Upper and lower solution method for n th-order BVPs on an infinite interval

Hairong Lian; Junfang Zhao; Ravi P Agarwal

doi:10.1186/1687-2770-2014-100

Research
Open Access

Upper and lower solution method for nth-order BVPs on an infinite interval

Hairong Lian¹Email author,
Junfang Zhao¹ and
Ravi P Agarwal²

Boundary Value Problems20142014:100

https://doi.org/10.1186/1687-2770-2014-100

Received: 9 January 2014
Accepted: 7 April 2014
Published: 7 May 2014

Abstract

This work is devoted to the study of n th-order ordinary differential equations on a half-line with Sturm-Liouville boundary conditions. The existence results of a solution, and triple solutions, are established by employing a generalized version of the upper and lower solution method, the Schäuder fixed point theorem, and topological degree theory. In our problem the nonlinearity depends on derivatives, and we allow solutions to be unbounded, which is an extra interesting feature.

MSC:34B15, 34B40.

Keywords

higher order
boundary value problem
half-line
upper solution
lower solution

1 Introduction

In this paper, we study n th-order ordinary differential equations on a half-line,

- u^{(n)} (t) = q (t) f (t, u (t), \dots, u^{(n - 1)} (t)), 0 < t < + \infty,

(1)

together with the Sturm-Liouville boundary conditions

{\begin{matrix} u^{(i)} (0) = A_{i}, i = 0, 1, \dots, n - 3, \\ u^{(n - 2)} (0) - a u^{(n - 1)} (0) = B, \\ u^{(n - 1)} (+ \infty) = C, \end{matrix}

(2)

where $q : (0, + \infty) \to (0, + \infty)$ , $f : [0, + \infty) \times R^{n} \to R$ are continuous, $a > 0$ , $A_{i}, B, C \in R$ , $i = 0, 1, \dots, n - 3$ , $u (+ \infty) = {lim}_{t \to + \infty} u (t)$ .

Higher-order boundary value problems (BVPs) have been studied in many papers, such as [1–3] for two-point BVP, [4, 5] for multipoint BVP, and [6–9] for infinite interval problem. However, most of these works have been done either on finite intervals, or for bounded solutions on an infinite interval. The authors in [1, 2, 5, 10–15] assumed one pair of well-ordered upper and lower solutions, and then applied some fixed point theorems or a monotone iterative technique to obtain a solution. In [5, 11, 16], the authors assumed two pairs of upper and lower solutions and showed the existence of three solutions.

Infinite interval problems occur in the study of radially symmetric solutions of nonlinear elliptic equations; see [6, 7, 17]. A principal source of such problems is fluid dynamics. In boundary layer theory, Blasius-type equations lead to infinite interval problems. Semiconductor circuits and soil mechanics are other applied fields. In addition, some singular boundary value problems on finite intervals can be converted into equivalent nonlinear problems on semi-infinite intervals [7]. During the last few years, fixed point theorems, shooting methods, upper and lower technique, etc. have been used to prove the existence of a single solution or multiple solutions to infinite interval problems; see [6–13, 17–24] and the references therein.

When applying the upper and lower solution method to infinite interval problems, the solutions are always assumed to be bounded. For example, in [10], Agarwal and O’Regan discussed the following second-order Sturm-Liouville boundary value problem:

{\begin{matrix} \frac{1}{p (t)} {(p (t) y^{'} (t))}^{'} = q (t) f (t, y (t)), t \in (0, + \infty), \\ - a_{0} y (0) + b_{0} {lim}_{t \to 0^{+}} p (t) y^{'} (t) = c_{0}, or {lim}_{t \to 0^{+}} p (t) y^{'} (t) = 0, \\ y (t) bounded on [0, + \infty), or {lim}_{t \to + \infty} y (t) = 0, \end{matrix}

where $a_{0} > 0$ , $b_{0} ⩾ 0$ . They established existence criteria by using a diagonalization argument and existence results of appropriate boundary value problems on finite intervals.

Eloe et al. [11] studied the BVP

{\begin{matrix} x^{″} (t) - a (t) x (t) + f (t, x (t)) = 0, t \in (0, + \infty), \\ x (0) = x_{0}, x (t) bounded on [0, + \infty) . \end{matrix}

They employed the technique of lower and upper solutions and the theory of fixed point index to obtain the existence of at least three solutions.

The problems related to global solutions, especially when the boundary data are prescribed asymptotically and the solutions may be unbounded, have been briefly discussed in [7, 9]. Recently, Yan et al. [14] developed the upper and lower solution theory for the boundary value problem

{\begin{matrix} y^{″} (t) + Φ (t) f (t, y (t), y^{'} (t)) = 0, t \in (0, + \infty), \\ a y (0) - b y^{'} (0) = y_{0} ⩾ 0, {lim}_{t \to + \infty} y^{'} (t) = k > 0, \end{matrix}

where $a > 0$ , $b > 0$ . By using the upper and lower solutions method and a fixed point theorem, they presented sufficient conditions for the existence of unbounded positive solutions; however, their results are suitable only to positive solutions. In [12, 13], Lian et al. generalized their existence results to unbounded solutions, and somewhat weakened the conditions in [14]. In 2012, Zhao et al. [15] similarly investigated the solutions to multipoint boundary value problems in Banach spaces on an infinite interval.

Inspired by the works listed above, in this paper, we aim to discuss the n th-order differential equation on a half-line with Sturm-Liouville boundary conditions. To the best of our knowledge, this is the first attempt to find the unbounded solutions to higher-order infinite interval problems by using the upper and lower solution technique. Since, the half-line is noncompact, the discussion is rather involved. We begin with the assumption that there exist a pair of upper and lower solutions for problem (1)-(2), and the nonlinear function f satisfies a Nagumo-type condition. Then, by using the truncation technique and the upper and lower solutions, we estimate a-priori bounds of modified problems. Next, the Schäuder fixed point theorem is used which guarantees the existence of solutions to (1)-(2). We also assume two pairs of upper and lower solutions and show that this infinite interval problem has at least three solutions. In the last section, an example is included which illustrates the main result.

2 Preliminaries

In this section, we present some definitions and lemmas to be used in the main theorem of this paper.

Consider the space X defined by

X = {u \in C^{n - 1} [0, + \infty), lim_{t \to + \infty} \frac{u^{(i)} (t)}{v_{i} (t)} exist, i = 0, 1, \dots, n - 1}

(3)

with the norm

∥ \cdot ∥

given by

∥ u ∥ = max {{∥ u ∥}_{0}, {∥ u ∥}_{1}, \dots, {∥ u ∥}_{n - 1}},

(4)

where

v_{i} (t) = 1 + t^{n - 1 - i}

and

{∥ u ∥}_{i} = sup_{t \in [0, + \infty)} | \frac{u^{(i)} (t)}{v_{i} (t)} |, i = 0, 1, \dots, n - 1 .

(5)

Then $(X, ∥ \cdot ∥)$ is a Banach space.

To obtain a solution of the BVP (1)-(2), we need a mapping whose kernel $G (t, s)$ is the Green function of $- u^{(n)} (t) = 0$ with the homogeneous boundary conditions (2), which is given in the following lemma.

