# Infinitely many solutions to superlinear second order *m*-point boundary value problems

- Ruyun Ma
^{1}Email author, - Chenghua Gao
^{1}and - Xiaoqiang Chen
^{2}

**2011**:14

https://doi.org/10.1186/1687-2770-2011-14

© Ma et al; licensee Springer. 2011

**Received: **28 April 2011

**Accepted: **15 August 2011

**Published: **15 August 2011

## Abstract

We consider the boundary value problem

where:

(1) *m* ≥ 3, *η*_{
i
} ∈ (0, 1) and *α*_{
i
} *>* 0 with $A:={\sum}_{i=1}^{m-2}{\alpha}_{i}<1$;

(2) *g* : ℝ → ℝ is continuous and satisfies

and

(3) *p* : [0, 1] × ℝ^{2} → ℝ is continuous and satisfies

for some *C >* 0 and *β* ∈ (0, 1/2).

We obtain infinitely many solutions having specified nodal properties by the bifurcation techniques.

**MSC(2000)**. 34B15, 58E05, 47J10

## Keywords

## 1 Introduction

where

*m*≥ 3,

*η*

_{ i }∈ (0, 1) and

*α*

_{ i }

*>*0 with

*g*: ℝ → ℝ is continuous and satisfies

*p*: [0, 1] × ℝ

^{2}→ ℝ is continuous and satisfies

for some *C >* 0 and *β* ∈ (0, 1/2).

*n*≥ 0,

*C*

^{ n }[0, 1] will denote the usual Banach space of

*n*-times continuously differentiable functions on [0, 1], with the usual sup-type norm, denoted by || · ||

*n*. Let

*X*:= {

*u*∈

*C*

^{2}[0, 1]:

*u*satisfies (1.2)},

*Y*:=

*C*

^{0}[0, 1], with the norms | · |

_{2}and | · |

_{0}, respectively. Let

with the norms | · |_{
E
}.

*L*:

*X*→

*Y*by

In addition, for any continuous function *g* : ℝ → ℝ and any *u* ∈ *Y*, let *g*(*u*) ∈ *Y* denote the function *g*(*u*(*x*)), *x* ∈ [0, 1].

*C*

^{1}function

*u*, if

*u*(

*x*

_{0}) = 0, then

*x*

_{0}is a simple zero of

*u*, if

*u*'(

*x*

_{0}) ≠ 0. Now, for any integer

*k*≥ 1 and any

*ν*∈ {+, -}, we define sets ${S}_{k}^{\nu},{\Gamma}_{k}^{\nu}\subset {C}^{2}\left[0,1\right]$ consisting of the set of functions

*u*∈

*C*

^{2}[0, 1] satisfying the following conditions:

- (i)
*u*(0) = 0,*νu*'(0)*>*0; (ii)*u*has only simple zeros in [0, 1] and has exactly*k*- 1 zeros in (0, 1).$\underset{}{{\Gamma}_{k}^{\nu}}$ - (i)
*u*(0) = 0,*νu*'(0)*>*0; (ii)*u*' has only simple zeros in (0, 1) and has exactly*k*such zeros; (iii)*u*has a zero strictly between each two consecutive zeros of*u*'.

**Remark 1.1** If we add the restriction *u*' (1) ≠ 0 on the functions in ${\Gamma}_{k}^{\nu}$ then ${\Gamma}_{k}^{\nu}$ becomes the set ${T}_{k}^{\nu}$, which used in [1]. The reason we use ${\Gamma}_{k}^{\nu}$ rather than ${T}_{k}^{\nu}$ is that the Equation (1.1) is not autonomous anymore.

- a.
If $u\in {T}_{k}^{\nu}$, then

*u*has exactly one zero between each two consecutive zeros of*u*', and all zeros of*u*are simple. Thus,*u*has at least*k*- 1 zeros in (0, 1), and at most*k*zeros in (0, 1]; - b.
The sets ${T}_{k}^{\nu}$ are open in

*X*and disjoint; - c.
When considering the multi-point boundary condition (1.2), the sets ${T}_{k}^{\nu}$ are in fact more appropriate than the sets ${S}_{k}^{\nu}$.

