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

- Patricia J. Y. Wong
^{1}Email author

**2014**:125

**DOI: **10.1186/1687-2770-2014-125

© Wong; licensee Springer. 2014

**Received: **25 June 2013

**Accepted: **15 August 2013

**Published: **20 May 2014

## Abstract

We consider the following complementary Lidstone boundary value problem:

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 ${y}^{\prime}$ and this derivative dependence is seldom investigated in the literature and a *new* technique is required to tackle the problem.

**MSC:**34B15, 34B18.

### Keywords

positive solutions complementary Lidstone boundary value problems derivative-dependent nonlinearity fixed point theorems## 1 Introduction

*complementary Lidstone*boundary value problem

where $m\ge 1$ and *F* is continuous at least in the interior of the domain of interest. It is noted that the nonlinear term *F* involves ${y}^{\prime}$, 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.

*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

*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 2

*m*th order differential equation and the

*Lidstone*boundary conditions

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 ${y}^{\prime}$ 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 $\parallel \cdot \parallel $.

**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:

- (a)
If $x\in C$ and $\alpha \ge 0$, then $\alpha x\in C$;

- (b)
If $x\in C$ and $-x\in C$, 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:

- (a)
$\psi :C\to [0,\mathrm{\infty})$ is continuous;

- (b)
$\psi (ty+(1-t)z)\ge t\psi (y)+(1-t)\psi (z)$ for all $y,z\in C$ and $0\le t\le 1$.

**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:

- (a)
$\beta :C\to [0,\mathrm{\infty})$ is continuous;

- (b)
$\beta (ty+(1-t)z)\le t\beta (y)+(1-t)\beta (z)$ for all $y,z\in C$ and $0\le t\le 1$.

*γ*,

*β*, Θ be nonnegative continuous convex functionals on

*C*and

*α*,

*ψ*be nonnegative continuous concave functionals on

*C*. For nonnegative numbers ${w}_{i}$, $1\le i\le 3$, we shall introduce the following notations:

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*$\psi (x)\le \parallel x\parallel $

*for all*$x\in \overline{C}({w}_{4})$,

*and let*$S:\overline{C}({w}_{4})\to \overline{C}({w}_{4})$

*be a continuous and completely continuous operator*.

*Suppose that there exist numbers*${w}_{1}$, ${w}_{2}$, ${w}_{3}$,

*where*$0<{w}_{1}<{w}_{2}<{w}_{3}\le {w}_{4}$,

*such that*

- (a)
$\{x\in C(\psi ,{w}_{2},{w}_{3})\mid \psi (x)>{w}_{2}\}\ne \mathrm{\varnothing}$,

*and*$\psi (Sx)>{w}_{2}$*for all*$x\in C(\psi ,{w}_{2},{w}_{3})$; - (b)
$\parallel Sx\parallel <{w}_{1}$

*for all*$x\in \overline{C}({w}_{1})$; - (c)
$\psi (Sx)>{w}_{2}$

*for all*$x\in C(\psi ,{w}_{2},{w}_{4})$*with*$\parallel Sx\parallel >{w}_{3}$.

*Then*

*S*

*has*(

*at least*)

*three fixed points*${x}_{1}$, ${x}_{2}$

*and*${x}_{3}$

*in*$\overline{C}({w}_{4})$.

*Furthermore*,

*we have*

**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*

*for all*$x\in \overline{P}(\gamma ,{w}_{5})$.

*Let*$S:\overline{P}(\gamma ,{w}_{5})\to \overline{P}(\gamma ,{w}_{5})$

*be a continuous and completely continuous operator*.

*Suppose that there exist nonnegative numbers*${w}_{i}$, $1\le i\le 4$,

*with*$0<{w}_{2}<{w}_{3}$

*such that*

- (a)
$\{x\in P(\gamma ,\mathrm{\Theta},\alpha ,{w}_{3},{w}_{4},{w}_{5})\mid \alpha (x)>{w}_{3}\}\ne \mathrm{\varnothing}$,

*and*$\alpha (Sx)>{w}_{3}$*for all*$x\in P(\gamma ,\mathrm{\Theta},\alpha ,{w}_{3},{w}_{4},{w}_{5})$; - (b)
$\{x\in Q(\gamma ,\beta ,\psi ,{w}_{1},{w}_{2},{w}_{5})\mid \beta (x)<{w}_{2}\}\ne \mathrm{\varnothing}$,

*and*$\beta (Sx)<{w}_{2}$*for all*$x\in Q(\gamma ,\beta ,\psi ,{w}_{1},{w}_{2},{w}_{5})$; - (c)
$\alpha (Sx)>{w}_{3}$

*for all*$x\in P(\gamma ,\alpha ,{w}_{3},{w}_{5})$*with*$\mathrm{\Theta}(Sx)>{w}_{4}$; - (d)
$\beta (Sx)<{w}_{2}$

*for all*$x\in Q(\gamma ,\beta ,{w}_{2},{w}_{5})$*with*$\psi (Sx)<{w}_{1}$.

