Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 28, 1-40; http://www.math.u-szeged.hu/ejqtde/

HIGHER ORDER MULTI-POINT BOUNDARY VALUE PROBLEMS WITH SIGN-CHANGING NONLINEARITIES AND NONHOMOGENEOUS BOUNDARY CONDITIONS

J. R. GRAEF, L. KONG, Q. KONG, AND J. S. W. WONG

Abstract. We study classes of nth order boundary value problems consisting of an equation having a sign-changing nonlinearity f(t, x) together with several different sets of nonhomogeneous multi-point boundary conditions. Criteria are established for the existence of nontrivial solutions, positive solutions, and negative solutions of the problems under consideration. Conditions are determined by the behavior of f(t, x)/x near 0 and ±∞ when compared to the smallest positive characteristic values of some associated linear integral operators. This work improves and extends a number of recent results in the literature on this topic. The results are illustrated with examples.

1. Introduction

Throughout this paper, let m ≥ 1 be an integer, and for any x = (x_{1}, …, x_{m}),

y = (y_{1}, …, y_{m}) ∈ ℝ^{m}, we write \overline{x} = ∑_{i=1}^{m} |x_{i}| and ⟨x, y⟩ = ∑_{i=1}^{m} x_{i}y_{i}. Let

α = (α_{1}, …, α_{m}), β = (β_{1}, …, β_{m}), γ = (γ_{1}, …, γ_{m}) ∈ ℝ_{+}^{m},

and

ξ = (ξ_{1}, …, ξ_{m}) ∈ (0, 1)^{m}

be fixed, where ℝ_{+} = [0, ∞) and ξ_{i}, i = 1, …, m, satisfy 0 < ξ_{1} < ξ_{2} < … < ξ_{m} < 1.

To clarify our notation, we wish to point out that while u(t) is a scalar valued function,

u(ξ) = (u(ξ_{1}), …, u(ξ_{m})) is a vector. In this paper, we are concerned with the

existence of nontrivial solutions of boundary value problems (BVPs) consisting of the

scalar nth order differential equation

u^{(n)} + g(t)f(t, u) = 0, t ∈ (0, 1),  (1.1)

and one of the three nonhomogeneous multi–point boundary conditions (BCs)

\{ \begin{matrix}u^{(i)}(0) = ⟨α, u^{(i)}(ξ)⟩ + λ_{i}, i = 0, …, n − 3, \\ u^{(n−1)}(0) = ⟨β, u^{(n−1)}(ξ)⟩ − λ_{n−2}, \\ u^{(n−2)}(1) = ⟨γ, u^{(n−2)}(ξ)⟩ + λ_{n−1}, \end{matrix}  (1.2)

EJQTDE, 2010 No. 28, p. 1

1991 Mathematics Subject Classification. primary 34B15; secondary 34B10.

Key words and phrases. Nontrivial solutions, boundary value problems, nonhomogeneous boundary conditions, Leray–Schauder degree, Krein–Rutman theorem.

\{ \begin{matrix}u^{(i)}(0) = ⟨α, u^{(i)}(ξ)⟩ + λ_{i}, i = 0, …, n − 3, \\ u^{(n−2)}(0) = ⟨β, u^{(n−2)}(ξ)⟩ + λ_{n−2}, \\ u^{(n−1)}(1) = ⟨γ, u^{(n−1)}(ξ)⟩ + λ_{n−1}, \end{matrix}  (1.3)

and

\{ \begin{matrix}u^{(i)}(0) = ⟨α, u^{(i)}(ξ)⟩ + λ_{i}, i = 0, …, n − 3, \\ u^{(n−2)}(0) = ⟨β, u^{(n−2)}(ξ)⟩ + λ_{n−2}, \\ u^{(n−2)}(1) = ⟨γ, u^{(n−2)}(ξ)⟩ + λ_{n−1}, \end{matrix}  (1.4)

where n ≥ 2 is an integer, f : [0, 1] × ℝ → ℝ and g : (0, 1) → ℝ_{+} are continuous, g ≢ 0 on any subinterval of (0, 1), λ_{i} ∈ ℝ_{+}, and u^{(i)}(ξ) = (u^{(i)}(ξ_{1}), …, u^{(i)}(ξ_{m})) for i = 0, …, n − 1. By a nontrivial solution of BVP (1.1), (1.2), we mean a function u ∈ C^{n−1}[0, 1] ∩ C^{n}(0, 1) such that u(t) ≢ 0 on (0, 1), u(t) satisfies Eq. (1.1) and BC (1.2). If u(t) > 0 on (0, 1), then u(t) is a positive solution. Similar definitions also apply for BVPs (1.1), (1.3) and (1.1), (1.4) as well as for negative solutions of these problems.

We remark that in case n = 2, the first equations in BCs (1.2), (1.3), and (1.4) vanish, and BVPs (1.1), (1.2) and (1.1), (1.3) and (1.1), (1.4) now reduce to the second order BVPs consisting of the equation

u′′ + g(t)f(t, u) = 0, t ∈ (0, 1),  (1.5)

one of the BCs

u′(0) = ⟨β, u′(ξ)⟩ − λ_{0}, u(1) = ⟨γ, u(ξ)⟩ + λ_{1},  (1.6)
u(0) = ⟨β, u(ξ)⟩ + λ_{0}, u′(1) = ⟨γ, u′(ξ)⟩ + λ_{1},  (1.7)

and

u(0) = ⟨β, u(ξ)⟩ + λ_{0}, u(1) = ⟨γ, u(ξ)⟩ + λ_{1}.  (1.8)

When f is positone (i.e., f ≥ 0), existence of solutions of the above second order BVPs, or some of their variations, has been extensively investigated in recent years. For instance, papers [10, 21, 22, 23, 25, 26, 29, 41, 42] studied BVPs with one– parameter BCs and [13, 14, 15, 16, 17] studied BVPs with two–parameter BCs. In particular, for one–parameter problems, Ma [25] studied BVP (1.5), (1.8) with m = 1 and β = λ_{0} = 0. Under certain assumptions, he showed that there exists λ_{1}^{∗} > 0 such that BVP (1.5), (1.8) has at least one positive solution for 0 < λ_{1} < λ_{1}^{∗} and has no positive solution for λ_{1} > λ_{1}^{∗}; later, Sun et al. [29] proved similar results for BVP (1.5), (1.6) with β = (0, …, 0) and λ_{0} = 0; Kwong and Wong [21] further significantly

EJQTDE, 2010 No. 28, p. 2 improved the results in [29] and also constructed a counterexample to point out that one of the main results in [29] is actually false; Zhang and Sun [41] recently obtained results, similar to those in [25], for BVP (1.5), (1.7) with λ_{0} = 0. Paper [21] does contain some optimal existence criteria. As for the second order two–parameter problems, Kong and Kong [13, 14, 15, 16] studied BVPs (1.5), (1.6) and (1.5), (1.8) with λ_{0}, λ_{1} ∈ ℝ and established many existence, nonexistence, and multiplicity results for positive solutions of the problems. Moreover, under some conditions, they proved that there exists a continuous curve Γ separating the (λ_{0}, λ_{1})–plane into two disjoint connected regions Λ^{E} and Λ^{N} with Γ ⊆ Λ^{E} such that BVPs (1.5), (1.6) and (1.5), (1.8) have at least two solutions for (λ_{0}, λ_{1}) ∈ Λ^{E} \ Γ, have at least one solution for (λ_{0}, λ_{1}) ∈ Γ, and have no solution for (λ_{0}, λ_{1}) ∈ Λ^{N}. The uniqueness of positive solutions and the dependence of positive solutions on the parameters λ_{0} and λ_{1} are investigated in [17] for BVP (1.5), (1.8). Recently, higher order positone BVPs with nonhomogeneous BCs have also been studied in the literature, for example, in [7, 8, 18, 19, 20, 28, 31, 32, 37]. In particular, paper [20] studied BVPs (1.1), (1.2) and (1.1), (1.3) and proved several optimal existence criteria for positive solutions of these problems under the assumption that f is nonnegative. In the present paper, we allow f to change sign.

However, very little has been done in the literature on BVPs with nonhomogeneous BCs when the nonlinearities are sign–changing functions. As far as we know, the only work to tackle this situation is the recent paper [6], where BVP (1.5), (1.8) is considered with m = 2, β = (β_{1}, 0), and γ = (0, γ_{2}), and where sufficient conditions for the existence of nontrivial solutions are obtained. Motivated partially by the recent papers [6, 12, 18, 20, 24], here we will derive several new criteria for the existence of nontrivial solutions, positive solutions, and negative solutions of BVPs (1.1), (1.2) and (1.1), (1.3) and (1.1), (1.4) when the nonlinear term f is a sign-changing function and not necessarily bounded from below. The proof uses topological degree theory together with a comparison between the behavior of the quotient f(t, x)/x for x near 0 and ±∞ and the smallest positive characteristic values (given by (3.1) below) of some related linear operators L, L̃, and L̂ (defined by (2.31)–(2.33) in Section 2). These characteristic values are known to exist by the Krein–Rutman theorem. Our

EJQTDE, 2010 No. 28, p. 3 results extend and improve many recent works on BVPs with nonhomogeneous BCs, especially those in papers [6, 10, 13, 14, 15, 16, 18, 19, 20, 25, 29, 41, 42]. We believe that our results are new even for homogeneous problems, i.e., when λ_{i} = 0, i = 0, …, n − 1, in BCs (1.2), (1.3), and (1.4). For other studies on optimal existence criteria on BVPs with homogeneous BCs, we refer the reader to [2, 3, 11, 27, 30, 33, 34, 35, 36, 38, 40] and the references therein. In particular, Webb and Infante [35] studied some higher order problems and obtained sharp results for the existence of one positive solution. They gave some non-existence results as well. The nonlocal boundary conditions in [35] are different from the ones studied here. Webb and Infante were mainly concerned with homogeneous boundary conditions but nonhomogeneous boundary conditions were also treated.

We assume the following condition holds throughout without further mention: (H) 0 ≤ \overline{α} < 1, 0 ≤ \overline{β} < 1, 0 ≤ \overline{γ} < 1, and ∫_{0}^{1} g(s)ds < ∞. The rest of this paper is organized as follows. Section 2 contains some preliminary

lemmas, Sections 3 contains the main results of this paper and several examples, and the proofs of the main results are presented in Section 4.

2. Preliminary results

In this section, we present some preliminary results that will be used in the statements and the proofs of the main results. In the rest of this paper, the bold 0 stands for the zero element in any given Banach space. We refer the reader to [9, Lemma 2.5.1] for the proof of the following well known lemma.

Lemma 2.1. Let Ω be a bounded open set in a real Banach space X with 0 ∈ Ω and T : \overline{Ω} → X be compact. If

T u ≠ τu for all u ∈ ∂Ω and τ ≥ 1,

then the Leray-Schauder degree

deg(I − T, Ω, 0) = 1.

Let (X, ||·||) be a real Banach space and L : X → X be a linear operator. We recall that λ is an eigenvalue of L with a corresponding eigenfunction φ if φ is nontrivial

EJQTDE, 2010 No. 28, p. 4 and Lφ = λφ. The reciprocals of eigenvalues are called the characteristic values of L. Recall also that a cone P in X is called a total cone if X = \overline{P − P}.

The following Krein-Rutman theorem can be found in either [1, Theorem 19.2] or [39, Proposition 7.26].

Lemma 2.2. Assume that P is a total cone in a real Banach space X. Let L : X → X be a compact linear operator with L(P) ⊆ P and the spectral radius, r_{L}, of L satisfy r_{L} > 0. Then r_{L} is an eigenvalue of L with an eigenfunction in P.

Let X^{∗} be the dual space of X, P be a total cone in X, and P^{∗} be the dual cone of P, i.e.,

P^{∗} = {l ∈ X^{∗} : l(u) ≥ 0 for all u ∈ P }.

Let L, M : X → X be two linear compact operators such that L(P) ⊆ P and M(P) ⊆ P. If their spectral radii r_{L} and r_{M} are positive, then by Lemma 2.2, there exist φ_{L}, φ_{M} ∈ P \ {0} such that

Lφ_{L} = r_{L}φ_{L} and Mφ_{M} = r_{M}φ_{M}.  (2.1)

Assume there exists h ∈ P^{∗} \ {0} such that

L^{∗}h = r_{M}h,  (2.2)

where L^{∗} is the dual operator of L. Choose δ > 0 and define

P (h, δ) = {u ∈ P : h(u) ≥ δ||u||}.  (2.3)

Then, P (h, δ) is a cone in X.

