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Symmetric positive solutions of higher-order boundary value problems
Boundary Value Problems volume 2014, Article number: 78 (2014)
We study the higher-order boundary value problems. The existence of symmetric positive solutions of the problem is discussed. Our results extend some recent work in the literature. The analysis of this paper mainly relies on the monotone iterative technique.
We study the boundary value problem (BVP)
where is an integer, is continuous, , are nonnegative constants, , . may be singular at , (and/or ). If a function is continuous and satisfies for , then we say that is symmetric on . By a symmetric positive solution of BVP (1.1) we mean a symmetric function such that for and satisfies (1.1).
In recent years, many authors have studied BVP (1.1), they only considered that f is nondecreasing or nonincreasing in u, or the boundary condition depends only on derivatives of even orders; see [1–8] and references cited therein. To the best of the author’s knowledge, there is no such results involving (1.1). In this note, we intend to fill in such gaps in the literature.
The organization of this paper is as follows. After this introduction, in Section 2, we state the assumptions and some preliminary lemmas. By applying the monotone iterative technique, we discuss the existence of symmetric positive solutions for (1.1) and obtain the main results in Section 3.
For convenience, in this paper we let , ,
where , and define
Remark 2.1 The set P is not a cone as it is not closed.
Throughout this paper, we assume the following:
(H1) , are nonnegative constants, , . is continuous and symmetric in t, i.e., f satisfies
(H2) For , is nondecreasing in u and there exists a constant such that if , then
(H2′) For , is nonincreasing in u and there exists a constant such that if , then
Example 2.1 Consider the equation
It is easy to see that the function f satisfies assumptions (H1) and (H2). In fact, if , there exists constant λ with such that .
Remark 2.2 It is easy to see that (H2) implies that if , then
and (H2′) implies that if , then
Now, we present several lemmas that will be used in the proof of our results. By routine calculations we have the following results.
Lemma 2.1 Let v be integrable on , then the BVP
has a unique solution
where are defined by (2.1).
Lemma 2.2 For any , we have
where , .
3 Main results
Define the operator by
where are defined by (2.1). It is clear that u is a solution of (1.1) if and only if u is a fixed point of T.
Theorem 3.1 Assume (H1)-(H3) hold. Then BVP (1.1) has at least one symmetric positive solution.
Claim 3.1 is completely continuous and nondecreasing.
In fact, for , it is obvious that , for and . (2.3), (2.9) and a change of variables imply
For any , from (2.4), (2.6), (2.8), and (H3), we have
for , where satisfies
Thus, it follows from (3.3) and (3.4) that , and so . Next by a standard method and the Ascoli-Arzela theorem one can prove that is completely continuous, we omit it here. From (H2), it is easy to see that T is nondecreasing in u. Hence, Claim 3.1 holds.
Claim 3.2 Let be fixed number satisfying
where λ is defined in (H2) in which , and assume
and there exists such that
In fact, since . So, from (3.5) and noting that , . From (3.6), we have and .
On the other hand, from (2.4) and (2.6), we have
Since and T is nondecreasing, by induction, (3.8) holds.
Let , then . It follows from
that, for any natural number n,
Thus, for all natural numbers n and p, we have
which implies that there exists such that (3.9) holds, and Claim 3.2 holds.
Letting in (3.7), we obtain , which is a symmetric positive solution of BVP (1.1), and this completes the proof of the theorem. □
Theorem 3.2 Assume (H1), (H2′) and (H3) hold. Then BVP (1.1) has at least one symmetric positive solution.
Claim 3.3 is completely continuous and nonincreasing.
The proof of Claim 3.3 is similar to the proof of Claim 3.1, so this is omitted.
Claim 3.4 Let be fixed number, be sufficiently large constant satisfying
where λ is defined in (H2′) in which , and assume
and there exists such that
In fact, since and . So from (3.11),
From (2.5), (3.10), (3.14), and noting that T is nonincreasing in u, we have
From (3.15), (3.17), (3.18), and noting that is nondecreasing, by induction, (3.12) holds.
On the other hand, from (2.5) and (2.7), for ,
Then from (3.16) and (3.19), we have
Therefore, for all natural numbers n and p, we have
From (3.20) and (3.21), there exists such that (3.13) holds, and Claim 3.4 holds.
Letting in (3.11), we obtain , which is a symmetric positive solution of BVP (1.1), and this completes the proof of the theorem. □
Example 3.1 Consider the BVP
where for , , , .
It is easy to see that function satisfies (H1) and (H3). If , there exists constant λ with such that , so (H2) is also satisfied. Therefore, from Theorem 3.1, (3.22) has at least one symmetric positive solution.
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Research supported by the Scientific Research Fund of Hunan Provincial Education Department (13C319).
The author declares that she has no competing interests.