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Positive solutions of higher-order Sturm–Liouville boundary value problems with fully nonlinear terms

Advances in Difference Equations20182018:184

https://doi.org/10.1186/s13662-018-1636-5

Received: 14 February 2018

Accepted: 7 May 2018

Published: 16 May 2018

Abstract

In this paper we consider the existence of positive solutions of nth-order Sturm–Liouville boundary value problems with fully nonlinear terms, in which the nonlinear term f involves all of the derivatives \(u',\ldots, u^{(n-1)}\) of the unknown function u. Such cases are seldom investigated in the literature. We present some inequality conditions guaranteeing the existence of positive solutions. Our inequality conditions allow that \(f(t, x_{0}, x_{1},\ldots, x_{n-1})\) is superlinear or sublinear growth on \(x_{0}, x_{1},\ldots, x_{n-1}\). Our discussion is based on the fixed point index theory in cones.

Keywords

  • Sturm–Liouville boundary value problem
  • Fully nonlinear term
  • Positive solution
  • Cone
  • Fixed point index

MSC

  • 34B18
  • 47H11
  • 47N20

1 Introduction

In this paper, we consider the existence of positive solutions of the nth-order Sturm–Liouville boundary value problem (BVP)
$$ \textstyle\begin{cases} u^{(n)}(t)+f(t, u(t), u'(t),\ldots, u^{(n-1)}(t))=0, \quad t\in [0, 1], \\ u^{(k)}(0)=0, \quad 0\le k\le n-3, \\ \alpha u^{(n-2)}(0)-\beta u^{(n-1)}(0)=0,\quad\quad \gamma u^{(n-2)}(1)+ \delta u^{(n-1)}(1)=0, \end{cases} $$
(1.1)
where \(n\ge 3\), \(f:[0, 1]\times {\mathbb{R}_{+}}^{n-1}\times {\mathbb{R}}\to \mathbb{R}_{+}\) is continuous, \(\mathbb{R}_{+}=[0, \infty )\), and α, β, γ and δ are constants and satisfy
$$ \beta \ge 0,\quad\quad \delta \ge 0,\quad\quad \alpha +\beta >0,\quad\quad \gamma + \delta >0, \quad\quad \rho :=\alpha \gamma +\alpha \delta +\beta \gamma >0, $$
(1.2)
which allow α and γ to be negative. This problem models various dynamic systems with n degrees of freedom in which n states are observed n times; see Meyer [1]. For some of the simple cases that the nonlinearity f does not contain a derivative term, the existence of positive solutions has been researched by many authors; see [211]. Zhou, Chu and Baleanu [12] studied a fractional differential equation boundary value problem and obtained existence results of positive solutions. For the cases of \(n=3\) or \(n=4\) and the nonlinearity f containing a derivative term \(u''\), the existence of positive solutions has also been discussed by some authors; see [1315]. Hajipour, Jajarmi and Baleanu [16] presented an accurate discretization method to solve some highly nonlinear boundary value problems. However, for the more general BVP (1.1) there are relatively few studies.
Wong [17] has considered the special case of BVP (1.1) that the nonlinearity f does not involve the derivative term \(u^{(n-1)}(t)\), namely the boundary value problem
$$ \textstyle\begin{cases} u^{(n)}(t)+f(t, u(t), u'(t),\ldots, u^{(n-2)}(t))=0, \quad t\in [0, 1], \\ u^{(k)}(0)=0, \quad 0\le k\le n-3, \\ \alpha u^{(n-2)}(0)-\beta u^{(n-1)}(0)=0,\quad\quad \gamma u^{(n-2)}(1)+ \delta u^{(n-1)}(1)=0, \end{cases} $$
(1.3)
and he has obtained the existence of a solution by assuming that BVP (1.3) has lower and upper solutions v and w such that \(v^{(n-2)}(t) \le w^{(n-2)}(t)\) on \([0, 1]\), and the nonlinearity f satisfies
$$\begin{aligned} f \bigl(t,v(t),v'(t),\ldots,v^{(n-3)}(t),x_{n-2} \bigr) \le &f(t,x_{0},x_{1},\ldots,x_{n-3},x_{n-2}) \\ \le& f \bigl(t,w(t),w'(t),\ldots,w^{(n-3)}(t),x_{n-2} \bigr), \end{aligned}$$
for any \((t,x_{0},x_{1},\ldots,x_{n-2})\in D\), where
$$D:=[0, 1]\times \bigl[v(t),w(t) \bigr]\times \bigl[v'(t),w'(t) \bigr]\times \cdots \times \bigl[v ^{(n-2)}(t), w^{(n-2)}(t) \bigr]. $$
Wong’s discussion is based on Schauder’s fixed point theorem and a truncating technique for the nonlinearity f. We note that, for BVP (1.3), there is a corresponding maximum principle.

Lemma 1.1

Let \(u\in C^{n}[0, 1]\) satisfy
$$ \textstyle\begin{cases} - u^{(n)}(t)\ge 0, \quad t\in [0, 1], \\ u^{(k)}(0)=0, \quad 0\le k\le n-3, \\ \alpha u^{(n-2)}(0)-\beta u^{(n-1)}(0)\ge 0,\quad\quad \gamma u^{(n-2)}(1)+ \delta u^{(n-1)}(1)\ge 0. \end{cases} $$
(1.4)
Then \(u^{(k)}(t)\ge 0\) for every \(t\in [0, 1]\), \(k=0, 1,\ldots, n-2\).
Therefore, Wong’s method is feasible to BVP (1.3). Since the maximum principle does not involve the derivative term \(u^{(n-1)}\), that is, Condition (1.4) does not entail that \(u^{(n-1)}(t)\ge 0\) on \([0, 1]\), this method is not applicable to BVP (1.1). A few years later, Grossinho and Minhós [18] developed Wong’s result and established the existence of a solution to the more general BVP (1.1) in the presence of lower and upper solutions. In the discussion, to obtain the estimate of the derivative \(u^{(n-1)}\), they require that the nonlinearity \(f(t, x_{0}, x_{1},\ldots, x_{n-1})\) satisfies the Nagumo-type growth condition on \(x_{n-1}\) in \(D\times \mathbb{R}\):
(NC): 
There is a continuous function \(h: [0, \infty )\to (0, \infty )\) satisfying \(\int_{0}^{\infty }\frac{r \,dr}{ h(r)}=\infty \), such that
$$| f(t, x_{0}, x_{1},\ldots, x_{n-1})\le h \bigl( \vert x_{n-1} \vert \bigr),\quad (t, x_{0}, x_{1},\ldots, x_{n-1})\in D\times \mathbb{R}. $$
Recently, Agarwal and Wong [19] discussed the existence of positive solutions of the special nth-order boundary value problem
$$ \textstyle\begin{cases} u^{(n)}(t)+f(t, u(t), u'(t),\ldots, u^{(q)}(t))=0, \quad t\in [0, 1], \\ u^{(k)}(0)=0, \quad 0\le k\le n-3, \\ \alpha u^{(n-2)}(0)-\beta u^{(n-1)}(0)=0,\quad\quad \gamma u^{(n-2)}(1)+ \delta u^{(n-1)}(1)=0, \end{cases} $$
(1.5)
where \(1\le q\le n-2\). They converted BVP (1.5) to an equivalent \((m-q)\)th-order Sturm–Liouville boundary value problem of integral-differential equations, and using Krasnoselskii’s fixed point theorem in cones they obtained existence results of one or more positive solutions. However, this method is not applicable to the more general BVP (1.1) owing to the presence of a derivative \(u^{(n-1)}\) in the nonlinearity f. Lately, the present author Li [20] considered the fully second-order boundary value
$$ \textstyle\begin{cases} -u''(t)=f(t, u(t), u'(t)), \quad t\in [0, 1], \\ u(0)=u(1)=0 , \end{cases} $$
(1.6)
which is a special form of BVP (1.1). Using the theory of fixed point index in cones Li obtained existence results of positive solutions under the nonlinearity \(f(t, x, y)\) showing superlinear or sublinear growth in x and y. But the discussion in [20] relies on the spatial boundary condition \(u(0)=u(1)=0\) and cannot be directly extended to the more general BVP (1.1).

Motivated by the research mentioned, in this paper we develop a different technique to discuss the fully nth-order boundary value problem (1.1). Our purpose is to obtain the existence of positive solution to BVP (1.1). By a positive solution u of BVP (1.1) we mean \(u\in C^{n}[0, 1]\) satisfying (1.1) and \(u(t)>0\) for \(t\in (0, 1)\). By using the theory of the fixed point index in cones we establish existence results of positive solutions for BVP (1.1). In our results, we present some inequality conditions on the nonlinearity \(f(t, x_{0}, x_{1},\ldots, x_{n-1})\) when \(\vert (x_{0}, x_{1},\ldots, x_{n-1}) \vert \) is small or large enough to guarantee the existence of positive solutions. These inequality conditions allow that \(f(t, x_{0}, x_{1},\ldots, x _{n-1})\) may be of superlinear or sublinear growth in \((x_{0}, x_{1},\ldots, x_{n-1})\) as \(\vert (x_{0}, x_{1},\ldots, x_{n-1}) \vert \to 0\) or ∞, and they are comparatively easy to check in applications. For the case that \(f(t, x_{0}, x_{1},\ldots, x_{n-1})\) has superlinear growth in \((x_{0}, x_{1},\ldots, x_{n-1})\) as \(\vert (x_{0}, x _{1},\ldots, x_{n-1}) \vert \to \infty \), similar to [18] we require that \(f(t, x_{0}, x_{1},\ldots, x_{n-1})\) satisfies a Nagumo-type growth condition in \(x_{n-1}\); see Assumption (F3) of Sect. 3. The Nagumo-type condition restricts f to have at most quadric growth on \(x_{n-1}\). Our work naturally generalizes and extends the known results for some special Sturm–Liouville boundary value problems [211] and complements the work of Refs. [1720].

