Dynamic Systems and Applications 15 (2006) 53-64

MULTIPLE SOLUTIONS OF A BOUNDARY VALUE PROBLEM ON AN UNBOUNDED DOMAIN

PAUL W. ELOE^{1}, ERIC R. KAUFMANN^{2}, AND CHRISTOPHER C. TISDELL^{3}

1Department of Mathematics, University of Dayton Dayton, OH 45469-2316 USA. E-mail: Paul.Eloe@notes.udayton.edu

2Department of Mathematics and Statistics, University of Arkansas at Little Rock Little Rock, AR 72204-1099 USA. E-mail: erkaufmann@ualr.edu

3School of Mathematics, The University of New South Wales Sydney 2052, Australia. E-mail: cct@maths.unsw.edu.au

Dedicated to John Baxley on the occasion of his retirement from Wake Forest University.

ABSTRACT. We are concerned with multiple solutions of a boundary value problem on an unbounded domain. We employ degree theory coupled with the method of upper and lower solutions on compact domains. We then extend solutions to the unbounded domain and apply sequential arguments.

AMS (MOS) Subject Classification. 34B10, 34B15.

1. INTRODUCTION

In this paper, we shall be interested in the existence of solutions of the boundary value problem (BVP) for the ordinary differential equation,

(1.1)  x′′(t) − a(t)x(t) + f(t, x(t)) = 0, 0 ≤ t,
(1.2)  x(0) = x_{0}, x(t) bounded on [0, ∞),

where x_{0} is real, f : [0, ∞)×(−∞, ∞) → (−∞, ∞) is continuous, a : [0, ∞) → (0, ∞) is continuous, a(t) ≥ α^{2} > 0, 0 ≤ t, for some α^{2} > 0.

Bebernes and Jackson [4] employed the method of subfunctions and superfunctions to study infinite interval boundary value problems for second order ordinary differential equations. More recently, Granas, Guenther, Lee and O’Regan [9], Baxley [3], and Bobisud [5], among others, have employed a variety of methods to study

Received October 22, 2004               1056-2176 $15.00 \begin{matrix}c \\ ◯ \end{matrix}Dynamic Publishers, Inc.

54 PAUL W. ELOE, ERIC R. KAUFMANN, AND CHRISTOPHER C. TISDELL

infinite interval boundary value problems for second order ordinary differential equations. Granas, Guenther, Lee and O’Regan [9] and related papers, [14], for example, employed sequential arguments in which the Arzela-Ascoli theorem is applied on compact domains. More bibliographic information is available in the recent monograph by Agarwal and O’Regan [1].

Recently, Eloe, Grimm and Mashburn [7] used the Green’s function to obtain sufficient conditions for the existence of solutions of the BVP, (1.1), (1.2). It can be observed that the BVP, (1.1), (1.2), is of limit-point case [6]. Hence, it is known that a unique Green’s function exists. Such Green’s functions have been constructed sequentially in [8].

In this paper we shall employ degree theory to address existence of solutions. On compact domains, we shall employ two pairs of upper and lower solutions and apply degree theory in a standard way to obtain two solutions of a boundary value problem on a compact domain. We shall then apply the additivity property of degree to obtain a third solution. We then extend these solutions to the unbounded domain with sequential arguments.

The argument given here is modelled on the recent work of Henderson and Thompson [10]. In that paper they develop the machinery described in the preceding paragraph for a family of BVPs on compact domains. They construct a particularly interesting example for an autonomous system. The sufficient conditions require growth conditions on f; the conditions are similar to and they compare them to conditions one requires if one applies Leggett-Williams type theorems [17]. We refer the reader to the Avery type development of the Leggett-Williams fixed point theorems [2].

In this paper, we, too, illustrate the main theorem with an example. Thus, the primary contribution of this paper is that we obtain sufficient conditions analogous to those one expects when applying Leggett-Williams [17] or Avery [2] type methods for singular BVPs in which the singularity is due to the unbounded domain. The Leggett-Williams method requires the corresponding fixed point map is compact.