*Then*

*S*

*has*(

*at least*)

*three fixed points*${x}_{1}$, ${x}_{2}$

*and*${x}_{3}$

*in*$\overline{P}(\gamma ,{w}_{5})$.

*Furthermore*,

*we have*

We also require the definition of an ${L}^{1}$-*Carathéodory function*.

**Definition 2.4** [42]

*Carathéodory function*if the following conditions hold:

- (a)
The map $t\to P(t,u)$ is measurable for all $u\in {\mathbb{R}}^{2}$.

- (b)
The map $u\to P(t,u)$ is continuous for almost all $t\in [0,1]$.

- (c)
For any $r>0$, there exists ${\mu}_{r}\in {L}^{1}[0,1]$ such that $|u|\le r$ implies that $|P(t,u)|\le {\mu}_{r}(t)$ for almost all $t\in [0,1]$.

*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

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*$(t,s)\in [0,1]\times [0,1]$,

*we have*

**Lemma 2.2** [38]

*Let*$\delta \in (0,\frac{1}{2})$

*be given*.

*For*$(t,s)\in [\delta ,1-\delta ]\times [0,1]$,

*we have*

## 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 $y\in C[0,1]$ satisfying (1.1) and $y(t)\ge 0$ for $t\in [0,1]$.

*complementary Lidstone*boundary value problem (1.1) reduces to the

*Lidstone*boundary value problem

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 $\parallel {y}^{\ast}\parallel \le \parallel {x}^{\ast}\parallel $; moreover if ${x}^{\ast}$ is positive, so is ${y}^{\ast}$. 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).

where ${g}_{m}(t,s)$ is the Green’s function given in (2.4). A fixed point ${x}^{\ast}$ of the operator *S* is clearly a solution of the boundary value problem (3.3), and as seen earlier ${y}^{\ast}(t)={\int}_{0}^{t}{x}^{\ast}(s)\phantom{\rule{0.2em}{0ex}}ds$ is a solution of (1.1).

*K*and $\tilde{K}$ are defined by

(C1) $F:[0,1]\times {\mathbb{R}}^{2}\to \mathbb{R}$ is an ${L}^{1}$-Carathéodory function.

*f*,

*ν*,

*μ*with $f:[0,\mathrm{\infty})\times [0,\mathrm{\infty})\to [0,\mathrm{\infty})$ and $\nu ,\mu :(0,1)\to [0,\mathrm{\infty})$ such that

*C*in

*B*as

where *θ* is given in (C4). Clearly, we have $C\subseteq \tilde{K}$.

**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 ${g}_{m}(t,s)\in C[0,1]\subseteq {L}^{\mathrm{\infty}}[0,1]$, $t\in [0,1]$ and the map $t\to {g}_{m}(t,s)$ is continuous from $[0,1]$ to $C[0,1]$. This together with $F:[0,1]\times {\mathbb{R}}^{2}\to \mathbb{R}$ is an ${L}^{1}$-Carathéodory function ensures (as in [[42], Theorem 4.2.2]) that *S* is continuous and completely continuous.

We have shown that $Sx\in C$. □

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

(C5) *the function* $\nu (s)sin\pi s>0$ *on a subset of* $[0,1]$ *of positive measure*.

*Suppose that there exists a number*$d>0$

*such that for*$u,v\in [0,d]$,

*Then*

*Proof*Let $x\in \overline{C}(d)$. So $\parallel x\parallel \le d$, which implies immediately that

This implies $\parallel Sx\parallel <d$. Together with the fact that $Sx\in C$ (Lemma 3.1), we have shown that $Sx\in C(d)$. 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*$u,v\in [0,d]$,

*Then*

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* $\delta \in (0,\frac{1}{2})$ *be fixed*. *Let* (C1)-(C5) *hold*, *and assume*

(C6) *for each* $t\in [\delta ,1-\delta ]$, *the function* $|{g}_{m}(t,s)|\mu (s)>0$ *on a subset of* $[\frac{1}{2},1-\delta ]$ *of positive measure*.

