Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 19, 1-15; http://www.math.u-szeged.hu/ejqtde/

Solvability for second-order nonlocal boundary value problems with a

p-Laplacian at resonance on a half-line∗

Aijun Yang^{1,⋆}, Chunmei Miao^{2,1}, Weigao Ge^{1}

1Department of Applied Mathematics, Beijing Institute of Technology, Beijing, 100081, P. R. China. 2College of Science, Changchun University, Changchun, 130024, P. R. China.

Abstract: This paper investigates the solvability of the second-order boundary value problems with the one-dimensional p-Laplacian at resonance on a half-line

\{ \begin{matrix}(c(t)ϕ_{p}(x′(t)))′ = f(t, x(t), x′(t)), 0 < t < ∞, \\ x(0) = ∑_{i=1}^{n} μ_{i}x(ξ_{i}), lim_{t→+∞} c(t)ϕ_{p}(x′(t)) = 0 \end{matrix}

and

\{ \begin{matrix}(c(t)ϕ_{p}(x′(t)))′ + g(t)h(t, x(t), x′(t)) = 0, 0 < t < ∞, \\ x(0) = ∫_{0}^{∞} g(s)x(s)ds, lim_{t→+∞} c(t)ϕ_{p}(x′(t)) = 0 \end{matrix}

with multi-point and integral boundary conditions, respectively, where ϕ_{p}(s) = |s|^{p−2}s, p > 1. The arguments are based upon an extension of Mawhin’s continuation theorem due to Ge. And examples are given to illustrate our results. Keywords: Boundary value problem; Multi-point boundary condition; Integral boundary condition; Resonance; Half-line; p-Laplacian MSC: 34B10; 34B15; 34B40

1. INTRODUCTION

In this paper, we consider the second-order boundary value problems with a p-Laplacian on a

half line

(c(t)ϕ_{p}(x′(t)))′ = f(t, x(t), x′(t)), 0 < t < ∞,  (1.1)
x(0) = ∑_{i=1}^{n} μ_{i}x(ξ_{i}), lim_{t→+∞} c(t)ϕ_{p}(x′(t)) = 0,  (1.2)

∗Supported by NNSF of China (10671012) and SRFDP of China (20050007011). ^{⋆}Corresponding author. E-mail address: yangaij2004@163.com

with 0 ≤ ξ_{i} < ∞, μ_{i} ∈ ℝ, i = 1, 2, ⋯, n,

∑_{i=1}^{n} μ_{i} = 1  (1.3)

and

(c(t)ϕ_{p}(x′(t)))′ + g(t)h(t, x(t), x′(t)) = 0, 0 < t < ∞,  (1.4)
x(0) = ∫_{0}^{∞} g(s)x(s)ds, lim_{t→+∞} c(t)ϕ_{p}(x′(t)) = 0  (1.5)

with g ∈ L^{1}[0, ∞), g(t) > 0 on [0, ∞) and

∫_{0}^{∞} g(s)ds = 1.  (1.6)

Throughout this paper, we assume (A1) c ∈ C[0, ∞) ∩ C^{1}(0, ∞) and c(t) > 0 on [0, ∞), ϕ_{q}(\frac{1}{c}) ∈ L^{1}[0, ∞).

(A2) ∑_{i=1}^{n} μ_{i} ∫_{0}^{ξ_{i}} ϕ_{q}(\frac{e^{−s}}{c(s)})ds ≠ 0.

Due to the conditions (1.3) and (1.6), the differential operator \frac{d}{dt}(cϕ_{p}(\frac{d}{dt}·)) in (1.1) and (1.4) is not invertible under the boundary conditions (1.2) and (1.5), respectively. In the literature, boundary value problems of this type are referred to problems at resonance.

The theory of boundary value problems (in short: BVPs) with multi-point and integral boundary conditions arises in a variety of different areas of applied mathematics and physics. For example, bridges of small size are often designed with two supported points, which leads to a standard two-point boundary condition and bridges of large size are sometimes contrived with multi-point supports, which corresponds to a multi-point boundary condition [1]. Heat conduction, chemical engineering, underground water flow, thermo-elasticity and plasma physics can be reduced to the nonlocal problems with integral boundary conditions [2,3]. The study of multi-point BVPs for linear second-order ordinary differential equations was initiated by Il’in and Moiseev [4] in 1987. Since then many authors have studied more nonlinear multi-point BVPs [7-14]. Recently, BVPs with integral boundary conditions have received much attention. To identify a few, we refer the readers to [17-22] and references therein.

Second-order BVPs on infinite intervals arising from the study of radially symmetric solutions of nonlinear elliptic equation and models of gas pressure in a semi-infinite porous medium, have received much attention. For an extensive collection of results on BVPs on unbounded domains, we refer the readers to a monograph by Agarwal and O’Regan [16]. Other recent results and methods for BVPs on a half-line can be found in [14,15] and the references therein.

From the existed results, we can see a fact: for the resonance case, only BVPs with linear differential operator on half-line were considered. The BVPs with multi-point and integral boundary conditions on a half-line have not investigated till now. Although some authors (see [5,9,10,12,17]) have studied BVPs with nonlinear differential operator, for example, with a p-Laplacian operator, the domains are bounded.

Motivated by the above works, we intend to discuss the BVPs (1.1)-(1.2) and (1.4)-(1.5) at resonance on a half-line. Due to the fact that the classical Mawhin’s continuation theorem can’t be directly used to discuss the BVP with nonlinear differential operator, in this paper, we investigate the BVPs (1.1)-(1.2) and (1.4)-(1.5) by applying an extension of Mawhin’s continuation theorem due to Ge [5]. Furthermore, examples are given to illustrate the results.

2. PRELIMINARIES

For the convenience of readers, we present here some definitions and lemmas.

Definition 2.1. We say that a mapping f : [0, ∞)×ℝ^{2} → ℝ satisfies the Carathéodory conditions, if the following two conditions are satisfied:

(B1) for each (u, v) ∈ ℝ^{2}, the mapping t ↦ f(t, u, v) is Lebesgue measurable;

(B2) for a.e. t ∈ [0, ∞), the mapping (u, v) ↦ f(t, u, v) is continuous on ℝ^{2}. In addition, f is called a L^{1}-Carathéodory function if (B1), (B2) and (B3) hold, f is called a g-Carathéodory function if (B1), (B2) and (B4) are satisfied.

(B3) for each r > 0, there exists α_{r} ∈ L^{1}[0, ∞) such that for a.e. t ∈ [0, ∞) and every (u, v) such that max{||u||_{∞}, ||v||_{∞}} ≤ r, we have |f(t, u, v)| ≤ α_{r}(t);

(B4) for each l > 0 and g ∈ L^{1}[0, ∞), there exists a function ψ_{l} : [0, ∞) → [0, ∞) satisfying ∫_{0}^{∞} g(s)ψ_{l}(s)ds < ∞ such that

max{|u|, |v|} ≤ l implies |f(t, u, v)| ≤ ψ_{l}(t) for a.e. t ∈ [0, ∞).