Lemma 2.1 Let

e \in L^{1} [0, + \infty)

. Then the linear boundary value problem

{\begin{matrix} - u^{(n)} (t) = e (t), 0 < t < + \infty, \\ u^{(i)} (0) = A_{i}, i = 0, 1, \dots, n - 3, \\ u^{(n - 2)} (0) - a u^{(n - 1)} (0) = B, \\ u^{(n - 1)} (+ \infty) = C \end{matrix}

(6)

has a unique solution given by

u (t) = l (t) + \int_{0}^{+ \infty} G (t, s) e (s) d s,

(7)

where

l (t) = \sum_{k = 0}^{n - 3} \frac{A_{k}}{k!} t^{k} + \frac{a C + B}{(n - 2)!} t^{n - 2} + \frac{C}{(n - 1)!} t^{n - 1},

and

G (t, s) = {\begin{matrix} \frac{a}{(n - 2)!} t^{n - 2} + \sum_{k = 0}^{n - 2} \frac{{(- 1)}^{k}}{(k + 1)! (n - 2 - k)!} s^{k + 1} t^{n - 2 - k}, & 0 ⩽ s ⩽ t < + \infty, \\ \frac{a}{(n - 2)!} t^{n - 2} + \frac{1}{(n - 1)!} t^{n - 1}, & 0 ⩽ t ⩽ s < + \infty . \end{matrix}

(8)

Proof Let

v (t) = u^{(n - 2)} (t)

. Then from (6), we obtain the Sturm-Liouville boundary value problem

{\begin{matrix} - v^{″} (t) = e (t), 0 < t < + \infty, \\ v (0) - a v^{'} (0) = B, \\ v^{'} (+ \infty) = C \end{matrix}

(9)

and the initial value problem

{\begin{matrix} u^{(n - 2)} (t) = v (t), 0 < t < + \infty, \\ u^{(i)} (0) = A_{i}, i = 0, 1, \dots, n - 3 . \end{matrix}

(10)

Clearly, (9) has a unique solution

v (t) = a C + B + C t + \int_{0}^{+ \infty} g (t, s) e (s) d s,

where

g (t, s) = {\begin{matrix} a + s, 0 ⩽ s ⩽ t < + \infty, \\ a + t, 0 ⩽ t ⩽ s < + \infty . \end{matrix}

Now integrating (10) and applying the initial conditions, we obtain (7). □

Lemma 2.2 The function $G (t, s)$ defined in (8) is $(n - 1)$ times continuously differentiable on $[0, + \infty) \times [0, + \infty)$ . For any $s \in [0, + \infty)$ , its ith derivative is uniformly continuous in t on any compact interval of $[0, + \infty)$ and is uniformly bounded on $[0, + \infty)$ .

Proof We denote

g_{i} (t, s) = \frac{\partial^{i} G (t, s)}{\partial t^{i}}

, which is also used in the later part of the paper. By direct calculations, we have

g_{i} (t, s) = {\begin{matrix} \frac{a t^{n - 2 - i}}{(n - 2 - i)!} + \sum_{k = 0}^{n - 2 - i} \frac{{(- 1)}^{k} s^{k + 1} t^{n - 2 - k - i}}{(k + 1)! (n - 2 - k - i)!}, & 0 ⩽ s ⩽ t < + \infty, \\ \frac{a}{(n - 2 - i)!} t^{n - 2 - i} + \frac{1}{(n - 1 - i)!} t^{n - 1 - i}, & 0 ⩽ t ⩽ s < + \infty . \end{matrix}

(11)

Obviously,

g_{i} (t, s)

is uniformly continuous in t on any compact interval of

[0, + \infty)

for any

s \in [0, + \infty)

. Now since, for all integers k and l,

sup_{t \in [0, + \infty)} \frac{t^{k}}{1 + t^{l}} = {\begin{matrix} \frac{l - k}{l} {(\frac{k}{l - k})}^{\frac{k}{l}}, & k < l, \\ 1, & k = l, \\ + \infty, & k > l \end{matrix}

from (11), when

s ⩽ t

, it follows that

\begin{array}{rcl} sup_{t \in [0, + \infty)} | \frac{g_{i} (t, s)}{v_{i} (t)} | & = & sup_{t \in [0, + \infty)} | \frac{a t^{n - 2 - i}}{(n - 2 - i)! v_{i} (t)} + \sum_{k = 0}^{n - 2 - i} \frac{{(- 1)}^{k} s^{k + 1} t^{n - 2 - k - i}}{(k + 1)! (n - 2 - k - i)! v_{i} (t)} | \\ ⩽ & sup_{t \in [0, + \infty)} (\frac{a t^{n - 2 - i}}{(n - 2 - i)! v_{i} (t)} + \sum_{k = 0}^{n - 2 - i} \frac{t^{k + 1} t^{n - 2 - k - i}}{(k + 1)! (n - 2 - k - i)! v_{i} (t)}) \\ ⩽ & \frac{a}{(n - 2 - i)!} sup_{t \in [0, + \infty)} \frac{t^{n - 2 - i}}{v_{i} (t)} + \frac{n - 1 - i}{(n - 2 - i)!} sup_{t \in [0, + \infty)} \frac{t^{n - 1 - i}}{v_{i} (t)} \\ = & \frac{a}{(n - 1 - i)!} {(n - 2 - i)}^{\frac{n - 2 - i}{n - 1 - i}} + \frac{n - 1 - i}{(n - 2 - i)!}, \end{array}

and, when

s ⩾ t

, we have

\begin{array}{rcl} sup_{t \in [0, + \infty)} | \frac{g_{i} (t, s)}{v_{i} (t)} | & = & sup_{t \in [0, + \infty)} | \frac{a t^{n - 2 - i}}{(n - 2 - i)! v_{i} (t)} + \frac{t^{n - 1 - i}}{(n - 1 - i)! v_{i} (t)} | \\ ⩽ & \frac{a}{(n - 2 - i)!} sup_{t \in [0, + \infty)} \frac{t^{n - 2 - i}}{v_{i} (t)} + \frac{1}{(n - 1 - i)!} sup_{t \in [0, + \infty)} \frac{t^{n - 1 - i}}{v_{i} (t)} \\ = & \frac{a}{(n - 1 - i)!} {(n - 2 - i)}^{\frac{n - 2 - i}{n - 1 - i}} + \frac{1}{(n - 1 - i)!} . \end{array}

Thus, we have

{∥ G (t, s) ∥}_{i} ⩽ \frac{a}{(n - 1 - i)!} {(n - 2 - i)}^{\frac{n - 2 - i}{n - 1 - i}} + \frac{n - 1 - i}{(n - 2 - i)!} : = K_{i},

(12)

for $i = 1, 2, \dots, n - 1$ . □

When applying the Schäuder fixed point theorem to prove the existence result, it is necessary to show that the operator is completely continuous. While the usual Arezà-Ascoli lemma fails here due to the non-compactness of $[0, + \infty)$ , the following generalization (see [6, 13]) will be used.

Lemma 2.3

M \subset X

is relatively compact if the following conditions hold:

1.

all the functions from M are uniformly bounded;
2.

all the functions from M are equi-continuous on any compact interval of $[0, + \infty)$ ;
3.

all the functions from M are euqi-convergent at infinity, that is, for any given $ϵ > 0$ , there exists a $T = T (ϵ) > 0$ such that for any $u \in M$ ,
$| \frac{u^{(i)} (t)}{v_{i} (t)} - \frac{u^{(i)} (+ \infty)}{v_{i} (+ \infty)} | < ϵ, t > T, i = 0, 1, \dots, n - 1 .$

Finally, we define lower and upper solutions of (1)-(2), and introduce the Nagumo-type condition.

Definition 2.1 A function

α (t) \in C^{n - 1} [0, + \infty) \cap C^{n} (0, + \infty)

satisfying

{\begin{matrix} - α^{(n)} (t) ⩽ q (t) f (t, α (t), \dots, α^{(n - 1)} (t)), 0 < t < + \infty, \\ α^{(i)} (0) ⩽ A_{i}, i = 0, 1, \dots, n - 3, \\ α^{(n - 2)} (0) - a α^{(n - 1)} (0) ⩽ B, \\ α^{(n - 1)} (+ \infty) < C \end{matrix}

(13)

is called a lower solution of (1)-(2). If the inequalities are strict, it is called a strict lower solution.

