Abstract
This work is devoted to the study of nthorder ordinary differential equations on a halfline with SturmLiouville 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; halfline; upper solution; lower solution1 Introduction
In this paper, we study nthorder ordinary differential equations on a halfline,
together with the SturmLiouville boundary conditions
where , are continuous, , , , .
Higherorder boundary value problems (BVPs) have been studied in many papers, such as [13] for twopoint BVP, [4,5] for multipoint BVP, and [69] 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 [1,2,5,1015] assumed one pair of wellordered 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, Blasiustype 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 semiinfinite 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 [613,1724] 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 secondorder SturmLiouville boundary value problem:
where , . 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 , . 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 nthorder differential equation on a halfline with SturmLiouville boundary conditions. To the best of our knowledge, this is the first attempt to find the unbounded solutions to higherorder infinite interval problems by using the upper and lower solution technique. Since, the halfline 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 Nagumotype condition. Then, by using the truncation technique and the upper and lower solutions, we estimate apriori 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.
2 Preliminaries
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
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.
Lemma 2.1Let. Then the linear boundary value problem
has a unique solution given by
where
and
Proof Let . Then from (6), we obtain the SturmLiouville 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 2.2The functiondefined in (8) istimes continuously differentiable on. For any, itsith derivative is uniformly continuous inton any compact interval ofand is uniformly bounded on.
Proof We denote , which is also used in the later part of the paper. By direct calculations, we have
Obviously, is uniformly continuous in t on any compact interval of for any . Now since, for all integers k and l,
from (11), when , it follows that
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 ArezàAscoli lemma fails here due to the noncompactness of , the following generalization (see [6,13]) will be used.
Lemma 2.3is relatively compact if the following conditions hold:
1. all the functions fromMare uniformly bounded;
2. all the functions fromMare equicontinuous on any compact interval of;
3. all the functions fromMare euqiconvergent at infinity, that is, for any given, there exists asuch that for any,
Finally, we define lower and upper solutions of (1)(2), and introduce the Nagumotype condition.
Definition 2.1 A function 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 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 . We say f satisfies a Nagumo condition with respect to α and β if there exist positive functions ψ and such that
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.
H_{1}: BVP (1)(2) has a pair of upper and lower solutions β, α in X with
and satisfies the Nagumo condition with respect to α and β.
H_{2}: For any fixed , , when , , the following inequality holds:
H_{3}: There exists a constant such that
where ψ is the function in Nagumo’s condition of f.
Lemma 3.1Suppose conditions (H_{1}) and (H_{3}) hold. Then there exists a constantsuch that every solutionuof (1)(2) with
Proof Set
where is any arbitrary constant. Choose where C is the nonhomogeneous boundary data, and satisfies
If holds for any , then the result follows immediately. If not, we claim that does not hold for all . Otherwise, without loss of generality, we suppose
But then, for any , it follows that
which is a contraction. So there must exist such that . Furthermore, if
just take and then the proof is completed. Finally, there exist such that , , or , , . Suppose that , , . Obviously,
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 3.1 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.
Theorem 3.2Suppose the conditions (H_{1})(H_{3}) hold. Then BVP (1)(2) has at least one solutionsatisfying
Moreover, there exists asuch that.
Proof Let be the same as in Lemma 3.1. Define the auxiliary functions , and as
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 . Set , then
Obviously, we have . By the boundary conditions, it follows that
which is a contradiction.
Case II. There exists a such that .
Clearly, we have
On the other hand,
Subcase i. If , from the definition of , we obtain
Subcase ii. If , from the conditions (H_{2}), we find
Similarly following the above argument, we could discuss the other two cases or , , and we have the following inequality:
Thus,
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.
Consider the operator 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 is completely continuous.
For any , by direct calculation, we find
where , the Lebesgue dominated convergent theorem implies that
For any convergent sequence in X, there exists such that . Thus, as in (21), we have
For this it suffices to show that T maps bounded subsets of X into relatively compact sets. Let B be any bounded subset of X, then there exists such that , . For any , , we have
where
where , and thus TB is uniformly bounded. Further, for any , if , we have
that is, TB is equicontinuous. From Lemma 2.3, it follows that if TB is equiconvergent at infinity, then TB is relatively compact. In fact, we have
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.
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.
H_{4}: BVP (1)(2) has two pairs of upper and lower solution, , inXwith, strict, and
for, , andsatisfies the Nagumo condition with respect toand.
Suppose further that conditions (H_{2}) and (H_{3}) hold withαandβreplaced byand, respectively. Then the problem (1)(2) has at least three solutions, , andsatisfying
Proof Define the truncated function , the same as F in Theorem 3.2 with α replaced by and β by , respectively. Consider the modified differential equation
with boundary conditions (2). Similarly to Theorem 3.2, it suffices to show that problem (22)(2) has at least three solutions. To this end, define the mapping as follows:
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).
Let
where and are defined as above, and . Set . Then for any , it follows that
and hence , which implies that .
Set
Now since , are strict lower and upper solutions, there is no solution in . Therefore
Next we will show that
For this, we define another mapping by
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,
Finally, using the properties of the degree, we conclude that has at least three fixed points , , and . □
5 An example
Example Consider the secondorder differential equation with SturmLiouville boundary conditions
Clearly, BVP (23) is a particular case of problem (1)(2) with
We let
Then , and , , , . Moreover, we have
and
Thus, and are strict lower solutions of problem (23).
Now we take
and
Thus, and are strict upper solutions of problem (23). Further, it follows that
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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgements
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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