The main result of this paper is the following

**Theorem 1.1** Let (H1)-(H3) hold. Then there exists an integer *k*_{0} ≥ 1 such that for all integers *k* ≥ *k*_{0} and each *ν* ∈ {+, -} the problem (1.1), (1.2) has at least one solution ${u}_{k}^{\nu}\in {\Gamma}_{k}^{\nu}$.

Superlinear problems with classical boundary value conditions have been considered in many papers, particularly in the second and fourth order cases, with either periodic or separated boundary conditions, see for example [2–11] and the references therein. Specifically, the second order periodic problem is considered in [2, 3], while [4–7] consider problems with separated boundary conditions, and results similar to Theorem 1.1 were obtained in each of these papers. The fourth order periodic problem is considered in [8–10]. Rynne [11] and De Coster [12] consider some general higher order problems with separated boundary conditions also.

(which is a nonlocal boundary value problem), under the assumptions:

(A0) *β* ∈ (0, 1) ∪ (1, ∞);

*g*: ℝ → ℝ is continuous and satisfies

*g*(

*s*)

*s >*0,

*s*≠ 0, $\frac{g\left(s\right)}{s}$

*is increasing and*

*p*: [0, 1] × ℝ

^{2}→ ℝ is a function satisfying the Carathéodory conditions and satisfies

where *M*_{1} : [0, 1] × [0, ∞) → [0, ∞) satisfies the condition: for each *s* ∈ [0, ∞), *M*_{1}(·, *s*) is integrable on [0, 1] and for each *t* ∈ [0, 1], *M*_{1}(*t*, ·) is increasing on [0, ∞) with ${s}^{-1}{\int}_{0}^{1}{M}_{1}\left(t,s\right)\mathsf{\text{d}}s\to 0$ as *s* → ∞.

Calvert and Gupta used Leray-Schauder degree and some ideas from Henrard [14] and Cappieto et al. [5] to prove the existence of infinity many solutions for (1.7), (1.8). Their results extend the main results in [14].

*m*-point boundary value problems (1.1), (1.2) under the assumptions (H1)-(H3). Obviously, our conditions (H2) and (H3) are much weaker than the corresponding restrictions imposed in [13]. Our paper uses some of ideas of Rynne [10], which deals with fourth order two-point boundary value problems. By the way, the proof [10, Lemma 2.8] contains a small error (since ||

*u*″|

_{0}≥

*ζ*

_{4}(0) ⇏ |

*u*″|

_{0}≥

*ζ*

_{4}(

*R*) there). So, we introduce a new function

*χ*(see (3.7)) with

which are required in applying Lemma 3.4.

## 2 Eigenvalues of the linear problem

Denote the spectrum of *L* by *σ*(*L*). The following spectrum results on (2.1) were established by Rynne [1], which extend the main result of Ma and O'Regan [16].

**Lemma 2.1**. [1, Theorem 3.1] The spectrum

*σ*(

*L*) consists of a strictly increasing sequence of eigenvalues

*λ*

_{ k }

*>*0,

*k*= 1, 2, ..., with corresponding eigenfunctions ${\varphi}_{k}\left(x\right)=sin\left({\lambda}_{k}^{1\u22152}x\right)$. In addition,

- (i)
lim

_{k→∞}*λ*_{ k }= ∞; - (ii)
$\varphi \in {T}_{k}^{\nu}$, for each

*k*≥ 1, and*ϕ*_{1}is strictly positive on (0, 1).

**Lemma 2.2** [1, Theorem 3.8] For each *k* ≥ 1, the algebraic multiplicity of the characteristic value *λ*_{
k
} of *L*^{-1} : *Y* → *Y* is equal to 1.

## 3 Proof of the main results

*u*∈

*X*, we define

*e*(

*u*)(·): [0, 1] → ℝ by

*s*∈ ℝ, let

*s*≥ 0, let

where *α* ∈ [0, 1] is an arbitrary fixed number and *λ* ∈ ℝ. In the following lemma (*λ*, *u*) ∈ ℝ × *X* will be an arbitrary solution of (3.2).