Definition 2.2 A function

β (t) \in C^{n - 1} [0, + \infty) \cap C^{n} (0, + \infty)

satisfying

{\begin{matrix} - β^{(n)} (t) ⩾ q (t) f (t, β (t), \dots, β^{(n - 1)} (t)), 0 < t < + \infty, \\ β^{(i)} (0) ⩾ A_{i}, i = 0, 1, \dots, n - 3, \\ β^{(n - 2)} (0) - a β^{(n - 1)} (0) ⩾ B, \\ β^{(n - 1)} (+ \infty) > C \end{matrix}

(14)

is called an upper solution of (1)-(2). If the inequalities are strict, it is called a strict upper solution.

Definition 2.3 Let α, β be the lower and upper solutions of BVP (1)-(2) satisfying

α^{(i)} (t) ⩽ β^{(i)} (t), i = 0, 1, \dots, n - 2,

on

[0, + \infty)

. We say f satisfies a Nagumo condition with respect to α and β if there exist positive functions ψ and

h \in C [0, + \infty)

such that

| f (t, u_{0}, u_{1}, \dots, u_{n - 1}) | ⩽ ψ (t) h (| u_{n - 1} |)

(15)

for all

(t, u_{0}, u_{1}, \dots, u_{n - 1}) \in [0, + \infty) \times [α (t), β (t)] \times \dots \times [α^{(n - 2)} (t), β^{(n - 2)} (t)] \times R

and

\int_{0}^{+ \infty} q (s) ψ (s) d s < + \infty, \int^{+ \infty} \frac{s}{h (s)} d s = + \infty .

3 The existence results

Our existence theory is based on using the unbounded lower and upper solution technique. Here we list some assumptions for convenience.

H₁: BVP (1)-(2) has a pair of upper and lower solutions β, α in X with

α^{(i)} (t) ⩽ β^{(i)} (t), i = 0, 1, \dots, n - 2, t \in [0, + \infty),

and $f \in C ([0, + \infty) \times R^{n}, R)$ satisfies the Nagumo condition with respect to α and β.

H₂: For any fixed

t \in [0, + \infty)

,

u_{n - 2}, u_{n - 1} \in R

, when

α^{(i)} (t) ⩽ u_{i} ⩽ β^{(i)} (t)

,

i = 0, 1, \dots, n - 3

, the following inequality holds:

\begin{array}{rcl} f (t, α (t), \dots, α^{(i)} (t), \dots, u_{n - 2}, u_{n - 1}) & ⩽ & f (t, u_{0}, \dots, u_{i}, \dots, u_{n - 2}, u_{n - 1}) \\ ⩽ & f (t, β (t), \dots, β^{(i)} (t), \dots, u_{n - 2}, u_{n - 1}) . \end{array}

H₃: There exists a constant

γ > 1

such that

sup_{0 ⩽ t < + \infty} {(1 + t)}^{γ} q (t) ψ (t) < + \infty,

where ψ is the function in Nagumo’s condition of f.

Lemma 3.1 Suppose conditions (H₁) and (H₃) hold. Then there exists a constant

R > 0

such that every solution u of (1)-(2) with

α^{(i)} (t) ⩽ u ⩽ β^{(i)} (t), i = 0, 1, \dots, n - 2, 0 ⩽ t < + \infty,

satisfies ${∥ u ∥}_{n - 1} ⩽ R$ .

Proof Set

\begin{matrix} M_{0} = sup_{0 ⩽ t < + \infty} {(1 + t)}^{γ} q (t) ψ (t), \\ M_{1} = sup_{0 ⩽ t < + \infty} \frac{β^{(n - 2)} (t)}{{(1 + t)}^{γ}} - inf_{0 ⩽ t < + \infty} \frac{α^{(n - 2)} (t)}{{(1 + t)}^{γ}}, \\ M_{2} = max {{∥ β ∥}_{n - 2}, {∥ α ∥}_{n - 2}}, \\ M_{3} = max {sup_{δ ⩽ t < + \infty} \frac{β^{(n - 2)} (t) - α^{(n - 2)} (0)}{t}, sup_{δ ⩽ t < + \infty} \frac{β^{(n - 2)} (0) - α^{(n - 2)} (t)}{t}}, \end{matrix}

where

δ > 0

is any arbitrary constant. Choose

R > C

where C is the nonhomogeneous boundary data, and

η ⩾ M_{3}

satisfies

\int_{η}^{R} \frac{s}{h (s)} d s ⩾ M_{0} (M_{1} + \frac{γ}{γ - 1} M_{2}) .

If

| u^{(n - 1)} (t) | ⩽ R

holds for any

t \in [0, + \infty)

, then the result follows immediately. If not, we claim that

| u^{(n - 1)} (t) | > η

does not hold for all

t \in [0, + \infty)

. Otherwise, without loss of generality, we suppose

u^{(n - 1)} (t) > η, t \in [0, + \infty) .

But then, for any

t ⩾ δ > 0

, it follows that

\begin{array}{rcl} \frac{β^{(n - 2)} (t) - α^{(n - 2)} (0)}{t} & ⩾ & \frac{u^{(n - 2)} (t) - u^{(n - 2)} (0)}{t} \\ = & \frac{1}{t} \int_{0}^{t} u^{(n - 1)} (s) d s \\ > & η ⩾ \frac{β^{(n - 2)} (t) - α^{(n - 2)} (0)}{t}, \end{array}

which is a contraction. So there must exist

t^{*} \in [0, + \infty)

such that

| u^{(n - 1)} (t^{*}) | ⩽ η

. Furthermore, if

| u^{(n - 1)} (t) | ⩽ η, \forall t \in [0, + \infty),

just take

R = η

and then the proof is completed. Finally, there exist

[t_{1}, t_{2}] \subset [0, + \infty)

such that

| u^{(n - 1)} (t_{1}) | = η

,

| u^{(n - 1)} (t) | > η

,

t \in (t_{1}, t_{2}]

or

| u^{(n - 1)} (t_{2}) | = η

,

| u^{(n - 1)} (t) | > η

,

t \in [t_{1}, t_{2})

. Suppose that

u^{(n - 1)} (t_{1}) = η

,

u^{(n - 1)} (t) > η

,

t \in (t_{1}, t_{2}]

. Obviously,

\begin{array}{rcl} \int_{u^{(n - 1)} (t_{1})}^{u^{(n - 1)} (t_{2})} \frac{s}{h (s)} d s & = & \int_{t_{1}}^{t_{2}} \frac{u^{(n - 1)} (s)}{h (u^{(n - 1)} (s))} u^{(n)} (s) d s \\ = & \int_{t_{1}}^{t_{2}} \frac{- q (s) f (s, u (s), \dots, u^{(n - 1)} (s))}{h (u^{(n - 1)} (s))} u^{(n - 1)} (s) d s \\ ⩽ & \int_{t_{1}}^{t_{2}} q (s) ψ (s) u^{(n - 1)} (s) d s ⩽ M_{0} \int_{t_{1}}^{t_{2}} \frac{u^{(n - 1)} (s)}{{(1 + s)}^{γ}} d s \\ = & M_{0} (\int_{t_{1}}^{t_{2}} {(\frac{u^{(n - 2)} (s)}{{(1 + s)}^{γ}})}^{'} d s - \int_{t_{1}}^{t_{2}} u^{(n - 2)} (s) {(\frac{1}{{(1 + s)}^{γ}})}^{'} d s) \\ ⩽ & M_{0} (M_{1} + M_{2} \int_{0}^{+ \infty} \frac{γ}{{(1 + s)}^{γ}} d s) ⩽ \int_{η}^{R} \frac{s}{h (s)} d s, \end{array}

from which one concludes that $u^{(n - 1)} (t_{2}) ⩽ R$ . Since $t_{1}$ and $t_{2}$ are arbitrary, we have $u^{(n - 1)} (t) ⩽ R$ if $u^{(n - 1)} (t) ⩾ η$ for $t \in [0, + \infty)$ . In a similarly way, we can show that $u^{(n - 1)} (t) ⩾ - R$ , if $u^{(n - 1)} (t) ⩽ - η$ for $t \in [0, + \infty)$ .

