This work is devoted to the study of nth-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.
Keywords:higher order; boundary value problem; half-line; upper solution; lower solution
In this paper, we study nth-order ordinary differential equations on a half-line,
together with the Sturm-Liouville boundary conditions
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 . 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 , Agarwal and O’Regan discussed the following second-order Sturm-Liouville boundary value problem:
Eloe et al. 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. developed the upper and lower solution theory for the boundary value problem
where , . 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 . In 2012, Zhao et al. 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 nth-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
has a unique solution given by
and the initial value problem
Clearly, (9) has a unique solution
Now integrating (10) and applying the initial conditions, we obtain (7). □
Thus, we have
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 , the following generalization (see [6,13]) will be used.
1. all the functions fromMare uniformly bounded;
Finally, we define lower and upper solutions of (1)-(2), and introduce the Nagumo-type condition.
is called a lower solution of (1)-(2). If the inequalities are strict, it is called a strict lower solution.
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
3 The existence results
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
where ψ is the function in Nagumo’s condition of f.
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 th derivative of the solution of BVP (1)-(2). Since the upper and lower solutions are in X, and may be asymptotic linearly at infinity.
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
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).
which is a contradiction.
Clearly, we have
On the other hand,
which contradicts (19).
Thus, , . Similarly, we can show that , . 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.
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 is completely continuous.
that is, TB is equi-continuous. From Lemma 2.3, it follows that if TB is equi-convergent at infinity, then TB is relatively compact. In fact, we have
Remark 3.3 The upper and lower solutions are more strict at infinity than usually assumed. For such right boundary conditions, it is not easy to estimate the sign of even though we have , , where or . It remains unsettled whether this strict inequality can be weakened.
4 The multiplicity results
In this section assuming two pairs of upper and lower solutions, we shall prove the existence of at least three solutions for our infinite interval problem.
Theorem 4.1Suppose that the following condition holds.
Next we will show that
where the function is similar to except changing to . Similar to the proof of Theorem 3.2, we find that u is a fixed point of only if . So . From the Schäuder fixed point theorem and , we have . Furthermore,
5 An example
Example Consider the second-order differential equation with Sturm-Liouville boundary conditions
Clearly, BVP (23) is a particular case of problem (1)-(2) with
Now we take
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
This research is supported by the National Natural Science Foundation of China (No. 11101385) and by the Fundamental Research Funds for the Central Universities.
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