hence, (4.9) reduces to

{ 0 < w 2 < w 3 < 31.42 w 3 ≤ w 4 ≤ w 5 < 162.4 w 2 , 0.95 w 2 < 0.4 w 3 and w 5 > 63.61 w 3

or equivalently (combining the first two inequalities)

{ 0 < w 2 < w 3 < 31.42 w 3 ≤ w 4 ≤ w 5 < 162.4 w 2 < 68.38 w 3 and w 5 > 63.61 w 3 .
(4.10)

It is clear that we can easily find numbers w i ’s that fulfill (4.10).

We shall check the conditions of Theorem 3.2. First, condition (P) is obviously satisfied. Next, since

r< p 1 < p 2 and w 3 r < w 5 q (i.e., w 5 >63.61 w 3 ),
(4.11)

we find for (u,v)∈[0, w 5 ]×[0, w 5 ],

f(u,v)≤ 1 2 ( w 3 r + w 3 p 1 ) < 1 2 ( w 3 r + w 3 r ) = w 3 r < w 5 q .

Hence, condition (Q) is met. Finally, (R) is satisfied since for (u,v)∈[ w 3 ( 1 2 − τ 2 ), w 4 ( 1 2 − τ 2 )]×[ w 3 , w 4 ], using (4.11) we get

f(u,v)= 1 2 ( w 3 r + w 3 p 1 ) > 1 2 ( w 3 p 1 + w 3 p 1 ) = w 3 p 1 .

By Theorem 3.2 (conclusion (b)), the boundary value problem (1.1) with m=3, (4.6), (4.7) and (4.8) has (at least) three positive solutions y 1 , y 2 , y 3 such that (from (3.26))

{ y 1 ( t ) < 0.95 w 2 , t ∈ [ 0 , 0.95 ] ; y 2 ( t ) > w 3 ( t − 0.1 ) , t ∈ [ 0.1 , 0.95 ] ; y 3 ( t ) > 1 10 π w 2 ( t − 0.05 ) , t ∈ [ 0.05 , 0.95 ] ,
(4.12)

where w i ’s satisfy (4.10).

Remark 4.1 In Example 4.2, we see that for (u,v)∈[ w 3 ( 1 2 − Ï„ 2 ), w 4 ( 1 2 − Ï„ 2 )]×[ w 3 , w 4 ],

f(u,v)= 1 2 ( w 3 r + w 3 p 1 ) < 1 2 ( w 3 r + w 3 r ) = w 3 r .

Thus, condition (R) of Corollary 3.1 is not satisfied and so Corollary 3.1 cannot be used to establish the existence of triple positive solutions in Example 4.2. Recalling that Corollary 3.1 is actually Theorem 3.1, this illustrates the case when Theorem 3.2 is applicable but not Theorem 3.1. Hence, this example shows that Theorem 3.2 is indeed more general than Theorem 3.1.

References

  1. Costabile FA, Dell’Accio F, Luceri R: Explicit polynomial expansions of regular real functions by means of even order Bernoulli polynomials and boundary values. J. Comput. Appl. Math. 2005, 175: 77-99. 10.1016/j.cam.2004.06.006

    Article  MathSciNet  Google Scholar 

  2. Agarwal RP, Pinelas S, Wong PJY: Complementary Lidstone interpolation and boundary value problems. J. Inequal. Appl. 2009., 2009: Article ID 624631

    Google Scholar 

  3. Agarwal RP, Wong PJY: Piecewise complementary Lidstone interpolation and error inequalities. J. Comput. Appl. Math. 2010, 234: 2543-2561. 10.1016/j.cam.2010.03.029

    Article  MathSciNet  Google Scholar 

  4. Agarwal RP, Wong PJY: Lidstone polynomials and boundary value problems. Comput. Math. Appl. 1989, 17: 1397-1421. 10.1016/0898-1221(89)90023-0

    Article  MathSciNet  Google Scholar 

  5. Agarwal RP, Wong PJY: Error Inequalities in Polynomial Interpolation and Their Applications. Kluwer Academic, Dordrecht; 1993.

