In this section, we establish criteria for the existence of one, two, or multiple nontrivial positive solutions of (1.1).
We rewrite (1.1) in a form suitable for investigation. To begin, we consider the initial value problem
$$ \left \{ \textstyle\begin{array}{@{}l} y^{(q)}(t)=x(t), \quad t\in[0,1],\\ y(0)=y'(0)=y''(0)=\cdots=y^{(q-1)}(0)=0. \end{array}\displaystyle \right . $$
(3.1)
Due to the initial conditions in (3.1), it is clear that
$$ y^{(k)}(t)= \int_{0}^{t} \int_{0}^{s_{1}} \int_{0}^{s_{2}}\cdots \int _{0}^{s_{q-k-1}}x (s_{q-k} )\,ds_{q-k} \cdots \,ds_{1},\quad 0\leq k\leq q-1. $$
(3.2)
We introduce the notation of the k-tuple integral
$$J^{k}x(t)= \int_{0}^{t} \int_{0}^{s_{1}} \int_{0}^{s_{2}}\cdots \int _{0}^{s_{k-1}}x (s_{k} )\,ds_{k} \cdots \,ds_{1},\quad k\geq1. $$
Then, it follows from (3.1) and (3.2) that
$$ y^{(k)}(t)= J^{q-k}x(t),\quad 0\leq k\leq q, $$
(3.3)
where \(J^{0}x(t)\equiv x(t)\).
Denote \(\tilde{J}x(t)= (J^{q}x(t),J^{q-1}x(t),\ldots,Jx(t),x(t) )\). Noting (3.1) and (3.3), we rewrite (1.1) as the following \((m-q)\)th-order Sturm-Liouville boundary value problem:
$$ \left \{ \textstyle\begin{array}{@{}l} x^{(m-q)}(t)+ F (t,\tilde{J}x(t) )=0,\quad t\in [0,1],\\ x^{(k)}(0)=0,\quad 0\leq k\leq m-q-3, \\ \zeta x^{(m-q-2)}(0)-\theta x^{(m-q-1)}(0)=0,\qquad \rho x^{(m-q-2)}(1)+\delta x^{(m-q-1)}(1)=0. \end{array}\displaystyle \right . $$
(3.4)
If (3.4) has a solution \(x^{*}\), then the boundary value problem (1.1) has a solution \(y^{*}\) given by
$$ y^{*(k)}(t)= J^{q-k}x^{*}(t),\quad 0\leq k\leq q, $$
(3.5)
and, in particular,
$$ y^{*}(t)=J^{q}x^{*}(t)= \int_{0}^{t} \int_{0}^{s_{1}} \int_{0}^{s_{2}}\cdots \int _{0}^{s_{q-1}}x^{*} (s_{q} )\,ds_{q}\cdots \,ds_{1}. $$
(3.6)
Hence, the existence of a solution of (1.1) follows from the existence of a solution of (3.4). Further, it is obvious from (3.5) that for \(0\leq k\leq q\), \(y^{*(k)}\) is positive if \(x^{*}\) is, and \(y^{*(k)}\) is nontrivial if \(x^{*}\) is. We study (1.1) via (3.4) and employ a new technique to tackle the nonlinear term F.
Let the Banach space
$$B=\bigl\{ x\in C^{(m-q)}[0,1] \mid x^{(k)}(0)=0, 0\leq k\leq m-q-3 \bigr\} $$
be equipped with the norm
$$\| x\|=\sup_{t\in[0,1]}\bigl| x^{(m-q-2)}(t)\bigr|. $$
Throughout the paper, let \(\eta\in (0,\frac{1}{2} )\) be fixed. Define the cone C in B by
$$ C= \Bigl\{ x\in B \bigm| x^{(m-q-2)}(t)\geq0, t\in [0,1]; \min _{t\in[\eta,1-\eta]} x^{(m-q-2)}(t)\geq\gamma\|x\| \Bigr\} , $$
(3.7)
where \(\gamma=K_{\eta}/L\) (L and \(K_{\eta}\) are defined in Lemma 2.3).
Lemma 3.1
[35, 36]
Let
\(x\in B\). For
\(0\leq i\leq m-q-2\), we have
$$ \bigl|x^{(i)}(t)\bigr|\leq\frac{t^{m-q-2-i}}{(m-q-2-i)!} \|x\|,\quad t\in[0,1]. $$
(3.8)
In particular,
$$ \bigl|x(t)\bigr|\leq\frac{1}{(m-q-2)!} \|x\|,\quad t\in[0,1]. $$
(3.9)
Lemma 3.2
[35, 36]
Let
\(x\in C\). For
\(0\leq i\leq m-q-2\), we have
$$ x^{(i)}(t)\geq0,\quad t\in[0,1], $$
(3.10)
and
$$ x^{(i)}(t)\geq(t-\eta)^{m-q-2-i}\frac{\gamma}{(m-q-2-i)!} \|x\|,\quad t \in[\eta,1-\eta]. $$
(3.11)
In particular, for fixed
\(z\in(\eta,1-\eta)\), we have
$$ x(t)\geq(z-\eta)^{m-q-2}\frac{\gamma}{(m-q-2)!} \|x\|,\quad t\in [z,1-\eta]. $$
(3.12)
Remark 3.1
-
(a)
A solution \(y^{*}\) of (1.1) can be obtained via (3.6), where \(x^{*}\) is a solution of (3.4). In view of (3.5), if \(x^{*}\) is nontrivial/positive, then so is \(y^{*(k)}\), \(0\leq k\leq q\).
-
(b)
If \(x^{*}\in C\) is a solution of (3.4), then (3.10) implies that \(x^{*}\) is a positive solution of (3.4).
The next result is useful in handling the nonlinear term F.
Lemma 3.3
-
(a)
Let
\(x\in B\). For
\(1\leq k\leq q\), we have
$$ \bigl|J^{k} x(t)\bigr|\leq\frac{t^{m-q-2+k}}{(m-q-2+k)!} \|x\|\leq\frac {1}{(m-q-2+k)!} \|x\|, \quad t\in[0,1]. $$
(3.13)
-
(b)
Let
\(x\in C\)
and
\(z\in(\eta,1-\eta)\)
be fixed. For
\(1\leq k\leq q\), we have
$$ J^{k} x(t)\geq(z-\eta)^{m-q-2+k}\frac{\gamma}{(m-q-2+k)!} \|x\|,\quad t \in [z,1-\eta]. $$
(3.14)
Proof
(a) Since \(x\in B\), using \(\mbox{(3.8)}|_{i=0}\), we obtain that, for \(1\leq k\leq q\) and \(t\in[0,1]\),
$$\begin{aligned} \bigl|J^{k}x(t)\bigr| \leq& \int_{0}^{t} \int _{0}^{s_{1}} \int _{0}^{s_{2}}\cdots \int_{0}^{s_{k-1}}\bigl|x (s_{k} )\bigr|\,ds_{k} \cdots \,ds_{1} \\ \leq& \int_{0}^{t} \int_{0}^{s_{1}} \int_{0}^{s_{2}}\cdots \int _{0}^{s_{k-1}} \frac{s_{k}^{m-q-2}\|x\|}{(m-q-2)!}\,ds_{k} \cdots \,ds_{1} \\ =& \frac{t^{m-q-2+k}\|x\|}{(m-q-2+k)!} \leq \frac{\|x\|}{(m-q-2+k)!}. \end{aligned}$$
(b) Since \(x\in C\), using \(\mbox{(3.11)}|_{i=0}\), we find that, for \(1\leq k\leq q\) and \(t\in[z,1-\eta]\),
$$\begin{aligned} J^{k} x(t) =& \int_{0}^{t} \int_{0}^{s_{1}} \int _{0}^{s_{2}}\cdots \int_{0}^{s_{k-1}}x (s_{k} )\,ds_{k} \cdots \,ds_{1} \\ \geq& \int_{\eta}^{z} \int_{\eta}^{s_{1}} \int_{\eta}^{s_{2}}\cdots \int_{\eta}^{s_{k-1}}x (s_{k} )\,ds_{k}\cdots \,ds_{1} \\ \geq& \int_{\eta}^{z} \int_{\eta}^{s_{1}} \int_{\eta}^{s_{2}}\cdots \int_{\eta}^{s_{k-1}}(s_{k}-\eta)^{m-q-2} \frac{\gamma\|x\|}{(m-q-2)!}\,ds_{k}\cdots \,ds_{1} \\ =& (z-\eta)^{m-q-2+k}\frac{\gamma\|x\|}{(m-q-2+k)!}. \end{aligned}$$
□
The next result gives the estimate of \(y^{*}=J^{q} x^{*}\) in terms of \(\|x^{*}\|\).