Definition 2.2^{[5]}. Let X and Z be two Banach spaces with norms || · ||_{X} and || · ||_{Z}, respectively. A continuous operator M : X∩ domM → Z is said to be quasi-linear if

(C1) ImM = M(X∩ domM) is a closed subset of Z;

(C2) ker M = {x ∈ X∩ domM : Mx = 0} is linearly homeomorphic to ℝ^{n}, n < ∞.

Definition 2.3^{[6]}. Let X be a Banach spaces and X_{1} ⊂ X a subspace. The operator P : X → X_{1} is said to be a projector provided P^{2} = P, P (λ_{1}x_{1} + λ_{2}x_{2}) = λ_{1}P x_{1} + λ_{2}P x_{2} for x_{1}, x_{2} ∈ X, λ_{1}, λ_{2} ∈ ℝ. The operator Q : X → X_{1} is said to be a semi-projector provided Q^{2} = Q and Q(λx) = λQx for x ∈ X, λ ∈ ℝ.

Let X_{1} = ker M and X_{2} be the complement space of X_{1} in X, then X = X_{1} ⊕ X_{2}. On the other hand, suppose Z_{1} is a subspace of Z and Z_{2} is the complement of Z_{1} in Z, then Z = Z_{1} ⊕Z_{2}. Let P : X → X_{1} be a projector and Q : Z → Z_{1} be a semi-projector, and Ω ⊂ X an open and bounded set with the origin θ ∈ Ω, where θ is the origin of a linear space. Suppose N_{λ} : \overline{Ω} → Z, λ ∈ [0, 1] is a continuous operator. Denote N_{1} by N. Let ∑_{λ} = \{x ∈ \overline{Ω} : Mx = N_{λ}x\}.

Definition 2.4^{[5]}. N_{λ} is said to be M-compact in \overline{Ω} if there is a vector subspace Z_{1} of Z with dimZ_{1}= dimX_{1} and an operator R : \overline{Ω} × [0, 1] → X_{2} being continuous and compact such that for λ ∈ [0, 1],

(I − Q)N_{λ}(\overline{Ω}) ⊂ ImM ⊂ (I − Q)Z,  (2.1)
QN_{λ}x = 0, λ ∈ (0, 1) ⟺ QNx = 0,  (2.2)
R(·, 0) is the zero operator and R(·, λ)|_{∑_{λ}} = (I − P)|_{∑_{λ}},  (2.3)
M[P + R(·, λ)] = (I − Q)N_{λ}.  (2.4)

Theorem 2.1^{[5]}. Let X and Z be two Banach spaces with norms || · ||_{X} and || · ||_{Z}, respectively, and Ω ⊂ X an open and bounded set. Suppose M : X ∩ domM → Z is a quasi-linear operator and N_{λ} : \overline{Ω} → Z, λ ∈ [0, 1] is M-compact. In addition, if

(D1) Mx ≠ N_{λ}x, for λ ∈ (0, 1), x ∈ domM ∩ ∂Ω;

(D2) deg{JQN, Ω ∩ ker M, 0} ≠ 0, where J : Z_{1} → X_{1} is a homeomorphism with J(θ) = θ. Then the abstract equation Mx = Nx has at least one solution in \overline{Ω}.

Proposition 2.1^{[6]}. ϕ_{p} has the following properties

(E1) ϕ_{p} is continuous, monotonically increasing and invertible. Moreover, ϕ_{p}^{−1} = ϕ_{q} with q > 1 satisfying \frac{1}{p} + \frac{1}{q} = 1;

(E2) ∀u, v ≥ 0, ϕ_{p}(u + v) ≤ ϕ_{p}(u) + ϕ_{p}(v),  if 1 < p < 2,
ϕ_{p}(u + v) ≤ 2^{p−2}(ϕ_{p}(u) + ϕ_{p}(v)), if p ≥ 2.

3. RELATED LEMMAS

Let AC[0, ∞) denote the space of absolutely continuous functions on the interval [0, ∞). In this paper, we work in the following spaces

X = {x : [0, ∞) → ℝ | x, cϕ_{p}(x′) ∈ AC[0, ∞), lim_{t→∞} x(t) and lim_{t→∞} x′(t) exist
and (cϕ_{p}(x′))′ ∈ L^{1}[0, ∞)},
Y = L^{1}[0, ∞) and Z = {z : [0, ∞) → ℝ : ∫_{0}^{∞} g(t)|z(t)|dt < ∞}

with norms ||x||_{X} = max{||x||_{∞}, ||x′||_{∞}}, where ||x||_{∞} = sup_{t∈[0,∞)} |x(t)|, ||y||_{1} = ∫_{0}^{∞} |y(t)|dt and

||z||_{Z} = ∫_{0}^{∞} g(t)|z(t)|dt for x ∈ X, y ∈ Y and z ∈ Z. By the standard arguments, we can prove that (X, || · ||_{X}), (Y, || · ||_{1}) and (Z, || · ||_{Z}) are all Banach spaces.

Define M_{1} : domM_{1} → Y and N_{λ}^{1} : X → Y with

domM_{1} = {x ∈ X : x(0) = ∑_{i=1}^{n} μ_{i}x(ξ_{i}), lim_{t→+∞} c(t)ϕ_{p}(x′(t)) = 0}

by M_{1}x(t) = (c(t)ϕ_{p}(x′(t)))′ and N_{λ}^{1}x(t) = λf(t, x(t), x′(t)), t ∈ [0, ∞).

Let M_{2} : domM_{2} → Z and N_{λ}^{2} : X → Z with

domM_{2} = {x ∈ X : gx ∈ L^{1}[0, ∞), x(0) = ∫_{0}^{∞} x(s)g(s)ds, lim_{t→∞} c(t)ϕ_{p}(x′(t)) = 0}

be defined by M_{2}x(t) = −\frac{1}{g(t)}(c(t)ϕ_{p}(x′(t)))′ and N_{λ}^{2}x(t) = λh(t, x(t), x′(t)), t ∈ [0, ∞). Then the BVPs (1.1)-(1.2) and (1.4)-(1.5) can be written as M_{1}x = N^{1}x and M_{2}x = N^{2}x, respectively, here denote N_{1}^{i} = N^{i}, i = 1, 2.

Lemma 3.1. The operators M_{1} : domM_{1} → Y and M_{2} : domM_{2} → Z are quasi-linear. Proof. It is clear that X_{1} = ker M_{1} = {x ∈ domM_{1} : x(t) ≡ a ∈ ℝ on [0, ∞)}.