*b*

_{1}≥ 1 such that

By (1.2), we have the following

**Lemma 3.1**. Let (H1) hold and let

*u*∈

*X*. Then

**Lemma 3.2**. Let

*u*be a solution of (3.2). Then for any

*x*

_{0},

*x*

_{1}∈ [0, 1],

**Proof**. Multiply (3.2) by *u*' and integrate from *x*_{0} to *x*_{1}, then we get the desired result. ■

*R*∈ (0, ∞) so large that

*R*≥

*b*

_{1}and

**Lemma 3.3**. There exists an increasing function

*ζ*

_{1}: [0, ∞) → [0, ∞), such that for any solution

*u*of (3.2) with 0 ≤

*λ*≤

*R*and |

*u*(

*x*0)| + |

*u*'(

*x*

_{0})| ≤

*R*for some

*x*

_{0}∈ [0, 1], we have

**Proof**. Choose

*x*

_{1}∈ [0, 1] such that |

*u*'|

_{0}= |

*u*'(

*x*

_{1})|. We obtain from Lemma 3.2 that

■

Clearly, the function is nondecreasing.

**Lemma 3.4** Let *u* be a solution of (3.2) with 0 ≤ *λ* ≤ *R* and |*u*'|_{0} ≥ *ζ*_{2}(*R*) for some *R >* 0. Then, for any *x* ∈ [0, 1] with |*u*(*x*)| ≤ *R*, we have |*u*'(*x*)| ≥ *R*^{2}.

**Proof**. Suppose, on the contrary that there exists

*x*

_{0}∈ (0, 1) such that |

*u*(

*x*

_{0})| ≤

*R*and |

*u*'(

*x*

_{0})|

*< R*

^{2}. Then

*λ*≤

*R < R*+

*R*

^{2}and using Lemma 3.3, it concludes that

However, this is impossible if |*u*'|_{0} ≥ *ζ*_{2}(*R*). ■

*R > b*

_{1}, let us define

where *θ* : ℝ → ℝ is a strictly increasing, *C*^{∞}-function with *θ*(*s*) = 0, *s* ≤ 1 and *θ*(*s*) = 1, *s* ≥ 2. The nonlinear term in (3.8) is a continuous function of (*λ*, *u*) ∈ ℝ × *X* and is zero for *λ* ∈ ℝ, |*u*'|_{0} ≤ *χ*(*λ*), so (3.8) becomes a linear eigenvalue problem in this region, and overall the problem can be regarded as a bifurcation (from *u* = 0) problem.

The next lemma now follows immediately.

**Lemma 3.5**The set of solutions (

*λ*,

*u*) of (3.8) with |

*u*'|

_{0}≤

*χ*(

*λ*) is

We also have the following global bifurcation result for (3.8).

**Lemma 3.6**For each

*k*≥ 1 and

*ν*∈ {+, -}, there exists a connected set ${\mathcal{C}}_{k}^{\nu}\subset \mathbb{R}\times E$ of nontrivial solutions of (3.8) such that ${\mathcal{C}}_{k}^{\nu}\cup \left({\lambda}_{k},0\right)$ is closed and connected and:

- (i)
there exists a neighborhood

*N*_{ k }of (*λ*_{ k }, 0) in ℝ ×*E*such that ${N}_{k}\cap {\mathcal{C}}_{k}^{\nu}\subset \mathbb{R}\times {\Gamma}_{k}^{\nu}$, - (ii)
${\mathcal{C}}_{k}^{\nu}$ meets infinity in ℝ ×

*E*(that is, there exists a sequence $\left({\lambda}_{n},{u}_{n}\right)\in {\mathcal{C}}_{k}^{\nu},n=1,2,\dots $, such that |*λ*_{ n }| + |*u*_{ n }|_{ E }→ ∞).

**Proof**. Since

*L*

^{-1}:

*Y*→

*X*exists and is bounded, (3.8) can be rewritten in the form

and since *L*^{-1} can be regarded as a compact operator from *Y* to *E*, it is clear that finding a solution (*λ*, *u*) of (3.8) in ℝ × *E* is equivalent to finding a solution of (3.9) in ℝ × *E*. Now, by the similar method used in the proof of [1, Theorem 4.2]), we may deduce the desired result.

■

Since *e*(*u*)(*t*) σ 0 in (3.8), nodal properties need not be preserved. However, we will rely on preservation of nodal properties for "large" solutions, encapsulated in the following result.

