Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 31, 1-13; http://www.math.u-szeged.hu/ejqtde/

Positive solutions of boundary value problems for systems of second-order differential equations with integral boundary condition on the half-line

Shouliang Xi^{∗}, Mei Jia^{†}, Huipeng Ji

College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

Abstract In this paper, we study the existence of positive solutions of boundary value problems for systems of second-order differential equations with integral boundary condition on the half-line. By using the fixed-point theorem in cones, we show the existence of at least one positive solution with suitable growth conditions imposed on the nonlinear terms. Moreover, the associated integral kernels for the boundary value problems are given.

MSC 34B10; 34B15; 34B18; 34B40

Keywords Boundary value problems; Positive solutions; Fixed-point theorem in cones; Integral boundary condition

1 Introduction

In this paper we consider the existence of positive solutions for second-order boundary value problems (BVPs) for systems of differential equations with integral boundary condition on the half-line:

\begin{matrix}u′′_{1}(t) + f_{1}(t, u_{1}(t), u_{2}(t)) = 0, t ∈ (0, ∞), \\ u′′_{2}(t) + f_{2}(t, u_{1}(t), u_{2}(t)) = 0, t ∈ (0, ∞), \\ u_{1}(0) = u_{2}(0) = 0, \\ u′_{1}(∞) = ∫_{0}^{+∞} g_{1}(s)u_{1}(s)ds, u′_{2}(∞) = ∫_{0}^{+∞} g_{2}(s)u_{2}(s)ds, \end{matrix}  (1.1)

where u′_{i}(∞) = lim_{t→+∞} u′_{i}(t), i = 1, 2.

Nonlocal BVPs have been well studied especially on a compact interval. Gupta and co-authors, for example [1], has made an extensive study of multipoint BVPs. The existence of positive solutions was studied by Ma in 1999 for a type of three-point boundary value problem [2]. Many authors have studied the existence of positive solutions for multi-point boundary value problems, and obtained many sufficient conditions for the existence of positive solutions. See [3] for a survey of some of the work done and for references to many contributions.

∗E-mail: xishouliang@163.com

†Corresponding author. E-mail: jiamei-usst@163.com

Boundary value problems with Riemann-Stieltjes integral boundary conditions (BCs) are now being studied since they include BVPs with two-point, multipoint and integral BCs as special cases. See [4]–[16] and the references therein.

In 2000, Ma [4] considered the existence of positive solutions for second order ordinary differential equations

\begin{matrix}u′′ + a(t)f(u) = 0 \\ u(0) = 0, u(1) = ∫_{α}^{β} h(t)u(t)dt, \end{matrix}

where [α, β] ⊂ (0, 1) and f is either superlinear or sublinear, by reducing this to a three point BVP. Karakostas and Tsamatos [5]–[8] weakened the restrictions on the function f and considered boundary conditions given by Riemann-Stieltjes integral which improved the result of Ma [4]. This was further improved in [15].

Yang [10]–[12] investigated the boundary value problems of differential equations with some nonlinear BCs

−(au′)′ + bu = g(t)f(t, u), t ∈ [0, 1]
cos γ_{0}u(0) − sin γ_{0}u′(0) = H_{1}( ∫_{0}^{1} u(τ)dα(τ)),
cos γ_{1}u(1) + sin γ_{1}u′(1) = H_{2}( ∫_{0}^{1} u(τ)dβ(τ)).

Afterward, see [14], he studied the boundary value problem for systems of second-order differential equations with integral boundary condition.

In [15, 16], Webb and Infante used fixed point index theory and gave a general method for solving problems with integral BCs of Riemann-Stieltjes type. In [16] they studied the existence of multiple positive solutions of nonlinear differential equations of the form

−u′′(t) = g(t)f(t, u(t)), t ∈ (0, 1)

with boundary conditions including the following

u(0) = α[u], u(1) = β[u],
u(0) = α[u], u′(1) = β[u],
u(0) = α[u], u′(1) + β[u] = 0,
u′(0) = α[u], u(1) = β[u],
u′(0) + α[u] = 0, u(1) = β[u],

where α, β are linear functionals on C[0, 1] given by

α[u] = ∫_{0}^{1} u(s)dA(s), β[u] = ∫_{0}^{1} u(s)dB(s),

with A, B functions of bounded variation. By giving a general approach to cover all of these boundary conditions (and others) in a unified way, this work includes much previous work as special cases and improves the previous results.

In particular, the work of Webb and Infante does not require the functionals α[u], β[u] to be positive for all positive u so some sign changing measures are allowed.

Feng [17] studied the existence of positive solutions for a class of boundaryvalue problem with integral boundary conditions in Banach spaces.