Let x ∈ domM_{1} and consider the equation (c(t)ϕ_{p}(x′(t)))′ = y(t). It follows from (1.2) that

c(t)ϕ_{p}(x′(t)) = − ∫_{t}^{∞} y(s)ds,

so that

x′(t) = −ϕ_{q}(\frac{1}{c(t)})ϕ_{q}(∫_{t}^{∞} y(s)ds),  (3.1)

and

x(t) = ∫_{t}^{∞} ϕ_{q}(\frac{1}{c(s)})ϕ_{q}(∫_{s}^{∞} y(τ)dτ)ds + C,  (3.2)

where C is a constant. In view of (1.2) and (1.3), we have

∑_{i=1}^{n} μ_{i} ∫_{0}^{ξ_{i}} ϕ_{q}(\frac{1}{c(s)})ϕ_{q}(∫_{s}^{∞} y(τ)dτ)ds = 0.  (3.3)

Thus,

ImM_{1} ⊂ \{y ∈ Y : ∑_{i=1}^{n} μ_{i} ∫_{0}^{ξ_{i}} ϕ_{q}(\frac{1}{c(s)})ϕ_{q}(∫_{s}^{∞} y(τ)dτ)ds = 0\}.

Conversely, if (3.3) holds for y ∈ Y, we take x ∈ dom M_{1} as given by (3.2), then (c(t)ϕ_{p}(x′(t)))′ = y(t) for t ∈ [0, ∞) and (1.2) is satisfied. Hence, we have

ImM_{1} = \{y ∈ Y : ∑_{i=1}^{n} μ_{i} ∫_{0}^{ξ_{i}} ϕ_{q}(\frac{1}{c(s)})ϕ_{q}(∫_{s}^{∞} y(τ)dτ)ds = 0\}.  (3.4)

So we have dimkerM_{1} = 1 < ∞, ImM_{1} ⊂ Y is closed. Therefore, M_{1} is a quasi-linear operator. Similarly, we can calculate that

ker M_{2} = {x ∈ domM_{2} : x(t) ≡ a ∈ ℝ on [0, ∞)}

and prove that

ImM_{2} = \{z ∈ Z : ∫_{0}^{∞} g(t) ∫_{0}^{t} ϕ_{q}(\frac{1}{c(s)})ϕ_{q}(∫_{s}^{∞} g(τ)z(τ)dτ)dsdt = 0\}.  (3.5)

Hence, M_{2} is also a quasi-linear operator.

In order to apply Theorem 2.1, we have to prove that R is completely continuous, and then to prove that N is M-compact. Because the Arzelà-Ascoli theorem fails to the noncompact interval case, we will use the following criterion.

Lemma 3.2^{[14]}. Let X be the space of all bounded continuous vector-valued functions on [0, ∞) and S ⊂ X. Then S is relatively compact if the following conditions hold:

(F1) S is bounded in X;

(F2) all functions from S are equicontinuous on any compact subinterval of [0, ∞); (F3) all functions from S are equiconvergent at infinity, that is, for any given ε > 0, there

exists a T = T (ε) > 0 such that ||χ(t) − χ(∞)||_{ℝ^{n}} < ε for all t > T and χ ∈ S.

Lemma 3.3. If f is a L^{1}-Carathéodory function, then the operator N_{λ}^{1} : \overline{U} → Y is M_{1}-compact in \overline{U}, where U ⊂ X is an open and bounded subset with θ ∈ U. Proof. We recall the condition (A2) and define the continuous operator Q_{1} : Y → Y_{1} by

Q_{1}y(t) = ω_{1}(t)ϕ_{p}(∑_{i=1}^{n} μ_{i} ∫_{0}^{ξ_{i}} ϕ_{q}(\frac{1}{c(s)})ϕ_{q}(∫_{s}^{∞} y(τ)dτ)ds),  (3.6)

where ω_{1}(t) = e^{−t}/ϕ_{p}(∑_{i=1}^{n} μ_{i} ∫_{0}^{ξ_{i}} ϕ_{q}(\frac{e^{−s}}{c(s)})ds). It is easy to check that Q_{1}^{2}y = Q_{1}y and Q_{1}(λy) = λQ_{1}y

for y ∈ Y, λ ∈ ℝ, that is, Q_{1} is a semi-projector and dimX_{1}=1=dimY_{1}. Moreover, (3.4) and (3.6) imply that ImM_{1}=ker Q_{1}.

It is easy to see that Q_{1}[(I − Q_{1})N_{λ}^{1}(x)] = 0, ∀x ∈ \overline{U}. So (I − Q_{1})N_{λ}^{1}(x) ∈ ker Q_{1} = ImM_{1}. For y ∈ ImM_{1}, we have Q_{1}y = 0. Thus, y = y − Q_{1}y = (I − Q_{1})y ∈ (I − Q_{1})Y. Therefore, (2.1) is satisfied. Obviously, (2.2) holds.

Define R_{1} : \overline{U} × [0, 1] → X_{2} by

R_{1}(x, λ)(t) = ∫_{t}^{∞} ϕ_{q}(\frac{1}{c(s)})ϕ_{q}(∫_{s}^{∞} λ(f(τ, x(τ), x′(τ)) − (Q_{1}f)(τ))dτ)ds,  (3.7)

where X_{2} is the complement space of X_{1} = ker M_{1} in X. Clearly, R_{1}(·, 0) = θ. Now we prove that R_{1} : \overline{U} × [0, 1] → X_{2} is compact and continuous.

We first assert that R_{1} is relatively compact for any λ ∈ [0, 1]. In fact, since U ⊂ X is a bounded set, there exists r > 0 such that \overline{U} ⊂ \{x ∈ X : ||x||_{X} ≤ r\}. Because the function f is L^{1}-Carathéodory, there exists α_{r} ∈ L^{1}[0, ∞) such that for a.e. t ∈ [0, ∞), |f(t, x(t), x′(t))| ≤ α_{r}(t) for x ∈ \overline{U}. Then for any x ∈ \overline{U}, λ ∈ [0, 1], we have

|R_{1}(x, λ)(t)| ≤ ∫_{t}^{∞} |ϕ_{q}(\frac{1}{c(s)})|ϕ_{q}(∫_{s}^{∞} λ|f(τ, x(τ), x′(τ)) − (Q_{1}f)(τ)|dτ)ds
≤ ∫_{0}^{∞} |ϕ_{q}(\frac{1}{c(s)})|dsϕ_{q}[∫_{0}^{∞} |α_{r}(s)|ds + ∫_{0}^{∞} |(Q_{1}f)(s)|ds]

= ||ϕ_{q}(\frac{1}{c})||_{1} · ϕ_{q}[||α_{r}||_{1} + ||Q_{1}f||_{1}] =: L_{1} < ∞.

