Symmetric positive solutions of higher-order boundary value problems

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.

MSC:34B15, 34B18.

We study the boundary value problem (BVP)

where n\ge 1 is an integer, f:(0,1)\times (0,\mathrm{\infty })\to [0,\mathrm{\infty }) is continuous, {\alpha }_i, {\beta }_i are nonnegative constants, {\alpha }_i^2+2{\alpha }_i{\beta }_i>0, i=1,\dots ,n. f(t,u) may be singular at u=0, t=0 (and/or t=1). If a function u:[0,1]\to R is continuous and satisfies u(t)=u(1-t) for t\in [0,1], then we say that u(t) is symmetric on [0,1]. By a symmetric positive solution of BVP (1.1) we mean a symmetric function u\in C^{2n}[0,1] such that u(t)>0 for t\in (0,1) and u(t) 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 E=C^{(2n)}[0,1], {\mathrm{\Delta }}_i={\alpha }_i^2+2{\alpha }_i{\beta }_i,

where i=1,\dots ,n, 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) {\alpha }_i, {\beta }_i are nonnegative constants, {\alpha }_i^2+2{\alpha }_i{\beta }_i>0, i=1,\dots ,n. f:(0,1)\times (0,\mathrm{\infty })\to [0,\mathrm{\infty }) is continuous and symmetric in t, i.e., f satisfies

(H2) For (t,u)\in (0,1)\times (0,\mathrm{\infty }), f(t,u) is nondecreasing in u and there exists a constant \lambda \in (0,1) such that if \sigma \in (0,1], then

(H2′) For (t,u)\in (0,1)\times (0,\mathrm{\infty }), f(t,u) is nonincreasing in u and there exists a constant \lambda \in (0,1) such that if \sigma \in (0,1], then

(H3) .

Example 2.1 Consider the equation

It is easy to see that the function f satisfies assumptions (H1) and (H2). In fact, if \sigma \in (0,1], there exists constant λ with such that f(t,\sigma u)\ge {\sigma }^{\lambda }f(t,u).

Remark 2.2 It is easy to see that (H2) implies that if \sigma \in [1,\mathrm{\infty }), then

and (H2′) implies that if \sigma \in [1,\mathrm{\infty }), 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 (0,1), then the BVP

has a unique solution

where G_i(t,s) are defined by (2.1).

Lemma 2.2 For any t,s\in [0,1], we have

where {\epsilon }_i=\frac{{\mathrm{\Delta }}_i}{{({\alpha }_i+{\beta }_i)}^2}, i=1,\dots ,n.

Define the operator T:E\to E by

where G_i(t,s) 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.

Proof

Claim 3.1 T:P\to P is completely continuous and nondecreasing.

In fact, for u\in P, it is obvious that Tu\in E, Tu(t)>0 for t\in (0,1) and Tu(0)\ge 0. (2.3), (2.9) and a change of variables imply

For any u\in P, from (2.4), (2.6), (2.8), and (H3), we have

for t\in [0,1], where l_{Tu} satisfies

Thus, it follows from (3.3) and (3.4) that Tu(t)\in P, and so T:P\to P. Next by a standard method and the Ascoli-Arzela theorem one can prove that T:P\to P 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 \sigma =\delta, and assume

Then

and there exists u^{\ast }\in P such that

In fact, since T{e}_1\in P. So, from (3.5) and noting that , . From (3.6), we have u_0,v_0\in P and u_0\le v_0.

On the other hand, from (2.4) and (2.6), we have

Since u_0\le v_0 and T is nondecreasing, by induction, (3.8) holds.

Let \varsigma ={\delta }^2, 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 u^{\ast }\in P such that (3.9) holds, and Claim 3.2 holds.

Letting n\to \mathrm{\infty } in (3.7), we obtain u^{\ast }(t)=T{u}^{\ast }(t), 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.

Proof

Claim 3.3 T:P\to P 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, \eta >1 be sufficiently large constant satisfying

where λ is defined in (H2′) in which \sigma ={\xi }^{\eta }, and assume

Then

and there exists u^{\ast }\in P such that

In fact, since and \eta >1. So from (3.11),

From (2.5), (3.10), (3.14), and noting that T is nonincreasing in u, we have

Therefore,

From (3.15), (3.17), (3.18), and noting that T^2 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

and thus

Therefore, for all natural numbers n and p, we have

From (3.20) and (3.21), there exists u^{\ast }\in P such that (3.13) holds, and Claim 3.4 holds.

Letting n\to \mathrm{\infty } in (3.11), we obtain u^{\ast }(t)=T{u}^{\ast }(t), which is a symmetric positive solution of BVP (1.1), and this completes the proof of the theorem.  □

Remark 3.1 [3, 5] only considered that f is nondecreasing or nonincreasing in u, and {\alpha }_i=1, {\beta }_i=0 in (1.1), so our results extend the work in the literature.

Example 3.1 Consider the BVP

where f(t,u)=u^{\alpha }sin\pi t for (t,u)\in (0,1)\times (0,\mathrm{\infty }), , {\alpha }_i={\beta }_i=1, i=1,2,3.

It is easy to see that function f(t,u) satisfies (H1) and (H3). If \sigma \in (0,1], there exists constant λ with such that f(t,\sigma u)\ge {\sigma }^{\lambda }f(t,u), 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).

Affiliations

1. Department of Mathematics, Hunan University of Science and Technology, Xiangtan, Hunan, 411201, China

   • Yan Luo

Corresponding author

Correspondence to Yan Luo.

Competing interests

The author declares that she has no competing interests.

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