The paper is organized as follows. In Sect. 2 some preliminaries to discussing BVP (1.1) are presented. We discuss the corresponding linear boundary value problem and present some properties of a positive solution of the linear boundary value problem, then we choose a cone K in work space \(C^{n-1}[0, 1]\) and convert BVP (1.1) into a fixed point problem of a completely continuous cone mapping \(A: K\to K\). Our main results are stated and proved in Sect. 3. Finally, in Sect. 4 we present some applications to illustrate the applicability of our main results.

2 Preliminaries

We use \(C(I)\) to denote the Banach space of all continuous function \(u(t)\) on \(I:=[0, 1]\) with norm \(\Vert u \Vert _{C}=\max_{t\in I} \vert u(t) \vert \). Generally, for \(m\in \mathbb{N}\), \(C^{m}(I)\) denotes the Banach space of all mth-order continuous differentiable function on I with the norm \(\Vert u \Vert _{C^{m}}=\max \{ \Vert u \Vert _{C}, \Vert u' \Vert _{C},\ldots, \Vert u^{(m)} \Vert _{C}\}\). Let \(C^{+}(I)\) be the cone of all nonnegative functions in \(C(I)\).

To discuss BVP (1.1), we first consider the corresponding linear boundary value problem (LBVP)
$$ \textstyle\begin{cases} -u^{(n)}(t)=h(t), \quad t\in [0, 1], \\ u^{(k)}(0)=0, \quad 0\le k\le n-3, \\ \alpha u^{(n-2)}(0)-\beta u^{(n-1)}(0)=0,\quad\quad \gamma u^{(n-2)}(1)+ \delta u^{(n-1)}(1)=0, \end{cases} $$
(2.1)
where \(h\in C(I)\). Setting \(v=u^{(n-2)}\), the LBVP (2.1) is rewritten as the second-order boundary value problem
$$ \textstyle\begin{cases} -v''(t)=h(t), \quad t\in [0, 1], \\ \alpha v(0)-\beta v'(0)=0,\quad\quad \gamma v(1)+\delta v'(1)=0, \end{cases} $$
(2.2)
and the \((n-2)\)th-order initial value problem (IVP)
$$ \textstyle\begin{cases} u^{(n-2)}(t)=v(t), \quad t\in [0, 1], \\ u^{(k)}(0)=0, \quad 0\le k\le n-3. \end{cases} $$
(2.3)
Let \(G(t, s)\) be the Green function corresponding to the linear boundary value problem (2.2). It is well known that [17, 19]
$$ G(t, s)=\frac{1}{\rho } \textstyle\begin{cases} (\beta +\alpha s) (\delta +\gamma (1-t)),& 0\le s\leq t\leq 1, \\ (\beta +\alpha t) (\delta +\gamma (1-s)),& 0\le t\le s\le 1. \end{cases} $$
(2.4)

Lemma 2.1

The Green function \(G(t, s)\) has the following properties:
  1. (a)

    \(G(t, s)\ge 0\) and \(G(t, s)=G(s, t)\) for \(t, s\in I\).

     
  2. (b)

    \(G(t, s)\le L G(s, s)\) for \(t, s\in I\), where \(L=\max \{ 1, \frac{\beta }{\alpha +\beta }, \frac{\delta }{\gamma +\delta } \} \ge 1\).

     
  3. (c)

    \(G(t, s)\ge \sigma G(t, t)G(s, s)\) for \(t, s\in I\), where \(\sigma =\frac{\rho }{\max \{\beta , \alpha +\beta \}\cdot \max \{ \delta , \gamma +\delta \}}>0\).

     

Proof

For the properties (a) and (b), see [13, Lemma 2.3], and we only need to show (c). For any \(t,s\in (0, 1)\), since \(G(t, s), G(s,s)>0\), by (2.4) we have
$$\begin{aligned} \frac{G(t, s)}{G(t, t)G(s, s)} &= \textstyle\begin{cases} \frac{\rho }{(\beta +\alpha t) (\delta +\gamma (1-s))},& 0 \le s\leq t\leq 1, \\ \frac{\rho }{(\beta +\alpha s) (\delta +\gamma (1-t))},& 0 \le t\le s\le 1, \end{cases}\displaystyle \\ &\ge \frac{\rho }{\max_{t\in I}(\beta +\alpha t)\cdot \max_{t\in I}( \delta +\gamma (1-t))} \\ &=\frac{\rho }{\max \{\beta , \alpha +\beta \}\cdot \max \{\delta , \gamma +\delta \}}:=\sigma >0. \end{aligned}$$
Hence, (c) holds. □

Lemma 2.2

For every \(h\in C(I)\), LBVP (2.1) has a unique solution \(u:=S h\in C^{n}(I)\). Moreover, the solution operator \(S: C(I)\to C ^{n-1}(I)\) is a completely continuous linear operator.

Proof

For any \(h\in C(I)\), by the Green function of the solution, BVP (2.2) has a unique solution
$$ v(t)= \int_{0}^{1} G(t, s)h(s)\,ds:=S_{2}h(t), \quad t\in I, $$
(2.5)
and the solution operator \(S_{2}: C(I)\to C^{2}(I)\) is continuous. Obviously, IVP (2.3) has a unique solution,
$$ u(t)= \int_{0}^{t} \int_{0}^{s_{1}}\cdots \int_{0}^{s_{n-3}}v(s_{n-2})\,ds _{n-2} \cdots ds_{2} \,ds_{1}:=J_{n-2}v(t),\quad t\in I, $$
(2.6)
and the solution operator \(J_{n-2}: C(I)\to C^{n-2}(I)\) is continuous. Consequently, \(u= J_{n-1}(S_{2}h)=(J_{n-2}\circ S_{2}) h\) is a unique solution of LBVP (2.1), and the solution operator \(S=J_{n-2}\circ S _{2}: C(I)\to C^{n}(I)\) is continuous. By the compactness of the embedding of \(C^{n}(I)\hookrightarrow C^{n-1}(I)\), \(S: C(I)\to C^{n-1}(I)\) is a completely continuous linear operator. □
Define a function on I by
$$ \theta (t):=\frac{\sigma }{L} G(t, t)=\frac{\sigma }{L}(\alpha + \beta t) \bigl(\gamma +\delta (1-t) \bigr),\quad t\in I, $$
(2.7)
where L and σ are positive constants in Lemma 2.1, then \(\theta (t)>0\) for \(t\in (0, 1)\). Define a positive constant by
$$ \Gamma_{0}:=\max_{t\in I} \int_{0}^{1} G(t, s) \,ds=\frac{\beta \delta +\alpha \delta +\alpha \gamma /2}{\rho }. $$
(2.8)

Lemma 2.3

Let \(h\in C^{+}(I)\). Then the unique solution \(u=S h\) of LBVP (2.1) has the following properties:
  1. (a)

    \(u^{(n-2)}(t)\ge \Vert u^{(n-2)} \Vert _{C}\theta (t)\) for every \(t\in I\).

     
  2. (b)

    \(u(t), u'(t),\ldots, u^{(n-3)}(t)\ge 0\) for every \(t\in I\).

     
  3. (c)

    There exists \(\xi \in (0, 1)\) such that \(u^{(n-1)}(\xi )=0\), \(u^{(n-1)}(t)\ge 0\) for \(t\in [0, \xi ]\) and \(u^{(n-1)}(t)\le 0\) for \(t\in [\xi , 1]\). Moreover, \(\Vert u^{(n-1)} \Vert _{C}=\max \{u^{(n-1)}(0), -u ^{(n-1)}(1)\}\).

     
  4. (d)

    \(\Vert u \Vert _{C}\le \Vert u' \Vert _{C}\le \cdots \le \Vert u^{(n-2)} \Vert _{C}\).

     
  5. (e)

    \(\Vert u^{(n-2)} \Vert _{C}\le \Gamma_{0} \Vert u^{(n)} \Vert _{C}\), \(\Vert u^{(n-1)} \Vert _{C}\le \Vert u^{(n)} \Vert _{C}\).