On the other hand, boundary value problems on infinite intervals occur in mathematical modeling of various applied problems, see the references of [18]. Several authors have studied problems on infinite intervals. See [18]–[27] and their references. In [21]–[25], they studied a similar problem

(p(t)x′(t))′ + λq(t)f(t, x(t), x′(t)) = 0, t ∈ [0, +∞)

α_{1}x(0) − β_{1}x′(0) − γ_{1} lim_{t→0} p(t)x′(t) = δ_{1},

α_{2} lim_{t→+∞} x(t) + β_{2} lim_{t→+∞} p(t)x′(t) = δ_{2}.

In [26], by applying fixed-point theorems, Tian, Ge, and Shan [26] considered the existence of positive solutions for the three-point boundary value problem on the half-line. Thereafter, Tian and Ge [27] studied the existence of positive solutions for the multi-point boundary value problem on the half-line.

Motivated by the papers mentioned above, in this paper we investigate the existence of positive solutions of boundary value problems for systems of secondorder differential equations (1.1) with integral boundary condition on the halfline.

2 Preliminaries

Throughout the paper, we denote ℝ^{+} = [0, +∞). Definition We say f_{i} : ℝ^{+} × ℝ^{+} × ℝ^{+} → ℝ^{+} is an L^{1}-Carathéodory function if

(1) f_{i}(·, x, y) is measurable for any (x, y) ∈ ℝ^{+} × ℝ^{+};

(2) f_{i}(t, ·, ·) is continuous for almost every t ∈ ℝ^{+};

(3) for each r_{1}, r_{2} > 0, there exists ϕ_{r_{1},r_{2}} ∈ L^{∞}[0, +∞) such that

0 ≤ f_{i}(t, (1 + t)x, (1 + t)y) ≤ ϕ_{r_{1},r_{2}}(t),

for all x ∈ [0, r_{1}], y ∈ [0, r_{2}] and almost every t ∈ [0, +∞).

We assume that the following conditions hold.

(H1) g_{i} ∈ L^{1}[0, +∞) is nonnegative and 1 − ∫_{0}^{+∞} sg_{i}(s)ds > 0, i = 1, 2;

(H2) f_{i} is an L^{1}-Carathéodory function, i = 1, 2.

For convenience, we denote

c_{i} = \frac{1}{1 − ∫_{0}^{+∞} sg_{i}(s)ds}, i = 1, 2.

Let

C(ℝ^{+}) = \{x : ℝ^{+} → ℝ : x is continuous and sup_{t∈ℝ^{+}} \frac{|x(t)|}{1 + t} < +∞\}.

Define ∥x∥_{1} = sup_{t∈ℝ^{+}} \frac{|x(t)|}{1+t}. Then (C(ℝ^{+}), ∥ · ∥_{1}) is a Banach space.

Let

X = \{(u_{1}, u_{2}) ∈ C(ℝ^{+}) × C(ℝ^{+}) : sup_{t∈ℝ^{+}} \frac{|u_{i}|}{1 + t} < +∞, i = 1, 2\}

with the norm ∥(u_{1}, u_{2})∥ = ∥u_{1}∥_{1} + ∥u_{2}∥_{1}. It is easy to prove that (X, ∥ · ∥) is a Banach space.

In this paper, the main tool we use is the following Krasnosel’skii’s fixed point theorem in cones, see for example [28].

Theorem 2.1 Let E be a Banach space and let K ⊂ E be a cone in E. Suppose that Ω_{1}, Ω_{2} are bounded open subsets of K with 0 ∈ Ω_{1}, \overline{Ω}_{1} ⊂ Ω_{2}, and A : K → K is a completely continuous operator such that either

∥Aw∥ ≤ ∥w∥, w ∈ ∂Ω_{1}, ∥Aw∥ ≥ ∥w∥, w ∈ ∂Ω_{2},

or

∥Aw∥ ≥ ∥w∥, w ∈ ∂Ω_{1}, ∥Aw∥ ≤ ∥w∥, w ∈ ∂Ω_{2}.

Then A has a fixed point in \overline{Ω}_{2} \ Ω_{1}.

Lemma 2.2 Assume that (H1) holds. Then for any y_{i} ∈ L^{1}[0, +∞) and y_{i} ≥ 0, the boundary value problem

u′′_{i}(t) + y_{i}(t) = 0  (2.1)

u_{i}(0) = 0, u′_{i}(∞) = ∫_{0}^{+∞} g_{i}(s)u_{i}(s)ds,  (2.2)

has a unique solution u_{i} ∈ C(ℝ^{+}), and

u_{i}(t) = ∫_{0}^{+∞} H_{i}(t, s)y_{i}(s)ds, i = 1, 2,  (2.3)

where

H_{i}(t, s) = G(t, s) + tc_{i} ∫_{0}^{+∞} g_{i}(r)G(s, r)dr, i = 1, 2,

and

G(t, s) = min{s, t}

Proof. In fact, for any y_{i} ∈ L^{1}[0, +∞) and y_{i} ≥ 0, by the methods of [15] and [16], it is easy to show that the unique solution of (2.1) and (2.2) is given by

u_{i}(t) = ∫_{0}^{+∞} H_{i}(t, s)y_{i}(s)ds, i = 1, 2,

where H_{i}(t, s) can be written

H_{i}(t, s) = G(t, s) + tc_{i}G_{i}(s), i = 1, 2,

with G_{i}(s) = ∫_{0}^{+∞} g_{i}(r)G(s, r)dr.