Lemma 3.4
Let
\(x^{*}\)
and
\(y^{*}\)
be related by (3.5) and (3.6).
-
(a)
Let
\(x^{*}\in B\). For
\(0\leq k\leq m-2\), we have
$$ \bigl|y^{*(k)}(t)\bigr|\leq\frac{t^{m-k-2}}{(m-k-2)!} \bigl\| x^{*}\bigr\| \leq\frac {1}{(m-k-2)!} \bigl\| x^{*} \bigr\| ,\quad t\in[0,1]. $$
(3.15)
-
(b)
Let
\(x^{*}\in C\). For
\(0\leq k\leq m-2\), we have
$$ y^{*(k)}(t)\geq(t-\eta)^{m-k-2}\frac{\gamma}{(m-k-2)!} \bigl\| x^{*}\bigr\| ,\quad t \in [\eta,1-\eta]. $$
(3.16)
Proof
(a) Since \(x^{*}\in B\), using (3.5) and (3.13), for \(0\leq k\leq q-1\), we obtain
$$\bigl|y^{*(k)}(t)\bigr|=\bigl|J^{q-k}x^{*}(t)\bigr|\leq\frac{t^{m-k-2}\bigl\| x^{*}\bigr\| }{(m-k-2)!}\leq \frac{\|x^{*}\|}{(m-k-2)!},\quad t\in[0,1]. $$
Further, since \(y^{*(q)}(t)=x^{*}(t)\), we have \(y^{*(q+i)}(t)=x^{*(i)}(t)\) for \(0\leq i\leq m-q-2\), and so from (3.8) it follows that
$$\begin{aligned} &\bigl|y^{*(q+i)}(t)\bigr|=\bigl|x^{*(i)}(t)\bigr|\leq\frac{t^{m-q-2-i}\bigl\| x^{*}\bigr\| }{(m-q-2-i)!}\leq \frac{\|x^{*}\|}{(m-q-2-i)!},\\ &\quad t\in[0,1], 0\leq i\leq m-q-2, \end{aligned}$$
which is the same as
$$\bigl|y^{*(k)}(t)\bigr|\leq\frac{t^{m-k-2}\|x^{*}\|}{(m-k-2)!}\leq\frac{\|x^{*}\| }{(m-k-2)!},\quad t \in[0,1], q\leq k\leq m-2. $$
Combining this with the inequality obtained earlier, we get (3.15).
(b) Since \(x^{*}\in C\), noting \(\mbox{(3.11)}|_{i=0}\), we find that, for \(0\leq k\leq q-1\) and \(t\in[\eta,1-\eta]\),
$$\begin{aligned} y^{*(k)}(t) =& J^{q-k} x^{*}(t)= \int _{0}^{t} \int _{0}^{s_{1}} \int_{0}^{s_{2}}\cdots \int _{0}^{s_{q-k-1}}x^{*}(s_{q-k})\,ds_{q-k} \cdots \,ds_{1} \\ \geq& \int_{\eta}^{t} \int_{\eta}^{s_{1}} \int_{\eta}^{s_{2}}\cdots \int_{\eta}^{s_{q-k-1}}x^{*}(s_{q-k})\,ds_{q-k} \cdots \,ds_{1} \\ \geq& \int_{\eta}^{t} \int_{\eta}^{s_{1}} \int_{\eta}^{s_{2}}\cdots \int_{\eta}^{s_{q-k-1}}(s_{q-k}- \eta)^{m-q-2} \frac{\gamma\|x^{*}\|}{(m-q-2)!}\,ds_{q-k}\cdots \,ds_{1} \\ =& (t-\eta)^{m-k-2}\frac{\gamma\|x^{*}\|}{(m-k-2)!}. \end{aligned}$$
Next, since \(y^{*(q)}(t)=x^{*}(t)\), we have \(y^{*(q+i)}(t)=x^{*(i)}(t)\) for \(0\leq i\leq m-q-2\), and so from (3.11) we have
$$y^{*(q+i)}(t)=x^{*(i)}(t)\geq\frac{(t-\eta)^{m-q-2-i}\gamma\|x^{*}\| }{(m-q-2-i)!},\quad t\in[\eta,1- \eta], 0\leq i\leq m-q-2, $$
or, equivalently,
$$y^{*(k)}(t)\geq\frac{(t-\eta)^{m-k-2}\gamma\|x^{*}\|}{(m-k-2)!},\quad t\in [\eta,1-\eta], q\leq k\leq m-2. $$
A combination with the earlier inequality yields (3.16). □
Let the operator \(S:B\rightarrow B\) be defined by
$$ Sx(t)= \int_{0}^{1} g_{m-q}(t,s)F \bigl(s,\tilde{J}x(s) \bigr)\,ds,\quad t\in[0,1]. $$
(3.17)
Noting that \(g_{m-q}(t,s)\) is the Green’s function of \((2.3)_{m-q}\) (see Lemma 2.3(d)), it is clear that a fixed point of S is a solution of (3.4). Moreover, (3.17) is equivalent to
$$ (Sx)^{(m-q-2)}(t)= \int_{0}^{1} G(t,s)F \bigl(s,\tilde{J}x(s) \bigr)\,ds, \quad t\in[0,1], $$
(3.18)
where \(G(t,s)\) is the Green’s function of (2.1). In view of Remark 3.1, to obtain a positive solution of (1.1), we shall seek a fixed point of the operator S in the cone C.
For easy reference, the conditions that will be used further are listed below. In these conditions, the sets K and K̃ are defined respectively by
$$\tilde{K}=\bigl\{ u\in C[0,1] \mid u(t)\geq0, t\in[0,1]\bigr\} $$
and
$$K=\bigl\{ u\in\tilde{K} \mid u(t)> 0 \mbox{ on some subset of }[0,1]\mbox{ of positive measure}\bigr\} . $$
-
(A1)
F is continuous on \([0,1]\times\tilde{K}^{q+1}\) with
$$F(t,u_{1},\ldots,u_{q+1})\geq0, \quad(t,u_{1}, \ldots,u_{q+1})\in[0,1]\times \tilde{K}^{q+1}, $$
and
$$F(t,u_{1},\ldots,u_{q+1})> 0, \quad(t,u_{1}, \ldots,u_{q+1})\in[0,1]\times K^{q+1}. $$
-
(A2)
There exist continuous functions \(\beta:[0,1]\to[0,\infty)\) and \(f:[0,\infty)^{q+1}\to[0,\infty)\) such that f is nondecreasing in each of its arguments and
$$F(t,u_{1},\ldots,u_{q+1})\leq\beta(t)f(u_{1},\ldots ,u_{q+1}), \quad(t,u_{1},\ldots,u_{q+1})\in[0,1] \times\tilde{K}^{q+1}. $$
-
(A3)
There exists \(a>0\) such that
$$a>M f \biggl(\frac{a}{(m-2)!}, \frac{a}{(m-3)!}, \ldots, \frac {a}{(m-q-2)!} \biggr), $$
where \(M=\sup_{t\in[0,1]}\int_{0}^{1}G(t,s)\beta(s)\,ds\).
