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

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

MSC:34B15, 34B40.

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

together with the Sturm-Liouville boundary conditions

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

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

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

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

where a_0>0, b_0⩾0. They established existence criteria by using a diagonalization argument and existence results of appropriate boundary value problems on finite intervals.

Eloe et al. [11] studied the BVP

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

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

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

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

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

Consider the space X defined by

with the norm \parallel \cdot \parallel given by

where v_i(t)=1+t^{n-1-i} and

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

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

Lemma 2.1 Let e\in L^1[0,+\mathrm{\infty }). Then the linear boundary value problem

has a unique solution given by

where

and

Proof Let v(t)=u^{(n-2)}(t). Then from (6), we obtain the Sturm-Liouville boundary value problem

and the initial value problem

Clearly, (9) has a unique solution

where

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

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

Proof We denote g_i(t,s)=\frac{{\partial }^{i}G(t,s)}{\partial t^i}, which is also used in the later part of the paper. By direct calculations, we have

Obviously, g_i(t,s) is uniformly continuous in t on any compact interval of [0,+\mathrm{\infty }) for any s\in [0,+\mathrm{\infty }). Now since, for all integers k and l,

from (11), when s⩽t, it follows that

and, when s⩾t, we have

Thus, we have

for i=1,2,\dots ,n-1. □

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

Lemma 2.3 M\subset X is relatively compact if the following conditions hold:

1. 1.

   all the functions from M are uniformly bounded;

2. 2.

   all the functions from M are equi-continuous on any compact interval of [0,+\mathrm{\infty });

3. 3.

   all the functions from M are euqi-convergent at infinity, that is, for any given ϵ>0, there exists a T=T(ϵ)>0 such that for any u\in M,

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

Definition 2.1 A function \alpha (t)\in C^{n-1}[0,+\mathrm{\infty })\cap C^n(0,+\mathrm{\infty }) satisfying

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

Definition 2.2 A function \beta (t)\in C^{n-1}[0,+\mathrm{\infty })\cap C^n(0,+\mathrm{\infty }) satisfying

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

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

on [0,+\mathrm{\infty }). We say f satisfies a Nagumo condition with respect to α and β if there exist positive functions ψ and h\in C[0,+\mathrm{\infty }) such that

for all (t,u_0,u_1,\dots ,u_{n-1})\in [0,+\mathrm{\infty })\times [\alpha (t),\beta (t)]\times \cdots \times [{\alpha }^{(n-2)}(t),{\beta }^{(n-2)}(t)]\times \mathbb{R} and

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

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

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

H₂: For any fixed t\in [0,+\mathrm{\infty }), u_{n-2},u_{n-1}\in \mathbb{R}, when {\alpha }^{(i)}(t)⩽u_i⩽{\beta }^{(i)}(t), i=0,1,\dots ,n-3, the following inequality holds:

H₃: There exists a constant \gamma >1 such that

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

Lemma 3.1 Suppose conditions (H₁) and (H₃) hold. Then there exists a constant R>0 such that every solution u of (1)-(2) with

satisfies {\parallel u\parallel }_{n-1}⩽R.

Proof Set

where \delta >0 is any arbitrary constant. Choose R>C where C is the nonhomogeneous boundary data, and \eta ⩾M_3 satisfies

If |u^{(n-1)}(t)|⩽R holds for any t\in [0,+\mathrm{\infty }), then the result follows immediately. If not, we claim that |u^{(n-1)}(t)|>\eta does not hold for all t\in [0,+\mathrm{\infty }). Otherwise, without loss of generality, we suppose

But then, for any t⩾\delta >0, it follows that

which is a contraction. So there must exist t^{\ast }\in [0,+\mathrm{\infty }) such that |u^{(n-1)}(t^{\ast })|⩽\eta. Furthermore, if

just take R=\eta and then the proof is completed. Finally, there exist [t_1,t_2]\subset [0,+\mathrm{\infty }) such that |u^{(n-1)}(t_1)|=\eta, |u^{(n-1)}(t)|>\eta, t\in (t_1,t_2] or |u^{(n-1)}(t_2)|=\eta, |u^{(n-1)}(t)|>\eta, t\in [t_1,t_2). Suppose that u^{(n-1)}(t_1)=\eta, u^{(n-1)}(t)>\eta, t\in (t_1,t_2]. Obviously,

from which one concludes that u^{(n-1)}(t_2)⩽R. Since t_1 and t_2 are arbitrary, we have u^{(n-1)}(t)⩽R if u^{(n-1)}(t)⩾\eta for t\in [0,+\mathrm{\infty }). In a similarly way, we can show that u^{(n-1)}(t)⩾-R, if u^{(n-1)}(t)⩽-\eta for t\in [0,+\mathrm{\infty }).

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

Remark 3.1 Similarly, we can prove that

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

Theorem 3.2 Suppose the conditions (H₁)-(H₃) hold. Then BVP (1)-(2) has at least one solution u\in C^{n-1}[0,+\mathrm{\infty })\cap C^n(0,+\mathrm{\infty }) satisfying

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

Proof Let R>0 be the same as in Lemma 3.1. Define the auxiliary functions f_0,f_1,\dots ,f_{n-2}, and F:[0,+\mathrm{\infty })\times {\mathbb{R}}^n\to \mathbb{R} as

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

Consider the modified differential equation with the truncated function

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

and

We divide the proof into the following two steps.

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

Suppose the right hand inequality in (17) does not hold for i=n-2. Set \omega (t)=u^{(n-2)}(t)-{\beta }^{(n-2)}(t), then

Case I. .

Obviously, we have {\omega }^{\prime }(0^{+})⩽0. By the boundary conditions, it follows that

which is a contradiction.

Case II. There exists a t^{\ast }\in (0,+\mathrm{\infty }) such that .

Clearly, we have

On the other hand,

Subcase i. If u^{(n-3)}(t^{\ast })>{\beta }^{(n-3)}(t^{\ast }), from the definition of f_{n-3}, we obtain

Subcase ii. If u^{(n-3)}(t^{\ast })⩽{\beta }^{(n-3)}(t^{\ast }), from the conditions (H₂), we find

Similarly following the above argument, we could discuss the other two cases u^{(i)}(t^{\ast })>{\beta }^{(i)}(t^{\ast }) or u^{(i)}(t^{\ast })⩽{\beta }^{(i)}(t^{\ast }), i=n-4,n-5,\dots ,1,0, and we have the following inequality:

Thus,

which contradicts (19).

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

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

Consider the operator T:X\to X defined by

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

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

For any u\in X, by direct calculation, we find

Obviously, Tu\in C^{n-1}[0,+\mathrm{\infty }). Further, because

where H={max}_{0⩽s⩽\parallel u\parallel }h(s), the Lebesgue dominated convergent theorem implies that

Thus, Tu\in X.

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

For any convergent sequence u_m\to u in X, there exists r_1>0 such that {sup}_{m\in N}\parallel u_m\parallel ⩽r_1. Thus, as in (21), we have