**Lemma 3.7** If (*λ*, *u*) is a solution of (3.8) with *λ* ≥ 0 and |*u*'|_{0}*> χ*(*λ*), then $u\in {\Gamma}_{k}^{\nu}$, for some *k* ≥ 1 and *ν* ∈ {+, -}.

**Proof**. If $u\notin {\Gamma}_{k}^{\nu}$ for any *k* ≥ 1 and *ν*, then one of the following cases must occur:

*Case 1. u*'(0) = 0;

*Case 2. u*' (*τ*) = *u*″(*τ*) = 0 for some *τ* ∈ (0, 1].

In the Case 1, *u*(*t*) ≡ 0 on [0, 1]. This contradicts the assumption |*u*'|_{0}*> χ*(*λ*) ≥ *ζ*_{2}(*λ*). So this case cannot occur.

*u*'|

_{0}

*> χ*(

*λ*), we have from the definition of

*θ*that

*u*(

*τ*)|

*> R*≥

*b*

_{1}. Combining this with (3.11) and (3.3), it concludes that

which contradicts (3.10). So, Case 2 cannot occur.

Therefore, $u\in {\Gamma}_{k}^{\nu}$ for any *k* ≥ 1 and *ν* ∈ {+, -}. ■

In view of Lemmas 3.5 and 3.7, in the following lemma, we suppose that (*λ*, *u*) is an arbitrary nontrivial solution of (3.8) with *λ* ≥ 0 and $u\in {\Gamma}_{k}^{\nu}$, for some *k* ≥ 1 and *ν*.

**Lemma 3.8**. There exists an integer

*k*

_{0}≥ 1 (depending only on

*χ*(0)) such that for any nontrivial solution

*u*of (3.8) with

*λ*= 0 and

*χ*(0) ≤ |

*u*'|

_{0}≤ 2

*χ*(0), we have

**Proof**. Let

*x*

_{1},

*x*

_{2}be consecutive zeros of

*u*. Then there exists

*x*

_{3}∈ (

*x*

_{1},

*x*

_{2}) such that

*u*'(

*x*

_{3}) = 0, and hence, Lemma 3.4, (3.3), and (3.7) yield that |

*u*(

*x*

_{3})|

*>*1. Since

*τ*

_{1}∈ (

*x*

_{3},

*x*

_{2}),

*τ*

_{2}∈ (

*x*

_{1},

*x*

_{3}), it follows that

*u*'|

_{0}

*> χ*(0) ≥

*ζ*

_{2}(

*R*) implies that $u\in {\Gamma}_{k}^{\nu}$ for some

*k*∈ ∞ and

*ν*∈ {+, -}, and subsequently, there exist 0

*< r*

_{1}

*< r*

_{2}

*<*· · ·

*< r*

_{k-1}, such that

and accordingly, *k <* |*u*'|_{0}/2 + 1 ≤ *χ*(0) + 1. ■

**Lemma 3.9**. Suppose that 0 ≤ *λ* ≤ *R* and |*u*'|_{0} ≥ *χ*(*R*). Then *W*_{
R
} (*u*) consists of at least *k* intervals and at most *k* + 1 intervals, each of length less than 2*/R*, and *V*_{
R
} (*u*) consists of at least *k* intervals and at most *k* + 1 intervals.

**Proof**. Lemma 3.4 implies that |

*u*'(

*x*)| ≥

*R*

^{2}for all

*x*∈

*W*

_{ R }(

*u*). For any interval

*I*⊂

*W*

_{ R }(

*u*),

*u*' does not change sign on

*I*, say,

We claim that the length of *I* is less than 2/*R*.

*x*,

*y*∈

*I*with

*x > y*, say,

can be treated by the similar method. Since *u* is monotonic in any subinterval containing in *W*_{
R
} (*u*), the desired result is followed. ■

**Lemma 3.10**. There exists

*ζ*

_{3}with lim

_{R→∞}

*ζ*

_{3}(

*R*) = 0, and

*η*

_{1}≥ 0 such that, for any

*R*≥

*η*

_{1}, if either

- (a)
0 ≤

*λ*≤*R*and |*u*'|_{0}= 2*χ*(*R*), or - (b)
*λ*=*R*and*χ*(*R*) ≤ |*u*'|_{0}≤ 2*χ*(*R*),

then the length of each interval of *V*_{
R
} (*u*) is less than *ζ*_{3}(*R*).