In the following, Lemma 2.3 is a generalization of [12, Theorem 2.1] and it is proved in [24, Lemma 2.5] for the case when L and M are two specific linear operators, but the proof there also works for any general linear operators L and M satisfying (2.1) and (2.2). Lemma 2.4 generalizes [4, Lemma 3.5] and it is proved in [5, Lemma 2.5]. From here on, for any R > 0, let B(0, R) = {u ∈ X : ||u|| < R} be the open ball of X centered at 0 with radius R.

Lemma 2.3. Assume that the following conditions hold: (A1) There exist φ_{L}, φ_{M} ∈ P \ {0} and h ∈ P^{∗} \ {0} such that (2.1) and (2.2) hold

and L(P) ⊆ P (h, δ); EJQTDE, 2010 No. 28, p. 5 (A2) H : X → P is a continuous operator and satisfies

lim_{||u||→∞} \frac{||Hu||}{||u||} = 0;

(A3) F : X → X is a bounded continuous operator and there exists u_{0} ∈ X such

that F u + Hu + u_{0} ∈ P for all u ∈ X; (A4) There exist v_{0} ∈ X and ϵ > 0 such that

LF u ≥ r_{M}^{−1}(1 + ϵ)Lu − LHu − v_{0} for all u ∈ X.

Let T = LF. Then there exists R > 0 such that the Leray-Schauder degree

deg(I − T, B(0, R), 0) = 0.

Lemma 2.4. Assume that (A1) and the following conditions hold: (A2)^{∗} H : X → P is a continuous operator and satisfies

lim_{||u||→0} \frac{||Hu||}{||u||} = 0;

(A3)^{∗} F : X → X is a bounded continuous operator and there exists r_{1} > 0 such

that

F u + Hu ∈ P for all u ∈ X with ||u|| < r_{1};

(A4)^{∗} There exist ϵ > 0 and r_{2} > 0 such that

LF u ≥ r_{M}^{−1}(1 + ϵ)Lu for all u ∈ X with ||u|| < r_{2}.

Let T = LF. Then there exists 0 < R < min{r_{1}, r_{2}} such that the Leray-Schauder degree

deg(I − T, B(0, R), 0) = 0.

Now let

G(t, s) = \{ \begin{matrix}1 − s, & 0 ≤ t ≤ s ≤ 1, \\ 1 − t, & 0 ≤ s ≤ t ≤ 1, \end{matrix}  (2.4)
G̃(t, s) = \{ \begin{matrix}t, & 0 ≤ t ≤ s ≤ 1, \\ s, & 0 ≤ s ≤ t ≤ 1, \end{matrix}  (2.5)

and

Ĝ(t, s) = \{ \begin{matrix}t(1 − s), & 0 ≤ t ≤ s ≤ 1, \\ s(1 − t), & 0 ≤ s ≤ t ≤ 1, \end{matrix}  (2.6)

EJQTDE, 2010 No. 28, p. 6 Then, it is well known that G(t, s) is the Green’s function of the BVP

−u′′ = 0 on (0, 1), u′(0) = u(1) = 0,

G̃(t, s) is the Green’s function of the BVP

−u′′ = 0 on (0, 1), u(0) = u′(1) = 0,

and Ĝ(t, s) is the Green’s function of the BVP

−u′′ = 0 on (0, 1), u(0) = u(1) = 0.

Recall that the characteristic function χ on an interval I is given by

χ_{I}(t) = \{ \begin{matrix}1, & t ∈ I, \\ 0, & t ∉ I. \end{matrix}

In the sequel, we write

G(ξ, s) = (G(ξ_{1}, s), …, G(ξ_{m}, s)),
χ_{[0,ξ]}(s) = (χ_{[0,ξ_{1}]}(s), …, χ_{[0,ξ_{m}]}(s)),
ξ(1 − ξ) = (ξ_{1}(1 − ξ_{1}), …, ξ_{m}(1 − ξ_{m})),

and for any v ∈ C[0, 1], we let

v(ξ) = (v(ξ_{1}), …, v(ξ_{m})) .

We also use some other similar notations that will be clear from the context and will not be listed here.

Define

H_{1}(t, s) = \frac{⟨α, χ_{[0,ξ]}(s)⟩}{1 − \overline{α}} + χ_{[0,t]}(s),  (2.7)
H_{0}(t, s) = G(t, s) + \frac{⟨γ, G(ξ, s)⟩}{1 − \overline{γ}} + \frac{[1 − ⟨γ, ξ⟩ − (1 − \overline{γ})t]⟨β, χ_{[0,ξ]}(s)⟩}{(1 − \overline{β})(1 − \overline{γ})},  (2.8)
H̃_{0}(t, s) = G̃(t, s) + \frac{⟨β, G̃(ξ, s)⟩}{1 − \overline{β}} + \frac{[⟨β, ξ⟩ + (1 − \overline{β})t]⟨γ, χ_{[ξ,1]}(s)⟩}{(1 − \overline{β})(1 − \overline{γ})},  (2.9)
Ĥ_{0}(t, s) = Ĝ(t, s) + \frac{t}{ρ}[(1 − \overline{β})⟨γ, Ĝ(ξ, s)⟩ − (1 − \overline{γ})⟨β, Ĝ(ξ, s)⟩]
+\frac{1}{ρ}[(1 − ⟨γ, ξ⟩)⟨β, Ĝ(ξ, s)⟩ + ⟨β, ξ⟩⟨γ, Ĝ(ξ, s)⟩],  (2.10)

where

ρ = (1 − \overline{β})(1 − ⟨γ, ξ⟩) + (1 − \overline{γ})⟨β, ξ⟩ > 0.  (2.11)

EJQTDE, 2010 No. 28, p. 7 For i = 1, …, n − 1, define K_{i}(t, s), K̃_{i}(t, s), and K̂_{i}(t, s) recursively as follows

K_{1}(t, s) = H_{0}(t, s), K_{i}(t, s) = ∫_{0}^{1} H_{1}(t, τ)K_{i−1}(τ, s)dτ, i = 2, …, n − 1, (2.12)

K̃_{1}(t, s) = H̃_{0}(t, s), K̃_{i}(t, s) = ∫_{0}^{1} H_{1}(t, τ)K̃_{i−1}(τ, s)dτ, i = 2, …, n − 1, and

K̂_{1}(t, s) = Ĥ_{0}(t, s), K̂_{i}(t, s) = ∫_{0}^{1} H_{1}(t, τ)K̂_{i−1}(τ, s)dτ, i = 2, …, n − 1.

Remark 2.1. It is easy to see that, for i = 1, …, n − 1, K_{i}(t, s) ≥ 0, K̃_{i}(t, s) ≥ 0, K̂_{i}(t, s) ≥ 0 for t, s ∈ [0, 1], and K_{i}(t, s) > 0, K̃_{i}(t, s) > 0, K̂_{i}(t, s) > 0 for t, s ∈ (0, 1).

The following lemma provides the equivalent integral forms for some BVPs.

Lemma 2.5. Let k ∈ L(0, 1) ∩ C(0, 1). Then we have the following:

(a) The function u(t) is a solution of the BVP consisting of the equation

u^{(n)} + k(t) = 0, t ∈ (0, 1),  (2.13)

and the BC

\{ \begin{matrix}u^{(i)}(0) = ⟨α, u^{(i)}(ξ)⟩, i = 0, …, n − 3, \\ u^{(n−1)}(0) = ⟨β, u^{(n−1)}(ξ)⟩, \\ u^{(n−2)}(1) = ⟨γ, u^{(n−2)}(ξ)⟩, \end{matrix}

if and only if

u(t) = ∫_{0}^{1} K_{n−1}(t, s)k(s)ds.

(b) The function u(t) is a solution of the BVP consisting of Eq. (2.13) and the BC

\{ \begin{matrix}u^{(i)}(0) = ⟨α, u^{(i)}(ξ)⟩, i = 0, …, n − 3, \\ u^{(n−2)}(0) = ⟨β, u^{(n−2)}(ξ)⟩, \\ u^{(n−1)}(1) = ⟨γ, u^{(n−1)}(ξ)⟩, \end{matrix}

if and only if

u(t) = ∫_{0}^{1} K̃_{n−1}(t, s)k(s)ds.

(c) The function u(t) is a solution of the BVP consisting of Eq. (2.13) and the BC

\{ \begin{matrix}u^{(i)}(0) = ⟨α, u^{(i)}(ξ)⟩, i = 0, …, n − 3, \\ u^{(n−2)}(0) = ⟨β, u^{(n−2)}(ξ)⟩, \\ u^{(n−2)}(1) = ⟨γ, u^{(n−2)}(ξ)⟩, \end{matrix}

if and only if

u(t) = ∫_{0}^{1} K̂_{n−1}(t, s)k(s)ds.

Parts (a) and (b) of Lemma 2.5 were proved in [20, Lemma 2.2], and part (c) can be proved similarly. We omit the proof of part (c) of the lemma.

Lemma 2.6 below obtains some useful estimates for K_{n−1}(t, s), K̃_{n−1}(t, s), and K̂_{n−1}(t, s).

Lemma 2.6. We have the following:

(a) The function K_{n−1}(t, s) satisfies

μ(t)(1 − s) ≤ K_{n−1}(t, s) ≤ ν(1 − s) for t, s ∈ [0, 1],  (2.14)

where

μ(t) = \{ \begin{matrix}1 − t + \frac{\overline{γ} − ⟨γ, ξ⟩}{1 − \overline{γ}}, & n = 2, \\ ∫_{0}^{t} \frac{(t − τ)^{n−3}}{(n − 3)!} (1 − τ + \frac{\overline{γ} − ⟨γ, ξ⟩}{1 − \overline{γ}}) dτ, & n ≥ 3, \end{matrix}  (2.15)

and

ν = (\frac{1}{1 − \overline{γ}} + \frac{1 − ⟨γ, ξ⟩}{(1 − \overline{β})(1 − \overline{γ})}) \frac{1 − ξ_{m} + \overline{β}}{1 − ξ_{m}} \frac{1}{(1 − \overline{α})^{n−2}} ,  (2.16)

(b) The function K̃_{n−1}(t, s) satisfies

μ̃(t)s ≤ K̃_{n−1}(t, s) ≤ ν̃s for t, s ∈ [0, 1],

where

μ̃(t) = \{ \begin{matrix}t + \frac{⟨β, ξ⟩}{1 − \overline{β}}, & n = 2, \\ ∫_{0}^{t} \frac{(t − τ)^{n−3}}{(n − 3)!} (τ + \frac{⟨β, ξ⟩}{1 − \overline{β}}) dτ, & n ≥ 3, \end{matrix}  (2.17)

and

ν̃ = (\frac{1}{1 − \overline{β}} + \frac{⟨β, ξ⟩ + 1 − \overline{β}}{(1 − \overline{β})(1 − \overline{γ})}) \frac{ξ_{1} + \overline{γ}}{ξ_{1}} \frac{1}{(1 − \overline{α})^{n−2}} ,  (2.18)

(c) The function K̂_{n−1}(t, s) satisfies

μ̂(t)s(1 − s) ≤ K̂_{n−1}(t, s) ≤ ν̂s(1 − s) for t, s ∈ [0, 1],

EJQTDE, 2010 No. 28, p. 9 where

μ̂(t) = \{ \begin{matrix}t(1 − t) + min \{â, b̂\}, & n = 2, \\ ∫_{0}^{t} \frac{(t − τ)^{n−3}}{(n − 3)!} (τ(1 − τ) + min \{â, b̂\}) dτ, & n ≥ 3, \end{matrix}  (2.19)

and

ν̂ = (1 + max \{ĉ, d̂\}) \frac{1}{(1 − \overline{α})^{n−2}} ,  (2.20)

with

â = \frac{1}{ρ}[(1 − ⟨γ, ξ)⟩)⟨β, ξ(1 − ξ)⟩ + ⟨β, ξ⟩⟨γ, ξ(1 − ξ)⟩],  (2.21)
b̂ = \frac{1}{ρ}[(1 − \overline{β} + ⟨β, ξ)⟩)⟨γ, ξ(1 − ξ)⟩ + (\overline{γ} − ⟨γ, ξ⟩)⟨β, ξ(1 − ξ)⟩],  (2.22)
ĉ = \frac{1}{ρ}[(1 − ⟨γ, ξ⟩)\overline{β} + ⟨β, ξ⟩\overline{γ}],  (2.23)
d̂= \frac{1}{ρ}[(1 − \overline{β} + ⟨β, ξ)⟩)\overline{γ} + (\overline{γ} − ⟨γ, ξ⟩)\overline{β}],  (2.24)

and ρ is defined by (2.11). Proof. We first prove part (a). From (2.4), it is clear that

(1 − t)(1 − s) ≤ G(t, s) ≤ (1 − s) for t, s ∈ [0, 1].