Lemma 2.3 Assume that (H1) holds. Let δ ∈ (0, 1), then for all t ∈ [δ, \frac{1}{δ}], τ, s ∈ ℝ^{+}, we have

H_{i}(τ, s) ≥ 0, H_{i}(t, s) ≥ \frac{δ}{1 + τ} H_{i}(τ, s).  (2.4)

Proof. It is clear that H_{i}(τ, s) ≥ 0.

For t ∈ [δ, \frac{1}{δ}], τ, s ∈ ℝ^{+}, noting that G(t, s) = min\{t, s\}, we have

\frac{δ}{1 + τ} G(τ, s) ≤ \frac{δ}{1 + τ} τ < δ ≤ t.

and

\frac{δ}{1 + τ} G(τ, s) ≤ \frac{δ}{1 + τ} s < s.

This proves \frac{δ}{1+τ} G(τ, s) ≤ min\{t, s\} = G(t, s). Therefore, it follows that

H_{i}(t, s) ≥ \frac{δ}{1 + τ} H_{i}(τ, s), for s, τ ∈ ℝ^{+}, t ∈ [δ, \frac{1}{δ}].

The lemma is proved.

Lemma 2.4 Assume that (H1) holds. If y_{i} ∈ L^{1}[0, +∞), y_{i} ≥ 0, then the unique solution u_{i}(t) of the boundary value problem (2.1) and (2.2) satisfies u_{i}(t) ≥ 0 and min_{δ≤t≤ \frac{1}{δ}} u_{i}(t) ≥ δ∥u_{i}∥_{1}, i = 1, 2.

Proof. It is clear that u_{i}(t) ≥ 0, for all t ∈ ℝ^{+}, i = 1, 2.

From (2.3) and (2.4), for any t ∈ [δ, \frac{1}{δ}], s, τ ∈ ℝ^{+}, i = 1, 2, we have

u_{i}(t) = ∫_{0}^{+∞} H_{i}(t, s)y_{i}(s)ds

≥ δ ∫_{0}^{+∞} \frac{1}{1 + τ} H_{i}(τ, s)y_{i}(s)ds

= δ \frac{1}{1 + τ} u_{i}(τ).

Hence,

min_{δ≤t≤ \frac{1}{δ}} u_{i}(t) ≥ δ∥u_{i}∥_{1}, i = 1, 2.

The lemma is proved.

Let

K = \{(u_{1}, u_{2}) ∈ X : u_{i} ≥ 0, min_{δ≤t≤ \frac{1}{δ}} u_{i}(t) ≥ δ∥u_{i}∥_{1}, i = 1, 2\}.

Then K ⊂ X is a cone in X.

By (H2), for (u_{1}, u_{2}) ∈ K, let \frac{u_{i}(s)}{1 + s} ≤ r_{i} for all s ≥ 0, then we have

∫_{0}^{+∞} f_{i}(s, u_{1}(s), u_{2}(s))ds = ∫_{0}^{+∞} f_{i}(s, (1 + s)\frac{u_{1}(s)}{1 + s}, (1 + s)\frac{u_{2}(s)}{1 + s})ds
≤ ∫_{0}^{+∞} ϕ_{r_{1},r_{2}}(s)ds < +∞.

So ∫_{0}^{+∞} f_{i}(s, u_{1}(s), u_{2}(s))ds is convergent and f_{i} ∈ L^{1}[0, +∞). Hence, from Lemma 2.2, we know that the boundary value problem (1.1) is equivalent to

u_{i}(t) = ∫_{0}^{+∞} H_{i}(t, s)f_{i}(s, u_{1}(s), u_{2}(s))ds, i = 1, 2.

Define T_{i} : K → C(ℝ^{+}) by

\begin{matrix}T_{1}(u_{1}, u_{2})(t) = ∫_{0}^{+∞} H_{1}(t, s)f_{1}(s, u_{1}(s), u_{2}(s))ds, \\ T_{2}(u_{1}, u_{2})(t) = ∫_{0}^{+∞} H_{2}(t, s)f_{2}(s, u_{1}(s), u_{2}(s))ds. \end{matrix}

Let

T (u_{1}, u_{2})(t) = (T_{1}(u_{1}, u_{2})(t), T_{2}(u_{1}, u_{2})(t)).