     

Proof

(a) Let \(h\in C^{+}(I)\) and \(u=Sh\) be the unique solution of LBVP (2.1). Set \(v(t)=u^{(n-2)}(t)\), then v is a unique solution of BVP (2.2) given by (2.5). By (2.5) and Lemma 2.1(a) and (b), we have
$$0\le v(t)= \int_{0}^{1} G(t, s)h(s)\,d s\le L \int_{0}^{1} G(s, s)h(s)\,d s,\quad t\in I. $$
This implies that
$$\Vert v \Vert _{C}\le L \int_{0}^{1} G(s, s)h(s)\,d s. $$
Therefore, by Lemma 2.1(c), we have
$$\begin{aligned} v(t)&= \int_{0}^{1} G(t, s)h(s)\,d s \ge \sigma \int_{0}^{1}G(t, t) G(s, s)h(s)\,d s \\ &=\theta (t)\cdot L \int_{0}^{1} G(s, s)h(s)\,d \ge \theta (t) \Vert v \Vert _{C}. \end{aligned}$$
Namely, the conclusion of Lemma 2.3(a) holds.
(b) Since \(u^{(n-2)}(t)\ge 0\) for \(t\in I\), integrating this inequality and using the boundary conditions \(u(0)=u'(0)=\cdots =u^{(n-3)}(0)=0\), we have
$$\begin{aligned} &u^{(n-3)}(t)= \int_{0}^{t} u^{(n-2)}(s)\,ds\ge 0, \quad t \in I, \\ &u^{(n-4)}(t)= \int_{0}^{t} u^{(n-3)}(s)\,ds\ge 0, \quad t \in I, \\ & \cdots \\ &u(t)= \int_{0}^{t} u'(s)\,ds\ge 0, \quad t \in I. \end{aligned}$$
Hence, the conclusion of Lemma 2.3(b) holds.
(c) Let \(v(t)=u^{(n-2)}(t)\). Since \(v''(t)=-h(t)\le 0\) for every \(t\in I\), it follows that \(v'(t)\) is a monotone nonincreasing function on I. Since v is a unique solution of BVP (2.2), by (2.4) and (2.5) we can obtain
$$\begin{aligned}& v'(0) = \frac{\beta }{\rho } \int_{0}^{1} \bigl(\gamma +\delta (1-s) \bigr)h(s) \,d s \ge 0, \\& v'(1) = -\frac{\delta }{\rho } \int_{0}^{1} (\alpha +\beta s)h(s)\,d s \le 0. \end{aligned}$$
From these facts we conclude that there exists \(\xi \in (0, 1)\) such that \(v'(\xi )=0\), \(v'(t)\ge 0\) for \(t\in [0, \xi ]\) and \(v'(t) \le 0\) for \(t\in [\xi , 1]\). Moreover,
$$\bigl\Vert v' \bigr\Vert _{C}=\max _{t\in I} \bigl\vert v'(t) \bigr\vert =\max \bigl\{ v'(0), -v'(1) \bigr\} . $$
Thus, the conclusion of (c) holds.
(d) By the boundary conditions of LBVP (2.1), we have
$$u^{(k-1)}(t)= \int_{0}^{t} u^{(k)}(s) \,ds,\quad t\in I, k=1, 2,\ldots, n-2. $$
Hence,
$$\bigl\vert u^{(k-1)}(t) \bigr\vert = \int_{0}^{t} \bigl\vert u^{(k)}(s) \bigr\vert \,ds\le t \bigl\Vert u^{(k)} \bigr\Vert _{C}, \quad t \in I, k=1, 2,\ldots, n-2. $$
So we have
$$\bigl\Vert u^{(k-1)} \bigr\Vert _{C} \le \bigl\Vert u^{(k)} \bigr\Vert _{C}, \quad k=1, 2,\ldots, n-2, $$
namely the conclusion of Lemma 2.3(d) holds.
(e) For every \(t\in I\), by (2.5) we have
$$\begin{aligned} 0&\le u^{(n-2)}(t) = v(t) = \int_{0}^{1} G(t, s)h(s)\,d s \le \int_{0} ^{1} G(t, s)\,d s\cdot \Vert h \Vert _{C} \\ & \le \max_{t\in I} \int_{0}^{1} G(t, s)\,d s\cdot \Vert h \Vert _{C}=\Gamma _{0} \bigl\Vert u^{(n)} \bigr\Vert _{C}, \end{aligned}$$
so we have \(\Vert u^{(n-2)} \Vert \le \Gamma_{0} \Vert u^{(n)} \Vert _{C}\). Furthermore, by (c) there exists \(\xi \in (0, 1)\) such that \(u^{(n-1)}(\xi )=0\). Hence
$$\bigl\vert u^{(n-1)}(t) \bigr\vert = \biggl\vert \int_{\xi }^{t} u^{(n)}(s) \,ds \biggr\vert \le \vert t- \xi \vert \bigl\Vert u^{(n)} \bigr\Vert _{C}, $$
this implies that \(\Vert u^{(n-1)} \Vert \le \Vert u^{(n)} \Vert _{C}\).

Hence, the conclusion of Lemma 2.3(e) holds. □

Next we consider the linear eigenvalue problem (EVP) corresponding to LBVP (2.1)
$$ \textstyle\begin{cases} -u^{(n)}(t)=\lambda u(t), \quad t\in [0, 1], \\ u^{(k)}(0)=0, \quad 0\le k\le n-3, \\ \alpha u^{(n-2)}(0)-\beta u^{(n-1)}(0)=0,\quad \quad \gamma u^{(n-2)}(1)+ \delta u^{(n-1)}(1)=0. \end{cases} $$
(2.9)
Similar to the second-order Sturm–Liouville boundary value problem [21], we have the following.

Lemma 2.4

EVP (2.9) has a minimum positive real eigenvalue \(\lambda_{1}>0\). Moreover, \(\lambda_{1}\) has a positive unit eigenfunction, namely there exists \(\phi_{1}\in C^{n}(I)\cap C^{+}(I)\) with \(\Vert \phi_{1} \Vert _{C}=1\) that satisfies the equation
$$ \textstyle\begin{cases} -\phi_{1}^{(n)}(t)=\lambda_{1} \phi_{1}(t), \quad t\in [0, 1], \\ \phi_{1}^{(k)}(0)=0, \quad 0\le k\le n-3, \\ \alpha \phi_{1}^{(n-2)}(0)-\beta \phi_{1}^{(n-1)}(0)=0,\quad\quad \gamma \phi_{1}^{(n-2)}(1)+\delta \phi_{1}^{(n-1)}(1)=0. \end{cases} $$
(2.10)

Proof

According to [21, Lemma 1.1], we show that the solution operator S of LBVP (2.1) has the strong positivity estimate
$$ Sh\ge \Vert Sh \Vert _{C} J_{n-2}\theta ,\quad h\in C^{+}(I). $$
(2.11)
Let \(h\in C^{+}(I)\) and \(u=Sh\), then \(u^{(n-2)}=S_{2}h\). By Lemma 2.3(a) and (d),
$$S_{2}h(t)=u^{(n-2)}\ge \bigl\Vert u^{(n-2)} \bigr\Vert _{C}\theta (t)\ge \Vert u \Vert _{C}\theta (t)= \Vert Sh \Vert _{C}\theta (t),\quad t\in I, $$
that is, \(S_{2}h\ge \Vert Sh \Vert _{C}\theta \). By the positivity of the operator \(J_{n-2}: C(I)\to C(I)\), we have
$$Sh=J_{n-2}(S_{2}h)\ge \Vert Sh \Vert _{C}J_{n-2}\theta , $$
namely, (2.11) holds. Therefore, by [21, Lemma 1.1] the conclusion of Lemma 2.4 holds. □
By [21, Lemma 2.3], the minimum positive real eigenvalue \(\lambda_{1}\) of EVP (2.9) is also the minimum positive real eigenvalue of the conjugate eigenvalue problem
$$ \textstyle\begin{cases} (-1)^{n-1}w^{(n)}(t)=\lambda w(t), \quad t\in [0, 1], \\ \alpha w(0)-\beta w'(0)=0,\quad \quad \gamma w(1)+\delta w'(1)=0, \\ w^{(k)}(1)=0, \quad 2\le k\le n-1, \end{cases} $$
(2.12)
and the conclusion of Lemma 2.4 also holds for EVP (2.12), that is, we have the following.

Lemma 2.5

The minimum positive real eigenvalue \(\lambda_{1}\) of EVP (2.9) is also a minimum positive real eigenvalue of EVP (2.12). Moreover, \(\lambda_{1}\) has a positive unit eigenfunction, namely there exists \(\psi_{1}\in C^{n}(I)\cap C^{+}(I)\) with \(\Vert \psi_{1} \Vert _{C}=1\) that satisfies the equation
$$ \textstyle\begin{cases} (-1)^{n-1}{\psi_{1}}^{(n)}(t)=\lambda {\psi_{1}}(t), \quad t\in [0, 1], \\ \alpha {\psi_{1}}(0)-\beta {\psi_{1}}'(0)=0,\quad\quad \gamma {\psi_{1}}(1)+ \delta {\psi_{1}}'(1)=0, \\ {\psi_{1}}^{(k)}(1)=0, \quad 2\le k\le n-1. \end{cases} $$
(2.13)
Now we consider BVP (1.1). Let \(f: I\times {\mathbb{R}_{+}}^{n-1}\times \mathbb{R}\to \mathbb{R}_{+}\) be continuous. Define a closed convex cone K in Banach space \(C^{n-1}(I)\) by
$$ K= \bigl\{ u\in C^{n-1}(I) | u^{(k)}(t)\ge 0 \mbox{ for } t \in I, k=0, 1,\ldots, n-2 \bigr\} . $$
(2.14)
By Lemma 2.3(a) and (b), \(S(C^{+}(I))\subset K\). For every \(u\in K\), set
$$ F(u) (t):= f \bigl(t, u(t), u'(t),\ldots, u^{(n-1)}(t) \bigr),\quad t\in I. $$
(2.15)
Then \(F: K\to C^{+}(I)\) is continuous. Define a mapping \(A: K \to K\) by
$$ A=S\circ F. $$
(2.16)
By Lemma 2.2, \(A: K \to K\) is a completely continuous mapping. By the definitions of S and the strong positivity estimate (2.11), the positive solution of BVP (1.1) is equivalent to the nonzero fixed point of A. We will find the nonzero fixed point of A by using the fixed point index theory in cones.

Let E be a Banach space and \(K\subset E\) be a closed convex cone in E. Assume Ω is a bounded open subset of E with boundary Ω and \(A:K\cap \overline{\Omega }\to K\) is a completely continuous mapping. If \(Au\ne u\) for any \(u\in K\cap \partial \Omega \), then the fixed point index \(i (A, K\cap \Omega , K)\) in Cone K is well defined. The following lemmas [22] are needed in our discussion.