-
(A4)
Let \(z\in(\eta,1-\eta)\) be fixed. There exists a continuous function \(\alpha:[z, 1-\eta]\to(0,\infty)\) such that
$$F(t,u_{1},\ldots,u_{q+1})\geq\alpha(t)f(u_{1}, \ldots,u_{q+1}),\quad (t,u_{1},\ldots,u_{q+1})\in[z, 1-\eta]\times K^{q+1}. $$
-
(A5)
Let \(z\in(\eta,1-\eta)\) be fixed. There exists \(b>0\) such that
$$b\leq N f \biggl(\frac{(z-\eta)^{m-2}\gamma b}{(m-2)!}, \frac {(z-\eta )^{m-3}\gamma b}{(m-3)!}, \ldots, \frac{(z-\eta)^{m-q-2}\gamma b}{(m-q-2)!} \biggr) , $$
where \(N=\sup_{t\in[0,1]}\int_{z}^{1-\eta}G(t,s)\alpha(s)\,ds\) and \(\gamma=K_{\eta}/L\).
Remark 3.2
The computation of the constants M and N in (A3) and (A5) can be avoided by using some upper bound of M and some lower bound of N. As a consequence, stricter inequalities are obtained. Indeed, using Lemma 2.3, we have
$$M=\sup_{t\in[0,1]} \int_{0}^{1}G(t,s)\beta(s)\,ds\leq \int_{0}^{1} LG(s,s)\beta(s)\,ds \equiv M' $$
and
$$\begin{aligned} N&=\sup_{t\in[0,1]} \int_{z}^{1-\eta}G(t,s)\alpha(s)\,ds \geq\sup _{t\in[\eta,1-\eta]} \int_{z}^{1-\eta}G(t,s)\alpha(s)\,ds \\ &\geq \int_{z}^{1-\eta}K_{\eta}G(s,s)\alpha(s)\,ds \equiv N'. \end{aligned}$$
Let (A3)′ denote condition (A3) with M replaced by \(M'\), and (A5)′ denote condition (A5) with N replaced by \(N'\). Obviously, (A3) is satisfied if the stronger condition (A3)′ is met; likewise, (A5) is satisfied if the stronger condition (A5)′ holds.
The first result below gives the existence of a solution, which may not be positive.
Theorem 3.5
Let
\(F:[0,1]\times \mathbb {R}^{q+1}\rightarrow \mathbb {R}\)
be continuous. Suppose that there exists a constant d, independent of
λ, such that
\(\|x\|\neq d \)
for any solution
\(x\in B\)
of the equation
$$ x(t)=\lambda \int_{0}^{1} g_{m-q}(t,s)F \bigl(s,\tilde{J}x(s) \bigr)\,ds,\quad t\in[0,1], \qquad (3.19)_{\lambda}$$
where
\(0<\lambda<1\). Then, (1.1) has at least one solution
\(y^{*}\in C^{(m)}[0,1]\)
such that, for
\(0\leq k\leq m-2\),
$$ \bigl|y^{*(k)}(t)\bigr|\leq\frac{t^{m-k-2}}{(m-k-2)!} d\leq\frac {d}{(m-k-2)!},\quad t\in[0,1]. $$
(3.20)
Proof
We recognize that a solution of \((3.19)_{\lambda}\) is a fixed point of the equation \(x=\lambda Sx\), where S is defined in (3.17). Using the Arzelà-Ascoli theorem, we see that S is continuous and completely continuous. Now, in the context of Theorem 2.1, let \(U=\{x\in B \mid \|x\|< d\}\). Noting that \(\|x\|\neq d\), where x is any solution of \((3.19)_{\lambda}\), we see that \(x\notin \partial U\), and so conclusion (b) of Theorem 2.1 is not valid. Hence, conclusion (a) of Theorem 2.1 must hold, that is, S has a fixed point in U̅. Hence, (3.4) has a solution \(x^{*}\in\overline{U}\) with \(\|x^{*}\|\leq d\).
By Remark 3.1(a), (1.1) has a solution \(y^{*}=J^{q} x^{*}\). Noting that \(\|x^{*}\|\leq d\), (3.20) is immediate from (3.15). □
Using Theorem 3.5, the next result gives the existence of a positive solution.
Theorem 3.6
Let (A1)-(A3) hold. Then, (1.1) has a positive solution
\(y^{*}\in C^{(m)}[0,1]\)
such that, for
\(0\leq k\leq m-2\),
$$ 0\leq y^{*(k)}(t)< \frac{t^{m-k-2}}{(m-k-2)!} a\leq\frac {a}{(m-k-2)!}, \quad t \in[0,1]. $$
(3.21)
Proof
Let \(\hat{F}:[0,1]\times \mathbb {R}^{q+1}\to \mathbb {R}\) be defined by
$$ \hat{F}(t,u_{1},\ldots,u_{q+1})=F\bigl(t,|u_{1}|, \ldots,|u_{q+1}|\bigr). $$
(3.22)
Noting (A1), we see that the function F̂ is well defined and continuous.
Since we plan to employ Theorem 3.5, we consider the equation
$$ x(t)= \lambda \int_{0}^{1} g_{m-q}(t,s)\hat{F} \bigl(s, \tilde{J}x(s) \bigr)\,ds,\quad t\in[0,1], \qquad (3.23)_{\lambda}$$
where \(0<\lambda<1\), and prove that any solution \(x\in B\) of \((3.23)_{\lambda}\) satisfies \(\|x\|\neq a\).
To proceed, let \(x\in B\) be any solution of \((3.23)_{\lambda}\). Using (3.22), Lemma 2.3(e), and (A1), we get
$$\begin{aligned} x(t) =& \lambda \int_{0}^{1} g_{m-q}(t,s)\hat{F} \bigl(s, \tilde{J}x(s) \bigr)\,ds \\ =& \lambda \int_{0}^{1} g_{m-q}(t,s)F \bigl(s,\bigl|J^{q}x(s)\bigr|,\ldots,\bigl|Jx(s)\bigr|,\bigl|x(s)\bigr| \bigr)\,ds \geq 0,\quad t \in[0,1]. \end{aligned}$$
Thus, x is a positive solution.
Similarly, it is easily seen that
$$x^{(m-q-2)}(t)=\lambda \int_{0}^{1} G(t,s)\hat{F} \bigl(s,\tilde{J}x(s) \bigr)\,ds\geq0,\quad t\in[0,1]. $$
Then, applying (A2), (3.13), and (3.9), we find that, for \(t\in[0,1]\),
$$\begin{aligned} \bigl|x^{(m-q-2)}(t)\bigr| =& x^{(m-q-2)}(t) \leq \int_{0}^{1} G(t,s)F \bigl(s,\bigl|J^{q}x(s)\bigr|, \ldots,\bigl|Jx(s)\bigr|,\bigl|x(s)\bigr| \bigr)\,ds \\ \leq& \int_{0}^{1}G(t,s) \beta(s)f \bigl(\bigl|J^{q}x(s)\bigr|, \ldots,\bigl|Jx(s)\bigr|,\bigl|x(s)\bigr| \bigr)\,ds \\ \leq& \int_{0}^{1}G(t,s) \beta(s)f \biggl( \frac{\|x\|}{(m-2)!}, \frac{\|x\|}{(m-3)!}, \ldots , \frac{\| x\|}{(m-q-2)!} \biggr)\,ds. \end{aligned}$$
Taking the suprema of both sides yields
$$ \|x\|\leq Mf \biggl(\frac{\|x\|}{(m-2)!}, \frac{\|x\|}{(m-3)!}, \ldots, \frac {\|x\|}{(m-q-2)!} \biggr). $$
(3.24)
Comparing (3.24) and (A3), it is clear that \(\|x\|\neq a\).