From (A1), we can see that ϕ_{q}(\frac{1}{c}) is bounded. Hence,

|R′_{1}(x, λ)(t)| ≤ |ϕ_{q}(\frac{1}{c(t)})|ϕ_{q}(∫_{t}^{∞} λ|f(s, x(s), x′(s)) − (Q_{1}f)(s)|ds
≤ ||ϕ_{q}(\frac{1}{c})||_{∞} · ϕ_{q}[||α_{r}||_{1} + ||Q_{1}f||_{1}] =: L_{2} < ∞,

that is, R_{1}(·, λ)\overline{U} is uniformly bounded. Meanwhile, for any t_{1}, t_{2} ∈ [0, T] with T a positive constant, one gets

|R_{1}(x, λ)(t_{2}) − R_{1}(x, λ)(t_{1})| = | ∫_{t_{1}}^{t_{2}} R′_{1}(x, λ)(s)ds| ≤ L_{2}|t_{2} − t_{1}| → 0, as |t_{2} − t_{1}| → 0

and

|ϕ_{p}(R′_{1}(x, λ)(t_{2})) − ϕ_{p}(R′_{1}(x, λ)(t_{1}))|

= |\frac{1}{c(t_{2})} ∫_{t_{2}}^{∞} λ[f(s, x(s), x′(s)) − (Q_{1}f)(s)]ds − \frac{1}{c(t_{1})} ∫_{t_{1}}^{∞} λ[f(s, x(s), x′(s)) − (Q_{1}f)(s)]ds|
≤ |\frac{1}{c(t_{2})}| · | ∫_{t_{2}}^{t_{1}} λ[f(s, x(s), x′(s)) − (Q_{1}f)(s)]ds|

+|[\frac{1}{c(t_{2})} − \frac{1}{c(t_{1})}] ∫_{t_{1}}^{∞} λ[f(s, x(s), x′(s)) − (Q_{1}f)(s)]ds|

≤ ||\frac{1}{c}||_{∞} · | ∫_{t_{1}}^{t_{2}}[α_{r}(s) + |(Q_{1}f)(s)|]ds| + [||α_{r}||_{1} + ||Q_{1}f||_{1}]|\frac{1}{c(t_{2})} − \frac{1}{c(t_{1})}| → 0,
as |t_{2} − t_{1}| → 0.

Then |R′_{1}(x, λ)(t_{2}) − R′_{1}(x, λ)(t_{1})| → 0, as |t_{2} − t_{1}| → 0. So, R_{1}(·, λ)\overline{U} is equicontinuous on [0,T]. In additional, we claim that R_{1}(·, λ)\overline{U} is equiconvergent at infinity. In fact,

|R_{1}(x, λ)(t) − R_{1}(x, λ)(+∞)| ≤ ∫_{t}^{∞} |ϕ_{q}(\frac{1}{c(s)})|ϕ_{q}(∫_{s}^{∞} λ|f(τ, x(τ), x′(τ)) − (Q_{1}f)(τ)|dτ)ds
≤ ∫_{t}^{∞} L_{2}ds → 0 uniformly as t → +∞.

|R′_{1}(x, λ)(t) − R′_{1}(x, λ)(+∞)| ≤ |ϕ_{q}(\frac{1}{c(t)})|ϕ_{q}(∫_{t}^{∞} λ|f(s, x(s), x′(s)) − (Q_{1}f)(s)|ds)
≤ ||ϕ_{q}(\frac{1}{c})||_{∞} · ϕ_{q}[∫_{t}^{∞}(α_{r}(s) + |(Q_{1}f)(s)|)ds] → 0

uniformly as t → +∞.

Thus, Lemma 3.2 implies that R_{1}(·, λ)\overline{U} is relatively compact. Since f is L^{1}-Carathéodory, the continuity of R_{1} on \overline{U} follows from the Lebesgue dominated convergence theorem.

Define a projector P_{1} : X → X_{1} by P_{1}x(t) = lim_{t→+∞} x(t). For any x ∈ ∑_{λ}^{1} = \{x ∈ \overline{U} : M_{1}x = N_{λ}^{1}x}, we have λf(t, x(t), x′(t)) = (c(t)ϕ_{p}(x′(t)))′ ∈ ImM_{1} = ker Q_{1}. Hence

R_{1}(x, λ)(t) = ∫_{t}^{∞} ϕ_{q}(\frac{1}{c(s)})ϕ_{q}(∫_{s}^{∞} λ(f(τ, x(τ), x′(τ)) − (Q_{1}f)(τ))dτ)ds
= ∫_{t}^{∞} ϕ_{q}(\frac{1}{c(s)})ϕ_{q}(∫_{s}^{∞}(c(τ)ϕ_{p}(x′(τ)))′dτ)ds
= − ∫_{t}^{∞} x′(s)ds = x(t) − lim_{t→+∞} x(t) = [(I − P_{1})x](t),

which implies (2.3). For any x ∈ \overline{U}, we have

M_{1}[P_{1}x + R_{1}(x, λ)](t)
= M_{1}[ lim_{t→+∞} x(t) + ∫_{t}^{∞} ϕ_{q}(\frac{1}{c(s)})ϕ_{q}(∫_{s}^{∞} λ(f(τ, x(τ), x′(τ)) − (Q_{1}f)(τ))dτ)ds]
= λ(f(t, x(t), x′(t)) − Q_{1}f(t, x(t), x′(t)))
= [((I − Q_{1})N_{λ}^{1})(x)](t),

which yields (2.4). As a result, N_{λ}^{1} is M_{1}-compact in \overline{U}.

Lemma 3.4. If h is a g-Carathéodory function, then the operator N_{λ}^{2} : \overline{Ω} → Z is M_{2}-compact, where Ω ⊂ X is an open and bounded subset with θ ∈ Ω. Proof. As in the proof of Lemma 3.3, we first define the semi-projection Q_{2} : Z → Z_{1} by

Q_{2}z(t) = ϕ_{p}(\frac{1}{ω_{2}} ∫_{0}^{∞} g(s) ∫_{0}^{s} ϕ_{q}(\frac{1}{c(τ)})ϕ_{q}(∫_{τ}^{∞} g(r)z(r)dr)dτds),  (3.8)

where ω_{2} = ∫_{0}^{∞} g(s) ∫_{0}^{s} ϕ_{q}(\frac{1}{c(τ)})ϕ_{q}(∫_{τ}^{∞} g(r)dr)dτds. (3.5) and (3.8) imply that ImM_{2}=ker Q_{2}. It is easy to check that the conditions (2.1) and (2.2) hold.

Let R_{2} : \overline{Ω} × [0, 1] → X′_{2} be defined by

R_{2}(x, λ)(t) = ∫_{0}^{t} ϕ_{q}(\frac{1}{c(s)})ϕ_{q}(∫_{s}^{∞} λg(τ)(h(τ, x(τ), x′(τ)) − (Q_{2}f)(τ))dτ)ds,  (3.9)

where X′_{2} is the complement space of X′_{1} = ker M_{2} in X. Clearly, R_{2}(·, 0) = θ.