In order that the paper is self-contained, we shall provide a short discussion on degree theory in Section 2. In Section 3, we shall apply the results of Section 2 to BVPs on compact domains. We believe these results are new. In Section 4, in the main theorem, Theorem 4.1, we shall then provide sequential arguments to extend the three distinct solutions to the unbounded domain and hence obtain three distinct solutions of the BVP, (1.1), (1.2). We shall close the paper in Section 5 with an example in which we illustrate the results in Theorem 4.1.

2. BACKGROUND MATERIAL FROM DEGREE THEORY

The material briefly presented here can be found in Lloyd [12] or Schwartz [15]. Let X denote a Banach space and let Ω ⊂ X be an open bounded set. Assume

T : \overline{Ω} → X is a compact map and let I : X → X denote the identity map. Set z = I − T. Let p ∈ X and p ∉ z(∂Ω). Let d(z, Ω, p) denote the degree of z at p relative to Ω [12]. Also, for λ ∈ [0, 1], let z_{λ} denote the operator I − λT.

Theorem 2.1. If d(z, Ω, p) ≠ 0, then the equation z(x) = p has at least one solution, x ∈ Ω.

Theorem 2.2. Let z, Ω, p be as above and define z_{λ} = I − λT, 0 ≤ λ ≤ 1, with d(z_{λ}, Ω, p) defined for all λ ∈ [0, 1]. Then d(z_{0}, Ω, p) = d(z_{1}, Ω, p).

Since d(z_{0}, Ω, p) can be readily calculated we have the following theorem.

Theorem 2.3. Let z, Ω, p be as above. Then

d(z, Ω, p) = \{ \begin{matrix}1, & p ∈ Ω, \\ 0, & p ∉ Ω. \end{matrix}

The last two results are crucial to the strategy to obtain three solutions.

Theorem 2.4. Let z, Ω, p be as above. Assume Ω = Ω_{1} ∪ Ω_{2} ∪ Ω_{3} where each Ω_{i} is open and nonempty and the family {Ω_{i}} is disjoint. Then

d(z, Ω, p) = d(z, Ω_{1}, p) + d(z, Ω_{2}, p) + d(z, Ω_{3}, p).

Theorem 2.5. let K ⊂ \overline{Ω}, assume K is closed, and assume p ∉ z(K). Then d(z, Ω, p) = d(z, Ω \ K, p).

3. AN ASSOCIATED BOUNDARY VALUE PROBLEM ON BOUNDED DOMAINS

We first address a right focal [13] boundary value problem on a bounded domain. Let B_{0} > 0 and let b > B_{0}. We shall first consider the boundary value problem

(3.1)  x′′(t) − a(t)x(t) + f(t, x(t)) = 0, 0 < t < b,

(3.2)  x(0) = x_{0}, x′(b) = 0.

Some of the work here is modelled by the work of Eloe, Grimm and Mashburn

56 PAUL W. ELOE, ERIC R. KAUFMANN, AND CHRISTOPHER C. TISDELL

Definition 3.1. We say the α_{b} is a C^{0}-lower solution of the BVP, (3.1), (3.2), if α_{b} ∈ C^{0}[0, b], α′_{b}(b) exists, and for each t ∈ (0, b) there exists an open interval I_{t} such that t ∈ I_{t} ⊂ (0, b) and an α_{t} ∈ C^{2}(I_{t}) such that

α_{t}(s) ≤ α_{b}(s), s ∈ I_{t}, α_{t}(t) = α_{b}(t),
(3.3)  α′′_{t}(s) − a(s)α_{t}(s) + f(s, α_{t}(s)) ≥ 0, s ∈ I_{t},
(3.4)  α_{b}(0) ≤ x_{0}, α′_{b}(b) < 0.