It now follows from the proof of Theorem 3.5 that \((3.23)|_{\lambda=1}\) has a solution \(x^{*}\in B\) with \(\|x^{*}\|\leq a\). Using a similar argument as before, it can be easily seen that \(x^{*}\) is a positive solution and \(\|x^{*}\|\neq a\). Thus, \(\|x^{*}\|< a\). Moreover, since \(x^{*}\) is positive, we have \(|J^{k} x^{*}(s)|=J^{k} x^{*}(s)\) for \(0\leq k\leq q\) and \(s\in[0,1]\). Using this we find that, for \(t\in[0,1]\),
$$\begin{aligned} x^{*}(t) =& \int_{0}^{1} g_{m-q}(t,s)\hat{F} \bigl(s, \tilde{J}x^{*}(s) \bigr)\,ds \\ =& \int_{0}^{1} g_{m-q}(t,s)F \bigl(s,\bigl|J^{q} x^{*}(s)\bigr|,\ldots,\bigl|Jx^{*}(s)\bigr|,\bigl|x^{*}(s)\bigr| \bigr)\,ds \\ =& \int_{0}^{1}g_{m-q}(t,s)F \bigl(s,J^{q} x^{*}(s),\ldots,Jx^{*}(s),x^{*}(s) \bigr)\,ds. \end{aligned}$$
Hence, \(x^{*}\) is actually a positive solution of (3.4) with \(\|x^{*}\|< a\). By Remark 3.1(a), \(y^{*}=J^{q} x^{*}\) is a positive solution of (1.1) satisfying (3.15), which, in view of \(\|x^{*}\|< a\), leads to (3.21) immediately. □
Remark 3.3
Note that the last inequality in (A1),
$$F(t,u_{1},\ldots,u_{q+1})> 0,\quad (t,u_{1}, \ldots,u_{q+1})\in[0,1]\times K^{q+1}, $$
is not needed in Theorem 3.6.
The positive solution guaranteed in Theorem 3.6 may be trivial. Our next result gives the existence of a nontrivial positive solution.
Theorem 3.7
Let (A1)-(A5) hold. Then, (1.1) has a nontrivial positive solution
\(y^{*}\in C^{(m)}[0,1]\)
such that, for
\(0\leq k\leq m-2\),
$$ 0\leq y^{*(k)}(t) \left \{ \textstyle\begin{array}{@{}l@{\quad}l} < \frac{t^{m-k-2}}{(m-k-2)!} a\leq \frac{a}{(m-k-2)!},\quad t\in[0,1], &\textit{if }a>b,\\ \leq \frac{t^{m-k-2}}{(m-k-2)!} b\leq \frac{b}{(m-k-2)!}, \quad t\in[0,1], &\textit{if }a< b, \end{array}\displaystyle \right . $$
(3.25)
and
$$ y^{*(k)}(t)\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \geq \frac{(t-\eta)^{m-k-2}}{(m-k-2)!} \gamma b,\quad t\in[\eta,1-\eta],& \textit{if }a>b,\\ > \frac{(t-\eta)^{m-k-2}}{(m-k-2)!} \gamma a,\quad t\in[\eta ,1-\eta],& \textit{if }a< b. \end{array}\displaystyle \right . $$
(3.26)
Proof
We apply Theorem 2.2 with the operator S and the cone C defined respectively in (3.17) and (3.7). To begin, note that the operator \(S:B\to B\) is continuous and completely continuous. Further, from (3.10) we see that if \(x\in C\), then x is nonnegative, and so \(J^{k} x\in\tilde{K}\) (or \(J^{k} x\in K\) if x is nontrivial) for \(0\leq k\leq q\).
First, we show that S maps C into C. Let \(x\in C\). Noting (3.18), Lemma 2.3(a), and (A1), it is clear that
$$ (Sx)^{(m-q-2)}(t)= \int_{0}^{1}G(t,s)F \bigl(s,\tilde{J}x(s) \bigr)\,ds \geq 0,\quad t\in[0,1]. $$
(3.27)
Using Lemma 2.3(b), we have that, for \(t\in[0,1]\),
$$\bigl|(Sx)^{(m-q-2)}(t)\bigr|=(Sx)^{(m-q-2)}(t)\leq \int_{0}^{1}LG(s,s) F \bigl(s,\tilde{J}x(s) \bigr)\,ds, $$
which immediately implies
$$ \|Sx\|\leq \int_{0}^{1}LG(s,s) F \bigl(s,\tilde{J}x(s) \bigr)\,ds. $$
(3.28)
Now, using Lemma 2.3(c) and (3.28), we find that, for \(t\in[\eta,1-\eta]\),
$$(Sx)^{(m-q-2)}(t) \geq \int_{0}^{1}K_{\eta}G(s,s) F \bigl(s, \tilde{J}x(s) \bigr)\,ds\geq\frac{K_{\eta}}{L} \|Sx\|=\gamma \|Sx\|. $$
It follows that
$$ \min_{t\in[\eta,1-\eta]}Sx(t)\geq\gamma\|Sx\|. $$
(3.29)
Inequalities (3.27) and (3.29) imply that \(S(C)\subseteq C\).
Next, let \(\Omega_{a}=\{x\in B \mid \|x\|< a\}\). Let \(x\in C \cap \partial\Omega_{a}\), so \(\|x\|=a\). Applying (A2), (3.13), and (3.9), we have, for \(t\in[0,1]\),
$$\begin{aligned} \bigl|(Sx)^{(m-q-2)}(t)\bigr| =&(Sx)^{(m-q-2)}(t) \\ \leq& \int_{0}^{1}G(t,s) \beta(s) f \bigl(\tilde{J}x(s) \bigr)\,ds \\ \leq& \int_{0}^{1}G(t,s) \beta(s) f \biggl( \frac{a}{(m-2)!}, \frac{a}{(m-3)!}, \ldots, \frac {a}{(m-q-2)!} \biggr)\,ds. \end{aligned}$$
Taking the suprema and using (A3), we get
$$ \|Sx\|\leq Mf \biggl(\frac{a}{(m-2)!}, \frac{a}{(m-3)!}, \ldots, \frac {a}{(m-q-2)!} \biggr)< a=\|x\|. $$
(3.30)
Hence, we have shown that \(\|Sx\| \leq\|x\|\) for \(x \in C \cap \partial\Omega_{a}\).
Next, let \(\Omega_{b}=\{x\in B \mid \|x\|< b\}\). Let \(x\in C \cap \partial\Omega_{b}\), so that \(\|x\|=b\). Noting (A4), we find that, for \(t\in [0,1]\),
$$\begin{aligned} \bigl|(Sx)^{(m-q-2)}(t)\bigr| \geq& \int_{z}^{1-\eta} G(t,s) F \bigl(s,\tilde{J}x(s) \bigr)\,ds \\ \geq& \int_{z}^{1-\eta} G(t,s) \alpha(s)f \bigl(\tilde{J}x(s) \bigr)\,ds \\ \geq& \int_{z}^{1-\eta} G(t,s) \alpha(s)f \biggl( \frac{(z-\eta)^{m-2}\gamma b}{(m-2)!}, \frac{(z-\eta)^{m-3}\gamma b}{(m-3)!}, \ldots,\\ &{} \frac{(z-\eta)^{m-q-2}\gamma b}{(m-q-2)!} \biggr)\,ds, \end{aligned}$$
where we have used (3.14) and (3.12) in the last inequality. Taking the suprema and using (A5) lead to
$$ \|Sx\|\geq Nf \biggl(\frac{(z-\eta)^{m-2}\gamma b}{(m-2)!}, \frac {(z-\eta )^{m-3}\gamma b}{(m-3)!}, \ldots, \frac{(z-\eta)^{m-q-2}\gamma b}{(m-q-2)!} \biggr)\geq b = \|x\| . $$
(3.31)
Hence, we have \(\|Sx\| \geq\|x\|\) for \(x \in C \cap\partial\Omega_{b}\).