Lemma 2.6

Let Ω be a bounded open subset of E with \(\theta \in \Omega \), and \(A: K\cap \overline{\Omega }\to K\) a completely continuous mapping. If \(\mu Au\neq u\) for every \(u\in K\cap \partial \Omega \) and \(0<\mu \le 1\), then \(i (A, K\cap \Omega , K)=1\).

Lemma 2.7

Let Ω be a bounded open subset of E and \(A:K\cap \overline{ \Omega }\to K\) a completely continuous mapping. If there exists \(e\in K\setminus \{\theta \}\) such that \(u-Au\ne \tau e\) for every \(u\in K\cap \partial \Omega \) and \(\tau \ge 0\), then \(i (A, K \cap \Omega , K)=0\).

Lemma 2.8

Let Ω be a bounded open subset of E, and \(A, A_{1}: K \cap \overline{\Omega }\to K\) be two completely continuous mappings. If \((1-s)Au+sA_{1}u\ne u\) for every \(u\in K\cap \partial \Omega \) and \(0\le s\le 1\), then \(i (A, K\cap \Omega , K)=i (A_{1}, K \cap \Omega , K)\).

3 Main results

In this section, we show the existence of positive solutions of BVP (1.1). Let \(f:[0, 1]\times {\mathbb{R}_{+}}^{n-1}\times \mathbb{R}\to \mathbb{R}_{+}\) be continuous and the constants α, β, γ and δ satisfy (1.2). We present some inequality conditions on the nonlinearity \(f(t, x_{0}, x_{1},\ldots, x_{n-1})\) when \(\vert (x_{0}, x _{1},\ldots, x_{n-1}) \vert \) is small or large enough to guarantee the existence of positive solutions. Here \(\vert (x_{0}, x_{1},\ldots, x_{n-1}) \vert =( \sum_{k=0}^{n-1} \vert x_{k} \vert ^{2})^{1/2}\) is the Euclidean norm of \((x_{0}, x_{1},\ldots, x_{n-1})\in \mathbb{R}^{n}\). Let \(G=I\times \mathbb{R}^{n-1} \times \mathbb{R}\) denote the definitional domain of f. Our main results are as follows.

Theorem 3.1

Assume that \(f:[0, 1]\times {\mathbb{R}_{+}}^{n-1}\times \mathbb{R}\to \mathbb{R}_{+}\) is continuous and satisfies the following conditions:
  1. (F1)
    there exist nonnegative constants \(a_{0}, a_{1},\ldots, a_{n-1}\) satisfying \(\Gamma_{0}(a_{0}+a_{1}+\cdots +a_{n-2})+a_{n-1}<1\) and \(\delta >0\) such that
    $$f(t, x_{0}, x_{1},\ldots, x_{n-1})\le a_{0} x_{0}+a_{1} x_{1}+ \cdots +a_{n-2}x_{n-2}+a_{n-1} \vert x_{n-1} \vert $$
    for \((t, x_{0}, x_{1},\ldots, x_{n-1})\in G\) with \(\vert (x_{0}, x _{1},\ldots, x_{n-1}) \vert <\delta \);
     
  2. (F2)
    there exist constants \(b_{0}>\lambda_{1}\), \(b_{1}, b_{2} \cdots ,b_{n-1}\ge 0\) and \(H>0\) such that
    $$f(t, x_{0}, x_{1},\ldots, x_{n-1})\ge b_{0} x_{0}+b_{1} x_{1}+ \cdots + b_{n-2}x_{n-2}+b_{n-1} \vert x_{n-1} \vert $$
    for \((t, x_{0}, x_{1},\ldots, x_{n-1})\in G\) with \(\vert (x_{0}, x _{1},\ldots, x_{n-1}) \vert >H\);
     
  3. (F3)
    for any \(M>0\), there is a continuous function \(h_{M}: \mathbb{R}_{+}\to (0, \infty )\) satisfying
    $$ \int_{0}^{\infty }\frac{r \,dr}{ h_{M}(r)+1}=\infty , $$
    (3.1)
    such that
    $$ f(t, x_{0}, x_{1},\ldots, x_{n-1})\le h_{M} \bigl( \vert x_{n-1} \vert \bigr) $$
    (3.2)
    for all \((t, x_{0}, x_{1},\ldots, x_{n-1})\in I\times [0, M]^{n-1} \times \mathbb{R}\).
     
Then BVP (1.1) has at least one positive solution.

Theorem 3.2

Assume that \(f:[0, 1]\times {\mathbb{R}_{+}}^{n-1}\times \mathbb{R}\to \mathbb{R}_{+}\) is continuous and satisfies the following conditions:
  1. (F4)
    there exist constants \(b_{0}>\lambda_{1}\), \(b_{1}, b_{2} \cdots ,b_{n-1}\ge 0\) and \(\delta >0\) such that
    $$f(t, x_{0}, x_{1},\ldots, x_{n-1})\ge b_{0} x_{0}+b_{1} x_{1}+ \cdots + b_{n-2}x_{n-2}+b_{n-1} \vert x_{n-1} \vert $$
    for \((t, x_{0}, x_{1},\ldots, x_{n-1})\in G\) with \(\vert (x_{0}, x _{1},\ldots, x_{n-1}) \vert <\delta \);
     
  2. (F5)
    there exist nonnegative constants \(a_{0}, a_{1},\ldots, a_{n-1}\) satisfying \(\Gamma_{0}(a_{0}+a_{1}+\cdots +a_{n-2})+a_{n-1}<1\) and \(H>0\) such that
    $$f(t, x_{0}, x_{1},\ldots, x_{n-1})\le a_{0} x_{0}+a_{1} x_{1}+ \cdots + a_{n-2}x_{n-2}+a_{n-1} \vert x_{n-1} \vert $$
    for \((t, x_{0}, x_{1},\ldots, x_{n-1})\in G\) with \(\vert (x_{0}, x _{1},\ldots, x_{n-1}) \vert >H\).
     
Then BVP (1.1) has at least one positive solution.
In Theorems 3.1 and 3.2, the conditions (F1), (F2), (F3) and (F4) are inequality conditions, in which the nonlinearity f compares with a linear growth function of \((x_{0}, x_{1},\ldots, x_{n-1})\) of the form of
$$\ell (t, x_{0}, x_{1},\ldots, x_{n-1})=c_{0} x_{0}+c_{1} x_{1}+ \cdots c_{n-2}x_{n-2}+c_{n-1} \vert x_{n-1} \vert $$
as \(\vert (x_{0}, x_{1},\ldots, x_{n-1}) \vert \) is small or large enough. These conditions are concise and applicable. See Sect. 4.

In Theorem 3.1, the conditions (F1) and (F2) allow that \(f(t, x_{0}, x_{1},\ldots, x_{n-1})\) has superlinear growth in \((x_{0}, x _{1},\ldots, x_{n-1})\) as \(\vert (x_{0}, x_{1},\ldots, x_{n-1}) \vert \to 0\) and ∞, respectively. In this case, we need the condition (F3) to restrict the growth of the nonlinearity \(f(t, x_{0}, x_{1},\ldots, x_{n-1})\) on \(x_{n-1}\). (F3) is a Nagumo-type growth condition. In Theorem 3.2, the conditions (F4) and (F5) allow that \(f(t, x_{0}, x_{1},\ldots, x_{n-1})\) has sublinear growth in \((x_{0}, x_{1},\ldots, x_{n-1})\) as \(\vert (x_{0}, x_{1},\ldots, x_{n-1}) \vert \to 0\) and ∞, respectively. In this case, the Nagumo-type condition (F3) is needless. In fact, we can easily show that (F5) implies (F3).