We say the β_{b} is a C^{0}-upper solution of the BVP, (3.1), (3.2), if β_{b} ∈ C^{0}[0, b], β′_{b}(b) exists, and for each t ∈ (0, b) there exists an open interval I_{t} such that t ∈ I_{t} ⊂ (0, b) and β_{t} ∈ C^{2}(I_{t}) such that

β_{t}(s) ≥ β_{b}(s), s ∈ I_{t}, β_{t}(t) = β_{b}(t),
β′′_{t}(s) − a(s)β_{t}(s) + f(s, β_{t}(s)) ≤ 0, s ∈ I_{t},
β_{b}(0) ≥ x_{0}, β′_{b}(b) > 0.

We will say α_{b} is a strict C^{0}-lower solution of the BVP, (3.1), (3.2), if the inequality in (3.3) is strict for each t ∈ (0, b). A strict C^{0}-upper solution is defined similarly.

Let G(b; t, s) denote the Green’s function for the right focal problem, −(x′′(t) − a(t)x(t)) = 0, 0 < t < b, with the boundary conditions, x(0) = 0, x′(b) = 0. Let p_{b}(t) denote the solution of the BVP, x′′(t) − a(t)x(t) = 0, 0 < t < b, with the boundary conditions, x(0) = x_{0}, x′(b) = 0. The uniqueness and existence of p is shown in [7]. Define the operator T_{b} : C[0, b] → C[0, b] by

(3.5)  T_{b}x_{b}(t) = p_{b}(t) + ∫_{0}^{b} G(b; t, s)f(s, x_{b}(s))ds.

Proving existence of a solution of the BVP, (3.1), (3.2), is equivalent to proving the existence of a fixed point of T_{b}.

Theorem 3.2. Assume f is bounded on (0, ∞)×(−∞, ∞). Then d(I −T_{b}, Ω_{b}, 0) = 1; in particular, T_{b} has a fixed point in C[0, b].

Proof. Let

N = sup |f(t, x)|, 0 < t, −∞ < x < ∞,

G_{b} = max_{[0,b]×[0,b]} |G(b; t, s)|,

||p_{b}||_{b} = max_{0≤t≤b} |p_{b}(t)|,

Ω_{b} = {x_{b} ∈ C[0, b] : ||x_{b}|| < ||p_{b}||_{b} + G_{b}N + 1}.

MULTIPLE SOLUTIONS ON AN UNBOUNDED DOMAIN 57

Consider the family of mappings

z_{λ} = I − λT_{b}, λ ∈ [0, 1],

where T_{b} is defined in (3.5). It follows by straight-forward applications of Theorems 2.2 and 2.3 that d(I − T_{b}, Ω_{b}, 0) = 1. This completes the proof of Theorem 3.2.

Theorem 3.3. Let α_{b}, β_{b} denote C^{0}-lower and C^{0}-upper solutions, respectively, of the BVP, (3.1), (3.2). Assume α_{b}(t) ≤ β_{b}(t), 0 ≤ t ≤ b. Assume f satisfies

(3.6)  f(t, β_{b}(t)) < f(t, x), for all β_{b}(t) < x,
(3.7)  f(t, α_{b}(t)) > f(t, x), for all α_{b}(t) > x.

If x_{b}(t) is a solution of BVP, (3.1), (3.2), then

α_{b}(t) ≤ x_{b}(t) ≤ β_{b}(t), t ∈ [0, b].

Proof. The proof is by contradiction and standard [11]. Assume for the sake of contradiction that x_{b}(t) ≤ β_{b}(t), t ∈ [0, b] is false. Then x_{b} − β_{b} has a positive maximum in [0, b]. If the positive maximum occurs at t_{0} ∈ (0, b) then x_{b} − β_{t_{0}} attains a positive maximum at t_{0}. Then (3.6) gives the contradiction

(x_{b} − β_{t_{0}})′′(t_{0}) ≥ f(t_{0}, x_{b}(t_{0})) − f(t_{0}, β_{t_{0}}(t_{0})) > 0.

