Symmetric positive solutions of higher-order boundary value problems

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.

Remark 2.1 The set P is not a cone as it is not closed.

Throughout this paper, we assume the following:

(H3) $0<{\int }_0^{1}f(t,e_1(t))dt<\mathrm{\infty }$.

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 $\frac{1}{4}\le \lambda <1$ such that $f(t,\sigma u)\ge {\sigma }^{\lambda }f(t,u)$.

Now, we present several lemmas that will be used in the proof of our results. By routine calculations we have the following results.

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

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

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.

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.

In fact, $0<l_{T{e}_1}<1$ since $T{e}_1\in P$. So, from (3.5) and noting that $0<\lambda <1$, $0<\delta <1$. From (3.6), we have $u_0,v_0\in P$ and $u_0\le v_0$.

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

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.

From (3.15), (3.17), (3.18), and noting that $T^2$ is nondecreasing, by induction, (3.12) holds.

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.

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

It is easy to see that function $f(t,u)$ satisfies (H1) and (H3). If $\sigma \in (0,1]$, there exists constant λ with $0<\alpha \le \lambda <1$ 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.