Therefore there exists a $R > 0$ , just related with α, β, and ψ, h under the Nagumo condition of f, such that ${∥ u ∥}_{n - 1} ⩽ R$ . □

Remark 3.1 Similarly, we can prove that

| β^{(n - 1)} (t) | ⩽ R, | α^{(n - 1)} (t) | ⩽ R, t \in [0, + \infty) .

Remark 3.2 The Nagumo condition plays a key role in estimating the prior bound for the $(n - 1)$ th derivative of the solution of BVP (1)-(2). Since the upper and lower solutions are in X, $α^{(n - 2)} (t)$ and $β^{(n - 2)} (t)$ may be asymptotic linearly at infinity.

Theorem 3.2 Suppose the conditions (H₁)-(H₃) hold. Then BVP (1)-(2) has at least one solution

u \in C^{n - 1} [0, + \infty) \cap C^{n} (0, + \infty)

satisfying

α^{(i)} (t) ⩽ u^{(i)} (t) ⩽ β^{(i)} (t), i = 0, 1, \dots, n - 2, t \in [0, + \infty) .

Moreover, there exists a $R > 0$ such that ${∥ u ∥}_{n - 1} ⩽ R$ .

Proof Let

R > 0

be the same as in Lemma 3.1. Define the auxiliary functions

f_{0}, f_{1}, \dots, f_{n - 2}

, and

F : [0, + \infty) \times R^{n} \to R

as

\begin{matrix} f_{0} (t, u_{0}, u_{1}, \dots, u_{n - 1}) = {\begin{matrix} f (t, β, u_{1}, \dots, u_{n - 1}), & u_{0} > β (t), t \in [0, + \infty), \\ f (t, u_{0}, u_{1}, \dots, u_{n - 1}), & α (t) ⩽ u_{0} ⩽ β (t), t \in [0, + \infty), \\ f (t, α, u_{1}, \dots, u_{n - 1}), & u_{0} < α (t), t \in [0, + \infty), \end{matrix} \\ f_{i} (t, u_{0}, \dots, u_{i}, \dots, u_{n - 1}) \\ = {\begin{matrix} f_{i - 1} (t, u_{0}, \dots, β^{(i)}, \dots, u_{n - 1}), & u_{i} > β^{(i)} (t), t \in [0, + \infty), \\ f_{i - 1} (t, u_{0}, \dots, u_{i}, \dots, u_{n - 1}), & α^{(i)} (t) ⩽ u_{i} ⩽ β^{(i)} (t), t \in [0, + \infty), \\ f_{i - 1} (t, u_{0}, \dots, α^{(i)}, \dots, u_{n - 1}), & u_{i} < α^{(i)} (t), t \in [0, + \infty), \end{matrix} \end{matrix}

for

i = 1, 2, \dots, n - 3

, and

\begin{matrix} f_{n - 2} (t, u_{0}, \dots, u_{n - 2}, u_{n - 1}) = {\begin{matrix} f_{n - 3} (t, u_{0}, \dots, β^{(n - 2)}, u_{n - 1}) - \frac{u_{n - 2} - β^{(n - 2)} (t)}{1 + | u_{n - 2} - β^{(n - 2)} (t) |}, \\ u_{n - 2} > β^{(n - 2)} (t), t \in [0, + \infty), \\ f_{n - 3} (t, u_{0}, \dots, u_{n - 2}, u_{n - 1}), \\ α^{(n - 2)} (t) ⩽ u_{n - 2} ⩽ β^{(n - 2)} (t), t \in [0, + \infty), \\ f_{n - 3} (t, u_{0}, \dots, α^{(n - 2)} (t), u_{n - 1}) + \frac{u_{n - 2} - α^{(n - 2)} (t)}{1 + | y - α^{(n - 2)} (t) |}, \\ u_{n - 2} < α^{(n - 2)} (t), t \in [0, + \infty), \end{matrix} \\ F (t, u_{0}, \dots, u_{n - 2}, u_{n - 1}) = {\begin{matrix} f_{n - 2} (t, u_{0}, \dots, u_{n - 2}, R), & u_{n - 1} > R, t \in [0, + \infty), \\ f_{n - 2} (t, u_{0}, \dots, u_{n - 2}, u_{n - 1}), & | u_{n - 1} | ⩽ R, t \in [0, + \infty), \\ f_{n - 2} (t, u_{0}, \dots, u_{n - 2}, - R), & u_{n - 1} < - R, t \in [0, + \infty) . \end{matrix} \end{matrix}

Consider the modified differential equation with the truncated function

- u^{(n)} (t) = q (t) F (t, u (t), \dots, u^{(n - 1)} (t)), t \in (0, + \infty),

(16)

with the boundary conditions (2). To complete the proof, it suffices to show that problem (16)-(2) has at least one solution u satisfying

α^{(i)} (t) ⩽ u^{(i)} (t) ⩽ β^{(i)} (t), i = 0, 1, \dots, n - 2, t \in [0, + \infty),

(17)

and

| u^{(n - 1)} (t) | ⩽ R, t \in [0, + \infty) .

(18)

We divide the proof into the following two steps.

Step 1: By contradiction we shall show that every solution u of problem (16)-(2) satisfy (17) and (18).

Suppose the right hand inequality in (17) does not hold for

i = n - 2

. Set

ω (t) = u^{(n - 2)} (t) - β^{(n - 2)} (t)

, then

sup_{0 ⩽ t < + \infty} ω (t) > 0 .

Case I. ${lim}_{t \to 0^{+}} ω (t) = {sup}_{0 ⩽ t < + \infty} ω (t) > 0$ .

Obviously, we have

ω^{'} (0^{+}) ⩽ 0

. By the boundary conditions, it follows that

ω^{'} (0^{+}) = u^{(n - 1)} (0) - β^{(n - 1)} (0) ⩾ \frac{1}{a} (u^{(n - 2)} (0) - β^{(n - 2)} (0)) = \frac{1}{a} ω (0) > 0,

which is a contradiction.

Case II. There exists a $t^{*} \in (0, + \infty)$ such that $ω (t^{*}) = {sup}_{0 ⩽ t < + \infty} ω (t) > 0$ .

Clearly, we have

ω (t^{*}) > 0, ω^{'} (t^{*}) = 0, ω^{″} (t^{*}) ⩽ 0 .