Proof of Theorem 3.1

Choose the Banach space \(E=C^{n-1}(I)\). For convenience, we denote the norm \(\Vert u \Vert _{C^{n-1}}\) of E by \(\Vert u \Vert _{E}\). Let \(K\subset E\) be the closed convex cone defined by (2.14) and \(A:K\to K\) be the completely continuous mapping defined by (2.16). By Lemma 2.3, the positive solution of BVP (1.1) is equivalent to the nontrivial fixed point of A. Let \(0< R_{1}< R_{2}<+\infty \) and set
$$ \Omega_{1}= \bigl\{ u\in E| \Vert u \Vert _{E}< R_{1} \bigr\} ,\qquad \Omega_{2}= \bigl\{ u\in E| \Vert u \Vert _{E}< R_{2} \bigr\} . $$
(3.3)
We prove that the A has a fixed point in \(K\cap (\Omega_{2}\setminus \overline{\Omega }_{1})\) when \(R_{1}\) is small enough and \(R_{2}\) large enough. The proof is separated into the following steps:
Step (1). Choosing \(R_{1}\in (0, \delta /\sqrt{n})\), where δ is the positive constant in Condition (F1), we prove that
$$ i (A, K\cap {\Omega }_{1}, K)=1. $$
(3.4)
To this end, we verify that A satisfies the condition of Lemma 2.6 in \(K\cap \partial \Omega_{1}\), namely
$$ \mu Au\ne u,\quad \forall u\in K\cap \partial \Omega_{1}, 0< \mu \leq 1. $$
(3.5)
If (3.8) does not hold, there exist \(u_{0}\in K\cap \partial \Omega _{1}\) and \(0<\mu_{0}\le 1\) such that \(\mu_{0} Au_{0}=u_{0}\). Since \(u_{0}=S(\mu_{0}F(u_{0}))\), by the definition of S, \(u_{0}\) is the unique solution of LBVP (2.1) for \(h=\mu_{0}F(u_{0})\in C^{+}(I)\). Hence, \(u_{0}\in C^{n}(I)\) satisfies the equation
$$ \textstyle\begin{cases} -u_{0}^{(n)}(t)=\mu_{0} f(t, u_{0}(t), u_{0}'(t),\ldots, u_{0}^{(n-1)}(t)), \quad t\in [0, 1], \\ u_{0}^{(k)}(0)=0, \quad 0\le k\le n-3, \\ \alpha u_{0}^{(n-2)}(0)-\beta u_{0}^{(n-1)}(0)=0,\quad\quad \gamma u_{0} ^{(n-2)}(1)+\delta u_{0}^{(n-1)}(1)=0. \end{cases} $$
(3.6)
Since \(u_{0}\in K\cap \partial \Omega_{1}\), by the definitions of K and \(\Omega_{1}\), we have
$$\begin{aligned}& \bigl(t, u_{0}(t), u_{0}'(t),\ldots, u_{0}^{(n-1)}(t) \bigr)\in G,\quad t \in I, \\& \bigl\vert \bigl(u_{0}(t), u_{0}'(t), \ldots, u_{0}^{(n-1)}(t) \bigr) \bigr\vert \le \sqrt{n} \Vert u _{0} \Vert _{C^{n-1}}< \delta ,\quad t\in I. \end{aligned}$$
Hence by Condition (F1) and Lemma 2.3(d) and (e), we have
$$\begin{aligned} f \bigl(t, u_{0}(t), u_{0}'(t), \cdots , u_{0}^{(n-1)}(t) \bigr) &\le a_{0} u _{0}(t)+\cdots +a_{n-2} u_{0}^{(n-2)}(t)+a_{n-1} \bigl\vert u_{0}^{(n-1)}(t) \bigr\vert \\ &\le a_{0} \Vert u_{0} \Vert _{C}+\cdots +a_{n-2} \bigl\Vert u_{0}^{(n-2)} \bigr\Vert +a_{n-1} \bigl\Vert u _{0}^{(n-1)} \bigr\Vert _{C} \\ &\le (a_{0}+\cdots +a_{n-2}) \bigl\Vert u_{0}^{(n-2)} \bigr\Vert _{C}+ a_{n-1} \bigl\Vert u_{0} ^{(n-1)} \bigr\Vert _{C} \\ &\le \bigl(\Gamma_{0}(a_{0}+a_{1}+\cdots +a_{n-2})+a_{n-1} \bigr) \bigl\Vert u_{0}^{(n)} \bigr\Vert _{C},\quad t\in I. \end{aligned}$$
By this inequality and Eq. (3.6) we obtain
$$\bigl\vert {u_{0}}^{(n)}(t) \bigr\vert \le \bigl( \Gamma_{0}(a_{0}+a_{1}+\cdots +a_{n-2})+a_{n-1} \bigr) \bigl\Vert u_{0}^{(n)} \bigr\Vert _{C},\quad t\in I. $$
So we have
$$ \bigl\Vert {u_{0}}^{(n)} \bigr\Vert _{C} \le \bigl(\Gamma_{0}(a_{0}+a_{1}+\cdots +a_{n-2})+a _{n-1} \bigr) \bigl\Vert u_{0}^{(n)} \bigr\Vert _{C}. $$
(3.7)
We say that \(\Vert {u_{0}}^{(n)} \Vert _{C} >0\). If it is false, \(u_{0}\) is the solution of LBVP (2.1) for \(h\equiv 0\), and by the uniqueness of solution of LBVP (2.1) \(u_{0}=0\). This contradicts \(u_{0}\in \partial \Omega_{1}\). Hence from (3.7) it follows that
$$\Gamma_{0}(a_{0}+a_{1}+\cdots +a_{n-2})+a_{n-1}\ge 1, $$
which contradicts the assumption in Condition (F1). Hence (3.5) holds, and by Lemma 2.5, (3.4) is proved.
Step (2). Let H be the positive constant in Condition (F2). Set
$$\begin{aligned} C_{0}&=\max \bigl\{ \bigl\vert f(t, x_{0}, x_{1},\ldots, x_{n-1})- \bigl(b_{0} x_{0}+\cdots + b _{n-2}x_{n-2}+b_{n-1} \vert x_{n-1} \vert \bigr) \bigr\vert : \\ &\quad\ (t, x_{0}, x_{1},\ldots, x_{n-1})\in G, \bigl\vert (x_{0}, x_{1},\ldots, x_{n-1}) \bigr\vert \le H \bigr\} +1, \end{aligned} $$
then, by Condition (F2), we have
$$ \begin{aligned}[b] &f(t, x_{0}, x_{1},\ldots, x_{n-1})\ge b_{0} x_{0}+\cdots + b _{n-2}x_{n-2}+b_{n-1} \vert x_{n-1} \vert -C_{0}, \\ &\quad \mbox{for every } (t, x_{0}, x_{1},\ldots, x_{n-1})\in G. \end{aligned} $$
(3.8)
Set \(F_{1}(u) = F(u)+C_{0}\) for every \(u\in K\), and define \(A_{1}: K \to K\) by
$$ A_{1}=S\circ F_{1}. $$
(3.9)
Then \(A_{1}: K \to K\) is a completely continuous mapping. Letting \(R_{2}>\delta /\sqrt{n}\), we prove that
$$ i (A_{1}, K\cap {\Omega }_{2}, K)=0. $$
(3.10)
Let \(\phi_{1}\) be the positive eigenvalue function of EVP (2.9) in Lemma 2.4. Since \(\phi_{1}=S(\lambda_{1}\phi_{1})\), by Lemma 2.4 \(\phi_{1} \in K\setminus \{\theta \}\). We show that \(A_{1}\) satisfies the condition of Lemma 2.7 in \(K\cap \partial \Omega_{2}\) for \(e=\phi_{1}\), namely
$$ u-A_{1}u\ne \tau \phi_{1},\quad \forall u\in K\cap \partial \Omega_{2}, \tau \ge 0. $$
(3.11)
If (3.11) is false, there exist \(u_{1}\in K\cap \partial \Omega_{2}\) and \(\tau_{0}\ge 0\) such that \(u_{1}-A_{1} u_{1}=\tau_{0} \phi_{1}\). Since \(u_{1}=A_{1} u_{1}+\tau_{0} \phi_{1}=S(F(u_{1})+C_{0}+\tau_{0}\lambda _{1} \phi_{1})\), by the definition of S, \(u_{1}\) is the unique solution of LBVP (2.1) for \(h= F(u_{1})+C_{0}+\tau_{0}\lambda_{1} \phi _{1} \in C^{+}(I)\). Hence \(u_{1}\in C^{n}(I)\) satisfies the equation
$$ \textstyle\begin{cases} -u_{1}^{(n)}(t)= f(t, u_{1}(t), u_{1}'(t),\ldots, u_{1}^{(n-1)}(t))+C _{0}+\tau_{0}\lambda_{1}\phi_{1}(t),\quad t\in I, \\ u_{1}^{(k)}(0)=0, \quad 0\le k\le n-3, \\ \alpha u_{1}^{(n-2)}(0)-\beta u_{1}^{(n-1)}(0)=0,\quad\quad \gamma u_{1} ^{(n-2)}(1)+\delta u_{1}^{(n-1)}(1)=0. \end{cases} $$
(3.12)
Since \({u_{1}}\in K\cap \partial \Omega_{2}\), by the definition of K, \((t, u_{1}(t), u_{1}'(t),\ldots, u_{1}^{(n-1)}(t))\in G\) for \(t\in I\). Hence from (3.8) we see that
$$\begin{aligned} f \bigl(t, u_{1}(t), u_{1}'(t),\ldots, u_{1}^{(n-1)}(t) \bigr) &\ge b_{0}u_{1}(t)+ \cdots +b_{n-2}u_{1}^{(n-2)}(t)+b_{n-1} \bigl\vert u_{1}^{(n-1)}(t) \bigr\vert -C_{0} \\ &\ge b_{0} u_{1}(t)-C_{0}, \quad t\in I. \end{aligned}$$
By this inequality and Eq. (3.12), we have
$$\begin{aligned} -u_{1}^{(n)}(t) &= f \bigl(t, u_{1}(t), u_{1}'(t),\ldots, u_{1}^{(n-1)}(t) \bigr)+C _{0}+\tau_{0}\lambda_{1} \phi_{1}(t) \\ &\ge b_{0} {u_{1}}(t)+\tau_{0} \lambda_{1}\phi_{1}(t) \\ &\ge b_{0} {u_{1}}(t),\quad t\in I. \end{aligned}$$
Let \(\psi_{1}(t)\) be the positive eigenvalue function of EVP (2.12) in Lemma 2.5. Multiplying the above inequality by \(\psi_{1}(t)\) and integrating on I, then using integration by parts for the left side, we obtain
$$ {\lambda_{1}} \int_{0}^{1}{u_{1}}(t) \psi_{1}(t) \,d t\ge b_{0} \int_{0} ^{1}{u_{1}}(t) \psi_{1}(t) \,dt. $$
(3.13)
Since \(u_{1}=Sh\), by (2.12),
$$u_{1}(t)=Sh(t)\ge \Vert Sh \Vert J_{n-2}\theta (t)= \Vert u_{1} \Vert _{C}J_{n-2}\theta (t)>0,\quad \forall t\in (0, 1), $$
so that \(\int_{0}^{1}{u_{1}}(t) \psi_{1}(t) \,dt>0\). Hence, from (3.13) we see that \(\lambda_{1}\ge b_{0}\), which contradicts the assumption in (F2). This means that (3.11) is true. By Lemma 2.7, (3.10) holds.
Step (3). We use Lemma 2.8 to prove that
$$ i (A, K\cap \Omega_{2}, K)=i (A_{1}, K\cap \Omega_{2}, K) $$
(3.14)
when R is large enough. For this purpose, we show that A and \(A_{1}\) satisfy the condition of Lemma 2.8 in \(K\cap \partial \Omega _{2}\) when R is large enough, namely
$$ (1-s)Au+s A_{1} u\ne u,\quad \forall u\in K\cap \partial \Omega _{2}, 0\le s\le 1. $$
(3.15)
If (3.15) is false, there exist \(u_{2}\in K\cap \partial \Omega_{2}\) and \(s_{0}\in [0, 1]\) such that \((1-s_{0})Au_{2}+s_{0} A_{1}u_{2}= u_{2}\). Since \(u_{2}=S((1-s_{0}) F(u_{2})+s_{0}F_{1}(u_{2}))\), by the definition of S, \(u_{2}\) is the unique solution of LBVP (2.1) for \(h=(1-s_{0}) F(u_{2})+s_{0}F_{1}(u_{2})\in C^{+}(I)\). Hence \(u_{2}\in C^{n}(I)\) satisfies the equation
$$ \textstyle\begin{cases} -u_{2}^{(n)}(t)= f(t, u_{2}(t), u_{2}'(t),\ldots, u_{2}^{(n-1)}(t))+s _{0} C_{0},\quad t\in I, \\ u_{2}^{(k)}(0)=0, \quad 0\le k\le n-3, \\ \alpha u_{2}^{(n-2)}(0)-\beta u_{2}^{(n-1)}(0)=0,\quad\quad \gamma u_{2} ^{(n-2)}(1)+\delta u_{2}^{(n-1)}(1)=0. \end{cases} $$
(3.16)
Since \({u_{2}}\in K\cap \partial \Omega_{2}\), by the definition of K, \((t, u_{2}(t), u_{2}'(t),\ldots, u_{2}^{(n-1)}(t))\in G\) for \(t\in I\). Hence by (3.8), we have
$$\begin{aligned} f \bigl(t, u_{2}(t), u_{2}'(t),\ldots, u_{2}^{(n-1)}(t) \bigr) &\ge b_{0}u_{2}(t)+ \cdots +b_{n-2}u_{2}^{(n-2)}(t)+b_{n-1} \bigl\vert u_{2}^{(n-1)}(t) \bigr\vert -C_{0} \\ &\ge b_{0} u_{2}(t)-C_{0},\quad t\in I. \end{aligned}$$
Hence by Eq. (3.16),
$$\begin{aligned} -u_{2}^{(n)}(t) &= f \bigl(t, u_{2}(t), u_{2}'(t),\ldots, u_{2}^{(n-1)}(t) \bigr)+s _{0} C_{0} \\ &\ge b_{0} {u_{2}}(t)-(1-s_{0}) C_{0}, \\ &\ge b_{0} u_{2}(t)-C_{0},\quad t\in I. \end{aligned}$$
Multiplying this inequality by \(\psi_{1}(t)\) and integrating on I, then using integration by parts for the left side, we obtain
$$\begin{aligned} {\lambda_{1}} \int_{0}^{1}{u_{2}}(t) \psi_{1}(t) \,d t &\ge b_{0} \int _{0}^{1}{u_{2}}(t) \psi_{1}(t) \,d t-C_{0} \int_{0}^{1}\psi_{1}(t)\,d t \\ &\ge b_{0} \int_{0}^{1}{u_{2}}(t) \psi_{1}(t)\,d t-C_{0}. \end{aligned}$$
From this inequality it follows that
$$ \int_{0}^{1}{u_{2}}(t) \psi_{1}(t)\,d t\le \frac{C_{0}}{b_{0}-\lambda _{1}}. $$
(3.17)
On the other hand, by Lemma 2.3(a),
$$u_{2}^{(n-2)}(t)\ge \bigl\Vert u_{2}^{(n-2)} \bigr\Vert _{C} \theta (t),\quad t\in I. $$
Since \(u_{2}=J_{n-2}u_{2}^{(n-2)}\) and \(J_{n-2}: C(I)\to C(I)\) is a positive operator, acting on the above inequality by \(J_{n-2}\), we have
$$u_{2}(t)\ge \bigl\Vert u_{2}^{(n-2)} \bigr\Vert _{C} J_{n-2}\theta (t),\quad t\in I. $$
Multiplying this inequality by \(\psi_{1}(t)\) and integrating on I, we obtain
$$ \int_{0}^{1}{u_{2}}(t) \psi_{1}(t) \,d t\ge M_{0} \bigl\Vert u_{2}^{(n-2)} \bigr\Vert _{C}, $$
(3.18)
where
$$M_{0}:= \int_{0}^{1}J_{n-2}\theta (t) \psi_{1}(t) \,d t $$
is a positive constant by the positivity of \(J_{n-2}\theta \) on \((0, 1)\). Combining (3.18) with (3.17), we obtain
$$ \bigl\Vert u_{2}^{(n-2)} \bigr\Vert _{C}\le \frac{C_{0}}{ M_{0}(b_{0}-\lambda_{1}) } :=M. $$
(3.19)
For this \(M>0\), by Condition (F3), there is a continuous function \(h_{M}: \mathbb{R}_{+}\to (0, \infty )\) satisfying (3.1) such that (3.2) holds. By Lemma 2.3(d), we have
$$0\le u_{2}^{(k)}(t)\le \bigl\Vert u_{2}^{(k)} \bigr\Vert \le \bigl\Vert u_{2}^{(n-2)} \bigr\Vert _{C}\le M, \quad t\in I, k=0, 1,\ldots,n-2. $$
Hence from (3.2) it follows that
$$f \bigl(t, u_{2}(t), u_{2}'(t),\ldots, u_{2}^{(n-1)}(t) \bigr)\le h_{M} \bigl( \bigl\vert u_{2} ^{(n-1)}(t) \bigr\vert \bigr),\quad t\in I. $$
By this inequality and Eq. (3.16), we obtain
$$ -u_{2}^{(n)}(t)\le h_{M} \bigl( \bigl\vert u_{2}^{(n-1)}(t) \bigr\vert \bigr)+C_{0},\quad t\in I. $$
(3.20)
By (3.1) we can easily obtain
$$\int_{0}^{\infty }\frac{r \,d r}{ h_{M}(r)+C_{0}}=\infty . $$
Hence there exists a positive constant \(M_{1}\ge M\) such that
$$ \int_{0}^{M_{1}}\frac{r \,d r}{ h_{M}(r)+C_{0}}>M. $$
(3.21)