*Suppose that there exist numbers*${w}_{1}$, ${w}_{2}$, ${w}_{3}$

*with*

*such that the following hold*:

- (P)
$f(u,v)<\frac{{w}_{1}}{q}$

*for*$u,v\in [0,{w}_{1}]$; - (Q)
*one of the following holds*:- (Q1)
${lim\hspace{0.17em}sup}_{u\to \mathrm{\infty},v\to \mathrm{\infty}}\frac{f(u,v)}{u}<\frac{1}{q}$

*or*${lim\hspace{0.17em}sup}_{u\to \mathrm{\infty},v\to \mathrm{\infty}}\frac{f(u,v)}{v}<\frac{1}{q}$; - (Q2)
*there exists a number**d*($\ge {w}_{3}$)*such that*$f(u,v)\le \frac{d}{q}$*for*$u,v\in [0,d]$;

- (Q1)
- (R)
$f(u,v)>\frac{{w}_{2}}{r}$

*for*$u\in [{w}_{2}(\frac{1}{2}-\delta ),{w}_{3}(\frac{1}{2}-\delta )]$*and*$v\in [{w}_{2},{w}_{3}]$.

*Then we have the following conclusions*:

- (a)
*The Lidstone boundary value problem*(3.3)*has*(*at least*)*three positive solutions*${x}_{1},{x}_{2},{x}_{3}\in C$ (*where**C**is defined in*(3.8))*such that*$\{\begin{array}{l}\parallel {x}_{1}\parallel <{w}_{1};\\ {x}_{2}(t)>{w}_{2},\phantom{\rule{1em}{0ex}}t\in [\delta ,1-\delta ];\\ \parallel {x}_{3}\parallel >{w}_{1}\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}{min}_{t\in [\delta ,1-\delta ]}{x}_{3}(t)<{w}_{2}.\end{array}$(3.14) - (b)
*The complementary Lidstone boundary value problem*(1.1)*has*(*at least*)*three positive solutions*${y}_{1}$, ${y}_{2}$, ${y}_{3}$*such that for*$i=1,2,3$,$\{\begin{array}{l}{y}_{i}(t)={\int}_{0}^{t}{x}_{i}(s)\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}t\in [0,1];\\ \parallel {y}_{i}\parallel \le \parallel {x}_{i}\parallel ;\phantom{\rule{2em}{0ex}}{y}_{i}(t)\ge \frac{2\delta \theta}{\pi}\parallel {x}_{i}\parallel (t-\delta ),\phantom{\rule{1em}{0ex}}t\in [\delta ,1-\delta ]\end{array}$(3.15)

*where*${x}_{i}$’

*s are those in conclusion*(a)).

*We further have*

*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 ${w}_{4}\ge {w}_{3}$, such that

Hence, $\parallel Sx\parallel <{w}_{4}$ and so $Sx\in C({w}_{4})\subset \overline{C}({w}_{4})$. Thus, (3.17) follows immediately. Note that the argument is similar if we assume that ${lim\hspace{0.17em}sup}_{u\to \mathrm{\infty},v\to \mathrm{\infty}}\frac{f(u,v)}{v}<\frac{1}{q}$ of (Q1) is satisfied.

Clearly, *ψ* is a nonnegative continuous concave functional on *C* and $\psi (x)\le \parallel x\parallel $ for all $x\in C$.

Therefore, we have shown that $\psi (Sx)>{w}_{2}$ for all $x\in C(\psi ,{w}_{2},{w}_{3})$.

Next, by condition (P) and Lemma 3.2 (with $d={w}_{1}$), we have $S(\overline{C}({w}_{1}))\subseteq C({w}_{1})$. Hence, condition (b) of Theorem 2.1 is fulfilled.

Hence, we have proved that $\psi (Sx)>{w}_{2}$ for all $x\in C(\psi ,{w}_{2},{w}_{4})$ with $\parallel Sx\parallel >{w}_{3}$.

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

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

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*$\delta \in (0,\frac{1}{2})$

*be fixed*.

*Let*(C1)-(C4)

*hold*.

*Assume that there exist numbers*${\tau}_{j}$, $1\le j\le 4$,

*with*

*such that*

(C7) *for each* $t\in [{\tau}_{2},{\tau}_{3}]$, *the function* $|{g}_{m}(t,s)|\mu (s)>0$ *on a subset of* $[\frac{1}{2},{\tau}_{3}]$ *of positive measure*;

(C8) *the function* $\nu (s)sin\pi s>0$ *on a subset of* $[{\tau}_{1},{\tau}_{4}]$ *of positive measure*.