Lemma 2.5 ([29]) Let M ⊂ C(ℝ^{+}), then M is a relatively compact if the following conditions hold:

(a) all functions from M are uniformly bounded in C(ℝ^{+});

(b) the functions from \{y : y = \frac{x}{1+t}, x ∈ M\} are equicontinuous on any compact interval of [0, +∞);

(c) the functions from \{y : y = \frac{x}{1+t}, x ∈ M\} and are equiconvergent at infinity, that is , for any ϵ > 0, there exists a T = T (ϵ) > 0, such that |y(t) − y(+∞)| < ϵ, for all t > T, and x ∈ M.

Lemma 2.6 Assume that (H1)(H2) hold. Then T : K → K is completely continuous.

Proof. (1) T : K → K. For any (u_{1}, u_{2}) ∈ K,

\frac{T_{i}(u_{1}, u_{2})(t)}{1 + t} = ∫_{0}^{+∞} \frac{H_{i}(t, s)}{1 + t}f_{i}(s, u_{1}(s), u_{2}(s))ds

= ∫_{0}^{+∞} \frac{G(t, s) + tc_{i}G_{i}(s)}{1 + t}f_{i}(s, u_{1}(s), u_{2}(s))ds

≤ (1 + c_{i} ∫_{0}^{+∞} rg_{i}(r)dr)(∫_{0}^{+∞} ϕ_{r_{1},r_{2}}(s)ds)

= c_{i} ∫_{0}^{+∞} ϕ_{r_{1},r_{2}}(s)ds < +∞.

So, T (u_{1}, u_{2}) ∈ X. From Lemma 2.4, we have T (u_{1}, u_{2}) ∈ K.

(2) T : K → K is continuous.

For any convergent sequence (u_{1,n}, u_{2,n}) → (u_{1}, u_{2}), then u_{1,n} → u_{1}, and u_{2,n} → u_{2}, as n → +∞, and there are constants r_{1}, r_{2} > 0, such that

∥u_{1,n}∥, ∥u_{1}∥ ≤ r_{1} and ∥u_{2,n}∥, ∥u_{2}∥ ≤ r_{2}.

By (H_{2}), for a.e. s ∈ [0, +∞), we have

|f_{i}(s, u_{1,n}(s), u_{2,n}(s)) − f_{i}(s, u_{1}(s), u_{2}(s))| → 0, as n → +∞,

and

|f_{i}(s, u_{1,n}(s), u_{2,n}(s)) − f_{i}(s, u_{1}(s), u_{2}(s))| ≤ 2ϕ_{r_{1},r_{2}}(s).

From the Lebesgue dominated convergence theorem, we have

∥T_{i}(u_{1,n}, u_{2,n}) − T_{i}(u_{1}, u_{2})∥_{1}

≤ sup_{t∈ℝ^{+}} ∫_{0}^{+∞} \frac{H_{i}(t, s)}{1 + t}|f_{i}(s, u_{1}(s), u_{2}(s)) − f_{i}(s, u_{1,n}(s), u_{2,n}(s))|ds → 0.

So,

∥T_{i}(u_{1,n}, u_{2,n}) − T_{i}(u_{1}, u_{2})∥_{1} → 0, n → +∞.

Hence, T : K → K continuous.

(3) T : K → K is relatively compact.

The proof of compactness follows from the Lemma 2.5 as in [29] So, T : K → K is completely continuous. The lemma is proved.

3 The main result

We assume

(H3) There exist nonnegative functions a_{i} ∈ L^{1}[0, +∞), and continuous functions h_{i} ∈ C[ℝ^{+} × ℝ^{+}, ℝ^{+}], i = 1, 2, such that

f_{i}(t, x, y) ≤ a_{i}(t)h_{i}(x, y), t, x, y ∈ ℝ^{+},
∫_{0}^{+∞} sa_{i}(s)ds < +∞.

Obviously, if (H3) holds then ∫_{0}^{+∞} a_{i}(s)ds < +∞ and if (H1) and (H3) hold then ∫_{0}^{+∞} \frac{H_{i}(t,s)}{1+t}(1 + s)a_{i}(s)ds < +∞ for t ∈ ℝ^{+} and i = 1, 2.