(19)

On the other hand,

\begin{array}{rcl} u^{(n)} (t^{*}) & = & - q (t^{*}) F (t^{*}, u (t^{*}), \dots, u^{(n - 2)} (t^{*}), u^{(n - 1)} (t^{*})) \\ = & - q (t^{*}) f_{n - 2} (t^{*}, u (t^{*}), \dots, u^{(n - 2)} (t^{*}), β^{(n - 1)} (t^{*})) \\ = & - q (t^{*}) (f_{n - 3} (t^{*}, u (t^{*}), \dots, β^{(n - 2)} (t^{*}), β^{(n - 1)} (t^{*})) \\ - \frac{u^{n - 2} (t^{*}) - β^{(n - 2)} (t^{*})}{1 + | u^{n - 2} (t^{*}) - β^{(n - 2)} (t^{*}) |}) . \end{array}

Subcase i. If

u^{(n - 3)} (t^{*}) > β^{(n - 3)} (t^{*})

, from the definition of

f_{n - 3}

, we obtain

\begin{array}{rcl} u^{(n)} (t^{*}) & = & - q (t^{*}) f_{n - 4} (t^{*}, u (t^{*}), \dots, β^{(n - 3)} (t^{*}), β^{(n - 2)} (t^{*}), β^{(n - 1)} (t^{*})) \\ + q (t^{*}) \frac{u^{n - 2} (t^{*}) - β^{(n - 2)} (t^{*})}{1 + | u^{n - 2} (t^{*}) - β^{(n - 2)} (t^{*}) |} . \end{array}

Subcase ii. If

u^{(n - 3)} (t^{*}) ⩽ β^{(n - 3)} (t^{*})

, from the conditions (H₂), we find

\begin{array}{rcl} u^{(n)} (t^{*}) & ⩾ & - q (t^{*}) f_{n - 4} (t^{*}, u (t^{*}), \dots, β^{(n - 3)} (t^{*}), β^{(n - 2)} (t^{*}), β^{(n - 1)} (t^{*})) \\ + q (t^{*}) \frac{u^{n - 2} (t^{*}) - β^{(n - 2)} (t^{*})}{1 + | u^{n - 2} (t^{*}) - β^{(n - 2)} (t^{*}) |} . \end{array}

Similarly following the above argument, we could discuss the other two cases

u^{(i)} (t^{*}) > β^{(i)} (t^{*})

or

u^{(i)} (t^{*}) ⩽ β^{(i)} (t^{*})

,

i = n - 4, n - 5, \dots, 1, 0

, and we have the following inequality:

u^{(n)} (t^{*}) ⩾ - q (t^{*}) (f (t^{*}, β (t^{*}), \dots, β^{(n - 2)} (t^{*}), β^{(n - 1)} (t^{*})) - \frac{u^{n - 2} (t^{*}) - β^{(n - 2)} (t^{*})}{1 + | u^{n - 2} (t^{*}) - β^{(n - 2)} (t^{*}) |}) .

Thus,

ω^{″} (t^{*}) ⩾ q (t^{*}) \frac{u^{n - 2} (t^{*}) - β^{(n - 2)} (t^{*})}{1 + | u^{n - 2} (t^{*}) - β^{(n - 2)} (t^{*}) |} > 0,

which contradicts (19).

Thus, $u^{(n - 2)} (t) ⩽ β^{(n - 2)} (t)$ , $t \in [0, + \infty)$ . Similarly, we can show that $u^{(n - 2)} (t) ⩾ α^{(n - 2)} (t)$ , $t \in [0, + \infty)$ . Integrating this inequality and using the boundary conditions in (2), (13), and (14), we obtain the inequality (17). Inequality (18) then follows from Lemma 3.1. In conclusion, u is the required solution to BVP (1)-(2).

Step 2. Problem (16)-(2) has a solution u.

Consider the operator

T : X \to X

defined by

T u (t) = l (t) + \int_{0}^{+ \infty} G (t, s) q (s) F (s, u (s), \dots, u^{(n - 1)} (s)) d s .

(20)

Lemma 2.1 shows that the fixed points of T are the solutions of BVP (16)-(2). Next we shall prove that T has at least one fixed point by using the Schäuder fixed point theorem. For this, it is enough to show that $T : X \to X$ is completely continuous.

(1) $T : X \to X$ is well defined.

For any

u \in X

, by direct calculation, we find

\begin{array}{rcl} {(T u)}^{(i)} (t) & = & \sum_{k = i}^{n - 3} \frac{A_{k} t^{k - i}}{(k - i)!} + \frac{(a C + B) t^{n - 2 - i}}{(n - 2 - i)!} + \frac{C t^{n - 1 - i}}{(n - 1 - i)!} \\ + \int_{0}^{+ \infty} g_{i} (t, s) q (s) F (s, u (s), \dots, u^{(n - 1)} (s)) d s, i = 0, 1, \dots, n - 1 . \end{array}

Obviously,

T u \in C^{n - 1} [0, + \infty)

. Further, because

| \int_{0}^{+ \infty} q (s) F (s, u (s), \dots, u^{(n - 1)} (s)) d s | ⩽ \int_{0}^{+ \infty} q (s) (H ψ (s) + 1) d s < + \infty,

(21)

where

H = {max}_{0 ⩽ s ⩽ ∥ u ∥} h (s)

, the Lebesgue dominated convergent theorem implies that

\begin{array}{rcl} lim_{t \to + \infty} \frac{{(T u)}^{(i)} (t)}{v_{i} (t)} & = & lim_{t \to + \infty} (\sum_{k = i}^{n - 3} \frac{A_{k} t^{k - i}}{(k - i)! v_{i} (t)} + \frac{(a C + B) t^{n - 2 - i}}{(n - 2 - i)! v_{i} (t)} + \frac{C t^{n - 1 - i}}{(n - 1 - i)! v_{i} (t)}) \\ + \int_{0}^{+ \infty} lim_{t \to + \infty} \frac{g_{i} (t, s)}{v_{i} (t)} q (s) F (s, u (s), \dots, u^{(n - 1)} (s)) d s \\ = & \frac{C}{(n - 1 - i)!} + \int_{0}^{+ \infty} \frac{1}{(n - 1 - i)!} q (s) F (s, u (s), \dots, u^{(n - 1)} (s)) d s \\ < & + \infty . \end{array}

Thus, $T u \in X$ .

(2) $T : X \to X$ is continuous.

For any convergent sequence

u_{m} \to u

in X, there exists

r_{1} > 0

such that

{sup}_{m \in N} ∥ u_{m} ∥ ⩽ r_{1}

. Thus, as in (21), we have

\begin{array}{rcl} {∥ T u_{m} - T u ∥}_{i} & = & sup_{t \in [0, + \infty)} | \frac{{(T u_{m})}^{(i)} (t)}{v_{i} (t)} - \frac{{(T u)}^{(i)} (t)}{v_{i} (t)} | \\ ⩽ & \int_{0}^{+ \infty} sup_{0 ⩽ t < + \infty} \frac{g_{i} (t, s)}{v_{i} (t)} \\ \cdot q (s) | F (s, u_{m} (s), \dots, u_{m}^{(n - 1)} (s)) - F (s, u (s), \dots, u^{(n - 1)} (s)) | d s \\ ⩽ & \int_{0}^{+ \infty} K_{i} q (s) | F (s, u_{m} (s), \dots, u_{m}^{(n - 1)} (s)) - F (s, u (s), \dots, u^{(n - 1)} (s)) | d s \\ \to & 0, as m \to + \infty, i = 0, 1, \dots, n - 1, \end{array}

and hence $T : X \to X$ is continuous.

(3) $T : X \to X$ is compact.