*Suppose that there exist numbers*${w}_{i}$, $2\le i\le 5$,

*with*

*such that the following hold*:

- (P)
$f(u,v)<\frac{1}{{p}_{2}}({w}_{2}-\frac{{w}_{5}{p}_{3}}{q})$

*for*$u\in [0,{\tau}_{1}{w}_{5}+({\tau}_{4}-{\tau}_{1}){w}_{2}]$*and*$v\in [0,{w}_{2}]$; - (Q)
$f(u,v)\le \frac{{w}_{5}}{q}$

*for*$u,v\in [0,{w}_{5}]$; - (R)
$f(u,v)>\frac{{w}_{3}}{{p}_{1}}$

*for*$u\in [{w}_{3}(\frac{1}{2}-{\tau}_{2}),{w}_{4}(\frac{1}{2}-{\tau}_{2})]$*and*$v\in [{w}_{3},{w}_{4}]$.

*Then we have the following conclusions*:

- (a)
*The Lidstone boundary value problem*(3.3)*has*(*at least*)*three positive solutions*${x}_{1},{x}_{2},{x}_{3}\in \overline{C}({w}_{5})$ (*where**C**is defined in*(3.8))*such that*$\{\begin{array}{l}{x}_{1}(t)<{w}_{2},\phantom{\rule{1em}{0ex}}t\in [{\tau}_{1},{\tau}_{4}];\\ {x}_{2}(t)>{w}_{3},\phantom{\rule{1em}{0ex}}t\in [{\tau}_{2},{\tau}_{3}];\\ {max}_{t\in [{\tau}_{1},{\tau}_{4}]}{x}_{3}(t)>{w}_{2}\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}{min}_{t\in [{\tau}_{2},{\tau}_{3}]}{x}_{3}(t)<{w}_{3}.\end{array}$(3.25) - (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*$\{\begin{array}{l}{y}_{1}(t)<{\tau}_{1}{max}_{s\in [0,{\tau}_{1}]}{x}_{1}(s)+({\tau}_{4}-{\tau}_{1}){w}_{2},\phantom{\rule{1em}{0ex}}t\in [{\tau}_{1},{\tau}_{4}];\\ {y}_{2}(t)>{w}_{3}(t-{\tau}_{2}),\phantom{\rule{1em}{0ex}}t\in [{\tau}_{2},{\tau}_{3}];\\ {y}_{3}(t)>\frac{2\delta \theta}{\pi}{w}_{2}(t-\delta ),\phantom{\rule{1em}{0ex}}t\in [\delta ,1-\delta ].\end{array}$(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*:

First, we shall show that the operator *S* maps $\overline{P}(\gamma ,{w}_{5})$ into $\overline{P}(\gamma ,{w}_{5})$. Note that $\overline{P}(\gamma ,{w}_{5})=\overline{C}({w}_{5})$. By (Q) and Lemma 3.3 (with $d={w}_{5}$), we immediately have $S(\overline{C}({w}_{5}))\subseteq \overline{C}({w}_{5})$.

Hence, $\alpha (Sx)>{w}_{3}$ for all $x\in P(\gamma ,\mathrm{\Theta},\alpha ,{w}_{3},{w}_{4},{w}_{5})$.

*i.e.*,

Therefore, $\beta (Sx)<{w}_{2}$ for all $x\in Q(\gamma ,\beta ,\psi ,{w}_{1},{w}_{2},{w}_{5})$.

*S*maps

*C*into

*C*, we find

Thus, $\alpha (Sx)>{w}_{3}$ for all $x\in P(\gamma ,\alpha ,{w}_{3},{w}_{5})$ with $\mathrm{\Theta}(Sx)>{w}_{4}$.

Finally, we shall prove that condition (d) of Theorem 2.2 is fulfilled. Let $x\in Q(\gamma ,\beta ,{w}_{2},{w}_{5})$ with $\psi (Sx)<{w}_{1}$. Then we have $\beta (x)\le {w}_{2}$ and $\gamma (x)\le {w}_{5}$ which give (3.29) and (3.30). As in proving condition (b), we get $\beta (Sx)<{w}_{2}$. 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 ${x}_{1},{x}_{2},{x}_{3}\in \overline{P}(\gamma ,{w}_{5})=\overline{C}({w}_{5})$ satisfying (2.2). Furthermore, (2.2) reduces to (3.25) immediately. This completes the proof of conclusion (a).

The proof of conclusion (b) is complete. □

In this case Theorem 3.2 yields the following corollary.

**Corollary 3.1** *Let* $\delta \in (0,\frac{1}{2})$ *be fixed*. *Let* (C1)-(C4) *hold*, *and assume*

(C7)′ *for each* $t\in [\delta ,1-\delta ]$, *the function* $|{g}_{m}(t,s)|\mu (s)>0$ *on a subset of* $[\frac{1}{2},1-\delta ]$ *of positive measure*;

(C8)′ *the function* $\nu (s)sin\pi s>0$ *on a subset of* $[0,1]$ *of positive measure*.

