For notational convenience, we denote
$$Q= \int^{1}_{0}q(s)\,ds, \qquad A=\max\bigl\{ 1,4(q-1) \bigr\} \biggl(\frac{1}{4}\lambda Q\biggr)^{q-1}. $$
Theorem 3.1
Assume that (H1)-(H3) hold, and there exists
\(a>0\), such that
- (H4)::
-
\(f(t,x_{1},y_{1},z_{1},\delta_{1})\leq f(t,x_{2},y_{2},z_{2},\delta_{2})\)
for any
\(0\leq t\leq1\), \(0\leq x_{1}\leq x_{2}\leq a\), \(0\leq|y_{1}|\leq |y_{2}|\leq a\), \(-a\leq z_{2}\leq z_{1}\leq0\), \(0\leq|\delta _{1}|\leq |\delta_{2}|\leq a\);
- (H5)::
-
\(\max_{0\leq t\leq1} f(t,a,a,-a,a)\leq(\frac {a}{A})^{p-1}\);
- (H6)::
-
\(f(t,0,0,0,0)\not\equiv0\)
for
\(0\leq t\leq1\).
Then the boundary value problem (1.1) has at least one positive symmetric concave solution
\(w^{\ast}\)
or
\(v^{\ast}\), such that
$$\begin{aligned}& 0\leq w^{\ast}\leq a, \qquad0\leq\bigl\vert \bigl(w^{\ast} \bigr)'\bigr\vert \leq a, \\& -a\leq\bigl(w^{\ast} \bigr)''\leq0, \qquad0\leq\bigl\vert \bigl(w^{\ast}\bigr)'''\bigr\vert \leq a, \\& w^{\ast}=\lim_{n\rightarrow\infty}w_{n}=\lim _{n\rightarrow\infty}T^{n}w_{0}, \\& \bigl(w^{\ast}\bigr)'=\lim_{n\rightarrow\infty}(w_{n})'= \lim_{n\rightarrow\infty }\bigl(T^{n}w_{0} \bigr)', \\& \bigl(w^{\ast}\bigr)''=\lim _{n\rightarrow\infty}(w_{n})''=\lim _{n\rightarrow \infty}\bigl(T^{n}w_{0} \bigr)'', \\& \bigl(w^{\ast} \bigr)'''=\lim_{n\rightarrow\infty}(w_{n})'''= \lim_{n\rightarrow \infty}\bigl(T^{n}w_{0} \bigr)''', \\& \quad\textit{where } w_{0}(t)=a \biggl(\frac{4}{3}t^{4}- \frac{8}{3}t^{3}+\frac {4}{3}t+\frac{3}{8} \biggr), 0 \leq t\leq1, \end{aligned}$$
and
$$\begin{aligned}& 0\leq v^{\ast}\leq a, \qquad0\leq\bigl\vert \bigl(v^{\ast} \bigr)'\bigr\vert \leq a, \\& -a\leq\bigl(v^{\ast} \bigr)''\leq0, \qquad0\leq\bigl\vert \bigl(v^{\ast}\bigr)'''\bigr\vert \leq a, \\& v^{\ast}=\lim_{n\rightarrow\infty}v_{n}=\lim _{n\rightarrow\infty}T^{n}v_{0}, \\& \bigl(v^{\ast}\bigr)'=\lim_{n\rightarrow\infty}(v_{n})'= \lim_{n\rightarrow\infty }\bigl(T^{n}v_{0} \bigr)', \\& \bigl(v^{\ast}\bigr)''=\lim _{n\rightarrow\infty}(v_{n})''=\lim _{n\rightarrow \infty}\bigl(T^{n}v_{0} \bigr)'', \\& \bigl(v^{\ast} \bigr)'''=\lim_{n\rightarrow\infty}(v_{n})'''= \lim_{n\rightarrow \infty}\bigl(T^{n}v_{0} \bigr)''', \\& \quad\textit{where } v_{0}(t)=0, 0\leq t\leq1, \end{aligned}$$
where
\((Tu)(t)\)
is defined by (2.5).
The successive iterative scheme in the theorem is \(w_{0}(t)=a(\frac{4}{3}t^{4}-\frac{8}{3}t^{3}+\frac{4}{3}t+\frac {3}{8})\), \(w_{n+1}=Tw_{n}=T^{n}w_{0}\), \(n=0,1,2,\ldots\) , which starts off with a known simple quartic function or \(v_{0}(t)=0\), \(v_{n+1}=Tv_{n}=T^{n}v_{0}\), \(n=0,1,2,\ldots \) , which starts off with the zero function.