Then, from (2.8), it is easy to see that

H_{0}(t, s) ≥ (1 − t)(1 − s) + \frac{⟨γ, G(ξ, s)⟩}{1 − \overline{γ}} ≥ (1 − t + \frac{\overline{γ} − ⟨γ, ξ⟩}{1 − \overline{γ}}) (1 − s) (2.25)

and

H_{0}(t, s) ≤ 1 − s + \frac{\overline{γ}}{1 − \overline{γ}} (1 − s) + \frac{1 − ⟨γ, ξ⟩}{(1 − \overline{β})(1 − \overline{γ})}\overline{β}χ_{[0,ξ_{m}]}(s)
= \frac{1}{1 − \overline{γ}} (1 − s) + \frac{1 − ⟨γ, ξ⟩}{(1 − \overline{β})(1 − \overline{γ})}\overline{β}χ_{[0,ξ_{m}]}(s)
≤ (\frac{1}{1 − \overline{γ}} + \frac{1 − ⟨γ, ξ⟩}{(1 − \overline{β})(1 − \overline{γ})}) (1 − s + \overline{β}χ_{[0,ξ_{m}]}(s))  (2.26)

for t, s ∈ [0, 1]. For s ∈ [0, ξ_{m}], since

(\frac{1 − s + \overline{β}}{1 − s} )′ = \frac{\overline{β}}{(1 − s)^{2}} ≥ 0,

we see that (1 − s + \overline{β})/(1 − s) is nondecreasing on [0, ξ_{m}], and so

\frac{1 − s + \overline{β}}{1 − s} ≤ \frac{1 − ξ_{m} + \overline{β}}{1 − ξ_{m}} for s ∈ [0, ξ_{m}].

EJQTDE, 2010 No. 28, p. 10 This in turn implies

1 − s + \overline{β}χ_{[0,ξ_{m}]}(s) = 1 − s + \overline{β} ≤ \frac{1 − ξ_{m} + \overline{β}}{1 − ξ_{m}} (1 − s) for s ∈ [0, ξ_{m}].

Note that

1 − s + \overline{β}χ_{[0,ξ_{m}]}(s) = 1 − s ≤ \frac{1 − ξ_{m} + \overline{β}}{1 − ξ_{m}} (1 − s) for s ∈ (ξ_{m}, 1].

Then,

1 − s + \overline{β}χ_{[0,ξ_{m}]}(s) ≤ \frac{1 − ξ_{m} + \overline{β}}{1 − ξ_{m}} (1 − s) for s ∈ [0, 1].

Combing the above inequality with (2.26) yields

H_{0}(t, s) ≤ (\frac{1}{1 − \overline{γ}} + \frac{1 − ⟨γ, ξ⟩}{(1 − \overline{β})(1 − \overline{γ})}) \frac{1 − ξ_{m} + \overline{β}}{1 − ξ_{m}} (1 − s).  (2.27)

When n = 2, note from (2.12) that K_{n−1}(t, s) = H_{0}(t, s), and by (2.15) and (2.16),

μ(t) = 1 − t + \frac{\overline{γ} − ⟨γ, ξ⟩}{1 − \overline{γ}} and ν = (\frac{1}{1 − \overline{γ}} + \frac{1 − ⟨γ, ξ⟩}{(1 − \overline{β})(1 − \overline{γ})}) \frac{1 − ξ_{m} + \overline{β}}{1 − ξ_{m}} .

Then, (2.14) follows from (2.25) and (2.27).

Now we assume that n ≥ 3. For t, s ∈ [0, 1], from (2.7),

χ_{[0,t]}(s) ≤ H_{1}(t, s) ≤ \frac{\overline{α}}{1 − \overline{α}} + 1 = \frac{1}{1 − \overline{α}}.  (2.28)

Then, from (2.12), (2.25), (2.27), (2.28), it follows that

K_{2}(t, s) ≥ ∫_{0}^{t} (1 − τ + \frac{\overline{γ} − ⟨γ, ξ⟩}{1 − \overline{γ}}) dτ (1 − s)

and

K_{2}(t, s) ≤ (\frac{1}{1 − \overline{γ}} + \frac{1 − ⟨γ, ξ⟩}{(1 − \overline{β})(1 − \overline{γ})}) \frac{1 − ξ_{m} + \overline{β}}{1 − ξ_{m}} \frac{1}{1 − \overline{α}} (1 − s).

Combining the above inequalities with (2.12) and (2.28), we see that

K_{3}(t, s) ≥ ∫_{0}^{t} ∫_{0}^{v} (1 − τ + \frac{\overline{γ} − ⟨γ, ξ⟩}{1 − \overline{γ}}) dτdv (1 − s)
= ∫_{0}^{t}(t − τ) (1 − τ + \frac{\overline{γ} − ⟨γ, ξ⟩}{1 − \overline{γ}}) dτ (1 − s)

and

K_{3}(t, s) ≤ (\frac{1}{1 − \overline{γ}} + \frac{1 − ⟨γ, ξ⟩}{(1 − \overline{β})(1 − \overline{γ})}) \frac{1 − ξ_{m} + \overline{β}}{1 − ξ_{m}} \frac{1}{(1 − \overline{α})^{2}} (1 − s).

An induction argument easily shows that

K_{n−1}(t, s) ≥ ∫_{0}^{t} \frac{(t − τ)^{n−3}}{(n − 3)!} (1 − τ + \frac{\overline{γ} − ⟨γ, ξ⟩}{1 − \overline{γ}}) dτ (1 − s) = μ(t)(1 − s)

EJQTDE, 2010 No. 28, p. 11 and

K_{n−1}(t, s) ≤ (\frac{1}{1 − \overline{γ}} + \frac{1 − ⟨γ, ξ⟩}{(1 − \overline{β})(1 − \overline{γ})}) \frac{1 − ξ_{m} + \overline{β}}{1 − ξ_{m}} \frac{1}{(1 − \overline{α})^{n−2}} (1 − s)
= ν(1 − s)

for t, s ∈ [0, 1]. Thus, (2.14) holds. This prove part (a).

Next, we show part (b). From (2.5), we have

ts ≤ G̃(t, s) ≤ s for t, s ∈ [0, 1].

Then, from (2.9), it is easy to see that

H̃_{0}(t, s) ≥ ts + \frac{⟨β, G̃(ξ, s)⟩}{1 − \overline{β}} ≥ (t + \frac{⟨β, ξ⟩}{1 − \overline{β}}) s

and

H̃_{0}(t, s) ≤ s + \frac{\overline{β}}{1 − \overline{β}} s + \frac{⟨β, ξ⟩ + 1 − \overline{β}}{(1 − \overline{β})(1 − \overline{γ})}\overline{γ}χ_{[ξ_{1},1]}(s)
= \frac{1}{1 − \overline{β}} s + \frac{⟨β, ξ⟩ + 1 − \overline{β}}{(1 − \overline{β})(1 − \overline{γ})}\overline{γ}χ_{[ξ_{1},1]}(s)
≤ (\frac{1}{1 − \overline{β}} + \frac{⟨β, ξ⟩ + 1 − \overline{β}}{(1 − \overline{β})(1 − \overline{γ})}) (s + \overline{γ}χ_{[ξ_{1},1]}(s))  (2.29)

for t, s ∈ [0, 1]. Note that (s + \overline{γ})/s is nonincreasing on [ξ_{1}, 1]. Then,

\frac{s + \overline{γ}}{s} ≤ \frac{ξ_{1} + \overline{γ}}{ξ_{1}} for s ∈ [ξ_{1}, 1].

Thus,

s + \overline{γ}χ_{[ξ_{1},1]}(s) = s + \overline{γ} ≤ \frac{ξ_{1} + \overline{γ}}{ξ_{1}} s for s ∈ [ξ_{1}, 1].

Since

s + \overline{γ}χ_{[ξ_{1},1]}(s) = s ≤ \frac{ξ_{1} + \overline{γ}}{ξ_{1}} s for s ∈ [0, ξ_{1}),

we have

s + \overline{γ}χ_{[ξ_{1},1]}(s) ≤ \frac{ξ_{1} + \overline{γ}}{ξ_{1}} s for s ∈ [0, 1].

Then, from (2.29),

H̃_{0}(t, s) ≤ (\frac{1}{1 − \overline{β}} + \frac{⟨β, ξ⟩ + 1 − \overline{β}}{(1 − \overline{β})(1 − \overline{γ})}) \frac{ξ_{1} + \overline{γ}}{ξ_{1}} s.

The rest of the proof is similar to the latter part of the one used in showing (2.14), and hence is omitted.

EJQTDE, 2010 No. 28, p. 12 Finally, we prove part (c). From (2.6), we have

t(1 − t)s(1 − s) ≤ Ĝ(t, s) ≤ s(1 − s) for t, s ∈ [0, 1].

Let â, b̂, ĉ, and d̂ be defined by (2.21)–(2.24), and define

p(t) = \frac{t}{ρ}[(1 − \overline{β})⟨γ, Ĝ(ξ, s)⟩ − (1 − \overline{γ})⟨β, Ĝ(ξ, s)⟩]

+\frac{1}{ρ}[(1 − ⟨γ, ξ⟩)⟨β, Ĝ(ξ, s)⟩ + ⟨β, ξ⟩⟨γ, Ĝ(ξ, s)⟩].

Then,

âs(1 − s) = \frac{1}{ρ}[(1 − ⟨γ, ξ⟩)⟨β, ξ(1 − ξ)⟩ + ⟨β, ξ⟩⟨γ, ξ(1 − ξ)⟩]s(1 − s)
≤ p(0) = \frac{1}{ρ}[(1 − ⟨γ, ξ⟩)⟨β, Ĝ(ξ, s)⟩ + ⟨β, ξ⟩⟨γ, Ĝ(ξ, s)⟩]
≤ \frac{1}{ρ}[(1 − ⟨γ, ξ⟩)\overline{β} + ⟨β, ξ⟩\overline{γ}]s(1 − s)
= ĉs(1 − s)

and

b̂s(1 − s) = \frac{1}{ρ}[(1 − \overline{β} + ⟨β, ξ)⟩)⟨γ, ξ(1 − ξ)⟩ + (\overline{γ} − ⟨γ, ξ⟩)⟨β, ξ(1 − ξ)⟩]s(1 − s)
≤ p(1) = \frac{1}{ρ}[(1 − \overline{β} + ⟨β, ξ⟩)⟨γ, Ĝ(ξ, s)⟩ + (\overline{γ} − ⟨γ, ξ⟩)⟨β, Ĝ(ξ, s)⟩]
≤ \frac{1}{ρ}[(1 − \overline{β} + ⟨β, ξ)⟩)\overline{γ} + (\overline{γ} − ⟨γ, ξ⟩)\overline{β}]s(1 − s)
= d̂s(1 − s),

i.e.,

âs(1 − s) ≤ p(0) ≤ ĉs(1 − s) and b̂s(1 − s) ≤ p(1) ≤ d̂s(1 − s).

Moreover, from (2.10), we see that Ĥ_{0}(t, s) = Ĝ(t, s) + p(t). Thus,

Ĥ_{0}(t, s) ≥ t(1 − t)s(1 − s) + min{p(0), p(1)}
≥ t(1 − t)s(1 − s) + min{â, b̂}s(1 − s)
= (t(1 − t) + min{â, b̂}) s(1 − s)

EJQTDE, 2010 No. 28, p. 13 and

Ĥ_{0}(t, s) ≤ s(1 − s) + max{p(0), p(1)}
≤ s(1 − s) + max{ĉ, d̂}s(1 − s)
= (1 + max{ĉ, d̂}) s(1 − s).