By Lemma 2.3(c), there exists \(\xi \in (0, 1)\) such that \(u_{2}^{(n-1)}( \xi )=0\), \(u_{2}^{(n-1)}(t)\ge 0\) for \(t\in [0, \xi ]\) and \(u_{2}^{(n-1)}(t)\le 0\) for \(t\in [\xi , 1]\), and \(\Vert u_{2}^{(n-1)} \Vert _{C}=\max \{u_{2}^{(n-1)}(0), -u_{2}^{(n-1)}(1)\}\). Hence \(\Vert u_{2} ^{(n-1)} \Vert _{C}=u_{2}^{(n-1)}(0)\) or \(\Vert u_{2}^{(n-1)} \Vert _{C}=-u_{2}^{(n-1)}(1)\). We only consider the case of \(\Vert u_{2}^{(n-1)} \Vert _{C}=u_{2}^{(n-1)}(0)\), and the other case can be treated with a quasi-way.

Since \(u_{2}^{(n-1)}(t)\ge 0\) for \(t\in [0, \xi ]\), multiplying both sides of the inequality (3.20) by \(u_{2}^{(n-1)}(t)\), we can obtain
$$-\frac{u_{2}^{(n-1)}(t) u_{2}^{(n)}(t)}{ h_{M}(u_{2}^{(n-1)}(t))+C _{0} }\le u_{2}^{(n-1)}(t),\quad t\in [0, \xi ]. $$
Integrating both sides of this inequality on \([0, \xi ]\) and making the variable transformation \(r=u_{2}^{(n-1)}(t)\) for the left side, we have
$$\int_{0}^{u_{2}^{(n-1)}(0)}\frac{r \,d r}{ h_{M}(r)+C_{0} }= u_{2} ^{(n-2)}(\xi )-u_{2}^{(n-2)}(0)\le \bigl\Vert u_{2}^{(n-2)} \bigr\Vert _{C}\le M. $$
Since \(u_{2}^{(n-1)}(0)= \Vert u_{2}^{(n-1)} \Vert _{C}\), it follows that
$$\int_{0}^{ \Vert u_{2}^{(n-1)} \Vert _{C}}\frac{r \,d r}{ h_{M}(r)+C_{0} } \le M. $$
Combining this inequality with (3.21), we conclude that
$$ \bigl\Vert u_{2}^{(n-1)} \bigr\Vert _{C}\le M_{1}. $$
(3.22)
By Lemma 2.3(d) and (3.19), we have
$$\bigl\Vert u_{2}^{(k)} \bigr\Vert _{C}\le \bigl\Vert u_{2}^{(n-2)} \bigr\Vert _{C}\le M\le M_{1},\quad k=0, 1,\ldots,n-2. $$
By these inequalities and (3.22), we obtain
$$ \Vert {u_{2}} \Vert _{E}=\max \bigl\{ \Vert u_{2} \Vert _{C}, \bigl\Vert u_{2}' \bigr\Vert _{C},\ldots, \bigl\Vert u _{2}^{(n-1)} \bigr\Vert _{C} \bigr\} \le M_{1}. $$
(3.23)

Now let \(R_{2}>\max \{M_{1}, \delta /\sqrt{n}\}\). Since \(u_{2} \in K\cap \partial \Omega_{2}\), by the definition of \(\Omega_{2}\), \(\Vert {u_{2}} \Vert _{E}=R_{2}>M_{1}\), which contradicts (3.23). This means that (3.15) is true. Hence by Lemma 2.8, (3.14) holds.

