In this section, we give some definitions and the fixed point theorem that will be used in this paper.
Definition 1
Let E be a real Banach space. A nonempty, closed, and convex set \(P\subset E\) is a cone if the following two conditions are satisfied:
-
(i)
if \(x \in P\) and \(\mu\geq0 \), then \(\mu x \in P\);
-
(ii)
if \(x \in P \) and \(-x \in P\), then \(x=0\).
Every cone \(P\subset E\) induces the ordering in E given by \(x_{1}\leq x_{2}\) if and only if \(x_{2}-x_{1} \in P\).
Definition 2
A map α is called a nonnegative continuous convex functional on a cone P in a real Banach space E if \(\alpha:P\rightarrow[0,+\infty)\) is continuous and
$$\alpha\bigl(\lambda x_{1}+(1-\lambda)x_{2}\bigr)\leq\lambda \alpha(x_{1})+(1-\lambda)\alpha(x_{2}) $$
for all \(x_{1},x_{2}\in P\) and \(0\leq\lambda\leq1\). Likewise, we know the map β is a nonnegative continuous concave functional on a cone P in a real Banach space E if \(\beta :P\rightarrow[0,+\infty)\) is continuous and
$$\beta\bigl(\lambda x_{1}+(1-\lambda)x_{2}\bigr)\geq\lambda \beta(x_{1})+(1-\lambda)\beta(x_{2}) $$
for all \(x_{1},x_{2}\in P\) and \(0\leq\lambda\leq1\).
We denote \(E=C(I)\), \(I=[0,1]\), with the maximum norm, and for all \(0<\widetilde{t}\leq\frac{1}{2}\), we define the cone \(P\subset E\) by
$$\begin{aligned} P=&\Bigl\{ u \in E: u(t) \mbox{ is concave, symmetric, and nonnegative-valued on }I, \\ &\mbox{and } \min_{t\in[\widetilde{t},1-\widetilde{t}]}u(t)\geq 2\widetilde{t}\Vert u \Vert \Bigr\} . \end{aligned}$$
Theorem 1
(Leggett-Williams fixed-point theorem [21])
Let
\(P \subset E\)
be a cone in a real Banach space
E. Let
\(l>0\)
and
\(N>0\), let
β
and
χ
be nonnegative continuous concave functionals on
P, and let
ζ, α, and
ρ
be nonnegative continuous convex functionals on
P
with
$$\beta(u)\leq\alpha(u),\qquad\Vert u \Vert\leq N\zeta(u), $$
for all
\(u \in\overline{\mathbb{P}(\zeta, l)}\). Suppose that
\(Y:\overline{\mathbb{P}(\zeta, l)}\rightarrow\overline{\mathbb {P}(\zeta, l)}\)
is a completely continuous operator and that there exist numbers
\(h>0\), \(d>0\), \(p, q>0\)
with
\(d< p\)
such that:
$$\begin{aligned} &u\in\mathbb{P}(\zeta,\rho, \beta, p,q, l):\quad \beta(u) >p\neq\emptyset \quad \textit{and}\quad\beta(Fu)>p \quad\textit{for } u\in\mathbb{P}(\zeta,\rho , \beta, p,q, l); \\ & u\in\mathbb{Q}(\zeta, \alpha,\chi, h, d, l):\quad \alpha(u)< d \neq \emptyset\quad \textit{and}\quad\alpha(Fu)< d \quad\textit{for } u\in\mathbb{Q}(\zeta, \alpha,\chi, h, d, l); \\ &\alpha(Fu)>p \quad\textit{for } u\in\mathbb{P}(\zeta, \beta, p, l) \textit { with } \rho(Fu)>q; \\ &\beta(Fu)< d \quad\textit{for } u\in\mathbb{Q}(\zeta, \alpha, d, l) \textit { with } \chi(Fu)< h. \end{aligned}$$
Then there exist at least three fixed points
\(u_{1}, u_{2}, u_{3} \in \overline{\mathbb{P}(\zeta, l)}\)
such that
$$\alpha(u_{1})< d,\qquad p < \beta(u_{2}), \quad\textit{and} \quad d< \alpha(u_{3}) \quad \textit{with } \beta(u_{3})< p. $$
Thereinto, some sets are as follows:
$$\begin{aligned} &\mathbb{P}(\zeta, l)=\bigl\{ u \in P:\zeta(u)< l\bigr\} , \\ &\mathbb{P}(\zeta,\beta, p, l)=\bigl\{ u \in P: p \leq\beta(u), \zeta (u)< l\bigr\} , \\ &\mathbb{P}(\zeta,\rho, \beta, p,q, l)=\bigl\{ u \in P: p \leq\beta (u),\rho(u) \leq q, \zeta(u)< l\bigr\} , \\ &\mathbb{Q}(\zeta, \alpha, d, l)=\bigl\{ u \in P: \alpha(u) \leq d, \zeta(u)< l \bigr\} , \\ &\mathbb{Q}(\zeta, \alpha,\chi, h, d, l)=\bigl\{ u \in P: h\leq\chi(u), \alpha(u) \leq d, \zeta(u)< l\bigr\} . \end{aligned}$$
Many other functionals on the cone P are defined by
$$\begin{aligned} &\beta(u)=\min_{t\in[t_{0},t_{1}]\cup[1-t_{1},1-t_{0}]}u(t)=u(t_{0}), \\ &\chi(u)=\min_{t\in[\frac{1}{\omega},\frac{\omega-1}{\omega }]}u(t)=u\biggl(\frac{1}{\omega}\biggr), \\ &\alpha(u)=\max_{t\in[\frac{1}{\omega},\frac{\omega-1}{\omega }]}u(t)=u\biggl(\frac{1}{2}\biggr), \\ &\rho(u)=\max_{t\in[t_{0},t_{1}]\cup[1-t_{1},1-t_{0}]}u(t)=u(t_{1}), \\ &\zeta(u)=\max_{t\in[0,\widetilde{t}]\cup[1-\widetilde {t},1]}u(t)=u(\widetilde{t}), \end{aligned}$$
where \(t_{0}\), \(t_{1}\), and ω are nonnegative numbers such that
$$0< t_{0}< t_{1}\leq\frac{1}{2} \quad\mbox{and}\quad \frac{1}{\omega}\leq t_{1}. $$
It is clear that, for all \(u\in P\),
$$\begin{aligned} &\beta(u)=u(t_{0})\leq u\biggl(\frac{1}{2}\biggr)= \alpha(u), \end{aligned}$$
(2)
$$\begin{aligned} &\Vert u \Vert=u\biggl(\frac{1}{2}\biggr)\leq\frac{1}{2\widetilde {t}}u( \widetilde{t})=\frac{1}{2\widetilde{t}}\zeta(u). \end{aligned}$$
(3)
Throughout the paper, we suppose that the following two conditions hold.
- (\(H_{0}\)):
-
\(a>1\), \(-1< b<0\);
- (\(H_{1}\)):
-
\(\varrho_{0}>0\) and \(\varrho_{1}>0\) are continuous on I, and the supplementary function \(\varphi(t, s)\), defined by
$$\varphi(t, s)=\frac{a+t}{1+a-b}\varrho_{1}(s)-\frac {b-1+t}{1+a-b} \varrho_{0}(s),\quad t, s \in I, $$
satisfies
$$0\leq m:=\min_{t, s \in I}\varphi(t, s)\leq M:=\max _{t, s \in I}\varphi(t, s)< 1. $$