In view of (3.30) and (3.31), we conclude from Theorem 2.2 that S has a fixed point \(x^{*}\in C \cap (\overline {\Omega}_{\max\{a,b\}} \backslash\Omega_{\min\{a,b\}} )\). Thus, \(\min\{a,b\}\leq\|x^{*}\|\leq\max\{a,b\}\). We further note that \(\|x^{*}\|\neq a\) follows from a similar argument as in the first part of the proof of Theorem 3.6. Hence, we obtain
$$ a< \bigl\| x^{*}\bigr\| \leq b \quad\mbox{if }a< b \quad\mbox{and}\quad b\leq\bigl\| x^{*}\bigr\| < a\quad \mbox{if }a>b. $$
(3.32)
By Remark 3.1, (1.1) has a nontrivial positive solution \(y^{*}=J^{q} x^{*}\). Since \(x^{*}\in B\), \(y^{*}\) satisfies (3.15) which, in view of (3.32), gives (3.25). Further, since \(x^{*}\in C\), using (3.32) in (3.16) leads to (3.26) immediately. □
The next result gives the existence of two positive solutions.
Theorem 3.8
Let (A1)-(A5) hold with
\(a< b\). Then, (1.1) has (at least) two positive solutions
\(y_{1}, y_{2}\in C^{(m)}[0,1]\)
such that, for
\(0\leq k\leq m-2\),
$$ \left \{ \textstyle\begin{array}{@{}l} 0\leq y_{1}^{(k)}(t)< \frac {t^{m-k-2}}{(m-k-2)!} a\leq\frac{a}{(m-k-2)!},\quad t\in[0,1],\\ 0\leq y_{2}^{(k)}(t)\leq\frac{t^{m-k-2}}{(m-k-2)!} b\leq \frac{b}{(m-k-2)!},\quad t\in[0,1],\\ y_{2}^{(k)}(t)>\frac{(t-\eta)^{m-k-2}}{(m-k-2)!} \gamma a,\quad t\in[\eta,1-\eta]. \end{array}\displaystyle \right . $$
(3.33)
Proof
From the proofs of Theorems 3.6 and 3.7 we see that (3.4) has two positive solutions \(x_{1}\in B\) and \(x_{2}\in C\) (\(x_{2}\) is nontrivial) such that
$$ 0\leq\|x_{1}\|< a< \|x_{2}\|\leq b. $$
(3.34)
By Remark 3.1, (1.1) has two positive solutions \(y_{1}=J^{q}x_{1}\) and \(y_{2}=J^{q} x_{2}\) (\(y_{2}\) is nontrivial). Using (3.34) in (3.15) and (3.16) gives (3.33) immediately. □
One of the solutions \((y_{1})\) may be trivial in Theorem 3.8. Our next result guarantees the existence of two nontrivial positive solutions.
Theorem 3.9
Let (A1)-(A5) and
\((\mathrm{A}5)|_{b=b'}\)
hold, where
\(0< b'< a< b\). Then, (1.1) has (at least) two nontrivial positive solutions
\(y_{1}, y_{2}\in C^{(m)}[0,1]\)
such that, for
\(0\leq k\leq m-2\),
$$ \left \{ \textstyle\begin{array}{@{}l} 0\leq y_{1}^{(k)}(t)< \frac {t^{m-k-2}}{(m-k-2)!} a\leq\frac{a}{(m-k-2)!},\quad t\in[0,1],\\ y_{1}^{(k)}(t)\geq\frac{(t-\eta)^{m-k-2}}{(m-k-2)!} \gamma b',\quad t\in[\eta,1-\eta],\\ 0\leq y_{2}^{(k)}(t)\leq\frac{t^{m-k-2}}{(m-k-2)!} b\leq \frac{b}{(m-k-2)!},\quad t\in[0,1],\\ y_{2}^{(k)}(t)>\frac{(t-\eta)^{m-k-2}}{(m-k-2)!} \gamma a,\quad t\in[\eta,1-\eta]. \end{array}\displaystyle \right . $$
(3.35)
Proof
From the proof of Theorem 3.7 (see (3.32)) we derive that (3.4) has two nontrivial positive solutions \(x_{1},x_{2}\in C\) such that
$$ 0< b'\leq\|x_{1}\|< a< \|x_{2}\|\leq b. $$
(3.36)
By Remark 3.1, (1.1) has two nontrivial positive solutions \(y_{1}=J^{q}x_{1}\) and \(y_{2}=J^{q} x_{2}\). Using (3.36) in (3.15) and (3.16) gives (3.35) immediately. □
Note that in Theorem 3.9, both (A3) and (A5) are required to obtain the existence of two nontrivial positive solutions. In the next two theorems, only one of (A3) and (A5) is used to ensure the existence of two nontrivial positive solutions. Define
$$\begin{aligned}& f_{0}=\lim_{u_{i}\to0+, 1\leq i\leq q+1}\frac{f(u_{1},\ldots ,u_{q+1})}{u_{q+1}} \quad\mbox{and}\\& f_{\infty}=\lim_{u_{i}\to\infty, 1\leq i\leq q+1}\frac{f(u_{1},\ldots,u_{q+1})}{u_{q+1}}. \end{aligned}$$
Theorem 3.10
Let (A1)-(A4) hold and
\(0<\int_{z}^{1-\eta} G(s,s)\alpha(s)\,ds<\infty\).
-
(a)
If
\(f_{0}=\infty\), then (1.1) has a nontrivial positive solution
\(y_{1}\in C^{(m)}[0,1]\)
such that, for
\(0\leq k\leq m-2\),
$$ 0\leq y_{1}^{(k)}(t)< \frac{t^{m-k-2}}{(m-k-2)!} a\leq \frac {a}{(m-k-2)!},\quad t\in[0,1]. $$
(3.37)
-
(b)
If
\(f_{\infty}=\infty\), then (1.1) has a nontrivial positive solution
\(y_{2}\in C^{(m)}[0,1]\)
such that, for
\(0\leq k\leq m-2\),
$$ y_{2}^{(k)}(t)>\frac{(t-\eta)^{m-k-2}}{(m-k-2)!} \gamma a,\quad t\in[\eta,1- \eta]. $$
(3.38)
-
(c)
If
\(f_{0}=f_{\infty}=\infty\), then (1.1) has (at least) two nontrivial positive solutions
\(y_{1},y_{2}\in C^{(m)}[0,1]\)
such that (3.37) and (3.38) hold for
\(0\leq k\leq m-2\).