Now we prove that R_{2} : \overline{Ω} × [0, 1] → X′_{2} is compact and continuous. We first assert that R_{2} is relatively compact for λ ∈ [0, 1]. In fact, there exists l > 0 such that \overline{Ω} ⊂ \{x ∈ X : ||x||_{X} ≤ l}. Again, since h is a g-Carathéodory function, there exists nonnegative function ψ_{l} satisfying ∫_{0}^{∞} g(s)ψ_{l}(s)ds < ∞ such that for a.e. t ∈ [0, ∞), |h(t, x(t), x′(t))| ≤ ψ_{l}(t) for x ∈ \overline{Ω}. Then for any x ∈ \overline{Ω}, λ ∈ [0, 1], we have

|R_{2}(x, λ)(t)| = | ∫_{0}^{t} ϕ_{q}(\frac{1}{c(s)})ϕ_{q}[∫_{s}^{∞} λg(τ)(h(τ, x(τ), x′(τ)) − (Q_{2}f)(τ))dτ]ds|
≤ ∫_{0}^{∞} ϕ_{q}(\frac{1}{c(s)})ds · ϕ_{q}(∫_{0}^{∞} g(s)|ψ_{l}(s)|ds + ∫_{0}^{∞} g(s)|(Q_{2}f)(s)|ds)

= ||ϕ_{q}(\frac{1}{c})||_{1} · ϕ_{q}(||ψ_{l}||_{Z} + ||Q_{2}f||_{Z}) =: L_{3} < ∞

and

|R′_{2}(x, λ)(t)| = |ϕ_{q}(\frac{1}{c(t)})ϕ_{q}[∫_{t}^{∞} λg(s)(h(s, x(s), x′(s)) − (Q_{2}f)(s))]ds|
≤ ||ϕ_{q}(\frac{1}{c})||_{∞} · ϕ_{q}(||ψ_{l}||_{Z} + ||Q_{2}f||_{Z}) =: L_{4} < ∞,

that is, R_{2}(·, λ)\overline{Ω} is uniformly bounded. Meanwhile, for any t_{1}, t_{2} ∈ [0, T] with T a positive constant, as in the proof of Lemma 3.3, we can also show that R_{2}(·, λ)\overline{Ω} is equicontinuous on [0,T] and equiconvergent at infinity. Thus, Lemma 3.2 yields that R_{2}(·, λ)\overline{Ω} is relatively compact. Since f is a g-Carathéodory function, the continuity of R_{1} on \overline{Ω} follows from the Lebesgue dominated convergence theorem.

Define P_{2} : X → X′_{1} by (P_{2}x)(t) = x(0). Similar to the proof of Lemma 3.3, we can check that the conditions (2.3) and (2.4) are satisfied. Therefore, N_{λ}^{2} is M_{2}-compact in \overline{Ω}.

4. EXISTENCE RESULT FOR (1.1)-(1.2)

Theorem 4.1. If f is a L^{1}-Carathéodory function and suppose that (G1) there exists a constant A > 0 such that

∑_{i=1}^{n} μ_{i} ∫_{0}^{ξ_{i}} ϕ_{q}(\frac{1}{c(s)})ϕ_{q}(∫_{s}^{∞} f(τ, x(τ), x′(τ))dτ)ds ≠ 0  (4.1)

for x ∈ domM_{1}\ ker M_{1} with |x(t)| > A on t ∈ [0, ∞); (G2) there exist functions α, β, γ ∈ L^{1}[0, ∞) such that

|f(t, x, y)| ≤ α(t)|x|^{p−1} + β(t)|y|^{p−1} + γ(t), ∀(x, y) ∈ ℝ^{2}, a.e. t ∈ [0, ∞),  (4.2)

here denote α_{1} = ||α||_{1}, β_{1} = ||β||_{1}, γ_{1} = ||γ||_{1}; (G3) there exist a constant B > 0 such that either

b · ∑_{i=1}^{n} μ_{i} ∫_{0}^{ξ_{i}} ϕ_{q}(\frac{1}{c(s)})ϕ_{q}(∫_{s}^{∞} f(τ, b, 0)dτ)ds < 0  (4.3)

or

b · ∑_{i=1}^{n} μ_{i} ∫_{0}^{ξ_{i}} ϕ_{q}(\frac{1}{c(s)})ϕ_{q}(∫_{s}^{∞} f(τ, b, 0)dτ)ds > 0  (4.4)

for all b ∈ ℝ with |b| > B. Then the BVP(1.1)-(1.2) has at least one solution provided

2^{q−2}β_{1}^{q−1}||ϕ_{q}(\frac{1}{c})||_{∞} + 2^{2(q−2)}α_{1}^{q−1}||ϕ_{q}(\frac{1}{c})||_{1} < 1 for p < 2,  (4.5)

β_{1}^{q−1}||ϕ_{q}(\frac{1}{c})||_{∞} + α_{1}^{q−1}||ϕ_{q}(\frac{1}{c})||_{1} < 1 for p ≥ 2.  (4.6)

Before the proof of the main result, we first prove two lemmas.

Lemma 4.1. U_{1} = {x ∈ domM_{1} : M_{1}x = N_{λ}^{1}x for some λ ∈ (0, 1)} is bounded. Proof. Since N_{λ}^{1}x ∈ ImM_{1} = ker Q_{1} for x ∈ U_{1}, Q_{1}N^{1}x = 0. It follows from (G1) that there exists t_{0} ∈ [0, ∞) such that |x(t_{0})| ≤ A. Now, |x(t)| = |x(t_{0}) + ∫_{t_{0}}^{t} x′(s)ds| ≤ A + ||x′||_{1}, that is,

||x||_{∞} ≤ A + ||x′||_{1}.  (4.7)

Also,

x′(t) = −ϕ_{q}(\frac{1}{c(t)})ϕ_{q}(∫_{t}^{∞} λf(s, x(s), x′(s))ds).

In the case 1 < p < 2, by (G2) and Proposition 2.1, one gets

||x′||_{∞} = sup_{t∈[0,∞)} |ϕ_{q}(\frac{1}{c(t)})ϕ_{q}(∫_{t}^{∞} λf(s, x(s), x′(s))ds)|
≤ ||ϕ_{q}(\frac{1}{c})||_{∞} · ϕ_{q}[α_{1}||x||_{∞}^{p−1} + β_{1}||x′||_{∞}^{p−1} + γ_{1}]
≤ ||ϕ_{q}(\frac{1}{c})||_{∞} · 2^{q−2}[ϕ_{q}(α_{1}||x||_{∞}^{p−1} + γ_{1}) + β_{1}^{q−1}||x′||_{∞}]
≤ ||ϕ_{q}(\frac{1}{c})||_{∞} · 2^{q−2}[2^{q−2}(α_{1}^{q−1}||x||_{∞} + γ_{1}^{q−1}) + β_{1}^{q−1}||x′||_{∞}].