    Book  Google Scholar 

  6. Davis PJ: Interpolation and Approximation. Blaisdell, Boston; 1961.

    Google Scholar 

  7. Varma AK, Howell G: Best error bounds for derivatives in two point Birkhoff interpolation problem. J. Approx. Theory 1983, 38: 258-268. 10.1016/0021-9045(83)90132-6

    Article  MathSciNet  Google Scholar 

  8. Lidstone GJ: Notes on the extension of Aitken’s theorem (for polynomial interpolation) to the Everett types. Proc. Edinb. Math. Soc. 1929, 2: 16-19.

    Article  Google Scholar 

  9. Boas RP: A note on functions of exponential type. Bull. Am. Math. Soc. 1941, 47: 750-754. 10.1090/S0002-9904-1941-07552-X

    Article  MathSciNet  Google Scholar 

  10. Boas RP: Representation of functions by Lidstone series. Duke Math. J. 1943, 10: 239-245. 10.1215/S0012-7094-43-01021-X

    Article  MathSciNet  Google Scholar 

  11. Poritsky H: On certain polynomial and other approximations to analytic functions. Trans. Am. Math. Soc. 1932, 34: 274-331. 10.1090/S0002-9947-1932-1501639-4

    Article  MathSciNet  Google Scholar 

  12. Schoenberg IJ: On certain two-point expansions of integral functions of exponential type. Bull. Am. Math. Soc. 1936, 42: 284-288. 10.1090/S0002-9904-1936-06293-2

    Article  MathSciNet  Google Scholar 

  13. Whittaker JM: On Lidstone’s series and two-point expansions of analytic functions. Proc. Lond. Math. Soc. 1933-1934, 36: 451-469.

    MathSciNet  Google Scholar 

  14. Whittaker JM: Interpolatory Function Theory. Cambridge University Press, Cambridge; 1935.

    Google Scholar 

  15. Widder DV: Functions whose even derivatives have a prescribed sign. Proc. Natl. Acad. Sci. USA 1940, 26: 657-659. 10.1073/pnas.26.11.657

    Article  MathSciNet  Google Scholar 

  16. Widder DV: Completely convex functions and Lidstone series. Trans. Am. Math. Soc. 1942, 51: 387-398.

    Article  MathSciNet  Google Scholar 

  17. Agarwal RP, Wong PJY: Explicit error bounds for the derivatives of piecewise-Lidstone interpolation. J. Comput. Appl. Math. 1995, 58: 67-81. 10.1016/0377-0427(93)E0262-K

    Article  MathSciNet  Google Scholar 

  18. Agarwal RP, Wong PJY: Error bounds for the derivatives of Lidstone interpolation and applications. In Approximation Theory: In Memory of A.K. Varma. Edited by: Govil NK, Mohapatra RN, Nashed Z, Sharma A, Szabados J. Dekker, New York; 1998:1-41.

    Google Scholar 

  19. Costabile FA, Dell’Accio F: Lidstone approximation on the triangle. Appl. Numer. Math. 2005, 52: 339-361. 10.1016/j.apnum.2004.08.003

    Article  MathSciNet  Google Scholar 

  20. Costabile FA, Serpe A: An algebraic approach to Lidstone polynomials. Appl. Math. Lett. 2007, 20: 387-390. 10.1016/j.aml.2006.02.034

    Article  MathSciNet  Google Scholar 

  21. Wong PJY: On Lidstone splines and some of their applications. Neural Parallel Sci. Comput. 1995, 1: 472-475.

    Google Scholar 

  22. Wong PJY, Agarwal RP: Sharp error bounds for the derivatives of Lidstone-spline interpolation. Comput. Math. Appl. 1994, 28(9):23-53. 10.1016/0898-1221(94)00176-6

    Article  MathSciNet  Google Scholar 

  23. Wong PJY, Agarwal RP: Sharp error bounds for the derivatives of Lidstone-spline interpolation II. Comput. Math. Appl. 1996, 31(3):61-90. 10.1016/0898-1221(95)00206-5

    Article  MathSciNet  Google Scholar 

  24. Agarwal RP, Akrivis G: Boundary value problems occurring in plate deflection theory. J. Comput. Appl. Math. 1982, 8: 145-154. 10.1016/0771-050X(82)90035-3

    Article  MathSciNet  Google Scholar 

  25. Agarwal RP, O’Regan D, Wong PJY: Positive Solutions of Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht; 1999.

    Book  Google Scholar 

  26. Agarwal RP, Wong PJY: Quasilinearization and approximate quasilinearization for Lidstone boundary value problems. Int. J. Comput. Math. 1992, 42: 99-116. 10.1080/00207169208804054