**Proof**. Define

*H*=

*H*(

*R*) by

and let *ζ*_{3}(*R*) := 2*π*/*H*(*R*). By (1.4), lim_{R→∞}*H*(*R*) = ∞, so lim_{R→∞}*ζ*_{3}(*R*) = 0, and we may choose *η*_{1} ≥ *b*_{1} sufficiently large that *H*(*R*) *>* 0 for all *R* ≥ *η*_{1}.

*u*(

*x*)| ≤

*R*on [0, 1], then Lemma 3.4 yields that either

However, these contradict the boundary conditions (1.2), since (H1) implies *u*'(*s*_{0}) = 0 for some *s*_{0} ∈ (0, 1). Therefore, (3.15) is valid.

*x*

_{0},

*x*

_{2}such that either

- (1)
*u*(*x*_{0}) =*u*(*x*_{2}) =*R*and*u > R*on (*x*_{0},*x*_{2}) or - (2)
*u*(*x*_{0}) =*R*,*x*_{2}= 1 and*u > R*on (*x*_{0}, 1].

*u <*0 is similar). Let

*H*(

*R*), if either (a) or (b) holds then

and by Lemma 3.4, *u*'(*x*_{0}) *>* 0, and *u*'(*x*_{2}) *<* 0, if *x*_{2}*<* 1.

*x*

_{2}-

*x*

_{0}

*> ζ*

_{3}(

*R*), that is,

*l*:= 2

*π*/(

*x*

_{2}-

*x*

_{0})

*< H*(

*R*). Defining

*x*

_{1}= (

*x*

_{0}+

*x*

_{2})/2 and

and this contradiction shows that *x*_{2} - *x*_{0} ≤ *ζ*_{3}(*R*), which proves the lemma.

■

Now, we are in the position to prove Theorem 1.1.

**Proof of Theorem 1.1**Now, choose an arbitrary integer

*k*≥

*k*

_{0}and

*ν*∈ {+, -}, and choose Λ

*>*max{

*η*

_{1},

*μ*

_{ k }} (Here, we assume Λ

*> η*

_{1}, so that Lemma 3.10 could be applied!) such that

*u*'|

_{0}≥

*χ*(Λ), then the length of each interval of

*W*

_{Λ}(

*u*) is less than $\frac{2}{\Lambda}$ for 0 ≤

*λ*≤ Λ. This together with (3.16) and Lemma 3.10 imply that there exists no solution (

*λ*,

*u*) of (3.8), which satisfies either

- (a)
0 ≤

*λ*≤ Λ and |*u*'|_{0}= 2*χ*(Λ) or - (b)
*λ*= Λ and*χ*(Λ) ≤ |*u*'|_{0}≤ 2*χ*(Λ).

*B*through the set

*D*

_{1}, while from Lemma 3.7, ${\mathcal{C}}_{k}^{\nu}\cap B\subset \mathbb{R}\times {\Gamma}_{k}^{\nu}$. Thus, by Lemma 3.6 and the fact

*B*. (Suppose, on the contrary that ${\mathcal{C}}_{k}^{\nu}$ does not "leave"

*B*, then

which contradicts the fact that ${\mathcal{C}}_{k}^{\nu}$ joins (*μ*_{
k
} , 0) to infinity in ℝ × *E*.) Since ${\mathcal{C}}_{k}^{\nu}$ is connected, it must intersect *∂B*. However, Lemmas 3.8-3.10 (together with (3.16)) show that the only portion of *∂B* (other than *D*_{1}), which ${\mathcal{C}}_{k}^{\nu}$ can intersect is *D*_{2}. Thus, there exists a point $\left(0,\phantom{\rule{2.77695pt}{0ex}}{u}_{k}^{\nu}\right)\in {\mathcal{C}}_{k}^{\nu}\cap {D}_{2}$, and clearly ${u}_{k}^{\nu}$ provides the desired solution of (1.1)-(1.2).

## Declarations

### Acknowledgements

The authors are grateful to the anonymous referee for his/her valuable suggestions. Supported by the NSFC(No.11061030), the Fundamental Research Funds for the Gansu Universities.

## Authors’ Affiliations

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