Denote

h_{i}^{0} = \overline{lim}_{x+y→0^{+}} \frac{h_{i}(x, y)}{x + y}, h_{i}^{∞} = \overline{lim}_{x+y→+∞} \frac{h_{i}(x, y)}{x + y};

f_{i}^{0} = \underline{lim}_{x+y→0^{+}} inf_{δ≤t≤ \frac{1}{δ}} \frac{f_{i}(t, x, y)}{x + y}, f_{i}^{∞} = \underline{lim}_{x+y→+∞} inf_{δ≤t≤ \frac{1}{δ}} \frac{f_{i}(t, x, y)}{x + y};

N_{i} = inf_{δ≤t≤ \frac{1}{δ}} ∫δ^{\frac{1}{δ}} \frac{H_{i}(t, s)}{1 + t}ds, n_{i} = sup_{t∈ℝ^{+}} ∫_{0}^{+∞} \frac{H_{i}(t, s)}{1 + t}(1 + s)a_{i}(s)ds.

Theorem 3.1 Suppose that hypotheses (H1) − (H3) hold and satisfy

(1) there exist positive numbers δ, m_{i}, M_{i}, 0 < δ < 1, such that m_{1}n_{1} + m_{2}n_{2} ≤ 1, M_{1}N_{1}δ + M_{2}N_{2}δ ≥ 1;

(2) 0 ≤ h_{i}^{0} < m_{i}, M_{i} < f_{i}^{∞} ≤ +∞.

Then, the boundary value problem (1.1) has at least one positive solution.

Proof. For 0 ≤ h_{i}^{0} < m_{i} and by continuity of h_{i}, there exists a δ_{1} > 0 (δ_{1} < 1), such that

h_{i}(u_{1}, u_{2}) ≤ m_{i}(u_{1} + u_{2}), u_{1} + u_{2} ∈ (0, δ_{1}].

Let

Ω_{1} = {(u_{1}, u_{2}) ∈ K : ∥(u_{1}, u_{2})∥ < δ_{1}}.

For any (u_{1}, u_{2}) ∈ ∂Ω_{1}, t ∈ ℝ^{+}, we have

\frac{T_{i}(u_{1}, u_{2})(t)}{1 + t} = ∫_{0}^{+∞} \frac{H_{i}(t, s)}{1 + t}f_{i}(s, u_{1}(s), u_{2}(s))ds

≤ ∫_{0}^{+∞} \frac{H_{i}(t, s)}{1 + t}a_{i}(s)h_{i}(u_{1}(s), u_{2}(s))ds

≤ m_{i}(∥u_{1}∥_{1} + ∥u_{2}∥_{1}) ∫_{0}^{+∞} \frac{H_{i}(t, s)}{1 + t}(1 + s)a_{i}(s)ds

≤ m_{i}n_{i}(∥u_{1}∥_{1} + ∥u_{2}∥_{1}).

So,

∥T (u_{1}, u_{2})∥ = ∥T_{1}(u_{1}, u_{2})∥_{1} + ∥T_{2}(u_{1}, u_{2})∥_{1}
≤ m_{1}n_{1}(∥u_{1}∥_{1} + ∥u_{2}∥_{1}) + m_{2}n_{2}(∥u_{1}∥_{1} + ∥u_{2}∥_{1})
≤ ∥u_{1}∥_{1} + ∥u_{2}∥_{1} = ∥(u_{1}, u_{2})∥.

Therefore

∥T (u_{1}, u_{2})∥ ≤ ∥(u_{1}, u_{2})∥, (u_{1}, u_{2}) ∈ ∂Ω_{1}.  (3.1)

For M_{i} < f_{i}^{∞} ≤ +∞, there exists δ_{2} > 1 such that

f_{i}(t, u_{1}, u_{2}) ≥ M_{i}(u_{1} + u_{2}), t ∈ [δ, \frac{1}{δ}], u_{1} + u_{2} > δδ_{2}.

Let

Ω_{2} = {(u_{1}, u_{2}) ∈ K : ∥(u_{1}, u_{2})∥ < δ_{2}}.

For (u_{1}, u_{2}) ∈ \overline{Ω}_{2} ⊂ K, by Lemma 2.4, we have

min_{δ≤t≤ \frac{1}{δ}} u_{i}(t) ≥ δ∥u_{i}∥_{1}, i = 1, 2.

Hence,

min_{δ≤t≤ \frac{1}{δ}}(u_{1}(t) + u_{2}(t)) ≥ δ(∥u_{1}∥_{1} + ∥u_{2}∥_{1}).

So for (u_{1}, u_{2}) ∈ ∂Ω_{2} and t ∈ ℝ^{+}, we get

\frac{T_{i}(u_{1}, u_{2})(t)}{1 + t} ≥ ∫δ^{\frac{1}{δ}} \frac{H_{i}(t, s)}{1 + t}f_{i}(s, u_{1}(s), u_{2}(s))ds

≥ δM_{i}(∥u_{1}∥_{1} + ∥u_{2}∥_{1}) ∫δ^{\frac{1}{δ}} \frac{H_{i}(t, s)}{1 + t}ds

≥ δM_{i}N_{i}(∥u_{1}∥_{1} + ∥u_{2}∥_{1}).