Proof
Now in order to investigate the properties of the operator T, let us denote \(\overline{P_{a}}=\{u\in P \mid\|u\|\leq a\}\). In the following, we will prove that \(T:\overline{P_{a}}\rightarrow\overline{P_{a}}\) firstly.
If \(u\in\overline{P_{a}}\), (H4), and (H5) implies that
$$\begin{aligned} 0 \leq& f\bigl(t,u(t),u'(t),u''(t),u'''(t) \bigr)\leq f(t,a,a,-a,a) \\ \leq&\max_{0\leq t\leq1}f(t,a,a,-a,a) \leq \frac{a}{A}\quad\mbox{for } 0\leq t\leq1. \end{aligned}$$
Since
$$\begin{aligned}& \max_{0\leq t\leq1}\bigl\vert (Tu) (t)\bigr\vert \leq \int^{1}_{0} \int^{1}_{0}\frac{1}{4}\phi_{q} \biggl[ \int^{1}_{0} \int^{1}_{0}\frac{1}{4} \lambda q(\zeta)f \bigl(\zeta,u(\zeta),u'(\zeta),u''(\zeta ),u'''(\zeta)\bigr) \,d\zeta\,dh(\tau) \\& \hphantom{\max_{0\leq t\leq1}\bigl\vert (Tu) (t)\bigr\vert \leq}{}+ \int^{1}_{0}\frac{1}{4}\lambda q(\zeta)f\bigl( \zeta,u(\zeta),u'(\zeta),u''( \zeta),u'''(\zeta)\bigr) \,d\zeta\biggr] \,d\tau\,dg(s) \\& \hphantom{\max_{0\leq t\leq1}\bigl\vert (Tu) (t)\bigr\vert \leq}{}+ \int^{1}_{0}\frac{1}{4}\phi_{q} \biggl[ \int^{1}_{0} \int^{1}_{0}\frac{1}{4} \lambda q(\tau)f \bigl(\tau,u(\tau),u'(\tau),u''(\tau ),u'''(\tau)\bigr) \,d\tau\,dh(s) \\& \hphantom{\max_{0\leq t\leq1}\bigl\vert (Tu) (t)\bigr\vert \leq}{}+ \int^{1}_{0}\frac{1}{4}\lambda q(\tau)f\bigl( \tau,u(\tau),u'(\tau),u''( \tau),u'''(\tau)\bigr) \,d\tau\biggr] \,ds \\& \hphantom{\max_{0\leq t\leq1}\bigl\vert (Tu) (t)\bigr\vert } = \frac {1}{4}\phi_{q} \biggl[ \int^{1}_{0}\frac{1}{4} \lambda q(\tau)f \bigl(\tau,u(\tau),u'(\tau),u''( \tau),u'''(\tau)\bigr) \,d\tau\biggr] \\& \hphantom{\max_{0\leq t\leq1}\bigl\vert (Tu) (t)\bigr\vert } \leq\frac {a}{A}\frac{1}{4}\biggl(\frac{1}{4} \lambda Q\biggr)^{q-1}< a, \\& \max_{0\leq t\leq1}\bigl\vert (Tu)'(t)\bigr\vert \leq \int^{1}_{t}(1-s)\phi_{q} \biggl[ \int^{1}_{0} \int^{1}_{0}G(s,\tau) \lambda q(\tau) \\& \hphantom{\max_{0\leq t\leq1}\bigl\vert (Tu)'(t)\bigr\vert \leq{}}{}\cdot f\bigl(\tau,u( \tau),u'(\tau),u''(\tau),u'''( \tau)\bigr) \,d\tau\,dh(s) \\& \hphantom{\max_{0\leq t\leq1}\bigl\vert (Tu)'(t)\bigr\vert \leq{}}{} + \int^{1}_{0}G(s,\tau)\lambda q(\tau) f\bigl(\tau,u( \tau),u'(\tau),u''(\tau),u'''( \tau)\bigr) \,d\tau\biggr] \,ds \\& \hphantom{\max_{0\leq t\leq1}\bigl\vert (Tu)'(t)\bigr\vert }\leq \int^{1}_{t}(1-s)\phi_{q} \biggl[ \int^{1}_{0} \int^{1}_{0}\frac{1}{4} \lambda q(\tau)f\bigl( \tau,u(\tau),u'(\tau),u''( \tau),u'''(\tau)\bigr) \,d\tau\,dh(s) \\& \hphantom{\max_{0\leq t\leq1}\bigl\vert (Tu)'(t)\bigr\vert \leq{}}{}+ \int^{1}_{0}\frac{1}{4} \lambda q(\tau)f\bigl( \tau,u(\tau),u'(\tau),u''( \tau),u'''(\tau)\bigr) \,d\tau\biggr] \,ds \\& \hphantom{\max_{0\leq t\leq1}\bigl\vert (Tu)'(t)\bigr\vert }\leq \int^{1}_{t}(1-s)\phi_{q} \biggl[ \int^{1}_{0}\frac{1}{4} \lambda q(\tau)f\bigl( \tau,u(\tau),u'(\tau),u''( \tau),u'''(\tau)\bigr) \,d\tau\biggr] \,ds \\& \hphantom{\max_{0\leq t\leq1}\bigl\vert (Tu)'(t)\bigr\vert }\leq\frac {a}{A}\frac{1}{2}\biggl(\frac{1}{4} \lambda Q\biggr)^{q-1}< a, \\& \max_{0\leq t\leq1}\bigl\vert (Tu)''(t) \bigr\vert \leq\phi_{q} \biggl[ \int^{1}_{0} \int^{1}_{0}\frac{1}{4} \lambda q(\tau)f \bigl(\tau,u(\tau),u'(\tau),u''( \tau),u'''(\tau)\bigr) \,d\tau\,dh(t) \\& \hphantom{\max_{0\leq t\leq1}\bigl\vert (Tu)''(t) \bigr\vert \leq{}}{}+ \int^{1}_{0}\frac{1}{4}\lambda q(\tau)f\bigl( \tau,u(\tau),u'(\tau),u''( \tau),u'''(\tau)\bigr) \,d\tau\biggr] \\& \hphantom{\max_{0\leq t\leq1}\bigl\vert (Tu)''(t) \bigr\vert } \leq\phi _{q} \biggl[ \int^{1}_{0}\frac{1}{4} \lambda q(\tau)f\bigl( \tau,u(\tau),u'(\tau),u''( \tau),u'''(\tau)\bigr) \,d\tau\biggr] \\& \hphantom{\max_{0\leq t\leq1}\bigl\vert (Tu)''(t) \bigr\vert } \leq\frac {a}{A}\biggl(\frac{1}{4}\lambda Q \biggr)^{q-1}\leq a, \end{aligned}$$
and
$$\begin{aligned} \max_{0\leq t\leq1}\bigl\vert (Tu)'''(t) \bigr\vert \leq&(q-1) \biggl[ \int^{1}_{0} \int^{1}_{0}\frac{1}{4} \lambda q(\tau)f \bigl(\tau,u(\tau),u'(\tau),u''( \tau),u'''(\tau)\bigr) \,d\tau\,dh(t) \\ &{}+ \int^{1}_{0}\frac{1}{4}\lambda q(\tau)f\bigl( \tau,u(\tau),u'(\tau),u''( \tau),u'''(\tau)\bigr) \,d\tau \biggr]^{q-2} \\ &{}\cdot \int^{1}_{t}(1-\tau)\lambda q(\tau)f\bigl(\tau,u( \tau),u'(\tau),u''(\tau),u'''( \tau)\bigr) \,d\tau \\ \leq&(q-1) \biggl[ \int^{1}_{0}\frac{1}{4}\lambda q(\tau)f\bigl( \tau,u(\tau),u'(\tau),u''( \tau),u'''(\tau)\bigr) \,d\tau \biggr]^{q-2} \\ &{}\cdot \int^{1}_{0}\lambda q(\tau)f\bigl(\tau,u( \tau),u'(\tau),u''(\tau),u'''( \tau)\bigr) \,d\tau \\ \leq& \frac{a}{A} 4(q-1) \biggl(\frac{1}{4}\lambda Q \biggr)^{q-1}\leq a, \end{aligned}$$
we get \(T:\overline{P_{a}}\rightarrow \overline{P_{a}}\).
Let \(w_{0}(t)=a(\frac{4}{3}t^{4}-\frac{8}{3}t^{3}+\frac{4}{3}t+\frac {3}{8})\), \(0\leq t\leq1\), then \(w_{0}(t)\in \overline{P_{a}}\). Let \(w_{1}=Tw_{0}\), then \(w_{1}\in \overline{P_{a}}\). We denote \(w_{n+1}=Tw_{n}\), \(n=0,1,2,\ldots\) . Then we have \(w_{n}\subseteq\overline{P_{a}}\), \(n=1,2,\ldots\) . Since T is completely continuous, we assert that \(\{w_{n}\}_{n=1}^{\infty}\) is a sequentially compact set.