*Suppose that there exist numbers*${w}_{i}$, $2\le i\le 5$,

*with*

*such that the following hold*:

- (P)
$f(u,v)<\frac{{w}_{2}}{q}$

*for*$u,v\in [0,{w}_{2}]$; - (Q)
$f(u,v)\le \frac{{w}_{5}}{q}$

*for*$u,v\in [0,{w}_{5}]$; - (R)
$f(u,v)>\frac{{w}_{3}}{r}$

*for*$u\in [{w}_{3}(\frac{1}{2}-\delta ),{w}_{4}(\frac{1}{2}-\delta )]$*and*$v\in [{w}_{3},{w}_{4}]$.

*Then we have the following conclusions*:

- (a)
*The Lidstone boundary value problem*(3.3)*has*(*at least*)*three positive solutions*${x}_{1},{x}_{2},{x}_{3}\in \overline{C}({w}_{5})$ (*where**C**is defined in*(3.8))*such that*$\{\begin{array}{l}\parallel {x}_{1}\parallel <{w}_{2};\\ {x}_{2}(t)>{w}_{3},\phantom{\rule{1em}{0ex}}t\in [\delta ,1-\delta ];\\ \parallel {x}_{3}\parallel >{w}_{2}\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}{min}_{t\in [\delta ,1-\delta ]}{x}_{3}(t)<{w}_{3}.\end{array}$(3.35) - (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*$\{\begin{array}{l}\parallel {y}_{1}\parallel <{w}_{2};\\ {y}_{2}(t)>{w}_{3}(t-\delta ),\phantom{\rule{1em}{0ex}}t\in [\delta ,1-\delta ];\\ {y}_{3}(t)>\frac{2\delta \theta}{\pi}{w}_{2}(t-\delta ),\phantom{\rule{1em}{0ex}}t\in [\delta ,1-\delta ].\end{array}$(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 ${w}_{1}$, ${\tau}_{1}$ and ${\tau}_{4}$ 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*$\delta \in (0,\frac{1}{2})$

*be fixed*.

*Let*(C1)-(C4)

*hold*.

*Assume that there exist numbers*${\tau}_{j}$, $1\le j\le 4$,

*with*

*such that*(C7)

*and*(C8)

*hold*.

*Suppose that there exist numbers*${w}_{i}$, $1\le i\le 5$,

*with*

*such that the following hold*:

- (P)
$f(u,v)<\frac{1}{{p}_{2}}({w}_{2}-\frac{{w}_{5}{p}_{3}}{q})$

*for*$u\in [0,{\tau}_{1}{w}_{5}+({\tau}_{4}-{\tau}_{1}){w}_{2}]$*and*$v\in [{w}_{1},{w}_{2}]$; - (Q)
$f(u,v)\le \frac{{w}_{5}}{q}$

*for*$u,v\in [0,{w}_{5}]$; - (R)
$f(u,v)>\frac{{w}_{3}}{{p}_{1}}$

*for*$u\in [{w}_{3}(\frac{1}{2}-{\tau}_{2}),{w}_{4}(\frac{1}{2}-{\tau}_{2})]$*and*$v\in [{w}_{3},{w}_{4}]$.

*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)):

As in the proof of Theorem 3.2, using (Q) and Lemma 3.3 we can show that $S:\overline{P}(\gamma ,{w}_{5})\to \overline{P}(\gamma ,{w}_{5})$.

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.

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 $\beta (Sx)<{w}_{2}$. 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.

*S*maps

*C*into

*C*, we find

Thus, $\beta (Sx)<{w}_{2}$ for all $x\in Q(\gamma ,\beta ,{w}_{2},{w}_{5})$ with $\psi (Sx)<{w}_{1}$.

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

*d*are in the context of Theorem 3.1 satisfying

For convenience, we take $r=1.171\times {10}^{-4}$ although this will lead to more stringent conditions.

and clearly we can easily find numbers ${w}_{i}$’s and *d* that satisfy (4.4).

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

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

where ${w}_{i}$’s satisfy (4.10).

**Remark 4.1**In Example 4.2, we see that for $(u,v)\in [{w}_{3}(\frac{1}{2}-{\tau}_{2}),{w}_{4}(\frac{1}{2}-{\tau}_{2})]\times [{w}_{3},{w}_{4}]$,

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.

## Declarations

## Authors’ Affiliations

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