The rest of the proof is similar to the latter part of the one used in showing (2.14), and hence is omitted. This completes the proof of the lemma. □

In the remainder of the paper, let X = C[0, 1] be the Banach space of continuous functions equipped with the norm ||u|| = max_{t∈[0,1]} |u(t)|. Define a cone P in X by

P = {u ∈ X : u(t) ≥ 0 on [0, 1]} .  (2.30)

Let the linear operators L, M, L̃, M̃, L̂, M̂ : X → X be defined by

Lu(t) = ∫_{0}^{1} K_{n−1}(t, s)g(s)u(s)ds, Mu(t) = ∫_{0}^{1} K_{n−1}(s, t)g(s)u(s)ds, (2.31)
L̃u(t) = ∫_{0}^{1} K̃_{n−1}(t, s)g(s)u(s)ds, M̃u(t) = ∫_{0}^{1} K̃_{n−1}(s, t)g(s)u(s)ds, (2.32)

and

L̂u(t) = ∫_{0}^{1} K̂_{n−1}(t, s)g(s)u(s)ds, M̂u(t) = ∫_{0}^{1} K̂_{n−1}(s, t)g(s)u(s)ds. (2.33)

Remark 2.2. When g ≡ 1, and the pair L and M are considered as operators in the space L^{2}(0, 1), then L and M are adjoints of each other, so r_{L} and r_{M} would be equal. Since the function g is present, they are not adjoints of each other. However, for g as in this paper, as can be seen from Proposition 5.1 in the Appendix, the proof of which is provided by J. R. L. Webb, we do have that r_{L} and r_{M} are equal. Similar statements hold for the other pairs of operators as well.

From here on, let r_{L}, r_{M}, rL̃, rM̃, rL̂, and rM̂ be the spectral radii of L, M, L̃, M̃, L̂, and M̂, respectively. The next lemma provides some information about the above operators.

Lemma 2.7. The operators L, M, L̃, M̃, L̂, M̂ map P into P and are compact. Moreover, we have

(a) r_{L} > 0 and r_{L} is an eigenvalue of L with an eigenfunction φ_{L} ∈ P; EJQTDE, 2010 No. 28, p. 14 (b) r_{M} > 0 and r_{M} is an eigenvalue of M with an eigenfunction φ_{M} ∈ P;

(c) rL̃ > 0 and rL̃ is an eigenvalue of L̃ with an eigenfunction φL̃ ∈ P;

(d) rM̃ > 0 and rM̃ is an eigenvalue of M̃ with an eigenfunction φM̃ ∈ P;

(e) rL̂ > 0 and rL̂ is an eigenvalue of L̂ with an eigenfunction φL̂ ∈ P;

(f) rM̂ > 0 and rM̂ is an eigenvalue of M̂ with an eigenfunction φM̂ ∈ P.

The proof of the compactness and cone invariance for these operators is standard. By virtue of Lemma 2.2, part (a) was proved in [20, Lemma 2.6] and parts (b)–(f) can be proved essentially by the same way. We omit the proof of the lemma.

3. Main results

For convenience, we will use the following notations.

f_{0} = lim inf_{x→0^{+}} min_{t∈[0,1]} \frac{f(t, x)}{x}, f_{0}^{∗} = lim inf_{x→0} min_{t∈[0,1]} \frac{f(t, x)}{x},

f_{∞} = lim inf_{x→∞} min_{t∈[0,1]} \frac{f(t, x)}{x}, f_{∞}^{∗} = lim inf_{|x|→∞} min_{t∈[0,1]} \frac{f(t, x)}{x},

F_{0} = lim sup_{x→0} max_{t∈[0,1]} |\frac{f(t, x)}{x}| , F_{∞} = lim sup_{|x|→∞} max_{t∈[0,1]} |\frac{f(t, x)}{x}| .

Let r_{M}, rM̃, rM̂, φ_{M}, φM̃, and φM̂ be given as in Lemma 2.7. Define

μ_{M} = \frac{1}{r_{M}} , μM̃ = \frac{1}{rM̃} , and μM̂ = \frac{1}{rM̂} .  (3.1)

Clearly, μ_{M} is the smallest positive characteristic value of M and satisfies φ_{M} = μ_{M}Mφ_{M}. Similar statements hold for μM̃ and μM̂.

We need the following assumptions. (H1) There exist three nonnegative functions a, b ∈ C[0, 1] and c ∈ C(ℝ) such that

c(x) is even and nondecreasing on ℝ_{+},

f(t, x) ≥ −a(t) − b(t)c(x) for all (t, x) ∈ [0, 1] × ℝ,  (3.2)

and

lim_{x→∞} \frac{c(x)}{x} = 0.  (3.3)

EJQTDE, 2010 No. 28, p. 15 (H2) There exist a constant r ∈ (0, 1) and two nonnegative functions d ∈ C[0, 1]

and e ∈ C(ℝ) such that e is even and nondecreasing on ℝ^{+},

f(t, x) ≥ −d(t)e(x) for all (t, x) ∈ [0, 1] × [−r, 0],  (3.4)

and

lim_{x→0} \frac{e(x)}{x} = 0.  (3.5)

(H3) xf(t, x) ≥ 0 for (t, x) ∈ [0, 1] × ℝ.

Remark 3.1. Here, we want to emphasize that, in (H1), we assume that f(t, x) is bounded from below by −a(t) − b(t)c(x) for all (t, x) ∈ [0, 1] × ℝ; however in (H2), we only require that f(t, x) is bounded from below by −d(t)e(x) for t ∈ [0, 1] and x in a small left-neighborhood of 0.

We first state our existence results for BVP (1.1), (1.2).

Theorem 3.1. Assume that (H1) holds and F_{0} < μ_{M} < f_{∞}. Then, for (λ_{0}, …, λ_{n−1}) ∈ ℝ_{+}^{n} with ∑_{i=0}^{n−1} λ_{i} sufficiently small, BVP (1.1), (1.2) has at least one nontrivial solution.

Theorem 3.2. Assume that (H2) holds and F_{∞} < μ_{M} < f_{0}. Then, for (λ_{0}, …, λ_{n−1}) ∈ ℝ_{+}^{n} with ∑_{i=0}^{n−1} λ_{i} sufficiently small, BVP (1.1), (1.2) has at least one nontrivial solution.

Theorem 3.3. Assume that (H3) holds, F_{0} < μ_{M} < f_{∞}^{∗}, and λ_{i} = 0 for i = 0, …, n− 1. Then BVP (1.1), (1.2) has at least one positive solution and one negative solution.

Theorem 3.4. Assume that (H3) holds, F_{∞} < μ_{M} < f_{0}^{∗}, and λ_{i} = 0 for i = 0, …, n−1. Then BVP (1.1), (1.2) has at least one positive solution and one negative solution.

Remark 3.2. If the nonlinear term f(t, x) is separable, say f(t, x) = f_{1}(t)f_{2}(x), then conditions such as μ_{M} < f_{∞} and μ_{M} < f_{0} imply that f_{1}(t) > 0 on [0, 1]. However, the function g(t) in Eq. (1.1) may have zeros on (0, 1).

We now present some applications of the above theorems. To this end, let EJQTDE, 2010 No. 28, p. 16

A = \frac{1}{ν ∫_{0}^{1}(1 − s)g(s)ds} and B = \frac{ν}{\underline{μ} ∫_{0}^{1}(1 − s)g(s)μ(s)ds},  (3.6)

where μ(t) and ν are defined by (2.15) and (2.16), respectively, and \underline{μ} = min_{t∈[θ_{1},θ_{2}]} μ(t) with 0 < θ_{1} < θ_{2} < 1 being fixed constants.

The following corollaries are immediate consequences of Theorems 3.1–3.4.

Corollary 3.1. Assume that (H1) holds and F_{0}/A < 1 < f_{∞}/B. Then the conclusion of Theorem 3.1 holds.

Corollary 3.2. Assume that (H2) holds and F_{∞}/A < 1 < f_{0}/B. Then the conclusion of Theorem 3.2 holds.

Corollary 3.3. Assume that (H3) holds, F_{0}/A < 1 < f_{∞}^{∗}/B, and λ_{i} = 0 for i = 0, …, n − 1. Then the conclusion of Theorem 3.3 holds.

Corollary 3.4. Assume that (H3) holds, F_{∞}/A < 1 < f_{0}^{∗}/B, and λ_{i} = 0 for i = 0, …, n − 1. Then the conclusion of Theorem 3.4 holds.

Replacing μ_{M} by μM̃ and μM̂ gives the following results for BVPs (1.1), (1.3) and (1.1), (1.4), respectively, that are analogous to Theorems 3.1–3.4 and Corollary 3.1–3.4 for BVP (1.1), (1.2).

Theorem 3.5. Assume that (H1) holds and F_{0} < μM̃ < f_{∞}. Then, for (λ_{0}, …, λ_{n−1}) ∈ ℝ_{+}^{n} with ∑_{i=0}^{n−1} λ_{i} sufficiently small, BVP (1.1), (1.3) has at least one nontrivial solution.

Theorem 3.6. Assume that (H2) holds and F_{∞} < μM̃ < f_{0}. Then, for (λ_{0}, …, λ_{n−1}) ∈ ℝ_{+}^{n} with ∑_{i=0}^{n−1} λ_{i} sufficiently small, BVP (1.1), (1.3) has at least one nontrivial solution.

Theorem 3.7. Assume that (H3) holds, F_{0} < μM̃ < f_{∞}^{∗}, and λ_{i} = 0 for i = 0, …, n− 1. Then BVP (1.1), (1.3) has at least one positive solution and one negative solution.

Theorem 3.8. Assume that (H3) holds, F_{∞} < μM̃ < f_{0}^{∗}, and λ_{i} = 0 for i = 0, …, n−1. Then BVP (1.1), (1.3) has at least one positive solution and one negative solution.

EJQTDE, 2010 No. 28, p. 17 Let

Ã = \frac{1}{ν̃ ∫_{0}^{1} sg(s)ds} and B̃ = \frac{ν̃}{\underline{μ̃} ∫_{0}^{1} sg(s)μ̃(s)ds},  (3.7)

where μ̃(t) and ν̃ are defined by (2.17) and (2.18), respectively, and \underline{μ̃} = min_{t∈[θ_{1},θ_{2}]} μ̃(t) with 0 < θ_{1} < θ_{2} < 1 being fixed constants.

Corollary 3.5. Assume that (H1) holds and F_{0}/Ã < 1 < f_{∞}/B̃. Then the conclusion of Theorem 3.5 holds.

Corollary 3.6. Assume that (H2) holds and F_{∞}/Ã < 1 < f_{0}/B̃. Then the conclusion of Theorem 3.6 holds.

Corollary 3.7. Assume that (H3) holds, F_{0}/Ã < 1 < f_{∞}^{∗}/B̃, and λ_{i} = 0 for i = 0, …, n − 1. Then the conclusion of Theorem 3.7 holds.

Corollary 3.8. Assume that (H3) holds, F_{∞}/Ã < 1 < f_{0}^{∗}/B̃, and λ_{i} = 0 for i = 0, …, n − 1. Then the conclusion of Theorem 3.8 holds.

Theorem 3.9. Assume that (H1) holds and F_{0} < μM̂ < f_{∞}. Then, for (λ_{0}, …, λ_{n−1}) ∈ ℝ_{+}^{n} with ∑_{i=0}^{n−1} λ_{i} sufficiently small, BVP (1.1), (1.4) has at least one nontrivial solution.

Theorem 3.10. Assume that (H2) holds and F_{∞} < μM̂ < f_{0}. Then, for (λ_{0}, …, λ_{n−1}) ∈ ℝ_{+}^{n} with ∑_{i=0}^{n−1} λ_{i} sufficiently small, BVP (1.1), (1.4) has at least one nontrivial solution.

Theorem 3.11. Assume that (H3) holds, F_{0} < μM̂ < f_{∞}^{∗}, and λ_{i} = 0 for i = 0, …, n−1. Then BVP (1.1), (1.4) has at least one positive solution and one negative solution.

Theorem 3.12. Assume that (H3) holds, F_{∞} < μM̂ < f_{0}^{∗}, and λ_{i} = 0 for i = 0, …, n−1. Then BVP (1.1), (1.4) has at least one positive solution and one negative solution.

Let

Â = \frac{1}{ν̂ ∫_{0}^{1} s(1 − s)g(s)ds} and B̂ = \frac{ν̂}{\underline{μ̂} ∫_{0}^{1} s(1 − s)g(s)μ̂(s)ds},  (3.8)

EJQTDE, 2010 No. 28, p. 18 where μ̂(t) and ν̂ are defined by (2.19) and (2.20), respectively, and \underline{μ̂} = min_{t∈[θ_{1},θ_{2}]} μ̂(t) with 0 < θ_{1} < θ_{2} < 1 being fixed constants.