Noticing (4.5), one arrives at

||x′||_{∞} ≤ \frac{2^{2(q−2)}(α_{1}^{q−1}||x||_{∞} + γ_{1}^{q−1})||ϕ_{q}(\frac{1}{c})||_{∞}}{1 − 2^{q−2}β_{1}^{q−1}||ϕ_{q}(\frac{1}{c})||_{∞}} =: W_{1} + W_{2}||x||_{∞},  (4.8)

where W_{1} = \frac{2^{2(q−2)}γ_{1}^{q−1}||ϕ_{q}( \frac{1}{c})||_{∞}}{1−2^{q−2}β_{1}^{q−1}||ϕ_{q}( \frac{1}{c})||_{∞}}, W_{2} = \frac{2^{2(q−2)}α_{1}^{q−1}||ϕ_{q}( \frac{1}{c})||_{∞}}{1−2^{q−2}β_{1}^{q−1}||ϕ_{q}( \frac{1}{c})||_{∞}}.

||x′||_{1} = ∫_{0}^{∞} |ϕ_{q}(\frac{1}{c(t)})ϕ_{q}(∫_{t}^{∞} λf(s, x(s), x′(s))ds)|dt
≤ ||ϕ_{q}(\frac{1}{c})||_{1} · ϕ_{q}[α_{1}||x||_{∞}^{p−1} + β_{1}||x′||_{∞}^{p−1} + γ_{1}]
≤ ||ϕ_{q}(\frac{1}{c})||_{1} · 2^{q−2}[ϕ_{q}(α_{1}||x||_{∞}^{p−1} + γ_{1}) + β_{1}^{q−1}||x′||_{∞}]
≤ ||ϕ_{q}(\frac{1}{c})||_{1} · 2^{q−2}[2^{q−2}(α_{1}^{q−1}||x||_{∞} + γ_{1}^{q−1}) + β_{1}^{q−1}(W_{1} + W_{2}||x||_{∞})]
≤ ||ϕ_{q}(\frac{1}{c})||_{1} · 2^{q−2}[(2^{q−2}α_{1}^{q−1} + W_{2}β_{1}^{q−1})||x||_{∞} + (2^{q−2}γ_{1}^{q−1} + W_{1}β_{1}^{q−1})]
=: W_{3} + W_{4}||x||_{∞},  (4.9)

where W_{3} = 2^{q−2}(2^{q−2}γ_{1}^{q−1} + W_{1}β_{1}^{q−1})||ϕ_{q}(\frac{1}{c})||_{1}, W_{4} = 2^{q−2}(2^{q−2}α_{1}^{q−1} + W_{2}β_{1}^{q−1})||ϕ_{q}(\frac{1}{c})||_{1}. Thus, from (4.7) and (4.9), we have ||x||_{∞} ≤ A + W_{3} + W_{4}||x||_{∞}.

In view of (4.5), we can see W_{4} = \frac{2^{2(q−2)}α_{1}^{q−1}||ϕ_{q}( \frac{1}{c})||_{1}}{1−2^{q−2}β_{1}^{q−1}||ϕ_{q}( \frac{1}{c})||_{∞}} < 1, then ||x||_{∞} ≤ \frac{A+W_{3}}{1−W_{4}} =: W_{5} and ||x′||_{∞} ≤ W_{1} + W_{2}W_{5} =: W_{6}.

Similarly, in the case p ≥ 2, it follows that

||x′||_{∞} ≤ ||ϕ_{q}(\frac{1}{c})||_{∞} · [α_{1}^{q−1}||x||_{∞} + β_{1}^{q−1}||x′||_{∞} + γ_{1}^{q−1}].

Again,

||x′||_{∞} ≤ \frac{(γ_{1}^{q−1} + α_{1}^{q−1}||x||_{∞})||ϕ_{q}(\frac{1}{c})||_{∞}}{1 − β_{1}^{q−1}||ϕ_{q}(\frac{1}{c})||_{∞}} =: V_{1} + V_{2}||x||_{∞},

where V_{1} = \frac{γ_{1}^{q−1}||ϕ_{q}( \frac{1}{c})||_{∞}}{1−β_{1}^{q−1}||ϕ_{q}( \frac{1}{c})||_{∞}}, V_{2} = \frac{α_{1}^{q−1}||ϕ_{q}( \frac{1}{c})||_{∞}}{1−β_{1}^{q−1}||ϕ_{q}( \frac{1}{c})||_{∞}}.

||x′||_{1} ≤ ||ϕ_{q}(\frac{1}{c})||_{1} · [α_{1}^{q−1}||x||_{∞} + β_{1}^{q−1}||x′||_{∞} + γ_{1}^{q−1}].
≤ ||ϕ_{q}(\frac{1}{c})||_{1} · [(α_{1}^{q−1} + V_{2}β_{1}^{q−1})||x||_{∞} + (γ_{1}^{q−1} + V_{1}β_{1}^{q−1})] =: V_{3} + V_{4}||x||_{∞},

where V_{3} = (γ_{1}^{q−1} + V_{1}β_{1}^{q−1})||ϕ_{q}(\frac{1}{c})||_{1}, V_{4} = (α_{1}^{q−1} + V_{2}β_{1}^{q−1})||ϕ_{q}(\frac{1}{c})||_{1}. Thus, ||x||_{∞} ≤ A + V_{3} + V_{4}||x||_{∞}, then ||x||_{∞} ≤ \frac{A+V_{3}}{1−V_{4}} := V_{5} and ||x′||_{∞} ≤ V_{1} + V_{2}V_{5} =: V_{6}. Therefore, U_{1} is bounded.

Lemma 4.2. If U_{2} = {x ∈ ker M_{1} : −λx + (1 − λ)JQ_{1}N^{1}x = 0, λ ∈ [0, 1]}, where J : ImQ_{1} → ker M_{1} is a homomorphism, then U_{2} is bounded. Proof. Define J : ImQ_{1} → ker M_{1} by J(bω_{1}(t)) = b. Then for all b ∈ U_{2},

λb = (1 − λ)ϕ_{p}(∑_{i=1}^{n} μ_{i} ∫_{0}^{ξ_{i}} ϕ_{q}(\frac{1}{c(s)})ϕ_{q}(∫_{s}^{∞} f(τ, b, 0)dτ)ds).

If λ = 1, then b = 0. In the case λ ∈ [0, 1), if |b| > B, then by (4.3), we have

0 ≤ λb^{2} = (1 − λ)bϕ_{p}(∑_{i=1}^{n} μ_{i} ∫_{0}^{ξ_{i}} ϕ_{q}(\frac{1}{c(s)})ϕ_{q}(∫_{s}^{∞} f(τ, b, 0)dτ)ds) < 0,

which is a contradiction. Thus, ||x||_{X} = |b| ≤ B, ∀x ∈ U_{2}, that is, U_{2} is bounded.