Step (4). Finally, from (3.10) and (3.14) it follows that
$$ i (A, K\cap \Omega_{2}, K)=0. $$
(3.24)
By the additivity of the fixed point index, (3.4) and (3.24), we have
$$i \bigl(A, K\cap (\Omega_{2}\setminus \overline{\Omega }_{1}), K \bigr)= i (A, K\cap {\Omega }_{2}, K)-i (A, K\cap {\Omega }_{1}, K)=-1. $$
Hence A has a fixed point in \(K\cap ({\Omega }_{2}\setminus \overline{ \Omega }_{1})\), which is a positive solution of BVP (1.1). The proof of Theorem 3.1 is completed. □

Proof of Theorem 3.2

Let \(E=C^{n-1}(I)\), \(K\subset E\) be the closed convex cone defined by (2.14) and \(A:K\to K\) the completely continuous mapping defined by (2.16). Let \(\Omega_{1}, \Omega_{2} \subset E\) be defined by (3.3). We prove that A has a fixed point in \(K\cap (\Omega_{2}\setminus \overline{\Omega }_{1})\) when \(R_{1}\) is small enough and \(R_{2}\) large enough.

Firstly, choosing \(R_{1}\in (0, \delta /\sqrt{n})\), where δ is the positive constant in Condition (F4), we prove that
$$ i (A, K\cap {\Omega }_{1}, K)=0. $$
(3.25)
Let \(\phi_{1}\) be the positive eigenvalue function of EVP (2.9) in Lemma 2.4. Then \(\phi_{1}\in K\setminus \{\theta \}\). We show that A satisfies the condition of Lemma 2.7 in \(K\cap \partial \Omega_{1}\) for \(e=\phi_{1}\), namely
$$ u-Au\ne \tau \phi_{1},\quad \forall u\in K\cap \partial \Omega _{1}, \tau \ge 0. $$
(3.26)
If (3.26) is false, there exist \(u_{4}\in K\cap \partial \Omega_{1}\) and \(\tau_{1}\ge 0\) such that \(u_{4}-Au_{4}=\tau_{1} \phi_{1}\). Since \(u_{4}=A u_{4}+\tau_{1} \phi_{1}=S(F(u_{4})+\tau_{1}\lambda_{1} \phi _{1})\), by the definition of S, \(u_{4}\) is the unique solution of LBVP (2.1) for \(h= F(u_{4})+\tau_{1}\lambda_{1} \phi_{1} \in C^{+}(I)\). Hence \(u_{4}\in C^{n}(I)\) satisfies the equation
$$ \textstyle\begin{cases} -u_{4}^{(n)}(t)= f(t, u_{4}(t), u_{4}'(t),\ldots, u_{4}^{(n-1)}(t))+ \tau_{1}\lambda_{1}\phi_{1}(t),\quad t\in I, \\ u_{4}^{(k)}(0)=0, \quad 0\le k\le n-3, \\ \alpha u_{4}^{(n-2)}(0)-\beta u_{4}^{(n-1)}(0)=0,\quad\quad \gamma u_{4} ^{(n-2)}(1)+\delta u_{4}^{(n-1)}(1)=0. \end{cases} $$
(3.27)
Since \(u_{4}\in K\cap \partial \Omega_{1}\), by the definitions of K and \(\Omega_{1}\), we have
$$\begin{aligned}& \bigl(t, u_{4}(t), u_{4}'(t),\ldots, u_{4}^{(n-1)}(t) \bigr)\in G,\quad t \in I, \\& \bigl\vert \bigl(u_{4}(t), u_{4}'(t), \ldots, u_{4}^{(n-1)}(t) \bigr) \bigr\vert \le \sqrt{n} \Vert u _{4} \Vert _{C^{n-1}}< \delta ,\quad t\in I. \end{aligned}$$
Hence by Condition (F4), we have
$$\begin{aligned} f \bigl(t, u_{4}(t), u_{4}'(t),\ldots, u_{4}^{(n-1)}(t) \bigr) &\ge b_{0}u_{4}(t)+ \cdots +b_{n-2} u_{4}^{(n-2)}(t)+b_{n-1} \bigl\vert u_{4}^{(n-1)}(t) \bigr\vert \\ &\ge b_{0} u_{4}(t), \quad t\in I. \end{aligned}$$
By this inequality and Eq. (3.27), we obtain
$$-u_{4}^{(n)}(t) \ge b_{0} {u_{4}}(t), \quad t\in I. $$
Let \(\psi_{1}(t)\) be the positive eigenvalue function of EVP (2.12) in Lemma 2.5. Multiplying the above inequality by \(\psi_{1}(t)\) and integrating on I, then using integration by parts for the left side, we have
$$ {\lambda_{1}} \int_{0}^{1}{u_{4}}(t) \psi_{1}(t) \,d t\ge b_{0} \int_{0} ^{1}{u_{4}}(t) \psi_{1}(t) \,dt. $$
(3.28)
Since \(u_{4}=Sh\), by (2.12) \(u_{4}(t) \ge \Vert u_{4} \Vert _{C}J_{n-2}\theta (t)>0\) for every \(t\in (0, 1)\), so we have \(\int_{0}^{1}{u_{4}}(t) \psi_{1}(t) \,dt>0\). Hence, from (3.28) it follows that \(\lambda_{1} \ge b_{0}\), which contradicts the assumption in (F4). This means that (3.26) is true. Hence by Lemma 2.7, (3.25) holds.
Secondly, we prove that
$$ i (A, K\cap {\Omega }_{2}, K)=1, $$
(3.29)
when \(R_{2}\) is large enough. For this purpose, we show that A satisfies the condition of Lemma 2.6 in \(K\cap \partial \Omega_{2}\), namely
$$ \mu Au\ne u,\quad \forall u\in K\cap \partial \Omega_{2}, 0< \mu \le 1. $$
(3.30)
If (3.30) is not true, there exist \(u_{5}\in K\cap \partial \Omega _{2}\) and \(0<\mu_{1}\le 1\) such that \(\mu_{1} Au_{5}=u_{5} \). Since \(u_{5}=S(\mu_{1}F(u_{5}))\), by the definition of S, \(u_{5}\in C^{n}(I)\) satisfies the equation
$$ \textstyle\begin{cases} -u_{5}^{(n)}(t)=\mu_{1} f(t, u_{5}(t), u_{5}'(t),\ldots, u_{5}^{(n-1)}(t)), \quad t\in [0, 1], \\ u_{5}^{(k)}(0)=0, \quad 0\le k\le n-3, \\ \alpha u_{5}^{(n-2)}(0)-\beta u_{5}^{(n-1)}(0)=0,\quad\quad \gamma u_{5} ^{(n-2)}(1)+\delta u_{5}^{(n-1)}(1)=0. \end{cases} $$
(3.31)
Let H be the positive constant in Condition (F5). Set
$$\begin{aligned} C_{1}&=\max \bigl\{ \bigl\vert f(t, x_{0}, x_{1},\ldots, x_{n-1})- \bigl(a_{0} x_{0}+\cdots + a _{n-2}x_{n-2}+a_{n-1} \vert x_{n-1} \vert \bigr) \bigr\vert : \\ &\quad\ (t, x_{0}, x_{1},\ldots, x_{n-1})\in G, \bigl\vert (x_{0}, x_{1},\ldots, x_{n-1}) \bigr\vert \le H \bigr\} +1. \end{aligned} $$
By Condition (F5), we have
$$ \begin{aligned}[b] &f(t, x_{0}, x_{1},\ldots, x_{n-1})\le a_{0} x_{0}+\cdots + a _{n-2}x_{n-2}+a_{n-1} \vert x_{n-1} \vert +C_{1}, \\ &\quad \mbox{for every } (t, x_{0}, x_{1},\ldots, x_{n-1})\in G. \end{aligned} $$
(3.32)
Since \({u_{5}}\in K\cap \partial \Omega_{2}\), by the definition of K, \((t, u_{5}(t), u_{5}'(t),\ldots, u_{5}^{(n-1)}(t))\in G\) for \(t\in I\). Hence by (3.32) and Lemma 2.3(d) and (e), we have
$$\begin{aligned} f \bigl(t, u_{5}(t), u_{5}'(t), \cdots , u_{5}^{(n-1)}(t) \bigr) &\le a_{0}u_{5}(t)+ \cdots +a_{n-2}u_{5}^{(n-2)}(t)+a_{n-1} \bigl\vert u_{5}^{(n-1)}(t) \bigr\vert +C_{1} \\ &\le a_{0} \Vert u_{5} \Vert _{C}+\cdots +a_{n-2} \bigl\Vert u_{5}^{(n-2)} \bigr\Vert +a_{n-1} \bigl\Vert u _{5}^{(n-1)} \bigr\Vert _{C}+C_{1} \\ &\le (a_{0}+\cdots +a_{n-2}) \bigl\Vert u_{5}^{(n-2)} \bigr\Vert _{C}+ a_{n-1} \bigl\Vert u_{5} ^{(n-1)} \bigr\Vert _{C}+C_{1} \\ &\le \bigl(\Gamma_{0}(a_{0}+a_{1}+\cdots +a_{n-2})+a_{n-1} \bigr) \bigl\Vert u_{5}^{(n)} \bigr\Vert _{C}+C_{1},\quad t\in I. \end{aligned}$$
By this inequality and Eq. (3.31) we obtain
$$\bigl\vert {u_{5}}^{(n)}(t) \bigr\vert \le \bigl( \Gamma_{0}(a_{0}+a_{1}+\cdots +a_{n-2})+a_{n-1} \bigr) \bigl\Vert u_{5}^{(n)} \bigr\Vert _{C}+C_{1},\quad t\in I. $$
So we have
$$\bigl\Vert {u_{5}}^{(n)} \bigr\Vert _{C} \le \bigl(\Gamma_{0}(a_{0}+a_{1}+\cdots +a_{n-2})+a _{n-1} \bigr) \bigl\Vert u_{5}^{(n)} \bigr\Vert _{C}+C_{1}. $$
From this it follows that
$$ \bigl\Vert {u_{5}}^{(n)} \bigr\Vert _{C}\le \frac{C_{1}}{ 1-(\Gamma_{0}(a_{0}+a_{1}+ \cdots +a_{n-2})+a_{n-1})}:=R_{0}. $$
(3.33)
Hence, by Lemma 2.3(d) and (e), we have
$$\begin{aligned} \Vert {u_{5}} \Vert _{E} &= \Vert {u_{5}} \Vert _{C^{n-1}}=\max \bigl\{ \Vert u_{5} \Vert _{C}, \bigl\Vert u _{5}' \bigr\Vert _{C},\ldots, \bigl\Vert u_{5}^{(n-1)} \bigr\Vert _{C} \bigr\} \\ &=\max \bigl\{ \bigl\Vert u_{5}^{(n-2)} \bigr\Vert _{C}, \bigl\Vert u_{5}^{(n-1)} \bigr\Vert _{C} \bigr\} \\ &\le \max \bigl\{ \Gamma_{0} \bigl\Vert u_{5}^{(n)} \bigr\Vert _{C}, \bigl\Vert u_{5}^{(n)} \bigr\Vert _{C} \bigr\} \\ &\le (\Gamma_{0}+1)R_{0}:=\overline{R}. \end{aligned}$$
(3.34)