For this it suffices to show that T maps bounded subsets of X into relatively compact sets. Let B be any bounded subset of X, then there exists

r_{2} > 0

such that

∥ u ∥ ⩽ r_{2}

,

\forall u \in B

. For any

u \in B

,

i = 0, 1, \dots, n - 1

, we have

\begin{array}{rcl} sup_{t \in [0, + \infty)} | \frac{{(T u)}^{(i)} (t)}{v_{i} (t)} | & ⩽ & \sum_{k = i}^{n - 3} \frac{| A_{k} |}{(k - i)!} sup_{t \in [0, + \infty)} | \frac{t^{k - i}}{v_{i} (t)} | + \frac{(a C + B)}{(n - 2 - i)!} sup_{t \in [0, + \infty)} | \frac{t^{n - 2 - i}}{v_{i} (t)} | \\ + \frac{C}{(n - 1 - i)!} sup_{t \in [0, + \infty)} | \frac{t^{n - 1 - i}}{v_{i} (t)} | \\ + \int_{0}^{+ \infty} sup_{t \in [0, + \infty)} | \frac{g_{i} (t, s)}{v_{i} (t)} | q (s) F (s, u (s), \dots, u^{(n - 1) (s)}) d s \\ ⩽ & L_{i} + \int_{0}^{+ \infty} K_{i} q (s) (r_{3} ψ (s) + 1) d s = r_{4, i} < + \infty, \end{array}

where

\begin{array}{rcl} L_{i} & = & \sum_{k = i}^{n - 3} \frac{| A_{k} | (n - 1 - k)}{(k - i)! (n - 1 - i)} \cdot {(\frac{k - i}{n - 1 - k})}^{\frac{k - i}{n - 1 - i}} \\ + \frac{(a C + B)}{(n - 3 - i)!} \cdot {(\frac{1}{n - 2 - i})}^{\frac{1}{n - 1 - i}} + \frac{| C |}{(n - 1 - i)!}, \end{array}

where

r_{3} = {max}_{0 ⩽ s ⩽ r_{2}} h (s)

, and thus TB is uniformly bounded. Further, for any

T > 0

, if

t_{1}, t_{2} \in [0, T]

, we have

\begin{matrix} | \frac{{(T u)}^{(i)} (t_{1})}{v_{i} (t_{1})} - \frac{{(T u)}^{(i)} (t_{2})}{v_{i} (t_{2})} | \\ ⩽ | \frac{1}{v_{i} (t_{1})} ({(T u)}^{(i)} (t_{1}) - {(T u)}^{(i)} (t_{2})) | + | {(T u)}^{(i)} (t_{2}) (\frac{1}{v_{i} (t_{1})} - \frac{1}{v_{i} (t_{2})}) | \\ ⩽ \sum_{k = i}^{n - 3} \frac{| A_{k} |}{(k - i)!} | t_{1}^{k - i} - t_{2}^{k - i} | + \frac{| a C + B |}{(n - 2 - i)!} | t_{1}^{n - 2 - i} - t_{2}^{n - 2 - i} | + | C | | t_{1}^{n - 2 - i} - t_{2}^{n - 2 - i} | \\ + \int_{0}^{+ \infty} | g_{i} (t_{1}, s) - g_{i} (t_{2}, s) | q (s) | F (s, u (s), \dots, u^{(n - 1)} (s)) | d s + r_{4, i} | t_{2}^{n - 1 - i} - t_{1} | \\ \to 0, uniformly as t_{1} \to t_{2}, \end{matrix}

that is, TB is equi-continuous. From Lemma 2.3, it follows that if TB is equi-convergent at infinity, then TB is relatively compact. In fact, we have

\begin{matrix} | \frac{{(T u)}^{i} (t)}{v_{i} (t)} - lim_{t \to + \infty} \frac{{(T u)}^{i} (t)}{v_{i} (t)} | \\ ⩽ | \frac{l^{(i)} (t)}{v_{i} (t)} - \frac{C}{(n - 1 - i)!} | + \int_{0}^{+ \infty} | \frac{g_{i} (t, s)}{v_{i} (t)} - \frac{1}{(n - 1 - i)!} | q (s) (r_{3} ψ (s) + 1) d s \\ \to 0, uniformly as t \to + \infty, \end{matrix}

and, therefore, $T : X \to X$ is completely continuous. The Schäuder fixed point theorem now ensures that the operator T has a fixed point, which is a solution of the BVP (1)-(2). □

Remark 3.3 The upper and lower solutions are more strict at infinity than usually assumed. For such right boundary conditions, it is not easy to estimate the sign of $ω^{″} (+ \infty)$ even though we have $ω (+ \infty) > 0$ , $ω^{'} (+ \infty) = 0$ , where $ω (t) = u^{n - 2} (t) - β^{n - 2} (t)$ or $ω (t) = u^{n - 2} (t) - α^{n - 2} (t)$ . It remains unsettled whether this strict inequality can be weakened.

4 The multiplicity results

In this section assuming two pairs of upper and lower solutions, we shall prove the existence of at least three solutions for our infinite interval problem.

Theorem 4.1 Suppose that the following condition holds.

H₄: BVP (1)-(2) has two pairs of upper and lower solution

β_{j}

,

α_{j}

,

j = 1, 2

in X with

α_{2}

,

β_{1}

strict, and

α_{1}^{(i)} (t) ⩽ α_{2}^{(i)} (t) ⩽ β_{2}^{(i)} (t), α_{1}^{(i)} (t) ⩽ β_{1}^{(i)} (t) ⩽ β_{2}^{(i)} (t), α_{2}^{(i)} (t) ≰ β_{1}^{(i)} (t),

for $i = 0, 1, \dots, n - 2$ , $t \in [0, + \infty)$ , and $f \in C ([0, + \infty) \times R^{n}, R)$ satisfies the Nagumo condition with respect to $α_{1}$ and $β_{2}$ .

Suppose further that conditions (H₂) and (H₃) hold with α and β replaced by

α_{1}

and

β_{2}

, respectively. Then the problem (1)-(2) has at least three solutions

u_{1}

,

u_{2}

, and

u_{3}

satisfying

α_{j}^{(i)} (t) ⩽ u_{j}^{(i)} (t) ⩽ β_{j}^{(i)} (t) (j = 1, 2), u_{3}^{(i)} (t) ≰ β_{1}^{(i)} (t) and u_{3}^{(i)} (t) ≱ α_{2}^{(i)} (t),

for $i = 0, 1, \dots, n - 2$ , $t \in [0, + \infty)$ .

Proof Define the truncated function

F_{1}

, the same as F in Theorem 3.2 with α replaced by

α_{1}

and β by

β_{2}

, respectively. Consider the modified differential equation

- u^{(n)} (t) = q (t) F_{1} (t, u (t), \dots, u^{(n - 1)} (t)), 0 < t < + \infty,

(22)

with boundary conditions (2). Similarly to Theorem 3.2, it suffices to show that problem (22)-(2) has at least three solutions. To this end, define the mapping

T_{1} : X \to X

as follows:

T_{1} u (t) = l (t) + \int_{0}^{+ \infty} G (t, s) q (s) F_{1} (s, u (s), \dots, u^{(n - 1)} (s)) d s .

Clearly, $T_{1}$ is completely continuous. By using the degree theory, we will show that $T_{1}$ has at least three fixed points which coincide with the solutions of (22)-(2).

Let

N > {max_{0 ⩽ i ⩽ n - 1} L_{i} + max_{0 ⩽ i ⩽ n - 1} K_{i} \int_{0}^{+ \infty} q (s) (H_{R} ψ (s) + 1) d s, ∥ α_{1} ∥, ∥ β_{2} ∥},

where

L_{i}

and

K_{i}

are defined as above, and

H_{r} = {max}_{0 ⩽ s ⩽ R} h (s)

. Set

Ω = {u \in X, ∥ u ∥ < N}

. Then for any

u \in \bar{Ω}

, it follows that

\begin{array}{rcl} ∥ T_{1} u ∥ & = & max_{0 ⩽ i ⩽ n - 1} {sup_{t \in [0, + \infty)} | \frac{{(T u)}^{(i)} (t)}{v_{i} (t)} |} \\ ⩽ & max_{0 ⩽ i ⩽ n - 1} {\sum_{k = i}^{n - 3} \frac{| A_{k} |}{(k - i)!} sup_{t \in [0, + \infty)} | \frac{t^{k - i}}{v_{i} (t)} | + \frac{(a C + B)}{(n - 2 - i)!} sup_{t \in [0, + \infty)} | \frac{t^{n - 2 - i}}{v_{i} (t)} | \\ + \frac{C}{(n - 1 - i)!} sup_{t \in [0, + \infty)} | \frac{t^{n - 1 - i}}{v_{i} (t)} | \\ + \int_{0}^{+ \infty} sup_{t \in [0, + \infty)} | \frac{g_{i} (t, s)}{v_{i} (t)} | q (s) F_{1} (s, u (s), \dots, u^{(n - 1) (s)}) d s} \\ ⩽ & max_{0 ⩽ i ⩽ n - 1} {L_{i} + \int_{0}^{+ \infty} K_{i} q (s) (H_{R} ψ (s) + 1) d s} < N \end{array}

and hence $T Ω \subset Ω$ , which implies that $deg (I - T_{1}, Ω, 0) = 1$ .

