- Open Access
Upper and lower solution method for nth-order BVPs on an infinite interval
© Lian et al.; licensee Springer. 2014
- Received: 9 January 2014
- Accepted: 7 April 2014
- Published: 7 May 2014
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.
- higher order
- boundary value problem
- upper solution
- lower solution
where , are continuous, , , , .
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.
where , . They established existence criteria by using a diagonalization argument and existence results of appropriate boundary value problems on finite intervals.
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.
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 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.
Then is a Banach space.
To obtain a solution of the BVP (1)-(2), we need a mapping whose kernel is the Green function of with the homogeneous boundary conditions (2), which is given in the following lemma.
Now integrating (10) and applying the initial conditions, we obtain (7). □
Lemma 2.2 The function defined in (8) is times continuously differentiable on . For any , its ith derivative is uniformly continuous in t on any compact interval of and is uniformly bounded on .
for . □
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.
all the functions from M are uniformly bounded;
all the functions from M are equi-continuous on any compact interval of ;
- 3.all the functions from M are euqi-convergent at infinity, that is, for any given , there exists a such that for any ,
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.
Our existence theory is based on using the unbounded lower and upper solution technique. Here we list some assumptions for convenience.
and satisfies the Nagumo condition with respect to α and β.
where ψ is the function in Nagumo’s condition of f.
from which one concludes that . Since and are arbitrary, we have if for . In a similarly way, we can show that , if for .
Therefore there exists a , just related with α, β, and ψ, h under the Nagumo condition of f, such 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.
Moreover, there exists a such that .
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).
Case I. .
which is a contradiction.
Case II. There exists a such that .
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.
(1) is well defined.
(2) is continuous.
and hence is continuous.
(3) is compact.
and, therefore, is completely continuous. The Schäuder fixed point theorem now ensures that the operator T has a fixed point, which is a solution of the BVP (1)-(2). □
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.
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 4.1 Suppose that the following condition holds.
for , , and satisfies the Nagumo condition with respect to and .
for , .
Clearly, is completely continuous. By using the degree theory, we will show that has at least three fixed points which coincide with the solutions of (22)-(2).
and hence , which implies that .
Finally, using the properties of the degree, we conclude that has at least three fixed points , , and . □
Thus, and are strict lower solutions of problem (23).
Moreover, for every , we find that is bounded. Finally, take , . Hence, all conditions in Theorem 4.1 are satisfied and therefore problem (23) has at least three solutions.
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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