We choose \(R>\max \{\overline{R}, \delta /\sqrt{n}\}\). Since \(u_{5}\in K\cap \partial \Omega_{2}\), by the definition of \(\Omega _{2}\), \(\Vert {u_{5}} \Vert _{C^{3}}=R>\overline{R}\), which contradicts (3.34). This means that (3.30) is true. Hence by Lemma 2.6, (3.29) holds.

Now by the additivity of the fixed point index, (3.25) and (3.29), we have
$$i \bigl(A, K\cap (\Omega_{2}\setminus \overline{\Omega }_{1}), K \bigr)= i (A, K\cap {\Omega }_{2}, K)-i (A, K\cap {\Omega }_{1}, K)=1. $$
Hence A has a fixed point in \(K\cap ({\Omega }_{2}\setminus \overline{ \Omega }_{1})\), which is a positive solution of BVP (1.1). The proof of Theorem 3.2 is completed. □

4 Applications

In this section, we present some applications of Theorems 3.1 and 3.2. For convenience, we introduce the following notation:
$$\begin{aligned} \begin{gathered} f_{0} = \liminf_{\substack{(x_{0}, x_{1},\ldots, x_{n-1})\in G,\\ \vert (x_{0}, x_{1},\ldots, x_{n-1}) \vert \to 0 }} \min _{t\in I} \frac{ f(t, x _{0}, x_{1},\ldots, x_{n-1}) }{ \vert (x_{0}, x_{1},\ldots, x_{n-1}) \vert } , \\ f^{0} = \limsup_{\substack{(x_{0}, x_{1},\ldots, x_{n-1})\in G,\\ \vert (x_{0}, x_{1},\ldots, x_{n-1}) \vert \to 0 } } \max_{t\in I} \frac{ f(t, x _{0}, x_{1},\ldots, x_{n-1}) }{ \vert (x_{0}, x_{1},\ldots, x_{n-1}) \vert } , \\ f_{\infty } = \liminf_{\substack{(x_{0}, x_{1},\ldots, x_{n-1})\in G,\\ \vert (x_{0}, x_{1},\ldots, x_{n-1}) \vert \to \infty }} \min_{t\in I} \frac{ f(t, x_{0}, x_{1},\ldots, x_{n-1}) }{ \vert (x_{0}, x_{1},\ldots, x _{n-1}) \vert } , \\ f^{\infty } = \limsup_{\substack{(x_{0}, x_{1},\ldots, x_{n-1})\in G,\\ \vert (x_{0}, x_{1},\ldots, x_{n-1}) \vert \to \infty }} \max_{t\in I} \frac{ f(t, x_{0}, x_{1},\ldots, x_{n-1}) }{ \vert (x_{0}, x_{1},\ldots, x _{n-1}) \vert } . \end{gathered} \end{aligned}$$
(4.1)

Theorem 4.1

Assume that \(f:[0, 1]\times {\mathbb{R}_{+}}^{n-1}\times \mathbb{R}\to \mathbb{R}_{+}\) is continuous and satisfies Assumption (F3) and the following condition:
  1. (F6)

    \(f^{0}<\frac{1}{(n-1)\Gamma_{0}+1}\), \(f_{\infty }> \lambda_{1}\),

     
then BVP (1.1) has at least one positive solution.

Proof

By the definitions of \(f^{0}\) and \(f_{\infty }\), we easily verify the following facts:
$$\begin{aligned}& f^{0}< \frac{1}{(n-1)\Gamma_{0}+1} \Longrightarrow \mbox{(F1) holds}, \\& f_{\infty }>\lambda_{1} \Longrightarrow \mbox{(F2) holds}. \end{aligned}$$
Hence, by Theorem 3.1, BVP (1.1) has at least one positive solution. □

Theorem 4.2

Assume that \(f:[0, 1]\times {\mathbb{R}_{+}}^{3}\times \mathbb{R}_{-}\to \mathbb{R}_{+}\) is continuous and satisfies the following condition:
  1. (F7)

    \(f_{0}>\lambda_{1}\), \(f^{\infty }<\frac{1}{(n-1)\Gamma _{0}+1}\).

     
Then BVP (1.1) has at least one positive solution.

Proof

By the definitions of \(f_{0}\) and \(f^{\infty }\), we can easily obtain
$$\begin{aligned}& f_{0}>\lambda_{1} \Longrightarrow \mbox{(F4) holds}, \\& f^{\infty }< \frac{1}{(n-1)\Gamma_{0}+1} \Longrightarrow \mbox{(F5) holds}. \end{aligned}$$
Hence, by Theorem 3.2, BVP (1.1) has at least one positive solution. □

Conditions (F6) and (F7) describe the growth state of f on \((x_{0}, x_{1},\ldots, x_{n-1})\) as \(\vert (x_{0}, x_{1},\ldots, x _{n-1}) \vert \to 0\) and \(\vert (x_{0}, x_{1},\ldots, x_{n-1}) \vert \to \infty \), and they contain the usual superlinear and sublinear growth conditions of f at 0 and ∞. Theorems 4.1 and 4.2 naturally extend some results in [211].

Example 4.1

Consider the third-order Sturm–Liouville boundary value problem
$$ \textstyle\begin{cases} u'''(t)+u^{4}(t)+u^{\prime 3}(t)+{u''}^{2}(t)=0, \quad t\in [0, 1], \\ u(0)=0,\quad\quad 2u'(0)-u''(0)=0,\quad\quad -u'(1)+3u''(1)=0, \end{cases} $$
(4.2)
corresponding to BVP (1.1), \(n=3\), the nonlinearity
$$ f(t, x_{0}, x_{1}, x_{2})={x_{0}}^{4}+{x_{1}}^{3}+{x_{2}}^{2}, $$
(4.3)
and the coefficients of the boundary condition
$$\alpha =2,\quad\quad \beta =1,\quad\quad \gamma =-1,\quad\quad \delta =3. $$
Clearly, α, β, γ and δ satisfy (1.2). By the definitions (4.1) and (4.3), we easily see that \(f(t, x_{0}, x _{1}, x_{2})\) satisfies the Nagumo-type condition (F3) on \(x_{2}\), and
$$f^{0}=0,\quad\quad f_{\infty }=\infty . $$
Hence, f also satisfies Condition (F6). By Theorem 4.1, BVP (4.2) has at least one positive solution.

Example 4.2

Consider the fourth-order Sturm–Liouville boundary value problem
$$ \textstyle\begin{cases} u^{(4)}(t)+ \sqrt[3]{{u}^{2}(t)+{u'}^{2}(t)+{u''}^{2}(t)+{u'''}^{2}(t)}, \quad t\in [0, 1], \\ u(0)=u'(0)=u''(0)=0,\quad\quad 2u''(1)+3u'''(1)=0, \end{cases} $$
(4.4)
corresponding to BVP (1.1), \(n=4\), the nonlinearity
$$ f(t, x_{0}, x_{1}, x_{2}, x_{3})= \sqrt[3]{{x_{0}}^{2}+{x_{1}} ^{2}+{x_{2}}^{2}+{x_{3}}^{2}}= \bigl\vert (x_{0}, x_{1}, x_{2}, x_{3}) \bigr\vert ^{2/3}, $$
(4.5)
and the coefficients of the boundary condition
$$\alpha =1,\quad\quad \beta =0,\quad\quad \gamma =2,\quad\quad \delta =3. $$
Obviously, α, β, γ and δ satisfy (1.2) and by (4.5) f satisfies
$$f_{0}=\infty ,\quad\quad f^{\infty }=0. $$
Hence f satisfies Condition (F7). By Theorem 4.2, BVP (4.4) has at least one positive solution.

Declarations

Acknowledgements

Research supported by NNSFs of China (11661071, 11761063).

Authors’ contributions

YL and QW carried out the first draft of this manuscript, YL prepared the final version of the manuscript. All authors read and approved the final version of the manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Northwest Normal University, Lanzhou, People’s Republic of China

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