    Article  Google Scholar 

  27. Baldwin P: Asymptotic estimates of the eigenvalues of a sixth-order boundary-value problem obtained by using global phase-integral method. Philos. Trans. R. Soc. Lond. Ser. A 1987, 322: 281-305. 10.1098/rsta.1987.0051

    Article  MathSciNet  Google Scholar 

  28. Baldwin P: A localized instability in a Bénard layer. Appl. Anal. 1987, 24: 117-156. 10.1080/00036818708839658

    Article  MathSciNet  Google Scholar 

  29. Boutayeb A, Twizell EH: Finite-difference methods for twelfth-order boundary value problems. J. Comput. Appl. Math. 1991, 35: 133-138. 10.1016/0377-0427(91)90202-U

    Article  MathSciNet  Google Scholar 

  30. Davis JM, Eloe PW, Henderson J: Triple positive solutions and dependence on higher order derivatives. J. Math. Anal. Appl. 1999, 237: 710-720. 10.1006/jmaa.1999.6500

    Article  MathSciNet  Google Scholar 

  31. Davis JM, Henderson J, Wong PJY: General Lidstone problems: multiplicity and symmetry of solutions. J. Math. Anal. Appl. 2000, 251: 527-548. 10.1006/jmaa.2000.7028

    Article  MathSciNet  Google Scholar 

  32. Eloe PW, Henderson J, Thompson HB: Extremal points for impulsive Lidstone boundary value problems. Math. Comput. Model. 2000, 32: 687-698. 10.1016/S0895-7177(00)00165-5

    Article  MathSciNet  Google Scholar 

  33. Forster P: Existenzaussagen und Fehlerabschätzungen bei gewissen nichtlinearen Randwertaufgaben mit gewöhnlichen Differentialgleichungen. Numer. Math. 1967, 10: 410-422. 10.1007/BF02162874

    Article  MathSciNet  Google Scholar 

  34. Guo Y, Ge W: Twin positive symmetric solutions for Lidstone boundary value problems. Taiwan. J. Math. 2004, 8: 271-283.

    MathSciNet  Google Scholar 

  35. Ma Y: Existence of positive solutions of Lidstone boundary value problems. J. Math. Anal. Appl. 2006, 314: 97-108. 10.1016/j.jmaa.2005.03.059

    Article  MathSciNet  Google Scholar 

  36. Twizell EH, Boutayeb A: Numerical methods for the solution of special and general sixth-order boundary value problems, with applications to Bénard layer eigenvalue problems. Proc. R. Soc. Lond. Ser. A 1990, 431: 433-450. 10.1098/rspa.1990.0142

    Article  MathSciNet  Google Scholar 

  37. Yao Q: On the positive solutions of Lidstone boundary value problems. Appl. Math. Comput. 2003, 137: 477-485. 10.1016/S0096-3003(02)00152-2

    Article  MathSciNet  Google Scholar 

  38. Agarwal RP, Wong PJY: Eigenvalues of complementary Lidstone boundary value problems. Bound. Value Probl. 2012., 2012: Article ID 49

    Google Scholar 

  39. Agarwal RP, Wong PJY: Positive solutions of complementary Lidstone boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2012., 2012: Article ID 60

    Google Scholar 

  40. Leggett RW, Williams LR: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J. 1979, 28: 673-688. 10.1512/iumj.1979.28.28046

    Article  MathSciNet  Google Scholar 

  41. Avery RI: A generalization of the Leggett-Williams fixed point theorem. Math. Sci. Res. Hot-Line 1998, 2: 9-14.

    MathSciNet  Google Scholar 

  42. O’Regan D, Meehan M: Existence Theory for Nonlinear Integral and Integrodifferential Equations. Kluwer Academic, Dordrecht; 1998.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Patricia J. Y. Wong.

Additional information

Competing interests

The author declares that she has no competing interests.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wong, P.J.Y. Triple solutions of complementary Lidstone boundary value problems via fixed point theorems. Bound Value Probl 2014, 125 (2014). https://doi.org/10.1186/1687-2770-2014-125

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-2770-2014-125

Keywords