Proof of Theorem 4.1. Let U ⊃ \overline{U}_{1} ∪ \overline{U}_{2} be a bounded and open set, then from Lemmas 4.1 and 4.2, we can obtain

(i) M_{1}x ≠ N_{λ}^{1}x for all (x, λ) ∈ [domM_{1} ∩ ∂U] × (0, 1);

(ii) Let H(x, λ) = −λx + (1 − λ)JQ_{1}N^{1}x, J is defined as in Lemma 4.2, we can see that H(x, λ) ≠ 0, ∀x ∈ domM ∩ ∂U. As a result, the homotopy invariance of Brouwer degree implies

deg\{JQ_{1}N^{1} |_{\overline{U}∩ker M_{1}}, U ∩ ker M_{1}, 0\} = deg\{H(·, 0), U ∩ ker M_{1}, 0\}
= deg{H(·, 1), U ∩ ker M_{1}, 0}
= deg{−I, U ∩ ker M_{1}, 0} ≠ 0.

Theorem 2.1 yields that M_{1}x = N^{1}x has at least one solution. The proof is completed. Remark 4.1. When the second part of condition (G3) holds, we choose Ũ_{2} = {x ∈ ker M_{1} : λx + (1 − λ)JQ_{1}N^{1}x = 0, λ ∈ [0, 1]} and take homotopy H̃(x, λ) = λx + (1 − λ)JQ_{1}N^{1}x. By a similar argument, we can also complete the proof. Example 4.1. Consider

\{ \begin{matrix}(e^{t+1}ϕ_{3}(x′(t)))′ = f(t, x(t), x′(t)), t ∈ (0, ∞), \\ x(0) = 2ex(\frac{1}{4}) + (1 − 2e)x(3), lim_{t→+∞} e^{t+1}ϕ_{p}(x′(t)) = 0. \end{matrix}  (4.10)

Corresponding to the BVP (1.1)-(1.2), we have p = 3, q = \frac{3}{2}, c(t) = e^{t+1}, μ_{1} = 2e, μ_{2} = 1 − 2e, ξ_{1} = \frac{1}{4}, ξ_{2} = 3 and

f(t, u, v) = \frac{1}{1 + t}e^{−t−1}u^{2} + e^{−t−2} sin t · v^{2} + \frac{1}{t^{2} + 1}.

It is easy to verify that (A1)-(A2) hold. Let α(t) = e^{−t−1}, β(t) = e^{−t−2}, γ(t) = \frac{1}{t^{2}+1}, then α_{1} = \frac{1}{e}, β_{1} = \frac{1}{e^{2}} , ||ϕ_{q}(\frac{1}{c})||_{∞} = \frac{1}{\sqrt{e}}, ||ϕ_{q}(\frac{1}{c})||_{1} = \frac{2}{\sqrt{e}}. Also, we can check that (G1)-(G3) and (4.6) are all satisfied. Thus, the BVP (4.10) has at least one solution, by using Theorem 4.1.

5. EXISTENCE RESULT FOR (1.4)-(1.5)

Theorem 5.1. If h is a g-Carathéodory function and suppose that (H1) there exists a constant A′ > 0 such that

∫_{0}^{∞} g(s) ∫_{0}^{s} ϕ_{q}(\frac{1}{c(τ)})ϕ_{q}(∫_{τ}^{∞} g(r)h(r, x(r), x′(r))dr)dτds ≠ 0  (5.1)

for x ∈ domM_{2}\ ker M_{2} with |x(t)| > A′ on t ∈ [0, ∞); (H2) there exist nonnegative functions δ, ζ, η ∈ Z such that

|h(t, u, v)| ≤ δ(t)|u|^{p−1} + ζ(t)|v|^{p−1} + η(t), ∀(u, v) ∈ ℝ^{2}, a.e. t ∈ [0, ∞),  (5.2)

here denote δ_{1} = ||δ||_{Z}, ζ_{1} = ||ζ||_{Z}, η_{1} = ||η||_{Z}; (H3) there exists a constant B′ > 0 such that either

d · ∫_{0}^{∞} g(s) ∫_{0}^{s} ϕ_{q}(\frac{1}{c(τ)})ϕ_{q}(∫_{τ}^{∞} g(r)h(r, d, 0)dr)dτds < 0  (5.3)

or

d · ∫_{0}^{∞} g(s) ∫_{0}^{s} ϕ_{q}(\frac{1}{c(τ)})ϕ_{q}(∫_{τ}^{∞} g(r)h(r, d, 0)dr)dτds > 0  (5.4)

for all d ∈ ℝ with |d| > B′. Then the BVP(1.4)-(1.5) has at least one solution on [0, ∞) provided

max\{2^{q−2}ζ_{1}^{q−1}||ϕ_{q}(\frac{1}{c})||_{∞}, \frac{2^{2(q−2)}δ_{1}^{q−1}||ϕ_{q}(\frac{1}{c})||_{1}}{1 − 2^{q−2}ζ_{1}^{q−1}||ϕ_{q}(\frac{1}{c})||_{∞}} \} < 1 for p < 2,  (5.5)

max\{ζ_{1}^{q−1}||ϕ_{q}(\frac{1}{c})||_{∞}, \frac{δ_{1}^{q−1}||ϕ_{q}(\frac{1}{c})||_{1}}{1 − ζ_{1}^{q−1}||ϕ_{q}(\frac{1}{c})||_{∞}} \} < 1 for p ≥ 2.  (5.6)

Proof. Let Ω_{1} = {x ∈ domM_{2} : M_{2}x = N_{λ}^{2}x for some λ ∈ (0, 1)}. As in the proof of Lemma 4.1, for x ∈ Ω_{1}, N_{λ}^{2}x ∈ ImM_{2} = ker Q_{2}, then Q_{2}N^{2}x = 0, i.e.,

∫_{0}^{∞} g(s) ∫_{0}^{s} ϕ_{q}(\frac{1}{c(τ)})ϕ_{q}(∫_{τ}^{∞} g(r)h(r, x(r), x′(r))dr)dτds = 0.

It follows from (H1) that there exists t_{0} ∈ [0, ∞) such that |x(t_{0})| ≤ A′. Thus, we can obtain

||x||_{∞} ≤ A′ + ||x′||_{1}.  (5.7)

Also,

x′(t) = ϕ_{q}(\frac{1}{c(t)})ϕ_{q}(∫_{t}^{∞} λg(s)h(s, x(s), x′(s))ds).