Proof
We apply Theorem 2.2 with the operator S and the cone C defined respectively in (3.17) and (3.7). As seen in the proof of Theorem 3.7, S maps C into C. Let \(\Omega_{a}=\{x\in B \mid \|x\|< a\}\). Using (A2) and (A3) as in the proof of Theorem 3.7, we obtain (3.30), and hence
$$ \|Sx\|\leq\|x\|,\quad x\in C\cap\partial\Omega_{a}. $$
(3.39)
(a) Define
$$ P= \biggl[\frac{(z-\eta)^{m-q-2}\gamma K_{\eta}}{(m-q-2)!} \int _{z}^{1-\eta} G(s,s)\alpha(s)\,ds \biggr]^{-1}. $$
(3.40)
Since \(f_{0}=\infty\), there exists \(0< r< a\) such that
$$ f(u_{1},\ldots,u_{q+1})\geq P u_{q+1},\quad 0< u_{i}\leq r, 1\leq i\leq q+1. $$
(3.41)
Let \(\Omega_{r}=\{x\in B \mid \|x\|< r\}\). Let \(x\in C\cap \partial\Omega_{r}\), so \(\|x\|=r\). Note that from (3.13) and (3.9) we have
$$ J^{k}x(s)\leq\frac{\|x\|}{(m-q-2+k)!}=\frac{r}{(m-q-2+k)!}< r,\quad s\in [0,1], 0\leq k\leq q . $$
(3.42)
For \(t\in[\eta,1-\eta]\), we use (A4), Lemma 2.3(c), (3.42), (3.41), (3.12), and (3.40) successively to get
$$\begin{aligned} \bigl|(Sx)^{(m-q-2)}(t)\bigr| \geq& \int_{z}^{1-\eta} G(t,s) F \bigl(s,\tilde{J}x(s) \bigr)\,ds \\ \geq& \int_{z}^{1-\eta}K_{\eta}G(s,s) \alpha(s)f \bigl(\tilde{J}x(s) \bigr)\,ds \\ \geq& \int_{z}^{1-\eta}K_{\eta}G(s,s) \alpha(s) Px(s)\,ds \\ \geq& \int_{z}^{1-\eta}K_{\eta}G(s,s) \alpha(s) P \frac{(z-\eta)^{m-q-2}\gamma\|x\|}{(m-q-2)!}\,ds = \|x\|. \end{aligned}$$
Hence, we have
$$ \|Sx\|\geq\|x\|,\quad x\in C\cap\partial\Omega_{r}. $$
(3.43)
Having established (3.39) and (3.43), by Theorem 2.2 we conclude that S has a fixed point \(x_{1}\in C\cap(\overline{\Omega}_{a}\backslash\Omega_{r})\) such that \(r\leq \|x_{1}\|\leq a\). Using a similar argument as in the first part of the proof of Theorem 3.6, we see that \(\|x_{1}\|\neq a\). Hence, we get \(r\leq\|x_{1}\|< a\) (\(x_{1}\) is nontrivial). By Remark 3.1, (1.1) has a nontrivial positive solution \(y_{1}=J^{q}x_{1}\). Since \(\|x_{1}\|< a\), (3.37) is immediate from (3.15).
(b) Since \(f_{\infty}=\infty\), we may choose \(w>a\) such that
$$ f(u_{1},\ldots,u_{q+1})\geq Pu_{q+1}, \quad u_{i}\geq w, 1\leq i\leq q+1, $$
(3.44)
where P is defined in (3.40). Let
$$w_{0}=\max \biggl\{ w \biggl[\frac{(z-\eta)^{m-q-2}\gamma }{(m-q-2)!} \biggr]^{-1}, w \biggl[\frac{(z-\eta)^{m-q-2+k}\gamma}{(m-q-2+k)!} \biggr]^{-1}, 1\leq k\leq q \biggr\} = \frac{w(m-2)!}{\gamma(z-\eta)^{m-2}}. $$
Clearly, \(w_{0}>w>a\). Let \(\Omega_{w_{0}}=\{x\in B \mid \|x\|< w_{0}\}\). Let \(x\in C\cap \partial\Omega_{w_{0}}\), so that \(\|x\|=w_{0}\). Note that from (3.12), (3.14), and the definition of \(w_{0}\) we have that, for \(s\in[z,1-\eta]\),
$$ \left \{ \textstyle\begin{array}{@{}l} x(s)\geq \frac{(z-\eta)^{m-q-2}\gamma}{(m-q-2)!} \|x\|=\frac{(z-\eta )^{m-q-2}\gamma}{(m-q-2)!} w_{0}\geq w, \\ J^{k}x(s)\geq \frac{(z-\eta)^{m-q-2+k}\gamma}{(m-q-2+k)!} \|x\|=\frac{(z-\eta )^{m-q-2+k}\gamma}{(m-q-2+k)!} w_{0}\geq w,\quad 1\leq k\leq q. \end{array}\displaystyle \right . $$
(3.45)
Using (A4), Lemma 2.3(c), (3.45), (3.44), (3.12), and (3.40) successively, we get that, for \(t\in[\eta, 1-\eta]\),
$$\begin{aligned} \bigl|(Sx)^{(m-q-2)}(t)\bigr| \geq& \int_{z}^{1-\eta}K_{\eta}G(s,s) \alpha(s)f \bigl(\tilde{J}x(s) \bigr)\,ds \\ \geq& \int_{z}^{1-\eta}K_{\eta}G(s,s) \alpha(s) Px(s)\,ds \\ \geq& \int_{z}^{1-\eta}K_{\eta}G(s,s) \alpha(s) P \frac{(z-\eta)^{m-q-2}\gamma\|x\|}{(m-q-2)!}\,ds = \|x\|. \end{aligned}$$
It follows that
$$ \|Sx\|\geq\|x\|,\quad x\in C\cap\partial\Omega_{w_{0}}. $$
(3.46)
With (3.39) and (3.46), by Theorem 2.2 we conclude that S has a fixed point \(x_{2}\in C\cap(\overline{\Omega}_{w_{0}}\backslash \Omega_{a})\) such that \(a\leq\|x_{2}\|\leq w_{0}\). Once again, as seen earlier, \(\|x_{2}\|\neq a\), so that \(a<\|x_{2}\|\leq w_{0}\) (\(x_{2}\) is nontrivial). By Remark 3.1, (1.1) has a nontrivial positive solution \(y_{2}=J^{q}x_{2}\). Since \(\|x_{2}\|>a\), (3.38) is immediate from (3.16).
(c) This follows from Cases (a) and (b). □
Theorem 3.11
Let (A1), (A2), (A4), (A5) hold, and
\(0<\int_{0}^{1} G(s,s)\beta(s)\,ds<\infty\).
-
(a)
If
\(f_{0}=0\), then (1.1) has a nontrivial positive solution
\(y_{1}\in C^{(m)}[0,1]\)
such that, for
\(0\leq k\leq m-2\),
$$ 0\leq y_{1}^{(k)}(t)\leq\frac{t^{m-k-2}}{(m-k-2)!} b\leq \frac {b}{(m-k-2)!},\quad t\in[0,1]. $$
(3.47)
-
(b)
If
\(f_{\infty}=0\), then (1.1) has a nontrivial positive solution
\(y_{2}\in C^{(m)}[0,1]\)
such that, for
\(0\leq k\leq m-2\),
$$ y_{2}^{(k)}(t)\geq\frac{(t-\eta)^{m-k-2}}{(m-k-2)!} \gamma b,\quad t\in[ \eta,1-\eta]. $$
(3.48)
-
(c)
If
\(f_{0}=f_{\infty}=0\), then (1.1) has (at least) two nontrivial positive solutions
\(y_{1},y_{2}\in C^{(m)}[0,1]\)
such that (3.47) and (3.48) hold for
\(0\leq k\leq m-2\).
