Symmetric positive solutions of higher-order boundary value problems
© Luo; licensee Springer. 2014
Received: 31 October 2013
Accepted: 20 March 2014
Published: 3 April 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.
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.
Remark 2.1 The set P is not a cone as it is not closed.
Throughout this paper, we assume the following:
It is easy to see that the function f satisfies assumptions (H1) and (H2). In fact, if , there exists constant λ with such that .
Now, we present several lemmas that will be used in the proof of our results. By routine calculations we have the following results.
where are defined by (2.1).
where , .
3 Main results
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.
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.
In fact, since . So, from (3.5) and noting that , . From (3.6), we have and .
Since and T is nondecreasing, by induction, (3.8) holds.
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.
From (3.15), (3.17), (3.18), and noting that is nondecreasing, by induction, (3.12) holds.
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. □
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.
Research supported by the Scientific Research Fund of Hunan Provincial Education Department (13C319).
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