In the case 1 < p < 2, by (H2), Proposition 2.1 and (5.5), one gets

||x′||_{∞} ≤ \frac{2^{2(q−2)}(δ_{1}^{q−1}||x||_{∞} + η_{1}^{q−1})||ϕ_{q}(\frac{1}{c})||_{∞}}{1 − 2^{q−2}ζ_{1}^{q−1}||ϕ_{q}(\frac{1}{c})||_{∞}} =: W′_{1} + W′_{2}||x||_{∞},  (5.8)

where W′_{1} = \frac{2^{2(q−2)}η_{1}^{q−1}||ϕ_{q}( \frac{1}{c})||_{∞}}{1−2^{q−2}ζ_{1}^{q−1}||ϕ_{q}( \frac{1}{c})||_{∞}}, W′_{2} = \frac{2^{2(q−2)}δ_{1}^{q−1}||ϕ_{q}( \frac{1}{c})||_{∞}}{1−2^{q−2}ζ_{1}^{q−1}||ϕ_{q}( \frac{1}{c})||_{∞}}.

||x′||_{1} = ∫_{0}^{∞} |ϕ_{q}(\frac{1}{c(t)})ϕ_{q}(∫_{t}^{∞} λg(s)h(s, x(s), x′(s))ds|dt
≤ ||ϕ_{q}(\frac{1}{c})||_{1} · ϕ_{q}[δ_{1}||x||_{∞}^{p−1} + ζ_{1}||x′||_{∞}^{p−1} + η_{1}]
≤ ||ϕ_{q}(\frac{1}{c})||_{1} · 2^{q−2}[(2^{q−2}δ_{1}^{q−1} + W′_{2}ζ_{1}^{q−1})||x||_{∞} + (2^{q−2}η_{1}^{q−1} + W′_{1}ζ_{1}^{q−1})]
=: W′_{3} + W′_{4}||x||_{∞},  (5.9)

where W′_{3} = 2^{q−2}(2^{q−2}η_{1}^{q−1} + W′_{1}ζ_{1}^{q−1})||ϕ_{q}(\frac{1}{c})||_{1}, W′_{4} = 2^{q−2}(2^{q−2}δ_{1}^{q−1} + W′_{2}ζ_{1}^{q−1})||ϕ_{q}(\frac{1}{c})||_{1}. Thus, from (5.7) and (5.9), we have ||x||_{∞} ≤ \frac{A′+W′_{3}}{1−W′_{4}} =: W′_{5}. Then, ||x′||_{∞} ≤ W′_{1} + W′_{2}W′_{5} =: W′_{6}.

Similarly, for p ≥ 2, it follows that

||x′||_{∞} ≤ \frac{(η_{1}^{q−1} + δ_{1}^{q−1}||x||_{∞})||ϕ_{q}(\frac{1}{c})||_{∞}}{1 − ζ_{1}^{q−1}||ϕ_{q}(\frac{1}{c})||_{∞}} =: V′_{1} + V′_{2}||x||_{∞},

where V′_{1} = \frac{η_{1}^{q−1}||ϕ_{q}( \frac{1}{c})||_{∞}}{1−ζ_{1}^{q−1}||ϕ_{q}( \frac{1}{c})||_{∞}}, V′_{2} = \frac{δ_{1}^{q−1}||ϕ_{q}( \frac{1}{c})||_{∞}}{1−ζ_{1}^{q−1}||ϕ_{q}( \frac{1}{c})||_{∞}}.

||x′||_{1} ≤ ||ϕ_{q}(\frac{1}{c})||_{1} · [δ_{1}^{q−1}||x||_{∞} + ζ_{1}^{q−1}||x′||_{∞} + η_{1}^{q−1}].
≤ ||ϕ_{q}(\frac{1}{c})||_{1} · [(δ_{1}^{q−1} + V′_{2}ζ_{1}^{q−1})||x||_{∞} + (η_{1}^{q−1} + V′_{1}ζ_{1}^{q−1})] =: V′_{3} + V′_{4}||x||_{∞},

where V′_{3} = (η_{1}^{q−1} + V′_{1}ζ_{1}^{q−1})||ϕ_{q}(\frac{1}{c})||_{1}, V′_{4} = (δ_{1}^{q−1} + V′_{2}ζ_{1}^{q−1})||ϕ_{q}(\frac{1}{c})||_{1}. Thus, ||x||_{∞} ≤ \frac{A′+V′_{3}}{1−V′_{4}} =: V′_{5} and ||x′||_{∞} ≤ V′_{1} + V′_{2}V′_{5} =: V′_{6}. As a result, Ω_{1} is bounded.

Define Ω_{2} = {x ∈ ker M_{2} : −μx + (1 − μ)JQ_{2}N^{2}x = 0, μ ∈ [0, 1]}, where J : ImQ_{2} → ker M_{2} is a homomorphism defined by J(d) = d. As in Lemma 4.2, we can prove that Ω_{2} is bounded.

Let Ω ⊃ \overline{Ω}_{1}∪\overline{Ω}_{2} be a bounded and open set. Then M_{2}x ≠ N_{λ}^{2}x, ∀(x, λ) ∈ (domM_{2}∩∂Ω)×(0, 1). Define a homotopy operator

T (x, μ) = −μx + (1 − μ)JQ_{2}N^{2}x.

We can see that T (x, μ) ≠ 0, ∀x ∈ domM_{2} ∩ ∂Ω. Therefore,

deg\{JQ_{2}N^{2} |_{\overline{Ω}∩ker M_{2}}, Ω ∩ ker M_{2}, 0\} = deg\{T (·, 0), Ω ∩ ker M_{2}, 0\}
= deg{T (·, 1), Ω ∩ ker M_{2}, 0}
= deg{−I, Ω ∩ ker M_{2}, 0} ≠ 0.

Theorem 2.1 implies that M_{2}x = N^{2}x has at least one solution. The proof is completed.

Remark 5.1. When the second part of condition (H3) holds, we may choose Ω̃_{2} = {x ∈ ker M_{2} : μx + (1 − μ)JQ_{2}N^{2}x = 0, μ ∈ [0, 1]} and take homotopy T̃(x, μ) = μx + (1 − μ)JQ_{2}N^{2}x.

Remark 5.2. Under the multi-point boundary conditions, we can obtain the existence of solutions on a half-line by assume the nonlinear function f is L^{1}-Carathéodory. When the boundary conditions involved in the integral condition, however, this assumption on the nonlinear term is invalid if the domain is unbounded. In this paper, we overcome this difficulty by introducing the definition of g-Carathéodory function and multiplying the g-Carathéodory function h by the function g ∈ L^{1}[0, ∞) in the equation (1.4).

Example 5.1. Consider

\{ \begin{matrix}3e^{t}(e^{t}ϕ_{3}(x′(t)))′ + h(t, x(t), x′(t)) = 0, t ∈ (0, ∞), \\ x(0) = ∫_{0}^{∞} e^{−t}x(t)dt, lim_{t→+∞} 3e^{t}ϕ_{3}(x′(t)) = 0. \end{matrix}  (5.10)

Corresponding to the BVP (1.4)-(1.5), we have p = 3, c(t) = 3e^{t}, g(t) = e^{−t} and h(t, u, v) = te^{−2t}u^{2} + e^{−t}v^{2}. It is easy to verify that (A1) holds. Let δ(t) = te^{−2t}, ζ(t) = e^{−t}, then δ_{1} = \frac{1}{9}, ζ_{1} = \frac{1}{2}, ||ϕ_{q}(\frac{1}{c})||_{∞} = \frac{1}{\sqrt{3}}, ||ϕ_{q}(\frac{1}{c})||_{1} = \frac{2}{\sqrt{3}}. Also, we can check that (H1)-(H3) and (5.6) are all satisfied. Thus, thanks to Theorem 5.1, the BVP (5.10) has at least one solution.

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