Set

\begin{matrix} Ω_{α_{2}} = {u \in Ω, u^{(n - 2)} (t) > α_{2}^{(n - 2)} (t), t \in [0, + \infty)}, \\ Ω^{β_{1}} = {u \in Ω, u^{(n - 2)} (t) < β_{1}^{(n - 2)} (t), t \in [0, + \infty)} . \end{matrix}

Because

α_{2}^{(n - 2)} ≰ β_{1}^{(n - 2)}

,

α_{1}^{(n - 2)} ⩽ α_{2}^{(n - 2)} ⩽ β_{2}^{(n - 2)}

and

α_{1}^{(n - 2)} ⩽ β_{1}^{(n - 2)} ⩽ β_{2}^{(n - 2)}

, we have

Ω_{α_{2}} \neq \emptyset, Ω^{β_{1}} \neq \emptyset, Ω ∖ \bar{Ω_{α_{2}} \cup Ω^{β_{1}}} \neq \emptyset, Ω_{α_{2}} \cap Ω^{β_{1}} = \emptyset .

Now since

α_{2}

,

β_{1}

are strict lower and upper solutions, there is no solution in

\partial Ω_{α_{2}} \cup \partial Ω^{β_{1}}

. Therefore

\begin{array}{rcl} deg (I - T_{1}, Ω, 0) & = & deg (I - T_{1}, Ω ∖ \bar{Ω_{α_{2}} \cup Ω^{β_{1}}}, 0) \\ + deg (I - T_{1}, Ω_{α_{2}}, 0) + deg (I - T_{1}, Ω^{β_{1}}, 0) . \end{array}

Next we will show that

deg (I - T_{1}, Ω_{α_{2}}, 0) = deg (I - T_{1}, Ω^{β_{1}}, 0) = 1 .

For this, we define another mapping

T_{2} : \bar{Ω} \to \bar{Ω}

by

T_{2} u (t) = l (t) + \int_{0}^{+ \infty} G (t, s) q (s) F_{2} (s, u (s), \dots, u^{(n - 1)} (s)) d s,

where the function

F_{2}

is similar to

F_{1}

except changing

α_{1}

to

α_{2}

. Similar to the proof of Theorem 3.2, we find that u is a fixed point of

T_{2}

only if

α_{2} (t) ⩽ u (t) ⩽ β_{2} (t)

. So

deg (I - T_{2}, Ω ∖ \bar{Ω_{α_{2}}}, 0) = 0

. From the Schäuder fixed point theorem and

T_{2} \bar{Ω} \subset Ω

, we have

deg (I - T_{2}, Ω, 0) = 1

. Furthermore,

\begin{array}{rcl} deg (I - T_{1}, Ω_{α_{2}}, 0) & = & deg (I - T_{2}, Ω_{α_{2}}, 0) \\ = & deg (I - T_{2}, Ω, 0) + deg (I - T_{2}, Ω ∖ \bar{Ω_{α_{2}}}, 0) = 1 . \end{array}

Similarly, we have

deg (I - T_{1}, Ω^{β_{1}}, 0) = 1

, and then

deg (I - T_{1}, Ω ∖ \bar{Ω_{α_{2}} \cup Ω^{β_{1}}}, 0) = - 1 .

Finally, using the properties of the degree, we conclude that $T_{1}$ has at least three fixed points $u_{1} \in Ω_{α_{2}}$ , $u_{2} \in Ω^{β_{1}}$ , and $u_{3} \in Ω ∖ \bar{Ω_{α_{2}} \cup Ω^{β_{1}}}$ . □

5 An example

Example Consider the second-order differential equation with Sturm-Liouville boundary conditions

{\begin{matrix} u^{″} (t) - \frac{3 (u^{'} (t) + 1) (u^{'} (t) - \frac{1}{3}) \sqrt[3]{u^{'} (t)}}{{(1 + t)}^{2}} - \frac{u^{'} (t) - \frac{1}{2}}{{(1 + t)}^{2}} = 0, 0 < t < + \infty, \\ u (0) - 3 u^{'} (0) = 0, u^{'} (+ \infty) = \frac{1}{2} . \end{matrix}

(23)

Clearly, BVP (23) is a particular case of problem (1)-(2) with

\begin{matrix} q (t) = \frac{1}{{(1 + t)}^{2}}, \\ f (t, u_{0}, u_{1}) = - 2 (u_{1} + 1) (u_{1} - \frac{1}{3}) \sqrt[3]{u_{1}} - (u_{1} - \frac{1}{2}), \\ a = 3 > 0, B = 0, and C = \frac{1}{2} . \end{matrix}

We let

α_{1} (t) = - t - 4, α_{2} (t) = \frac{t}{3}, t \in [0, + \infty) .

Then

α_{1}, α_{2} \in C^{2} [0, + \infty)

, and

α_{1}^{'} (t) = - 1

,

α_{1}^{″} (t) = 0

,

α_{2}^{'} (t) = \frac{1}{3}

,

α_{2}^{″} (t) = 0

. Moreover, we have

{\begin{matrix} α_{1}^{″} (t) + f (t, α_{1} (t), α_{1}^{'} (t)) = - \frac{- 1 - \frac{1}{2}}{{(1 + t)}^{2}} > 0, t \in (0, + \infty), \\ α_{1} (0) - 3 α_{1}^{'} (0) = - 1 < 0, α_{1}^{'} (+ \infty) = - 1 < \frac{1}{2}, \end{matrix}

and

{\begin{matrix} α_{2}^{″} (t) + f (t, α_{2} (t), α_{2}^{'} (t)) = - \frac{\frac{1}{3} - \frac{1}{2}}{{(1 + t)}^{2}} > 0, t \in (0, + \infty), \\ α_{2} (0) - 3 α_{2}^{'} (0) = - 1 < 0, α_{2}^{'} (+ \infty) = \frac{1}{3} < \frac{1}{2} . \end{matrix}

Thus, $α_{1}$ and $α_{2}$ are strict lower solutions of problem (23).

Now we take

β_{1} (t) = {\begin{cases} - \frac{t}{4}, & 0 ⩽ t ⩽ 1, \\ \frac{3}{4} t - 1, & t > 1, \end{cases} β_{2} (t) = t + 4, t \in [0, + \infty) .

Then

β_{1} \in C^{2} [0, 1) \cup C^{2} (1, + \infty)

,

β_{2} \in C^{2} [0, + \infty)

,

{\begin{matrix} β_{1}^{″} (t) + f (t, β_{1} (t), β_{1}^{'} (t)) = \frac{- \frac{21}{16} \cdot \sqrt[3]{\frac{1}{4}} + \frac{3}{4}}{{(1 + t)}^{2}} < 0, t \in (0, 1), \\ β_{1}^{″} (t) + f (t, β_{1} (t), β_{1}^{'} (t)) = \frac{- \frac{35}{16} \cdot \sqrt[3]{\frac{3}{4}} - \frac{1}{4}}{{(1 + t)}^{2}} < 0, t \in (1, + \infty), \\ β_{1} (0) - 3 β_{1}^{'} (0) = \frac{3}{4} > 0, \\ β_{1}^{'} (+ \infty) = \frac{3}{4} > \frac{1}{2}, \end{matrix}

and

{\begin{matrix} β_{2}^{″} (t) + f (t, β_{2} (t), β_{2}^{'} (t)) = - \frac{9}{2 {(1 + t)}^{2}} < 0, t \in (0, + \infty), \\ β_{2} (0) - 3 β_{2}^{'} (0) = 1 > 0, \\ β_{2}^{'} (+ \infty) = 1 > \frac{1}{2} . \end{matrix}

Thus,

β_{1}

and

β_{2}

are strict upper solutions of problem (23). Further, it follows that

α_{1} (t) ⩽ α_{2} (t) ⩽ β_{2} (t), α_{1} (t) ⩽ β_{1} (t) ⩽ β_{2} (t), α_{2} (t) ≰ β_{1} (t), t \in [0, + \infty) .