Corollary 3.9. Assume that (H1) holds and F_{0}/Â < 1 < f_{∞}/B̂. Then the conclusion of Theorem 3.9 holds.

Corollary 3.10. Assume that (H2) holds and F_{∞}/Â < 1 < f_{0}/B̂. Then the conclusion of Theorem 3.10 holds.

Corollary 3.11. Assume that (H3) holds, F_{0}/Â < 1 < f_{∞}^{∗}/B̂, and λ_{i} = 0 for i = 0, …, n − 1. Then the conclusion of Theorem 3.11 holds.

Corollary 3.12. Assume that (H3) holds, F_{∞}/Â < 1 < f_{0}^{∗}/B̂, and λ_{i} = 0 for i = 0, …, n − 1. Then the conclusion of Theorem 3.12 holds.

Although not explicitly discussed in this paper since we ask that m ≥ 1, it is interesting to examine our results in the case of two-point boundary conditions, that is, if α = β = γ = 0. All theorems and corollaries stated in this section remain valid in this case. Notice that the quantities ξ_{1} and ξ_{m} do not affect the values in (2.16) and (2.18).

Remark 3.3. We wish to point out that the optimal existence results given in this section, Theorems 3.1–3.12, are all related to the smallest positive characteristic values of the operators M, M̃, and M̂ as given by (3.1). In [20], the authors proved optimal existence results for positive solutions of BVPs (1.1), (1.2) and (1.1), (1.3), in terms of the smallest positive characteristic values of the operators L and L̃ as defined by (2.31) and (2.32). In that paper it was assumed that f was nonnegative. Other optimal type results for existence of positive solutions under different types of boundary conditions from the ones used in this paper can be found in [2, 35, 38].

Remark 3.4. In this paper, we do not study the multiplicity and nonexistence of solutions of the problems under consideration. Since this paper is somewhat long, we will leave such investigations to future work. Other papers investigating these kinds of questions for different boundary conditions include [34, 38].

EJQTDE, 2010 No. 28, p. 19 Remark 3.5. In Theorems 3.1 and 3.2 (as well as Theorems 3.5, 3.6, 3.9, 3.10), we have the requirement that ∑_{i=0}^{n−1} λ_{i} be sufficiently small. It is worth mentioning that this “smallness” can be estimated. For example, for n = 2, m = 3, β = (β_{1}, 0), and γ = (0, γ_{2}), this was described in the paper [6] (see [6, Remark 3.2 and Example 3.2]).

We conclude this section with several examples.

Example 3.1. In equations (1.1) and (1.2), let m = 1, n = 3, α = β = γ = ξ = 1/2, g(t) ≡ 1 on [0, 1],

f(t, x) = \{ \begin{matrix}∑_{i=1}^{k} a_{i}(t)x^{i}, & x ∈ [−1, ∞), \\ ∑_{i=1}^{k}(−1)^{i}a_{i}(t) − b̄(t)|x|^{κ} + b̄(t), & x ∈ (−∞, −1), \end{matrix}  (3.9)

where k > 1 is an integer, a_{i}, b̄ ∈ C[0, 1] with 0 ≤ ||a_{1}|| < 1/10 and a_{k}(t) > 0 on [0, 1], and 0 ≤ κ < 1. Then, for (λ_{0}, …, λ_{n−1}) ∈ ℝ_{+}^{n} with ∑_{i=0}^{n−1} λ_{i} sufficiently small, BVP (1.1), (1.2) has at least one nontrivial solution.

To see this, we first note that f ∈ C([0, 1] × ℝ) and assumption (H) is satisfied. Let

a(t) = ∑_{i=1}^{k} |a_{i}(t)| + |b̄(t)|, b(t) = |b̄(t)|, and c(x) = |x|^{κ}.

Then, it is easy to see that (H1) holds.

From (3.6) with θ_{1} = 1/4 and θ_{2} = 3/4, and by a simple calculation, we have

A = 1/10 and B = 3072/11.

Moreover, (3.9) implies that

F_{0} = lim sup_{x→0} max_{t∈[0,1]} |\frac{f(t, x)}{x}| = ||a_{1}|| < \frac{1}{10} and f_{∞} = lim inf_{x→∞} min_{t∈[0,1]} \frac{f(t, x)}{x} = ∞.

Hence, F_{0}/A < 1 < f_{∞}/B. The conclusion then follows from Corollary 3.1.

Example 3.2. In equations (1.1)–(1.4), let m ≥ 1 and n ≥ 2 be any integers, ξ = (ξ_{1}, …, ξ_{m}) ∈ (0, 1)^{m} with 0 < ξ_{1} < … < ξ_{m} < 1, α, β, γ ∈ ℝ_{+}^{m} with 0 ≤ \overline{α}, \overline{β}, \overline{γ} < 1. Also let g : (0, 1) → ℝ_{+} be continuous, g ≢ 0 on any subinterval of (0, 1), ∫_{0}^{1} g(s)ds < ∞, and

f(t, x) = \{ \begin{matrix}−16t^{2} + 13 + (|x|^{1/2} − 2)x^{1/3}, & x < −4, \\ −t^{2}x^{2} + 3|x| + 1, & −4 ≤ x ≤ 0, \\ 1 − tx^{1/2}, & x > 0. \end{matrix}  (3.10)

Then, for (λ_{0}, …, λ_{n−1}) ∈ ℝ_{+}^{n} with ∑_{i=0}^{n−1} λ_{i} sufficiently small, we have (i) BVP (1.1), (1.2) has at least one nontrivial solution. (ii) BVP (1.1), (1.3) has at least one nontrivial solution. (iii) BVP (1.1), (1.4) has at least one nontrivial solution.

To see this, we first note that f ∈ C([0, 1] × ℝ) and assumption (H) is satisfied. Now with d(t) = t^{2} and e(x) = x^{2}, from (3.10), we see that (3.4) and (3.5) hold for any r ∈ (0, 1), and so (H2) holds. Moreover, from (3.10), we have f_{0} = ∞ and F_{∞} = 0. Thus,

F_{∞} < μ_{M} < f_{0}, F_{∞} < μM̃ < f_{0}, and F_{∞} < μM̂ < f_{0},

where μ_{M}, μM̃, and μM̂ are defined in (3.1). The conclusions (i), (ii), and (iii) then follow from Theorems 3.2, 3.6, and 3.10, respectively.

Example 3.3. In equations (1.1)–(1.4), let m ≥ 1 and n ≥ 2 be any integers, λ_{i} = 0 for i = 0, …, n − 1, ξ = (ξ_{1}, …, ξ_{m}) ∈ (0, 1)^{m} with 0 < ξ_{1} < … < ξ_{m} < 1, α, β, γ ∈ ℝ_{+}^{m} with 0 ≤ \overline{α}, \overline{β}, \overline{γ} < 1. Also let f(t, x) = x^{3} and g(t) = t^{−1/2}. Then, we have

(i) BVP (1.1), (1.2) has at least one positive solution and one negative solution. (ii) BVP (1.1), (1.3) has at least one positive solution and one negative solution. (iii) BVP (1.1), (1.4) has at least one positive solution and one negative solution.

To see this, we first note that assumptions (H) and (H3) are satisfied. Moreover, we have F_{0} = 0 and f_{∞}^{∗} = ∞. Thus,

F_{0} < μ_{M} < f_{∞}^{∗}, F_{0} < μM̃ < f_{∞}^{∗}, and F_{0} < μM̂ < f_{∞}^{∗},

where μ_{M}, μM̃, and μM̂ are defined in (3.1). The conclusions (i), (ii), and (iii) then follow from Theorems 3.3, 3.7, and 3.11, respectively.

Additional examples may also be readily given to illustrate the other results. We leave the details to the interested reader.

4. Proofs of the main results

Let ϕ(t) be the unique solution of the BVP

u′′ = 0, t ∈ (0, 1), EJQTDE, 2010 No. 28, p. 21

u′(0) = ⟨β, u′(ξ)⟩ − 1, u(1) = ⟨γ, u(ξ)⟩,

and let ψ(t) be the unique solution of the BVP

u′′ = 0, t ∈ (0, 1),

u′(0) = ⟨β, u′(ξ)⟩, u(1) = ⟨γ, u(ξ)⟩ + 1.

Then,

ϕ(t) = −\frac{t}{1 − \overline{β}} + \frac{1 − ⟨γ, ξ⟩}{(1 − \overline{β})(1 − \overline{γ})} and ψ(t) = \frac{1}{1 − \overline{γ}} on [0, 1].

Let ϕ̃(t) be the unique solution of the BVP

u′′ = 0, t ∈ (0, 1),

u(0) = ⟨β, u(ξ)⟩, u′(1) = ⟨γ, u′(ξ)⟩ + 1,

and let ψ̃(t) be the unique solution of the BVP

u′′ = 0, t ∈ (0, 1),

u(0) = ⟨β, u(ξ)⟩ + 1, u′(1) = ⟨γ, u′(ξ)⟩.

Then, we have

ϕ̃(t) = \frac{t}{1 − \overline{γ}} + \frac{⟨β, ξ⟩}{(1 − \overline{β})(1 − \overline{γ})} and ψ̃(t) = \frac{1}{1 − \overline{β}} on [0, 1].

Similarly, let ϕ̂(t) be the unique solution of the BVP

u′′ = 0, t ∈ (0, 1),

u(0) = ⟨β, u(ξ)⟩, u(1) = ⟨γ, u(ξ)⟩ + 1,

and let ψ̂(t) be the unique solution of the BVP

u′′ = 0, t ∈ (0, 1),

u(0) = ⟨β, u(ξ)⟩ + 1, u(1) = ⟨γ, u(ξ)⟩.

Then, we have

ϕ̂(t) = \frac{1}{ρ}[(1 − \overline{β})t + ⟨β, ξ⟩] and ψ̂(t) = \frac{1}{ρ}[(\overline{γ} − 1)t + (1 − ⟨γ, ξ⟩)] on [0, 1].

Clearly, ϕ(t), ψ(t), ϕ̃(t), ψ̃(t), ϕ̂(t), and ψ̂(t) are nonnegative on [0, 1]. EJQTDE, 2010 No. 28, p. 22

Let J_{1}(t, s) = H_{1}(t, s), where H_{1}(t, s) is defined by (2.7). If n ≥ 4, then we recursively define

J_{k}(t, s) = ∫_{0}^{1} H_{1}(t, τ)J_{k−1}(τ, s)dτ, k = 2, …, n − 2.

We now define several functions y_{i}(t), ỹ_{i}(t), and ŷ_{i}(t), i = 0, …, n − 1, as follows: when n = 2, let

y_{0}(t) = ϕ(t) and y_{1}(t) = ψ(t),  (4.1)
ỹ_{0}(t) = ϕ̃(t) and ỹ_{1}(t) = ψ̃(t),
ŷ_{0}(t) = ϕ̂(t) and ŷ_{1}(t) = ψ̂(t);

and when n ≥ 3, let

y_{0}(t) = \frac{1}{1 − \overline{α}}, y_{k}(t) = \frac{1}{1 − \overline{α}} ∫_{0}^{1} J_{k}(t, s)ds, k = 1, …, n − 3,  (4.2)
y_{n−2}(t) = ∫_{0}^{1} J_{n−2}(t, s)ϕ(s)ds,  (4.3)
y_{n−1}(t) = ∫_{0}^{1} J_{n−2}(t, s)ψ(s)ds,  (4.4)

and

ỹ_{k}(t) = y_{k}(t), k = 0, …, n − 3,
ỹ_{n−2}(t) = ∫_{0}^{1} J_{n−2}(t, s)ϕ̃(s)ds,
ỹ_{n−1}(t) = ∫_{0}^{1} J_{n−2}(t, s)ψ̃(s)ds,

and

ŷ_{k}(t) = y_{k}(t), k = 0, …, n − 3,
ŷ_{n−2}(t) = ∫_{0}^{1} J_{n−2}(t, s)ϕ̂(s)ds,
ŷ_{n−1}(t) = ∫_{0}^{1} J_{n−2}(t, s)ψ̂(s)ds.