Thus, the positive maximum does not occur in (0, b). The positive maximum does not occur at t_{0} = 0, since (x_{b} − β_{b})(0) ≤ 0. So the only other possibility is that the positive maximum occurs as t_{0} = b. Then (x_{b} − β_{b})′(b) ≥ 0. But x′_{b}(b) = 0 and β′_{b}(b) > 0 contradicts this inequality. This completes the proof of Theorem 3.3.

We have need for stronger inequalities which are given in the next theorem.

Theorem 3.4. Let α_{b}, β_{b} denote strict C^{0}-lower and strict C^{0}-upper solutions, respectively, of the BVP, (3.1), (3.2). Assume α_{b}(t) ≤ β_{b}(t), 0 ≤ t ≤ b. Assume f satisfies

(3.8)  f(t, β_{b}(t)) < f(t, x), for all β_{b}(t) < x,
(3.9)  f(t, α_{b}(t)) > f(t, x), for all α_{b}(t) > x.

If x(t) is a solution of BVP, (3.1), (3.2), then

α_{b}(t) < x(t) < β_{b}(t), t ∈ (0, b].

Proof. The proof is similar to the proof of Theorem 3.3. The strict lower and upper solutions give strict inequalities in the conclusion.

At this point we change notation for more clear exposition. Let α_{b}(t) be denoted by α(b; t). Let β_{b}(t) be denoted by β(b; t).

58 PAUL W. ELOE, ERIC R. KAUFMANN, AND CHRISTOPHER C. TISDELL

Theorem 3.5. Assume f : [0, b] × (−∞, ∞) → (−∞, ∞) is continuous and assume there exist strict C^{0}-lower and strict C^{0}-upper solutions of BVP, (3.1), (3.2), respectively, α_{2}(b; t) and β_{1}(b; t), and lower and upper solutions of BVP, (3.1), (3.2), (not necessarily strict) respectively, α_{1}(b; t) and β_{2}(b; t), all satisfying

(3.10)  α_{1}(b; t) ≤ α_{2}(b; t) ≤ β_{2}(b; t), 0 ≤ t ≤ b,
(3.11)  α_{1}(b; t) ≤ β_{1}(b; t) ≤ β_{2}(b; t), 0 ≤ t ≤ b,
(3.12)  α_{2}(b; t) ≰ β_{1}(b; t), 0 ≤ t ≤ b.

Then the BVP, (3.1), (3.2) has at least three solutions, x_{1}(b; t), x_{2}(b; t), x_{3}(b; t) satisfying

α_{1}(b; t) ≤ x_{1}(b; t) ≤ β_{1}(b; t), α_{2}(b; t) ≤ x_{2}(b; t) ≤ β_{2}(b; t), 0 ≤ t ≤ b,
x_{3}(b; t) ≰ β_{1}(b; t), α_{2}(b; t) ≰ x_{3}(b; t), 0 ≤ t ≤ b.

Proof. We modify the BVP, (3.1), (3.2), in a standard way [11]. Define a bounded function, F_{1}, by

F_{1}(t, x) = \{ \begin{matrix}f(t, β_{2}(b; t)) + \frac{x−β_{2}(b;t)}{1+|x−β_{2}(b;t)|}, & β_{2}(b; t) ≤ x, \\ f(t, x), & α_{1}(b; t) ≤ x ≤ β_{2}(b; t), \\ f(t, α_{1}(b; t)) + \frac{x−α_{1}(b;t)}{1+|x−α_{1}(b;t)|}, & x ≤ α_{1}(b; t). \end{matrix}

Clearly F_{1} is bounded and F_{1} satisfies (3.6) and (3.7). Consider the modified BVP,

(3.13)  x′′(t) − a(t)x(t) + F_{1}(t, x(t)) = 0, 0 < t < b,

satisfying boundary conditions (3.2).