Proof
Once again, we apply Theorem 2.2 with the operator S and the cone C defined respectively in (3.17) and (3.7). Let \(\Omega_{b}=\{x\in B \mid \|x\|< b\}\). Using (A4) and (A5) as in the proof of Theorem 3.7, we obtain (3.31), and so
$$ \|Sx\|\geq\|x\|,\quad x\in C\cap\partial\Omega_{b}. $$
(3.49)
(a) Let
$$ T= \biggl[\frac{L}{(m-q-2)!} \int_{0}^{1} G(s,s)\beta(s)\,ds \biggr]^{-1}. $$
(3.50)
Since \(f_{0}=0\), there exists \(0< r< b\) such that
$$ f(u_{1},\ldots,u_{q+1})\leq Tu_{q+1},\quad 0< u_{i}\leq r, 1\leq i\leq q+1. $$
(3.51)
Let \(\Omega_{r}=\{x\in B \mid \|x\|< r\}\). Let \(x\in C\cap \partial\Omega_{r}\), so \(\|x\|=r\). Note that (3.42) holds. Using (A2), Lemma 2.3(b), (3.42), (3.51), (3.9), and (3.50) successively, we find that, for \(t\in[0,1]\),
$$\begin{aligned} \bigl|(Sx)^{(m-q-2)}(t)\bigr| \leq& \int_{0}^{1} LG(s,s) \beta(s)f \bigl(\tilde{J}x(s) \bigr)\,ds \\ \leq& \int_{0}^{1} LG(s,s)\beta(s) Tx(s)\,ds \leq \int_{0}^{1} LG(s,s)\beta(s)T \frac{\|x\|}{(m-q-2)!}\,ds = \|x\|. \end{aligned}$$
Hence, we have
$$ \|Sx\|\leq\|x\|,\quad x\in C\cap \partial\Omega_{r}. $$
(3.52)
Noting (3.49) and (3.52), it follows from Theorem 2.2 that S has a fixed point \(x_{1}\in C\cap(\overline{\Omega}_{b}\backslash\Omega_{r})\) such that \(r\leq\|x_{1}\|\leq b\) (\(x_{1}\) is nontrivial). Hence, we see from Remark 3.1 that (1.1) has a nontrivial positive solution \(y_{1}=J^{q}x_{1}\). Using \(\|x_{1}\|\leq b\) in (3.15) yields (3.47) immediately.
(b) Since \(f_{\infty}=0\), we may choose \(w>b\) such that
$$ f(u_{1},\ldots,u_{q+1})\leq Tu_{q+1},\quad u_{i}\geq w, 1\leq i\leq q+1, $$
(3.53)
where T is defined in (3.50). To proceed, we consider two cases, when f is bounded and when f is unbounded.
Case 1. Suppose that f is bounded. Then, for some \(A>0\),
$$ f(u_{1},\ldots,u_{q+1})\leq A,\quad u_{i}\in[0,\infty), 1\leq i\leq q+1. $$
(3.54)
Let
$$w_{0}=\max \biggl\{ b+1, LA \int_{0}^{1}G(s,s)\beta(s)\,ds \biggr\} . $$
Clearly, \(w_{0}>b\). Let \(\Omega_{w_{0}}=\{x\in B \mid \|x\|< w_{0}\}\). Let \(x\in C\cap\partial\Omega_{w_{0}}\), so \(\|x\|=w_{0}\). Using (A2), Lemma 2.3(b), and (3.54) provides, for \(t\in[0,1]\),
$$\begin{aligned} \bigl|(Sx)^{(m-q-2)}(t)\bigr| \leq& \int_{0}^{1}LG(s,s)\beta(s)f \bigl(\tilde{J}x(s) \bigr)\,ds \\ \leq& \int_{0}^{1}LG(s,s)\beta(s)A \,ds \leq w_{0} = \|x\|. \end{aligned}$$
Hence, we have
$$ \|Sx\|\leq\|x\|,\quad x\in C\cap \partial\Omega_{w_{0}}. $$
(3.55)
Case 2. Suppose that f is unbounded. Then, there exists \(w_{0}>w(m-2)! \) (>b) such that
$$\begin{aligned} &f(u_{1},\ldots,u_{q+1})\leq f \biggl(\frac{w_{0}}{(m-2)!}, \frac {w_{0}}{(m-3)!}, \ldots, \frac{w_{0}}{(m-q-2)!} \biggr), \\ &\quad 0\leq u_{i}\leq w_{0}, 1\leq i\leq q+1. \end{aligned}$$
(3.56)
Let \(\Omega_{w_{0}}=\{x\in B \mid \|x\|< w_{0}\}\). Let \(x\in C\cap \partial\Omega_{w_{0}}\), so \(\|x\|=w_{0}\). It follows from (3.13) and (3.9) that
$$ J^{k}x(s)\leq\frac{\|x\|}{(m-q-2+k)!}=\frac{w_{0}}{(m-q-2+k)!}< w_{0},\quad s \in [0,1], 0\leq k\leq q . $$
(3.57)
Now, we apply (A2), Lemma 2.3(b), (3.57), (3.56), (3.53), and (3.50) successively to obtain, for \(t\in[0,1]\),
$$\begin{aligned} \bigl|(Sx)^{(m-q-2)}(t)\bigr| \leq& \int_{0}^{1}LG(s,s)\beta(s)f \bigl(\tilde{J}x(s) \bigr)\,ds \\ \leq& \int_{0}^{1}LG(s,s)\beta(s)f \biggl( \frac{w_{0}}{(m-2)!}, \frac {w_{0}}{(m-3)!}, \ldots, \frac{w_{0}}{(m-q-2)!} \biggr)\,ds \\ \leq& \int_{0}^{1}LG(s,s)\beta(s)T \frac{w_{0}}{(m-q-2)!}\,ds = w_{0} = \|x\|. \end{aligned}$$
It follows that \(\|Sx\|\leq\|x\|\) for \(x\in C\cap \partial\Omega_{w_{0}}\), that is, (3.55) holds.
Having established (3.49) and (3.55), by Theorem 2.2 we see that S has a fixed point \(x_{2}\in C\cap(\overline{\Omega}_{w_{0}}\backslash \Omega_{b})\) such that \(b\leq\|x_{2}\|\leq w_{0}\) (\(x_{2}\) is nontrivial). It follows from Remark 3.1 that (1.1) has a nontrivial positive solution \(y_{2}=J^{q}x_{2}\). Using \(\|x_{2}\|\geq b\) in (3.16) leads to (3.48) immediately.
(c) This follows from Cases (a) and (b). □
Remark 3.4
Comparing Theorem 3.9 with Theorems 3.10(c) and 3.11(c), we note that all of them guarantee the existence of two nontrivial positive solutions of (1.1); also, conclusion (3.35) in Theorem 3.9 gives more details than the conclusions in Theorems 3.10(c) and 3.11(c). This might be explained by the fact that condition (A5) is required in Theorem 3.9
twice but not at all in Theorems 3.10(c) and 3.11(c); further, more effort might be needed to check (A5). Therefore, the ‘more’ details in (3.35) require possibly greater efforts.
Using the earlier results, we now give the existence of multiple positive solutions of (1.1).
Theorem 3.12
Let (A1), (A2), and (A4) hold. Suppose that (A3) is satisfied for
\(a=a_{\ell}\), \(\ell=1,2,\ldots,k\), and (A5) is satisfied for
\(b=b_{\ell}\), \(\ell=1,2,\ldots,n\).