Moreover, for every $(t, u_{0}, u_{1}) \in [0, + \infty) \times [- t - 4, t + 4] \times [- 1, 1]$ , we find that $f (t, u_{0}, u_{1})$ is bounded. Finally, take $ψ (t) = 1$ , $h (s) = {max}_{- 1 ⩽ r ⩽ 1} | 3 (r + 1) (r - \frac{1}{3}) \sqrt[3]{r} + (r - \frac{1}{2}) |$ . Hence, all conditions in Theorem 4.1 are satisfied and therefore problem (23) has at least three solutions.

Declarations

Acknowledgements

This research is supported by the National Natural Science Foundation of China (No. 11101385) and by the Fundamental Research Funds for the Central Universities.

Authors’ Affiliations

(1)

School of Science, China University of Geosciences, Beijing, 100083, PR China

(2)

Department of Mathematics, Texas A&M University-Kingsville, Kingsville, Texas 78363, USA

References

Graef JR, Kong L, Mihós FM: Higher order ϕ -Laplacian BVP with generalized Sturm-Liouville boundary conditions. Differ. Equ. Dyn. Syst. 2010, 18(4):373-383. 10.1007/s12591-010-0071-1MathSciNetView ArticleGoogle Scholar
Graef JR, Kong L, Mihós FM, Fialho J: On the lower and upper solution method for higher order functional boundary value problems. Appl. Anal. Discrete Math. 2011, 5: 133-146. 10.2298/AADM110221010GMathSciNetView ArticleMATHGoogle Scholar
Grossinho MR, Minhós FM: Existence result for some third order separated boundary value problems. Nonlinear Anal. 2001, 47: 2407-2418. 10.1016/S0362-546X(01)00364-9MathSciNetView ArticleMATHGoogle Scholar
Bai Z: Positive solutions of some nonlocal fourth-order boundary value problem. Appl. Math. Comput. 2010, 215: 4191-4197. 10.1016/j.amc.2009.12.040MathSciNetView ArticleMATHGoogle Scholar
Du Z, Liu W, Lin X: Multiple solutions to a three-point boundary value problem for higher-order ordinary differential equations. J. Math. Anal. Appl. 2007, 335: 1207-1218. 10.1016/j.jmaa.2007.02.014MathSciNetView ArticleMATHGoogle Scholar
Agarwal RP, O’Regan D: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht; 2001.View ArticleMATHGoogle Scholar
Countryman M, Kannan R: Nonlinear boundary value problems on semi-infinite intervals. Comput. Math. Appl. 1994, 28: 59-75. 10.1016/0898-1221(94)00186-3MathSciNetView ArticleMATHGoogle Scholar
Philos CG: A boundary value problem on the half-line for higher-order nonlinear differential equations. Proc. R. Soc. Edinb. 2009, 139A: 1017-1035.MathSciNetView ArticleMATHGoogle Scholar
Umamahesearam S, Venkata Rama M: Multipoint focal boundary value problems on infinite interval. J. Appl. Math. Stoch. Anal. 1992, 5: 283-290. 10.1155/S1048953392000236View ArticleMATHGoogle Scholar
Agarwal RP, O’Regan D: Nonlinear boundary value problems on the semi-infinite interval: an upper and lower solution approach. Mathematika 2002, 49: 129-140. 10.1112/S0025579300016120MathSciNetView ArticleMATHGoogle Scholar
Eloe PW, Kaufmann ER, Tisdell CC: Multiple solutions of a boundary value problem on an unbounded domain. Dyn. Syst. Appl. 2006, 15(1):53-63.MathSciNetMATHGoogle Scholar
Lian H, Wang P, Ge W: Unbounded upper and lower solutions method for Sturm-Liouville boundary value problem on infinite intervals. Nonlinear Anal. 2009, 70: 2627-2633. 10.1016/j.na.2008.03.049MathSciNetView ArticleMATHGoogle Scholar
Lian H, Zhao J: Existence of unbounded solutions for a third-order boundary value problem on infinite intervals. Discrete Dyn. Nat. Soc. 2012. 10.1155/2012/357697Google Scholar
Yan B, O’Regan D, Agarwal RP: Unbounded solutions for singular boundary value problems on the semi-infinite interval: upper and lower solutions and multiplicity. J. Comput. Appl. Math. 2006, 197: 365-386. 10.1016/j.cam.2005.11.010MathSciNetView ArticleMATHGoogle Scholar
Zhao Y, Chen H, Xu C: Existence of multiple solutions for three-point boundary-value problems on infinite intervals in Banach spaces. Electron. J. Differ. Equ. 2012, 44: 1-11.MathSciNetView ArticleMATHGoogle Scholar
Ehme J, Eloe PW, Henderson J: Upper and lower solution methods for fully nonlinear boundary value problems. J. Differ. Equ. 2002, 180: 51-64. 10.1006/jdeq.2001.4056MathSciNetView ArticleMATHGoogle Scholar
Agarwal RP, O’Regan D: Infinite interval problems modeling phenomena which arise in the theory of plasma and electrical potential theory. Stud. Appl. Math. 2003, 111: 339-358. 10.1111/1467-9590.t01-1-00237MathSciNetView ArticleMATHGoogle Scholar
Baxley JV: Existence and uniqueness for nonlinear boundary value problems on infinite interval. J. Math. Anal. Appl. 1990, 147: 122-133. 10.1016/0022-247X(90)90388-VMathSciNetView ArticleMATHGoogle Scholar
Chen S, Zhang Y: Singular boundary value problems on a half-line. J. Math. Anal. Appl. 1995, 195: 449-468. 10.1006/jmaa.1995.1367MathSciNetView ArticleMATHGoogle Scholar
Gross OA: The boundary value problem on an infinite interval, existence, uniqueness and asymptotic behavior of bounded solutions to a class of nonlinear second order differential equations. J. Math. Anal. Appl. 1963, 7: 100-109. 10.1016/0022-247X(63)90080-5MathSciNetView ArticleMATHGoogle Scholar
Jiang D, Agarwal RP: A uniqueness and existence theorem for a singular third-order boundary value problem on $[0, + \infty)$ . Appl. Math. Lett. 2002, 15: 445-451. 10.1016/S0893-9659(01)00157-4MathSciNetView ArticleMATHGoogle Scholar
Liu Y: Boundary value problems for second order differential equations on infinite intervals. Appl. Math. Comput. 2002, 135: 211-216.View ArticleGoogle Scholar
Ma R: Existence of positive solution for second-order boundary value problems on infinite intervals. Appl. Math. Lett. 2003, 16: 33-39. 10.1016/S0893-9659(02)00141-6MathSciNetView ArticleMATHGoogle Scholar
Bai C, Li C: Unbounded upper and lower solution method for third-order boundary-value problems on the half-line. Electron. J. Differ. Equ. 2009, 119: 1-12.MathSciNetView ArticleMATHGoogle Scholar

Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.