Apply Theorem 3.2 and obtain a solution, x, of the BVP, (3.13), (3.2). Since F_{1}(t, α_{1}(t)) = f(t, α_{1}(t)) and F_{1}(t, β_{2}(t)) = f(t, β_{2}(t)) it follows that α_{1} and β_{2} serve as lower and upper solutions, respectively, of the BVP, (3.13), (3.2), as well. Apply Theorem 3.3 and note that

α_{1}(t) ≤ x(t) ≤ β_{2}(t), 0 ≤ t ≤ b,

and so, x is a solution of the original BVP, (3.1), (3.2).

Let N denote an upper bound on F_{1} and define Ω_{b} as in Theorem 3.2. Define

Ω_{1} = {x ∈ Ω_{b} : x(t) > α_{2}(t), 0 ≤ t ≤ b},
Ω_{2} = {x ∈ Ω_{b} : x(t) < β_{1}(t), 0 ≤ t ≤ b}.

Since α_{2} ≰ β_{1}, α_{2} > −(||p_{b}||_{b} + G_{b}N + 1) and β_{1} < (||p_{b}||_{b} + G_{b}N + 1), it follows that each of the sets, Ω_{1}, Ω_{2}, \overline{Ω}_{1} ∩ \overline{Ω}_{2}, and Ω \ \overline{Ω_{1} ∪ Ω_{2}} are all non-empty.

MULTIPLE SOLUTIONS ON AN UNBOUNDED DOMAIN 59

The sets Ω_{1}, Ω_{2}, and Ω \ \overline{Ω_{1} ∪ Ω_{2}} are mutually disjoint. Moreover, since α_{2} and β_{1} are strict lower and upper solutions, respectively, of the BVP, (3.1), (3.2), there are no solutions x ∈ ∂Ω_{1} ∪ ∂Ω_{2} of the BVP, (3.1), (3.2), by Theorem 3.4.

To apply Theorem 3.2 define

T_{1}x(t) = p_{b}(t) + ∫_{0}^{b} G(b; t, s)F_{1}(s, x(s))ds, 0 ≤ t ≤ b.

Apply Theorems 2.4 and 2.5 to obtain

1 =d(I − T_{1}, Ω, 0)
=d(I − T_{1}, Ω_{1}, 0) + d(I − T_{1}, Ω_{2}, 0) + d(I − T_{1}, Ω \ \overline{Ω_{1} ∪ Ω_{2}}, 0).

Consider a second modified BVP,

(3.14)  x′′(t) − a(t)x(t) + F_{2}(t, x(t)) = 0, 0 < t < b,

satisfying boundary conditions (3.2) where

F_{2}(t, x) = \{ \begin{matrix}f(t, β_{2}(t)) + \frac{x−β_{2}(t)}{1+|x−β_{2}(t)|}, & β_{2}(t) ≤ x, \\ f(t, x), & α_{2}(t) ≤ x ≤ β_{2}(t), \\ f(t, α_{2}(t)) + \frac{x−α_{2}(t)}{1+|x−α_{2}(t)|}, & x ≤ α_{2}(t). \end{matrix}

Define

T_{2}x(t) = p_{b}(t) + ∫_{0}^{b} G(b; t, s)F_{2}(s, x(s))ds, 0 ≤ t ≤ b.

It follows by Theorem 3.2 that d(I − T_{2}, Ω, 0) = 1.

Apply Theorem 2.4 and the contrapositive of Theorem 2.1 to see that

d(I − T_{2}, Ω \ \overline{Ω}_{1}, 0) = 0.

Thus,

d(I − T_{1}, Ω_{1}, 0) = d(I − T_{2}, Ω_{1}, 0)
= d(I − T_{2}, Ω_{1}, 0) + d(I − T_{2}, Ω \ \overline{Ω}_{1}, 0)
= d(I − T_{2}, Ω, 0) = 1.