Clearly, y_{i}(t) ≥ 0, ỹ_{i}(t) ≥ 0, and ŷ_{i}(t) ≥ 0 for t ∈ [0, 1] and i = 0, …, n − 1. The following lemma gives some properties of y_{i}(t), ỹ_{i}(t), and ŷ_{i}(t) when n ≥ 3.

Lemma 4.1. Assume that n ≥ 3. Then, we have the following:

EJQTDE, 2010 No. 28, p. 23 (a) For any k ∈ {0, …, n − 3}, y_{k}(t) is the unique solution of the BVP consisting

of the equation

u^{(n)} = 0, t ∈ (0, 1),  (4.5)

and the BC

\{ \begin{matrix}u^{(k)}(0) = ⟨α, u^{(k)}(ξ)⟩ + 1, \\ u^{(i)}(0) = ⟨α, u^{(i)}(ξ)⟩, i = 0, …, n − 3, i ≠ k, \\ u^{(n−1)}(0) = ⟨β, u^{(n−1)}(ξ)⟩, \\ u^{(n−2)}(1) = ⟨γ, u^{(n−2)}(ξ)⟩, \end{matrix}

and y_{n−2}(t) is the unique solution of the BVP consisting of Eq. (4.5) and the BC

\{ \begin{matrix}u^{(i)}(0) = ⟨α, u^{(i)}(ξ)⟩, i = 0, …, n − 3, \\ u^{(n−1)}(0) = ⟨β, u^{(n−1)}(ξ)⟩ − 1, \\ u^{(n−2)}(1) = ⟨γ, u^{(n−2)}(ξ)⟩, \end{matrix}

and y_{n−1}(t) is the unique solution of the BVP consisting of Eq. (4.5) and the BC

\{ \begin{matrix}u^{(i)}(0) = ⟨α, u^{(i)}(ξ)⟩, i = 0, …, n − 3, \\ u^{(n−1)}(0) = ⟨β, u^{(n−1)}(ξ)⟩, \\ u^{(n−2)}(1) = ⟨γ, u^{(n−2)}(ξ)⟩ + 1. \end{matrix}

(b) For any k ∈ {0, …, n − 3}, ỹ_{k}(t) is the unique solution of the BVP consisting of Eq. (4.5) and the BC

\{ \begin{matrix}u^{(k)}(0) = ⟨α, u^{(k)}(ξ)⟩ + 1, \\ u^{(i)}(0) = ⟨α, u^{(i)}(ξ)⟩, i = 0, …, n − 3, i ≠ k, \\ u^{(n−2)}(0) = ⟨β, u^{(n−2)}(ξ)⟩, \\ u^{(n−1)}(1) = ⟨γ, u^{(n−1)}(ξ)⟩, \end{matrix}

and ỹ_{n−2}(t) is the unique solution of the BVP consisting of Eq. (4.5) and the BC

\{ \begin{matrix}u^{(i)}(0) = ⟨α, u^{(i)}(ξ)⟩, i = 0, …, n − 3, \\ u^{(n−2)}(0) = ⟨β, u^{(n−2)}(ξ)⟩ + 1, \\ u^{(n−1)}(1) = ⟨γ, u^{(n−1)}(ξ)⟩, \end{matrix}

and ỹ_{n−1}(t) is the unique solution of the BVP consisting of Eq. (4.5) and the BC

\{ \begin{matrix}u^{(i)}(0) = ⟨α, u^{(i)}(ξ)⟩, i = 0, …, n − 3, \\ u^{(n−2)}(0) = ⟨β, u^{(n−2)}(ξ)⟩, \\ u^{(n−1)}(1) = ⟨γ, u^{(n−1)}(ξ)⟩ + 1. \end{matrix}

(c) For any k ∈ {0, …, n − 3}, ŷ_{k}(t) is the unique solution of the BVP consisting of Eq. (4.5) and the BC

\{ \begin{matrix}u^{(k)}(0) = ⟨α, u^{(k)}(ξ)⟩ + 1, \\ u^{(i)}(0) = ⟨α, u^{(i)}(ξ)⟩, i = 0, …, n − 3, i ≠ k, \\ u^{(n−2)}(0) = ⟨β, u^{(n−2)}(ξ)⟩, \\ u^{(n−2)}(1) = ⟨γ, u^{(n−2)}(ξ)⟩, \end{matrix}

and ŷ_{n−2}(t) is the unique solution of the BVP consisting of Eq. (4.5) and the BC

\{ \begin{matrix}u^{(i)}(0) = ⟨α, u^{(i)}(ξ)⟩, i = 0, …, n − 3, \\ u^{(n−2)}(0) = ⟨β, u^{(n−2)}(ξ)⟩ + 1, \\ u^{(n−2)}(1) = ⟨γ, u^{(n−2)}(ξ)⟩, \end{matrix}

and ŷ_{n−1}(t) is the unique solution of the BVP consisting of Eq. (4.5) and the BC

\{ \begin{matrix}u^{(i)}(0) = ⟨α, u^{(i)}(ξ)⟩, i = 0, …, n − 3, \\ u^{(n−2)}(0) = ⟨β, u^{(n−2)}(ξ)⟩, \\ u^{(n−2)}(1) = ⟨γ, u^{(n−2)}(ξ)⟩ + 1. \end{matrix}

Parts (a) and (b) of Lemma 4.1 were proved in [20, Lemma 2.4], and part (c) can be proved similarly. We omit the proof of part (c) of the lemma.

For any λ = (λ_{0}, …, λ_{n−1}) ∈ ℝ_{+}^{n}, let

u(t) = v(t) + ∑_{i=0}^{n−1} λ_{i}y_{i}(t), t ∈ [0, 1].  (4.6)

Here, the reader is reminded that y_{i}(t), i = 0, …, n − 1, are defined by (4.1) if n = 2, and are given by (4.2)–(4.4) if n ≥ 3. Then, by Lemma 4.1 (a), BVP (1.1), (1.2) is equivalent to the BVP consisting of the equation

v^{(n)} + g(t)f (t, v + ∑_{i=0}^{n−1} λ_{i}y_{i}(t)) , t ∈ (0, 1),  (4.7)

and the homogeneous BC

\{ \begin{matrix}v^{(i)}(0) = ⟨α, v^{(i)}(ξ)⟩, i = 0, …, n − 3, \\ v^{(n−1)}(0) = ⟨β, v^{(n−1)}(ξ)⟩, \\ v^{(n−2)}(1) = ⟨γ, v^{(n−2)}(ξ)⟩. \end{matrix}  (4.8)

Moreover, if v(t) is a solution of BVP (4.7), (4.8), then u(t) given by (4.6) is a solution of BVP (1.1), (1.2).

EJQTDE, 2010 No. 28, p. 25

Let P, L, and M be defined as in (2.30) and (2.31). By Lemma 2.7, L and M map P into P and are compact. Define operators F_{λ}, T : X → X by

F_{λ}v(t) = f (t, v + ∑_{i=0}^{n−1} λ_{i}y_{i}(t))  (4.9)

and

T v(t) = LF_{λ}v(t) = ∫_{0}^{1} K_{n−1}(t, s)g(s)F_{λ}v(s)ds,  (4.10)

where K_{n−1} is defined by (2.12) with i = n − 1. Then, F_{λ} is bounded, and a standard argument shows that T is compact. Moreover, by Lemma 2.5 (a), a solution of BVP (4.7), (4.8) is equivalent to a fixed point of T in X.

Proof of Theorem 3.1. We first verify that conditions (A1)–(A4) of Lemma 2.3 are satisfied. By Lemma 2.7 (a) and (b), there exist φ_{L}, φ_{M} ∈ P \ {0} such that (2.1) holds. To show (2.2), we let

h(v) = ∫_{0}^{1} φ_{M}(t)g(t)v(t)dt, v ∈ X.  (4.11)

Then h ∈ P^{∗} \ {0}, and from (2.1) and (2.31),

(L^{∗}h)(v) = h(Lv) = ∫_{0}^{1} φ_{M}(t)g(t)Lv(t)dt
= ∫_{0}^{1} φ_{M}(t)g(t) (∫_{0}^{1} K_{n−1}(t, s)g(s)v(s)ds) dt
= ∫_{0}^{1} g(s)v(s) (∫_{0}^{1} K_{n−1}(t, s)g(t)φ_{M}(t)dt) ds
= ∫_{0}^{1} g(s)v(s)Mφ_{M}(s)ds
= r_{M} ∫_{0}^{1} g(s)v(s)φ_{M}(s)ds = r_{M}h(v),

i.e., h satisfies (2.2).

From the fact that φ_{M} = μ_{M}Mφ_{M}, (2.31), and (3.1),

r_{M}φ_{M}(s) = ∫_{0}^{1} K_{n−1}(t, s)g(t)φ_{M}(t)dt.  (4.12)

EJQTDE, 2010 No. 28, p. 26 Then, by Lemma 2.6 (a) (see (2.14)), we have

r_{M}φ_{M}(s) ≥ (1 − s) ∫_{0}^{1} μ(t)g(t)φ_{M}(t)dt
= (\frac{1}{ν} ∫_{0}^{1} μ(t)g(t)φ_{M}(t)dt) ν(1 − s)
≥ (\frac{1}{ν} ∫_{0}^{1} μ(t)g(t)φ_{M}(t)dt) K_{n−1}(t, s)
= δK_{n−1}(t, s) for t, s ∈ [0, 1],  (4.13)

where μ(t) and ν are defined by (2.15) and (2.16), and

δ = \frac{1}{ν} ∫_{0}^{1} μ(t)g(t)φ_{M}(t)dt.

To see that δ > 0, first note that μ(t) > 0 on (0, 1) and ν > 0. If δ = 0, then g(t)φ_{M}(t) ≡ 0 on (0, 1). Since r_{M} > 0, this implies φ_{M}(t) ≡ 0 by (4.12), which is a contradiction.

Let P (h, δ) be defined by (2.3). For any v ∈ P and t ∈ [0, 1], from (2.31), (4.11), and (4.13), it follows that

h(Lv) = r_{M}h(v) = r_{M} ∫_{0}^{1} φ_{M}(s)g(s)v(s)ds
≥ δ ∫_{0}^{1} K_{n−1}(t, s)g(s)v(s)ds
= δLv(t).

Hence, h(Lv) ≥ δ||Lv||, i.e., L(P) ⊆ P (h, δ). Therefore, (A1) of Lemma 2.3 holds. Let λ = (λ_{0}, …, λ_{n−1}) ∈ ℝ_{+}^{n} and

H_{λ}v(t) = b̄c (|v(t)| + ∑_{i=0}^{n−1} λ_{i}||y_{i}||)

for v ∈ X, where b̄ = max_{t∈[0,1]} b(t). Since c is nondecreasing on ℝ^{+}, we have

H_{λ}v(t) ≤ b̄c (||v|| + ∑_{i=0}^{n−1} λ_{i}||y_{i}||) for all v ∈ P and t ∈ [0, 1].

Then, from the fact that c is even, it follows that

H_{λ}v(t) ≤ b̄c (||v|| + ∑_{i=0}^{n−1} λ_{i}||y_{i}||) for all v ∈ X and t ∈ [0, 1].

EJQTDE, 2010 No. 28, p. 27 Thus,

||H_{λ}v|| ≤ b̄c (||v|| + ∑_{i=0}^{n−1} λ_{i}||y_{i}||) for all v ∈ X.

From (3.3), we see that

lim_{||v||→∞} \frac{||H_{λ}v||}{||v||} = 0 for any v ∈ X.

Thus, (A2) of Lemma 2.3 holds with H = H_{λ}.

Let F_{λ} be defined by (4.9), and u_{0}(t) = a(t). Then, from (H1), we have F_{λ}v + H_{λ}v + u_{0} ∈ P for all v ∈ X. Hence, (A3) of Lemma 2.3 holds with F = F_{λ} and H = H_{λ}.

Since f_{∞} > μ_{M}, there exist ϵ > 0 and N > 0 such that

f(t, x) ≥ μ_{M}(1 + ϵ)x for (t, x) ∈ [0, 1] × [N, ∞).

In view of (3.2), we see that there exists ζ > 0 large enough so that

f(t, x) ≥ μ_{M}(1 + ϵ)x − b̄c(x) − ζ for (t, x) ∈ [0, 1] × ℝ.