-
(a)
If
\(n=k+1\)
and
\(0< b_{1}< a_{1}<\cdots<b_{k} <a_{k}<b_{k+1}\), then (1.1) has (at least) 2k
nontrivial positive solutions
\(y_{1},\ldots, y_{2k}\in C^{(m)}[0,1]\)
such that, for
\(0\leq i\leq m-2\)
and
\(\ell=1,2,\ldots,k\),
$$ \left \{ \textstyle\begin{array}{@{}l} 0\leq y_{2\ell-1}^{(i)}(t)< \frac {t^{m-i-2}}{(m-i-2)!} a_{\ell}\leq\frac{a_{\ell}}{(m-i-2)!},\quad t\in[0,1],\\ y_{2\ell-1}^{(i)}(t)\geq\frac{(t-\eta )^{m-i-2}}{(m-i-2)!} \gamma b_{\ell}, \quad t\in[\eta,1-\eta],\\ 0\leq y_{2\ell}^{(i)}(t)\leq \frac{t^{m-i-2}}{(m-i-2)!} b_{\ell+1}\leq \frac{b_{\ell+1}}{(m-i-2)!},\quad t\in[0,1],\\ y_{2\ell}^{(i)}(t)>\frac{(t-\eta)^{m-i-2}}{(m-i-2)!} \gamma a_{\ell},\quad t\in[\eta,1-\eta]. \end{array}\displaystyle \right . $$
(3.58)
-
(b)
If
\(n=k\)
and
\(0< b_{1}< a_{1}<\cdots<b_{k} <a_{k}\), then (1.1) has (at least) \(2k-1\)
nontrivial positive solutions
\(y_{1},\ldots, y_{2k-1}\in C^{(m)}[0,1]\)
such that, for
\(0\leq i\leq m-2\), \(\ell=1,2,\ldots,k\), and
\(j=1,2,\ldots,k-1\),
$$ \left \{ \textstyle\begin{array}{@{}l} 0\leq y_{2\ell-1}^{(i)}(t)< \frac {t^{m-i-2}}{(m-i-2)!} a_{\ell}\leq\frac{a_{\ell}}{(m-i-2)!},\quad t\in[0,1],\\ y_{2\ell-1}^{(i)}(t)\geq\frac{(t-\eta )^{m-i-2}}{(m-i-2)!} \gamma b_{\ell},\quad t\in[\eta,1-\eta],\\ 0\leq y_{2j}^{(i)}(t)\leq\frac{t^{m-i-2}}{(m-i-2)!} b_{j+1}\leq \frac{b_{j+1}}{(m-i-2)!},\quad t\in[0,1],\\ y_{2j}^{(i)}(t)>\frac{(t-\eta)^{m-i-2}}{(m-i-2)!} \gamma a_{j},\quad t\in[\eta,1-\eta]. \end{array}\displaystyle \right . $$
(3.59)
-
(c)
If
\(k=n+1\)
and
\(0< a_{1}< b_{1}<\cdots<a_{n}<b_{n} <a_{n+1}\), then (1.1) has (at least) \(2n+1\)
positive solutions
\(y_{0},\ldots, y_{2n}\in C^{(m)}[0,1]\), where
\(y_{1},\ldots,y_{2n}\)
are nontrivial, such that, for
\(0\leq i\leq m-2\)
and
\(\ell=1,2,\ldots,n\),
$$ \left \{ \textstyle\begin{array}{@{}l} 0\leq y_{0}^{(i)}(t)< \frac{t^{m-i-2}}{(m-i-2)!} a_{1}\leq \frac{a_{1}}{(m-i-2)!},\quad t\in[0,1],\\ 0\leq y_{2\ell-1}^{(i)}(t)\leq \frac{t^{m-i-2}}{(m-i-2)!} b_{\ell}\leq \frac{b_{\ell}}{(m-i-2)!},\quad t\in[0,1],\\ y_{2\ell-1}^{(i)}(t)>\frac{(t-\eta)^{m-i-2}}{(m-i-2)!} \gamma a_{\ell},\quad t\in[\eta,1-\eta],\\ 0\leq y_{2\ell}^{(i)}(t)< \frac{t^{m-i-2}}{(m-i-2)!} a_{\ell+1}\leq \frac{a_{\ell+1}}{(m-i-2)!},\quad t\in[0,1],\\ y_{2\ell}^{(i)}(t)\geq\frac{(t-\eta )^{m-i-2}}{(m-i-2)!} \gamma b_{\ell},\quad t\in[\eta,1-\eta]. \end{array}\displaystyle \right . $$
(3.60)
-
(d)
If
\(k=n\)
and
\(0< a_{1}< b_{1}<\cdots<a_{k}<b_{k}\), then (1.1) has (at least) 2k
positive solutions
\(y_{0},\ldots, y_{2k-1}\in C^{(m)}[0,1]\), where
\(y_{1},\ldots, y_{2k-1}\)
are nontrivial, such that, for
\(0\leq i\leq m-2\), \(\ell=1,2,\ldots,k\), and
\(j=1,2,\ldots,k-1\),
$$ \left \{ \textstyle\begin{array}{@{}l} 0\leq y_{0}^{(i)}(t)< \frac{t^{m-i-2}}{(m-i-2)!} a_{1}\leq \frac{a_{1}}{(m-i-2)!},\quad t\in[0,1],\\ 0\leq y_{2\ell-1}^{(i)}(t)\leq \frac{t^{m-i-2}}{(m-i-2)!} b_{\ell}\leq \frac{b_{\ell}}{(m-i-2)!}, \quad t\in[0,1],\\ y_{2\ell-1}^{(i)}(t)>\frac{(t-\eta)^{m-i-2}}{(m-i-2)!} \gamma a_{\ell},\quad t\in[\eta,1-\eta], \\ 0\leq y_{2j}^{(i)}(t)< \frac{t^{m-i-2}}{(m-i-2)!} a_{j+1}\leq \frac{a_{j+1}}{(m-i-2)!},\quad t\in[0,1],\\ y_{2j}^{(i)}(t)\geq\frac{(t-\eta)^{m-i-2}}{(m-i-2)!} \gamma b_{j}, \quad t\in[\eta,1-\eta]. \end{array}\displaystyle \right . $$
(3.61)
Proof
The proof involves repeated usage of Theorems 3.6 and 3.7. In (a) and (b), we apply (3.32) repeatedly to get multiple positive solutions of (3.4) as follows.
-
(a)
If \(n=k+1\) and \(0< b_{1}< a_{1}<\cdots<b_{k} <a_{k}<b_{k+1}\), then (3.4) has (at least) 2k nontrivial positive solutions \(x_{1},\ldots, x_{2k}\in C\) such that
$$ 0< b_{1}\leq\|x_{1}\|< a_{1}< \|x_{2}\|\leq b_{2}\leq \cdots< a_{k}< \|x_{2k}\|\leq b_{k+1}. $$
(3.62)
-
(b)
If \(n=k\) and \(0< b_{1}< a_{1}<\cdots<b_{k} <a_{k}\), then (3.4) has (at least) \(2k-1\) nontrivial positive solutions \(x_{1},\ldots, x_{2k-1}\in C\) such that
$$ 0< b_{1}\leq\|x_{1}\|< a_{1}< \|x_{2}\|\leq b_{2}\leq\cdots \leq b_{k}\leq\|x_{2k-1} \|< a_{k}. $$
(3.63)
Hence, conclusions (a) and (b) follow from Remark 3.1. Inequalities (3.58) and (3.59) are obtained by using (3.62) and (3.63) in (3.15) and (3.16).
Next, in (c) and (d), from the proof of Theorem 3.6 we see that (3.4) has a positive solution \(x_{0}\in B\) with \(0\leq\|x_{0}\|< a_{1}\). Applying (3.32) repeatedly again, we get more solutions as follows.
-
(c)
If \(k=n+1\) and \(0< a_{1}< b_{1}<\cdots<a_{n}<b_{n} <a_{n+1}\), then (3.4) has (at least) \(2n+1\) positive solutions \(x_{0}\in B\), \(x_{1},\ldots, x_{2n}\in C\) such that
$$ 0\leq\|x_{0}\|< a_{1}< \|x_{1}\|\leq b_{1} \leq\|x_{2}\|< a_{2}< \cdots \leq b_{n}\leq \|x_{2n}\|< a_{n+1}. $$
(3.64)
-
(d)
If \(k=n\) and \(0< a_{1}< b_{1}<\cdots<a_{k}<b_{k}\), then (3.4) has (at least) 2k positive solutions \(x_{0}\in B\), \(x_{1},\ldots, x_{2k-1}\in C\) such that
$$ 0\leq\|x_{0}\|< a_{1}< \|x_{1}\|\leq b_{1} \leq\|x_{2}\|< a_{2}< \cdots < a_{k}< \|x_{2k-1} \|\leq b_{k}. $$
(3.65)
Hence, conclusions (c) and (d) follow from Remark 3.1. Inequalities (3.60) and (3.61) are obtained by using (3.64) and (3.65) in (3.15) and (3.16). □