Similarly, d(I − T_{1}, Ω_{2}, 0) = 1. Thus, apply Theorem 2.4 to obtain

d(I − T_{1}, Ω \ \overline{Ω_{1} ∪ Ω_{2}}, 0), 0) = −1.

It now follows by Theorem 2.1 that there exist three solutions x_{b1} ∈ Ω_{b1}, x_{b2} ∈ Ω_{b2}, and x_{b3} ∈ Ω_{b}\\overline{Ω_{b1} ∪ Ω_{b2}}, of the BVP, (3.1), (3.2). This completes the proof of Theorem 3.5.

4. EXTENDING THREE SOLUTIONS TO THE UNBOUNDED DOMAIN

In this section, we show that limiting arguments can be used to construct three distinct solutions of the singular BVP, (1.1), (1.2). The technical details are provided in [7] and we only outline those details here.

Theorem 4.1. Assume f : [0, ∞)×(−∞, ∞) → (−∞, ∞) is continuous. Let B > 0. Assume the existence of α_{2} and β_{1} such that, for each b > B, α_{2} is a strict C^{0}-lower solutions and β_{1} is a strict C^{0}-upper solutions of BVP, (3.1), (3.2), respectively; assume the existence of α_{1} and β_{2} such that, for each b > B, α_{1} is a C^{0}-lower solutions and β_{2} is a C^{0}-upper solutions of BVP, (3.1), (3.2), respectively. Assume

α_{1}(t) ≤ α_{2}(t) ≤ β_{2}(t), 0 ≤ t,
α_{1}(t) ≤ β_{1}(t) ≤ β_{2}(t), 0 ≤ t,
α_{2}(t) ≰ β_{1}(t), 0 ≤ t.

Then the BVP, (1.1), (1.2) has at least three solutions, x_{1}(t), x_{2}(t), x_{3}(t) satisfying

α_{1}(t) ≤ x_{1}(t) ≤ β_{1}(t), α_{2}(t) ≤ x_{2}(t) ≤ β_{2}(t), 0 ≤ t,
x_{3}(t) ≰ β_{1}(t), α_{2}(t) ≰ x_{3}(t), 0 ≤ t.

Outline of Proof. Let {b_{i}} denote an unbounded increasing sequence of reals. Assume without loss of generality that B ≤ b_{1}. For each b_{i}, apply Theorem 3.5 and obtain the three distinct functions, x_{1}(b_{i}; t), x_{2}(b_{i}; t), x_{3}(b_{i}; t). Define for each j = 1, 2, 3,

u_{j}(b_{i}; t) = \{ \begin{matrix}x_{j}(b_{i}; t), & 0 ≤ t ≤ b_{i}, \\ x_{j}(b_{i}; b), & b_{i} < t. \end{matrix}

Consider the family of functions, {u_{j}(b_{i}; t)|_{[0,b_{l}]}} where we mean the restriction of u_{j}(b_{i}; t) to the interval [0, b_{l}] for some l. There is a subsequence (for details see [7]) that converges in C^{2}[0, b_{l}] to say w_{j}(b_{l}; t). Perform this construction inductively for l = 1, 2, …. By the inductive construction,

w_{j}(b_{l+1}; t) = w_{j}(b_{l}; t), 0 ≤ t ≤ b_{l}.

Define

x_{j}(t) = ∪_{l}w_{j}(b_{l}; t), j = 1, 2, 3.

Then x_{1}(t), x_{2}(t) and x_{3}(t) denote the three distinct solutions of the BVP, (1.1), (1.2), and the outline of the details is complete.

5. AN EXAMPLE

We show that the autonomous boundary value problem,

x(t)′′ − x(t) + f(x(t)) = 0, 0 ≤ t, (5.1) (5.2) x(0) = 2, x(t) bounded on [0, ∞),

has three positive solutions under certain growth conditions on f.