We have

∥T (u_{1}, u_{2})∥ = ∥T_{1}(u_{1}, u_{2})∥_{1} + ∥T_{2}(u_{1}, u_{2})∥_{1}
≥ δM_{1}N_{1}(∥u_{1}∥_{1} + ∥u_{2}∥_{1}) + δM_{2}N_{2}(∥u_{1}∥_{1} + ∥u_{2}∥_{1})
≥ (∥u_{1}∥_{1} + ∥u_{2}∥_{1}) ≥ ∥(u_{1}, u_{2})∥.

Therefore

∥T (u_{1}, u_{2})∥ ≥ ∥(u_{1}, u_{2})∥, (u_{1}, u_{2}) ∈ ∂Ω_{2}.  (3.2)

From (3.1) and (3.2) and Theorem 2.1, we know that T has at least one positive solution in K ∩ (\overline{Ω}_{2} \ Ω_{1}). Therefore the boundary value problem (1.1) has at least one positive solution.

The theorem is proved.

Theorem 3.2 Suppose that hypotheses (H1) − (H3) hold and satisfy

(1) there exist positive numbers δ, p_{i}, P_{i}, 0 < δ < 1, such that 0 ≤ h_{i}^{∞} < p_{i}, P_{i} < f_{i}^{0} ≤ +∞;

(2) p_{1}n_{1} + p_{2}n_{2} ≤ 1, P_{1}N_{1}δ + P_{2}N_{2}δ ≥ 1.

Then, the boundary value problem (1.1) has at least one positive solution.

Proof. For P_{i} < f_{i}^{0} ≤ +∞, there exists a δ_{3} > 0 (δ_{3} < 1), such that

f_{i}(t, u_{1}, u_{2}) ≥ P_{i}(u_{1} + u_{2}), t ∈ [δ, \frac{1}{δ}], 0 < u_{1} + u_{2} < δ_{3}.

Let

Ω_{3} = {(u_{1}, u_{2}) ∈ K : ∥(u_{1}, u_{2})∥ < δ_{3}}.

For (u_{1}, u_{2}) ∈ \overline{Ω}_{3} ⊂ K, from Lemma 2.4 we have

min_{δ≤t≤ \frac{1}{δ}} u_{i}(t) ≥ δ∥u_{i}∥_{1}, i = 1, 2.

Hence,

min_{δ≤t≤ \frac{1}{δ}}(u_{1}(t) + u_{2}(t)) ≥ δ(∥u_{1}∥_{1} + ∥u_{2}∥_{1}).

For (u_{1}, u_{2}) ∈ ∂Ω_{3} and t ∈ ℝ^{+},

\frac{T_{i}(u_{1}, u_{2})(t)}{1 + t} ≥ ∫δ^{\frac{1}{δ}} \frac{H_{i}(t, s)}{1 + t}f_{i}(s, u_{1}(s), u_{2}(s))ds

≥ δP_{i}(∥u_{1}∥_{1} + ∥u_{2}∥_{1}) ∫δ^{\frac{1}{δ}} \frac{H_{i}(t, s)}{1 + t}ds

≥ δP_{i}N_{i}(∥u_{1}∥_{1} + ∥u_{2}∥_{1}).

We can get

∥T (u_{1}, u_{2})∥ = ∥T_{1}(u_{1}, u_{2})∥_{1} + ∥T_{2}(u_{1}, u_{2})∥_{1}
≥ δP_{1}N_{1}(∥u_{1}∥_{1} + ∥u_{2}∥_{1}) + δP_{2}N_{2}(∥u_{1}∥_{1} + ∥u_{2}∥_{1})
≥ (∥u_{1}∥_{1} + ∥u_{2}∥_{1}) ≥ ∥(u_{1}, u_{2})∥.

Therefore

∥T (u_{1}, u_{2})∥ ≥ ∥(u_{1}, u_{2})∥, (u_{1}, u_{2}) ∈ ∂Ω_{3}.  (3.3)

For 0 ≤ h_{i}^{∞} < p_{i}, there exists a R_{0} > 0, such that

h_{i}(u_{1}, u_{2}) ≤ p_{i}(∥u_{1}∥_{1} + ∥u_{2}∥_{1}), ∥u_{1}∥_{1} + ∥u_{2}∥_{1} > R_{0}.

Denote q_{i} = max_{0≤∥u_{1}∥_{1}+∥u_{2}∥_{1}≤R_{0}} h_{i}(u_{1}, u_{2}), then we have

h_{i}(u_{1}, u_{2}) ≤ q_{i} + p_{i}(∥u_{1}∥_{1} + ∥u_{2}∥_{1}), (u_{1}, u_{2}) ∈ X.