From (3.1) and (4.9), we have

F_{λ}v(t) ≥ μ_{M}(1 + ϵ) (v(t) + ∑_{i=0}^{n−1} λ_{i}y_{i}(t)) − b̄c (v(t) + ∑_{i=0}^{n−1} λ_{i}y_{i}(t)) − ζ
≥ μ_{M}(1 + ϵ)v(t) − b̄c (|v(t)| + ∑_{i=0}^{n−1} λ_{i}||y_{i}||) − ζ
= r_{M}^{−1}(1 + ϵ)v(t) − H_{λ}v(t) − ζ for all v ∈ X.

Thus,

LF_{λ}v(t) ≥ r_{M}^{−1}(1 + ϵ)Lv(t) − LH_{λ}v(t) − Lζ for all v ∈ X.

Therefor, (A4) of Lemma 2.3 holds with F = F_{λ}, H = H_{λ}, and v_{0} = Lζ.

We have verified that all the conditions of Lemma 2.3 hold, so there exists R_{1} > 0 such that

deg(I − T, B(0, R_{1}), 0) = 0.  (4.14)

Next, since F_{0} < μ_{M}, there exist 0 < q < 1 and 0 < R_{2} < R_{1} such that

|f(t, x)| ≤ μ_{M}(1 − q)|x| for (t, x) ∈ [0, 1] × [−2R_{2}, 2R_{2}].  (4.15)

EJQTDE, 2010 No. 28, p. 28 In what follows, let λ = (λ_{0}, …, λ_{n−1}) ∈ ℝ_{+}^{n} be small enough so that

∑_{i=0}^{n−1} λ_{i}||y_{i}|| < R_{2}  (4.16)

and

C_{1} := μ_{M}(1 − q) (∑_{i=0}^{n−1} λ_{i}||y_{i}||) max_{t∈[0,1]} ∫_{0}^{1} K_{n−1}(t, s)g(s)ds < qR_{2}.  (4.17)

We claim that

T v ≠ τv for all v ∈ ∂B(0, R_{2}) and τ ≥ 1.  (4.18)

If this is not the case, then there exist v̄ ∈ ∂B(0, R_{2}) and τ̄ ≥ 1 such that T v̄ = τ̄v̄. It follows that v̄ = s̄T v̄, where s̄ = 1/τ̄. Clearly, s̄ ∈ (0, 1]. From (4.9), (4.15), and (4.16), we have

|F_{λ}v̄(t)| ≤ μ_{M}(1 − q) |v̄(t) + ∑_{i=0}^{n−1} λ_{i}y_{i}(t)|
≤ μ_{M}(1 − q) (|v̄(t)| + ∑_{i=0}^{n−1} λ_{i}||y_{i}||) .  (4.19)

Assume R_{2} = ||v̄|| = |v̄(t̄)| for some t̄∈ [0, 1]. Then, from (2.31), (3.1), (4.10), (4.17), and (4.19), we obtain

R_{2} = |v̄(t̄)| = s̄|T v̄(t̄)| ≤ ∫_{0}^{1} K_{n−1}(t̄, s))g(s)|F_{λ}v̄(s)|ds
≤ μ_{M}(1 − q) ∫_{0}^{1} K_{n−1}(t̄, s)g(s) (v(s) + ∑_{i=0}^{n−1} λ_{i}||y_{i}||) ds
= μ_{M}(1 − q) ∫_{0}^{1} K_{n−1}(t̄, s)g(s)|v(s)|ds

+μ_{M}(1 − q) (∑_{i=0}^{n−1} λ_{i}||y_{i}||) ∫_{0}^{1} K_{n−1}(t̄, s)g(s)ds

≤ μ_{M}(1 − q)L|v(t̄)| + C_{1} = r_{M}^{−1}(1 − q)LR_{2} + C_{1}.

EJQTDE, 2010 No. 28, p. 29 Consequently,

h(R_{2}) ≤ r_{M}^{−1}(1 − q)h(LR_{2}) + h(C_{1})
= r_{M}^{−1}(1 − q)(L^{∗}h)(R_{2}) + h(C_{1})
= r_{M}^{−1}(1 − q)r_{M}h(R_{2}) + h(C_{1})
= (1 − q)h(R_{2}) + h(C_{1}).

Thus,

(C_{1} − qR_{2})h(1) ≥ 0.

Since h(1) > 0, we have C_{1} ≥ qR_{2}. But this contradicts (4.17). Thus, (4.18) holds.

Now, Lemma 2.1 implies

deg(I − T, B(0, R_{2}), 0) = 1.  (4.20)

By the additivity property of the Leray-Schauder degree, (4.14), and (4.20), we have

deg(I − T, B(0, R_{1}) \ \overline{B(0, R_{2})}) = −1.

Then, from the solution property of the Leray-Schauder degree, T has at least one fixed point v in B(0, R_{1})\\overline{B(0, R_{2})}, which is a solution of BVP (4.7), (4.8). Therefore, we have shown that, for λ = (λ_{0}, …, λ_{n−1}) ∈ ℝ_{+}^{n} satisfying (4.16) and (4.17), BVP (4.7), (4.8) has at least one solution v(t) satisfying ||v|| ≥ R_{2}. Thus, for each λ = (λ_{0}, …, λ_{n−1}) ∈ ℝ_{+}^{n} with ∑_{i=0}^{n−1} λ_{i} sufficiently small, BVP (1.1), (1.2) has at least one solution u(t) = v(t) + ∑_{i=0}^{n−1} λ_{i}y_{i}(t) satisfying

||u|| ≥ ||v|| − ∑_{i=0}^{n−1} λ_{i}||y_{i}|| ≥ R_{2} − ∑_{i=0}^{n−1} λ_{i}||y_{i}|| > 0.

This completes the proof of the theorem. □ Proof of Theorem 3.2. We first verify that conditions (A1) and (A2)^{∗}–(A4)^{∗} of Lemma 2.4 are satisfied. As in the proof of Theorem 3.1, there exist φ_{L}, φ_{M} ∈ P \ {0} and h ∈ P^{∗} \ {0} defined by (4.11) such that (A1) holds.

From the fact that e is even and nondecreasing on ℝ^{+}, it is easy to see that

e(v(t)) ≤ e(||v||) for all v ∈ X and t ∈ [0, 1].

Thus,

||e(v)|| ≤ e(||v||) for all v ∈ X.

EJQTDE, 2010 No. 28, p. 30 This, together with (3.5), implies that

lim_{||v||→0} \frac{||e(v)||}{||v||} = 0 for any v ∈ X.

Let Hv(t) = d̄e(v(t)) for v ∈ X, where d̄= max_{t∈[0,1]} d(t). Then, (A2)^{∗} of Lemma 2.4 holds.

Since f_{0} > μ_{M}, there exist ϵ > 0 and 0 < ζ_{1} < 1 such that

f(t, x) ≥ μ_{M}(1 + ϵ)x = r_{M}^{−1}(1 + ϵ)x ≥ 0 for (t, x) ∈ [0, 1] × [0, 2ζ_{1}].  (4.21)

Let λ = (λ_{0}, …, λ_{n−1}) ∈ ℝ_{+}^{n} be small enough so that

∑_{i=0}^{n−1} λ_{i}||y_{i}|| ≤ ζ_{1}  (4.22)

and F_{λ} be defined by (4.9). Then, from (4.21), we have

F_{λ}v(t) ≥ μ_{M}(1 + ϵ) (v(t) + ∑_{i=0}^{n−1} λ_{i}y_{i}(t))

≥ μ_{M}(1 + ϵ)v(t) = r_{M}^{−1}(1 + ϵ)v(t) for all v ∈ P with ||v|| ≤ ζ_{1}. (4.23)

Let r be given in (H2). Now, in view of (3.4) and (4.23), we see that (A3)^{∗} of Lemma 2.4 holds with F = F_{λ} and r_{1} = min{r, ζ_{1}}.

From (3.5), there exists 0 < ζ_{2} < min{r, ζ_{1}} such that

−e(x) ≥ d̄^{−1}r_{M}^{−1}(1 + ϵ)x for x ∈ [−ζ_{2}, 0].

Then, from (3.4),

f(t, x) ≥ d(t)d̄^{−1}r_{M}^{−1}(1 + ϵ)x ≥ r_{M}^{−1}(1 + ϵ)x for (t, x) ∈ [0, 1] × [−ζ_{2}, 0]. (4.24)

From (4.21) and (4.24), it is easy to see that

F_{λ}v(t) ≥ μ_{M}(1 + ϵ) (v(t) + ∑_{i=0}^{n−1} λ_{i}y_{i}(t))

≥ μ_{M}(1 + ϵ)v(t) = r_{M}^{−1}(1 + ϵ)v(t) for all v ∈ X with ||v|| ≤ ζ_{2},

which clearly implies that

LF_{λ}v(t) ≥ r_{M}^{−1}(1 + ϵ)Lv(t) for all v ∈ X with ||u|| < ζ_{2}.

Hence, (A4)^{∗} of Lemma 2.4 holds with F = F_{λ} and r_{2} = ζ_{2}.

EJQTDE, 2010 No. 28, p. 31

We have verified that all the conditions of Lemma 2.4 hold, so there exists R_{3} > 0 such that

deg(I − T, B(0, R_{3}), 0) = 0.  (4.25)

Next, since F_{∞} < μ_{M}, there exist 0 < z < 1 and R > R_{3} such that

|f(t, x)| ≤ μ_{M}(1 − z)|x| = r_{M}^{−1}(1 − z)|x| for (t, |x|) ∈ [0, 1] × (R, ∞). (4.26)

Let

C_{2} = r_{M}^{−1}(1 − z) (∑_{i=0}^{n−1} λ_{i}||y_{i}||) max_{t∈[0,1]} ∫_{0}^{1} K_{n−1}(t, s)g(s)ds
+ max_{t∈[0,1],|x|≤R} |f(t, x)| max_{t∈[0,1]} ∫_{0}^{1} K_{n−1}(t, s)g(s)ds.  (4.27)

Then 0 < C_{2} < ∞. Choose R_{4} large enough so that

R_{4} > max{R, z^{−1}C_{2}}.  (4.28)

We claim that

T v ≠ τv for all v ∈ ∂B(0, R_{4}) and τ ≥ 1.  (4.29)

If this is not the case, then there exist v̄ ∈ ∂B(0, R_{4}) and τ̄ ≥ 1 such that T v̄ = τ̄v̄. It follows that v̄ = s̄T v̄, where s̄ = 1/τ̄. Clearly, s̄ ∈ (0, 1]. Assume R_{4} = ||v̄|| = |v̄(t̄)| for some t̄∈ [0, 1]. Let

J_{1}(v̄) = {t ∈ [0, 1] : |v̄(t) + ∑_{i=0}^{n−1} λ_{i}y_{i}(t)| > R} ,

J_{2}(v̄) = [0, 1] \ J_{1}(v̄),

and

p(v̄(t)) = min {|v̄(t) + ∑_{i=0}^{n−1} λ_{i}y_{i}(t)|, R} for t ∈ [0, 1].

EJQTDE, 2010 No. 28, p. 32 Then, from (2.31), (4.10), (4.26), and (4.27), it follows that

R_{4} = |v̄(t̄)| = s̄|T v̄(t̄)|
≤ ∫_{0}^{1} K_{n−1}(t̄, s)g(s)|F_{λ}v̄(s)|ds
= ∫_{J_{1}(v̄)} K_{n−1}(t̄, s)g(s)|F_{λ}v̄(s)|ds + ∫_{J_{2}(v̄)} K_{n−1}(t̄, s)g(s)|F_{λ}v̄(s)|ds
≤ r_{M}^{−1}(1 − z) ∫_{J_{1}(v̄)} K_{n−1}(t̄, s)g(s)|v̄(s) + ∑_{i=0}^{n−1} λ_{i}y_{i}(s)|ds

+ ∫_{J_{2}(v̄)} K_{n−1}(t̄, s)g(s)|F_{λ}p(v̄(s))|ds

≤ r_{M}^{−1}(1 − z) ∫_{0}^{1} K_{n−1}(t̄, s)g(s)|v̄(s)|ds

+r_{M}^{−1}(1 − z) (∑_{i=0}^{n−1} λ_{i}||y_{i}||) ∫_{0}^{1} K_{n−1}(t̄, s)g(s)ds

+ ∫_{0}^{1} K_{n−1}(t̄, s)g(s)|F_{λ}p(v̄(s))|ds

≤ r_{M}^{−1}(1 − z)L|v(t̄)| + C_{2} = r_{M}^{−1}(1 − z)LR_{4} + C_{2}.