Theorem 5.1. Let f : [0, +∞) → [0, +∞) be such that f(x) > 0 if x > 0. Assume there exist real numbers a and c such that

(i) 0 < a < cosh(1), 2 cosh(2) − 1 < c, (ii) f(x) < 2 + a for x ∈ [0, a], (iii) f(x) > 2 cosh(2) for x ∈ [ cosh(1), 2 cosh(2) − 1], (iv) f(x) ≤ 2 + c for x ∈ [0, c].

Then the BVP, (5.1), (5.2), has three positive solutions x_{1}, x_{2}, and x_{3} satisfying

e^{−t} ≤ x_{1}(t) ≤ 2 + a(1 − e^{−t}), α_{2}(t) ≤ x_{2}(t) ≤ 2 + c(1 − e^{−t})

and

x_{3}(t) ≰ 2 + a(1 − e^{−t}), α_{2}(t) ≰ x_{3}(t),

for all t > 0, where α_{2} is given by

α_{2}(t) = \{ \begin{matrix}cosh(t), & for all 0 ≤ t ≤ 2, \\ 2 cosh(2) − cosh(t − 4), & for all 2 < t < 6, \\ cosh(2)e^{6−t}, & for all 6 ≤ t. \end{matrix}

Proof. Set B > 6 and let b > B. It is clear that β_{1}(t) = 2 + a(1 − e^{−t}) and β_{2}(t) = 2+c(1−e^{−t}) are upper solutions for (5.1), (5.2) with β_{1} being a strict upper solution. Also, it is clear α_{1}(t) = e^{−t} is a lower solution for (5.1), (5.2).

We show that α_{2}(t) is a strict C^{0}-lower solution. First note that α_{2}(4) = 2 cosh(2) − 1 > β_{1}(4), and hence α_{2}(t) ≰ β_{1}(t), for 0 ≤ t ≤ b. Since α_{2}(2^{+}) = α_{2}(2^{−}) and α_{2}(6^{+}) = α_{2}(6^{−}), then α_{2} ∈ C^{0}[0, b]. Furthermore, for t ∈ (0, 2) and t ∈ (6, +∞) it is easy to see that α_{2}(t)′′ − α_{2}(t) + f(α_{2}(t)) = f(α_{2}(t)) > 0. If t ∈ (2, 6), then α_{2}(t) > cosh(1), and so α_{2}(t)′′ − α_{2}(t) + f(α_{2}(t)) = −2 cosh(2) + f(α_{2}(t)) > 0. Now let t_{0} = 2 and let I_{t_{0}} = (2 − ε, 2 + ε), where ε > 0 is chosen so that, for all t ∈ I_{t_{0}}, we have 2 cosh(2) − cosh(t − 4) > cosh(1) and 2 − ε > 1. Then a_{2}(t) ≥ 2 cosh(2) − cosh(t − 4) := α_{2,t_{0}}(t), and for all t ∈ I_{t_{0}},

α′′_{2,t_{0}}(t) − α_{2,t_{0}}(t) + f(α_{2,t_{0}}) = −2 cosh(2) + f(α_{2,t_{0}}) > 0.

A similar argument holds at t_{0} = 6 and so α_{2} is a C^{0}-lower solution. An application of Theorem 4.1 yields the existence of the three solutions and the proof is complete.

Remark: The function

f(x) = \{ \begin{matrix}2x, & 0 ≤ x ≤ 2, \\ 2 + \frac{6}{cosh(1)−1}(x − 1), & 2 ≤ x ≤ cosh(1), \\ 8, & cosh(1) ≤ x ≤ 2 cosh(2) − 1, \\ 8 + \frac{2}{9−2 cosh(2)}, & 2 cosh(2) − 1 ≤ x ≤ 8, \\ 10, & x ≥ 8, \end{matrix}

satisfies conditions (ii) - (iv) of Theorem 5.1 when a = 1 and c = 8.

ACKNOWLEDGMENTS

C. C. Tisdell gratefully acknowledges the research support of the Australian Research Council’s Discovery Projects (DP0450752) and The Australian Academy of Science’s Scientific Visits Program.

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