There exists a δ_{4} (δ_{4} > max{1, δ_{3}, (q_{1}n_{1} + q_{2}n_{2})(1 − p_{1}n_{1} − p_{2}n_{2})^{−1}}).

Let

Ω_{4} = {(u_{1}, u_{2}) ∈ K : ∥(u_{1}, u_{2})∥ < δ_{4}}.

For any (u_{1}, u_{2}) ∈ ∂Ω_{4} and t ∈ ℝ^{+}, we have

\frac{T_{i}(u_{1}, u_{2})(t)}{1 + t} ≤ ∫_{0}^{+∞} \frac{H_{i}(t, s)}{1 + t}a_{i}(s)h_{i}(u_{1}(s), u_{2}(s))ds

≤ ∫_{0}^{+∞} \frac{H_{i}(t, s)}{1 + t}a_{i}(s)(q_{i} + p_{i}(u_{1} + u_{2}))ds

< p_{i}(∥u_{1}∥_{1} + ∥u_{2}∥_{1}) ∫_{0}^{+∞} \frac{H_{i}(t, s)}{1 + t}(1 + s)a_{i}(s)ds

+ q_{i} ∫_{0}^{+∞} \frac{H_{i}(t, s)}{1 + t}(1 + s)a_{i}(s)ds

≤ p_{i}n_{i}(∥u_{1}∥_{1} + ∥u_{2}∥_{1}) + q_{i}n_{i}.

So,

∥T (u_{1}, u_{2})∥ = ∥T_{1}(u_{1}, u_{2})∥_{1} + ∥T_{2}(u_{1}, u_{2})∥_{1}
≤ (p_{1}n_{1} + p_{2}n_{2})(∥u_{1}∥_{1} + ∥u_{2}∥_{1}) + q_{1}n_{1} + q_{2}n_{2}
≤ (∥u_{1}∥_{1} + ∥u_{2}∥_{1}) = ∥(u_{1}, u_{2})∥.

Therefore

∥T (u_{1}, u_{2})∥ ≤ ∥(u_{1}, u_{2})∥, (u_{1}, u_{2}) ∈ ∂Ω_{4}.  (3.4)

From (3.3) and (3.4) and Theorem 2.1, we know T has at least one positive solution at K ∩ (\overline{Ω}_{4} \ Ω_{3}). Therefore the boundary value problem (1.1) has at least one positive solution.

The theorem is proved.

4 Illustration

We give an example to illustrate our results. Example Consider the following the boundary value problem:

u′′_{1}(t) + (u_{1} + u_{2})^{2}e^{−t} = 0,
u′′_{2}(t) + (u_{1} + u_{2})^{\frac{3}{2}} e^{−t} = 0,
u_{1}(0) = u_{2}(0) = 0,
u′_{1}(∞) = ∫_{0}^{+∞} e^{−2s}u_{1}(s)ds, u′_{2}(∞) = ∫_{0}^{+∞} e^{−2s}u_{2}(s)ds.

In this example,

f_{1}(t, u_{1}, u_{2}) = (u_{1}+u_{2})^{2}e^{−t}, f_{2}(t, u_{1}, u_{2}) = (u_{1}+u_{2})^{\frac{3}{2}} e^{−t}, g_{1}(s) = g_{1}(s) = e^{−2s},

a_{1}(t) = a_{2}(t) = e^{−t}, h_{1}(u_{1}, u_{2}) = (u_{1} + u_{2})^{2}, h_{2}(u_{1}, u_{2}) = (u_{1} + u_{2})^{\frac{3}{2}} .

For the constants r_{1}, r_{2} > 0, we take ϕ_{r_{1},r_{2}}(t) = (1 + r_{1} + r_{2})^{2}(1 + t)^{2}e^{−t}. Let δ = \frac{1}{2}.

Since n_{i} ≤ \frac{68}{27} and N_{i} ≥ \frac{1}{9}, we take m_{i} = \frac{1}{6}, M_{i} = 10. Then the conditions in Theorem 3.1 are all satisfied. By Theorem 3.1, the boundary value problem mentioned in the example above has at least one positive solution.

Acknowledgement We are grateful to Professor J. R.L. Webb for his recommendations of many useful and helpful references and his help with this paper. Also we appreciate the referee’s valuable comments and suggestions.

References

[1] C. P. Gupta, S. K. Ntouyas and P. Ch. Tsamatos; Solvability of an m-point boundary-value problem for second-order ordinary differential equations, J. Math Anal.Appl. 189 (1995) 575-584.

[2] Ruyun Ma; Positive solutions for a nonlinear three-point boundary value problem, Electron. J. Differential Equations 1999, No. 34., 8pp.