Hence, for h defined by (4.11), we have

h(R_{4}) ≤ r_{M}^{−1}(1 − z)h(LR_{4}) + h(C_{2})
= r_{M}^{−1}(1 − z)(L^{∗}h)(R_{4}) + h(C_{2})
= r_{M}^{−1}(1 − z)r_{M}h(R_{4}) + h(C_{2})
= (1 − z)h(R_{4}) + h(C_{2}),

which implies

(zR_{4} − C_{2})h(1) ≤ 0.

In view of the fact that h(1) > 0, it follows that R_{4} ≤ z^{−1}C_{2}. This contradicts (4.28) and so (4.29) holds. By Lemma 2.1, we have

deg(I − T, B(0, R_{4}), 0) = 1.  (4.30)

By the additivity property of the Leray-Schauder degree, (4.25), and (4.30), we obtain

deg(I − T, B(0, R_{4}) \ \overline{B(0, R_{3})}) = 1.

EJQTDE, 2010 No. 28, p. 33 Thus, from the solution property of the Leray-Schauder degree, T has at least one fixed point v in B(0, R_{4})\\overline{B(0, R_{3})}, which is a solution of BVP (4.7), (4.8). Therefore, we have shown that, for λ = (λ_{0}, …, λ_{n−1}) ∈ ℝ_{+}^{n} satisfying (4.22), BVP (4.7), (4.8) has at least one solution v(t) satisfying ||v|| ≥ R_{3}. Thus, for each λ = (λ_{0}, …, λ_{n−1}) ∈ ℝ_{+}^{n} with ∑_{i=0}^{n−1} λ_{i} sufficiently small, BVP (1.1), (1.2) has at least one solution u(t) = v(t) + ∑_{i=0}^{n−1} λ_{i}y_{i}(t) satisfying

||u|| ≥ ||v|| − ∑_{i=0}^{n−1} λ_{i}||y_{i}|| ≥ R_{3} − ∑_{i=0}^{n−1} λ_{i}||y_{i}|| > 0.

This completes the proof of the theorem. □

Proof of Theorem 3.3. For (t, x) ∈ [0, 1] × ℝ, let

f_{1}(t, x) = \{ \begin{matrix}f(t, x), & x ≥ 0, \\ −f(t, x), & x < 0. \end{matrix}  (4.31)

In virtue of (H3), we see that f_{1} : [0, 1]×ℝ → ℝ is continuous and nonnegative. Then, (H1) with f = f_{1} is trivially satisfied. Moreover, from F_{0} < μ_{M} < f_{∞}^{∗}, it follows that F_{1,0} < μ_{M} < f_{1,∞}, where

F_{1,0} = lim sup_{x→0} max_{t∈[0,1]} |\frac{f_{1}(t, x)}{x}| and f_{1,∞} = lim inf_{x→∞} min_{t∈[0,1]} \frac{f_{1}(t, x)}{x}.

Thus, by Theorem 3.1, we know that the BVP consisting of the equation

u^{(n)} + g(t)f_{1}(t, u) = 0, t ∈ (0, 1),

and BC (1.2) has at least one nontrivial solution u_{1}(t). By Lemma 2.5 (a), we have

u_{1}(t) = ∫_{0}^{1} K_{n−1}(t, s)g(s)f_{1}(s, u_{1}(s))ds.

Then, by Remark 2.1, u_{1}(t) > 0 on (0, 1). Therefore, from (4.31), f_{1}(t, u(t)) = f(t, u(t)), and so u_{1}(t) is a positive solution of BVP (1.1), (1.2).

For (t, x) ∈ [0, 1] × ℝ, let

f_{2}(t, x) = \{ \begin{matrix}−f(t, −x), & x ≥ 0, \\ f(t, −x), & x < 0. \end{matrix}  (4.32)

In virtue of (H3), we see that f_{2} : [0, 1]×ℝ → ℝ is continuous and nonnegative. Then, (H1) with f = f_{2} is trivially satisfied. Moreover, from F_{0} < μ_{M} < f_{∞}^{∗}, it follows that

EJQTDE, 2010 No. 28, p. 34 F_{2,0} < μ_{M} < f_{2,∞}, where

F_{2,0} = lim sup_{x→0} max_{t∈[0,1]} |\frac{f_{2}(t, x)}{x}| and f_{2,∞} = lim inf_{x→∞} min_{t∈[0,1]} \frac{f_{2}(t, x)}{x}.

Thus, as above, we know that the BVP consisting of the equation

u^{(n)} + g(t)f_{2}(t, u) = 0, t ∈ (0, 1),

and BC (1.2) has at least one solution v(t) satisfying v(t) > 0 on (0, 1) and

v(t) = ∫_{0}^{1} K_{n−1}(t, s)g(s)f_{2}(s, v(s))ds.

Then, from (4.32),

−v(t) = ∫_{0}^{1} K_{n−1}(t, s)g(s)f(s, −v(s))ds.

Therefore, u_{2}(t) := −v(t) is a negative solution of BVP (1.1), (1.2), and the theorem is proved. □

Using Theorem 3.2, Theorem 3.4 can be proved by similar ideas as those given in the proof of Theorem 3.3. We omit the details here.

Lemma 4.2. Let μ_{M}, μM̃, and μM̂ be given in (3.1). Then we have the following: (a) A ≤ μ_{M} ≤ B, where A and B are defined in (3.6).

(b) Ã ≤ μM̃ ≤ B̃, where Ã and B̃ are defined in (3.7).

(c) Â ≤ μM̂ ≤ B̂, where Â and B̂ are defined in (3.8).

Proof. We first prove part (a). Let φ_{M} be given as in Lemma 2.7 (b). Then,

φ_{M}(t) = μ_{M} ∫_{0}^{1} K_{n−1}(t, s)g(s)φ_{M}(s)ds for t ∈ [0, 1].

By Lemma 2.6 (a), we have

φ_{M}(t) ≤ μ_{M}ν ∫_{0}^{1}(1 − s)g(s)φ_{M}(s)ds on [0, 1]  (4.33)

and

φ_{M}(t) ≥ μ_{M}μ(t) ∫_{0}^{1}(1 − s)g(s)φ_{M}(s)ds on [0, 1].  (4.34)

Thus,

φ_{M}(t) ≥ \frac{1}{ν}μ(t)||φ_{M}|| on [0, 1].  (4.35)

EJQTDE, 2010 No. 28, p. 35 From (4.33), we have

φ_{M}(t) ≤ μ_{M}ν||φ_{M}|| ∫_{0}^{1}(1 − s)g(s)ds on [0, 1].

Hence,

μ_{M} ≥ \frac{1}{ν ∫_{0}^{1}(1 − s)g(s)ds} = A.

From (4.34) and (4.35), we see that

φ_{M}(t) ≥ \frac{1}{ν}μ_{M}μ(t)||φ_{M}|| ∫_{0}^{1}(1 − s)g(s)μ(s)ds
≥ \frac{1}{ν}μ_{M} \underline{μ} ||φ_{M}|| ∫_{0}^{1}(1 − s)g(s)μ(s)ds for t ∈ [θ_{1}, θ_{2}].

Hence,

μ_{M} ≤ \frac{ν}{\underline{μ} ∫_{0}^{1}(1 − s)g(s)μ(s)ds} = B.

This proves part (a).

By Lemma 2.6 (b) and (c), and a similar argument as above, parts (b) and (c) can be proved. This completes the proof of the lemma. □

Proof of Corollary 3.1. The conclusion follows readily from Theorem 3.1 and Lemma 4.1 (a). □ Proof of Corollary 3.2. The conclusion follows readily from Theorem 3.2 and Lemma 4.1 (a). □ Proof of Corollary 3.3. The conclusion follows readily from Theorem 3.3 and Lemma 4.1 (a). □ Proof of Corollary 3.4. The conclusion follows readily from Theorem 3.4 and Lemma 4.1 (a). □

Finally, we remark that, by virtue of Lemma 2.5 (b) and (c), Lemma 4.1 (b) and (c), and Lemma 4.2 (b) and (c), Theorems 3.5–3.12 and Corollaries 3.5–3.12 can be proved by techniques similar to those in the proofs of Theorems 3.1–3.4 and Corollaries 3.1–3.4. To avoid redundancy and save space, we omit the details.

Acknowledgment. The authors would like to thank the referee for doing a very careful reading of the manuscript and making many suggestions that have improved the paper.

EJQTDE, 2010 No. 28, p. 36

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Appendix: Equality of two eigenvalues

The authors wish to thank professor Jeff R. L. Webb for the following proof.

Let P be the cone of non-negative functions in C[0, 1],

P = {u ∈ C[0, 1] : u(t) ≥ 0, t ∈ [0, 1]}.

We consider linear operators defined by

Lu(t) = ∫_{0}^{1} C(t, s)g(s)u(s) ds, Mu(t) = ∫_{0}^{1} C(s, t)g(s)u(s) ds,

where C(t, s) ≥ 0 for almost all t, s ∈ [0, 1] together with some implicit conditions that ensure that L, M are well defined bounded linear operators on C[0, 1]. For example, we might ask that C be continuous and g belong to L^{1}.

Proposition 5.1. Let g ∈ L^{1}(0, 1) be such that g(s) ≥ 0 for almost all s ∈ [0, 1]. Suppose that L and M are well defined bounded linear operators on C[0, 1] and that λ_{L} and λ_{M} are positive eigenvalues of L, and M with respective eigenfunctions φ, ψ in P \ {0}. Suppose that ∫_{0}^{1} g(s)φ(s)ψ(s) ds > 0. Then λ_{L} = λ_{M}.

Proof. We have

λ_{L}φ(t) = ∫_{0}^{1} C(t, s)g(s)φ(s) ds and λ_{M}ψ(t) = ∫_{0}^{1} C(s, t)g(s)ψ(s) ds.

Then, for almost every t ∈ (0, 1),

λ_{L}g(t)φ(t)ψ(t) = ∫_{0}^{1} g(t)ψ(t)C(t, s)g(s)φ(s) ds.

Integrating over [0, 1] gives

λ_{L} ∫_{0}^{1} g(t)φ(t)ψ(t) dt = ∫_{0}^{1}(∫_{0}^{1} C(t, s)g(s)φ(s) ds)g(t)ψ(t) dt
= ∫_{0}^{1}(∫_{0}^{1} C(t, s)g(t)ψ(t) dt)g(s)φ(s) ds
= λ_{M} ∫_{0}^{1} g(s)φ(s)ψ(s) ds,

where changing the order of integration is justified by Tonelli’s theorem since

∫_{0}^{1} g(t)φ(t)ψ(t) dt ≤ ∥φ∥ ∥ψ∥ ∫_{0}^{1} g(t) dt < ∞.

As ∫_{0}^{1} g(t)φ(t)ψ(t) dt > 0, this proves λ_{L} = λ_{M}. □

Remark 5.1. The argument here is similar to one used in [38]. In the proof, L and M do not need to be compact operators, but they are compact if G is continuous and g ∈ L^{1}. Compactness is useful so that the Krein-Rutman theorem can be applied to assert that r_{L} and r_{M} are positive eigenvalues with eigenfunctions in P \ {0}.

Remark 5.2. The positivity condition ∫_{0}^{1} g(t)φ(t)ψ(t) dt > 0 is satisfied if there is a subinterval [t_{0}, t_{1}] such that φ(t)ψ(t) > 0 for t ∈ [t_{0}, t_{1}] and g is positive almost everywhere on [t_{0}, t_{1}], say g is continuous on (0, 1) and g ≢ 0 on any subinterval of

EJQTDE, 2010 No. 28, p. 39 (0, 1). Under quite general conditions, it is known that φ(t) > 0 for t ∈ [t_{0}, t_{1}] for

an arbitrarily chosen [t_{0}, t_{1}] in (0, 1) (see, for example, [36]), and then [t_{0}, t_{1}] can be

chosen so that φ(t)ψ(t) > 0 for t ∈ [t_{0}, t_{1}].

(Received April 1, 2010)

Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA.

E-mail address: John-Graef@utc.edu

Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA.

E-mail address: Lingju-Kong@utc.edu

Department of Mathematics, Northern Illinois University, DeKalb, IL 60115, USA.

E-mail address: kong@math.niu.edu

The University of Hong Kong, City University of Hong Kong, and Chinney Investment Ltd., Hong Kong.

E-mail address: jsww@chinneyhonkwok.com