[3] Ruyun Ma; A survey on nonlocal boundary value problems, Appl. Math. E-notes 7 (2007) 257-279.

[4] Ruyun Ma; Positive solutions for second order functional differential equations, Dynam. Systems Appl. 10 (2001), 215-23.

[5] G. L. Karakostas, P. Ch. Tsamatos; Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems, Electron. J. Differential Equations 2002, No. 30, 17pp.

[6] G. L. Karakostas, P. Ch. Tsamatos; Nonlocal boundary vector value problems for ordinary differential systems of higher order, Nonlinear Anal., 51 (2002), 1421-1427.

[7] G. L. Karakostas, P. Ch. Tsamatos; Positive solutions for a nonlocal boundary-value problem with increasing response, Electron. J. Differential Equations 2000, No. 73, 8pp.

[8] G. L. Karakostas, P. Ch. Tsamatos; Existence of multiple solutions for a nonlocal boundary value problem, Topol. Methods Nonlinear Anal. 19 (2002), 109-121.

[9] Abdelkader Boucherif; Second-order boundary value problems with integral boundary conditions, Nonlinear Anal. 70 (2009), 364-371.

[10] Zhilin Yang; Existence and nonexistence results for positive solutions of an integral boundary value problem, Nonlinear Anal. 65 (2006), 1489-1511.

[11] Zhilin Yang; Existence of nontrivial solutions for a nonlinear Sturm-Liouville problem with integral boundary conditions, Nonlinear Anal. 68 (2008), 216-225.

[12] Zhilin Yang; Positive solutions of a second-order integral boundary value problem, J. Math. Anal. Appl. 321 (2006), 751-765.

[13] Zhilin Yang, Donal O’Regan; Positive solvability of systems of nonlinear Hammerstein integral equations, J. Math. Anal. Appl. 311 (2005), 600-614.

[14] Zhilin Yang; Positive solutions to a system of second-order nonlocal boundary value problems, Nonlinear Anal. 62 (2005), 1251-1265.

[15] J. R. L Webb, Gennaro Infante; Positive solutions of nonlocal boundary value problems involving integral conditions, NoDEA Nonlinear Differential Equations Appl. 15 (2008), 45-67.

[16] J. R. L Webb, Gennaro Infante; Positive solutions of nonlocal boundary value problems: a unified approach, J. London Math. Soc. 74 (2006), 673-693.

[17] Meiqiang Feng, Dehong Ji, Weigao Ge; Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces, J. Comput. Appl. Math. 222 (2008), 561-573.

[18] R. P. Agarwal, O. G. Mustafa, Yu. V. Rogovchenko; Existence and asymptotic behavior of solutions of a boundary value problem on an infinite interval, Math. Comput. Modelling 41 (2005), 135-157.

[19] Yansheng Liu; Boundary value problems for second order differential equations on unbounded domains in a Banach space, Applied Math. Comput. 135 (2003), 569-583.

[20] Hairong Lian, Peiguang Wang, Weigao Ge; Unbounded upper and lower solutions method for Sturm-Liouville boundary value problem on infinite intervals, Nonlinear Anal. 70 (2009), 2627-2633.

[21] Yansheng Liu; Existence and unboundedness of positive solutions for singular boundary value problems on half-line, Applied Math. Comput. 144 (2003), 543-556.

[22] Baoqiang Yan, Donal O’Regan, Ravi Agarwal; Unbounded solutions for singular boundary value problems on the semi-infinite interval: Upper and lower solutions and multiplicity, J. Comput. Appl. Math. 197 (2006), 365-386.

[23] Ruyun Ma; Existence of positive solutions for second-order boundary value problems on infinity intervals, Applied Math. Lett. 16 (2003), 33-39.

[24] Hairong Lian, Weigao Ge; Existence of positive solutions for Sturm-Liouville boundary value problems on the half-line, J. Math. Anal. Appl. 321 (2006), 781-792.

[25] Yan Sun, Yongping Sun, Lokenath Debnath; On the existence of positive solutions for singular boundary value problems on the half-line, Applied Math. Lett. 22 (2009), 806-812

[26] Yu Tian, Weigao Ge, Wenrui Shan; Positive solution for three-point boundary-value problem on the half-line, Comput. Math. Appl. 53 (2007), 1029–1039.

[27] Yu Tian, Weigao Ge; Positive solutions for multi-point boundary value problem on the half-line, J. Math. Anal. Appl. 325 (2007), 1339-1349.

[28] D. Guo, V. Lakshmikantham; Nonlinear problems in abstract cones, Academic press, 1988.

[29] C. Corduneanu; Integral equations and stability of feedback systems, Academic press, New York, 1973.

(Received March 23, 2009)