Next, we investigate the convergence property of the iterative scheme; since
$$\begin{aligned}& w_{1}(t)=Tw_{0}(t) \\& \hphantom{w_{1}(t)}= \int^{1}_{0} \int^{1}_{0}G(s,\tau)\phi_{q} \biggl[ \int^{1}_{0} \int^{1}_{0}G(\tau,\zeta) \lambda q(\zeta)f\bigl( \zeta,w_{0}(\zeta),w_{0}'(\zeta ),w_{0}''(\zeta),w_{0}'''( \zeta)\bigr) \,d\zeta\,dh(\tau) \\& \hphantom{w_{1}(t)={}}{}+ \int^{1}_{0}G(\tau,\zeta)\lambda q(\zeta)f \bigl(s,w_{0}(\zeta),w_{0}'( \zeta),w_{0}''(\zeta),w_{0}'''( \zeta)\bigr) \,d\zeta\biggr] \,d\tau\,dg(s) \\& \hphantom{w_{1}(t)={}}{}+ \int^{1}_{0}G(t,s)\phi_{q} \biggl[ \int^{1}_{0} \int^{1}_{0}G(s,\tau) \lambda q(\tau)f\bigl( \tau,w_{0}(\tau),w_{0}'(\tau ),w_{0}''(\tau),w_{0}'''( \tau)\bigr) \,d\tau\,dh(s) \\& \hphantom{w_{1}(t)={}}{}+ \int^{1}_{0}G(s,\tau)\lambda q(\tau)f\bigl( \tau,w_{0}(\tau),w_{0}'(\tau),w_{0}''( \tau),w_{0}'''(\tau)\bigr) \,d \tau\biggr] \,ds \\& \hphantom{w_{1}(t)}\leq \int^{1}_{0}G(t,s)\phi_{q} \biggl[ \int^{1}_{0}G(s,\tau)\lambda q(\tau)f\bigl( \tau,w_{0}(\tau),w_{0}'(\tau),w_{0}''( \tau),w_{0}'''(\tau)\bigr) \,d \tau\biggr] \,ds \\& \hphantom{w_{1}(t)}\leq a \biggl(\frac{4}{3}t^{4}- \frac{8}{3}t^{3}+\frac{4}{3}t+\frac {3}{8} \biggr), \quad0\leq t\leq1, \\& \bigl\vert w_{1}'(t)\bigr\vert =\bigl\vert (Tw_{0})'(t)\bigr\vert \\& \hphantom{\bigl\vert w_{1}'(t)\bigr\vert }=\biggl\vert \int^{1}_{t}(1-s)\phi_{q} \biggl[ \int^{1}_{0} \int^{1}_{0}G(s,\tau) \lambda q(\tau)f\bigl( \tau,w_{0}(\tau),w_{0}'(\tau),w_{0}''( \tau),w_{0}'''(\tau)\bigr) \,d \tau\,dh(s) \\& \hphantom{\bigl\vert w_{1}'(t)\bigr\vert ={}}{}+ \int^{1}_{0}G(s,\tau)\lambda q(\tau)f \bigl(s,w_{0}(\tau),w_{0}'( \tau),w_{0}''(\tau),w_{0}'''( \tau)\bigr) \,d\tau\biggr] \,ds \\& \hphantom{\bigl\vert w_{1}'(t)\bigr\vert ={}}{}- \int^{t}_{0}s\phi_{q} \biggl[ \int^{1}_{0} \int^{1}_{0}G(s,\tau) \lambda q(\tau)f\bigl( \tau,w_{0}(\tau),w_{0}'(\tau),w_{0}''( \tau),w_{0}'''(\tau)\bigr) \,d \tau\,dh(s) \\& \hphantom{\bigl\vert w_{1}'(t)\bigr\vert ={}}{}+ \int^{1}_{0}G(s,\tau)\lambda q(\tau)f\bigl( \tau,w_{0}(\tau),w_{0}'(\tau),w_{0}''( \tau),w_{0}'''(\tau)\bigr) \,d \tau\biggr] \,ds \biggr\vert \\& \hphantom{\bigl\vert w_{1}'(t)\bigr\vert }\leq\biggl\vert \int^{1}_{t}(1-s)\phi_{q} \biggl[ \int^{1}_{0}G(s,\tau)\lambda q(\tau)f \bigl(s,w_{0}(\tau),w_{0}'( \tau),w_{0}''(\tau),w_{0}'''( \tau)\bigr) \,d\tau\biggr] \,ds\biggr\vert \\& \hphantom{\bigl\vert w_{1}'(t)\bigr\vert }\leq a \biggl(\frac{16}{3}t^{3}-8t^{2}+ \frac{4}{3} \biggr), \quad0\leq t\leq1, \\& w_{1}''(t)=(Tw_{0})''(t) \\& \hphantom{w_{1}''(t)}=-\phi_{q} \biggl[ \int^{1}_{0} \int^{1}_{0}G(t,\tau) \lambda q(\tau)f\bigl(\tau,u( \tau),u'(\tau),u''(\tau),u'''( \tau)\bigr) \,d\tau\,dh(t) \\& \hphantom{w_{1}''(t)={}}{}+ \int^{1}_{0}G(t,\tau)\lambda q(\tau)f\bigl(s,u( \tau),u'(\tau),u''(\tau),u'''( \tau)\bigr) \,d\tau\biggr] \\& \hphantom{w_{1}''(t)}\geq-\phi_{q} \biggl[ \int^{1}_{0}G(t,\tau)\lambda q(\tau)f\bigl(s,u(\tau ),u'(\tau),u''(\tau),u'''( \tau)\bigr) \,d\tau\biggr] \\& \hphantom{w_{1}''(t)}\geq a\bigl(16t^{2}-16t\bigr), \quad0\leq t\leq1, \end{aligned}$$
and
$$\begin{aligned} \bigl\vert w_{1}'''(t)\bigr\vert =&\bigl\vert (Tw_{0})'''(t) \bigr\vert \\ \leq&(q-1) \biggl[ \int^{1}_{0} \int^{1}_{0}G(t,\tau) \lambda q(\tau)f\bigl(\tau,u( \tau),u'(\tau),u''(\tau),u'''( \tau)\bigr) \,d\tau\,dh(t) \\ &{}+ \int^{1}_{0}G(t,\tau)\lambda q(\tau)f\bigl(s,u( \tau),u'(\tau),u''(\tau),u'''( \tau)\bigr) \,d\tau\biggr]^{q-2} \\ &{}\cdot\biggl( \int^{1}_{t}(1-\tau)\lambda q(\tau)f\bigl(s,u( \tau),u'(\tau),u''(\tau),u'''( \tau)\bigr) \,d\tau \\ &{}- \int^{t}_{0}\tau\lambda q(\tau)f\bigl(s,u(\tau ),u'(\tau),u''(\tau),u'''( \tau)\bigr) \,d\tau\biggr) \\ \leq&(q-1) \biggl[ \int^{1}_{0}G(t,\tau)\lambda q(\tau)f\bigl(s,u( \tau),u'(\tau),u''(\tau),u'''( \tau)\bigr) \,d\tau\biggr]^{q-2} \\ &{}\cdot \int^{1}_{t}(1-\tau)\lambda q(\tau)f\bigl(s,u( \tau),u'(\tau),u''(\tau),u'''( \tau)\bigr) \,d\tau \\ \leq& \frac{a}{A} (q-1) \biggl(\frac{1}{4}\lambda Q \biggr)^{q-1}16t\leq32at, \quad0\leq t\leq1. \end{aligned}$$
We get
$$\begin{aligned}& w_{2}(t)=Tw_{1}(t)\leq Tw_{0}(t)=w_{1}(t), \\& \bigl\vert w_{2}'(t)\bigr\vert =\bigl\vert (Tw_{1})'(t)\bigr\vert \leq\bigl\vert (Tw_{0})'(t)\bigr\vert =\bigl\vert w'_{1}(t)\bigr\vert , \\& w_{2}''(t)=(Tw_{1})''(t) \geq(Tw_{0})''(t)=w''_{1}(t), \\& \bigl\vert w_{2}'''(t)\bigr\vert =\bigl\vert (Tw_{1})'''(t) \bigr\vert \leq\bigl\vert (Tw_{0})'''(t) \bigr\vert =\bigl\vert w'''_{1}(t) \bigr\vert , \quad0\leq t\leq1. \end{aligned}$$
By induction, the iterative scheme is clear, then
$$\begin{aligned}& w_{n+1}\leq w_{n}, \qquad\bigl\vert w_{n+1}'(t) \bigr\vert \leq\bigl\vert w_{n}'(t)\bigr\vert , \qquad w_{n+1}''(t)\geq w_{n}''(t), \\& \bigl\vert w_{n+1}'''(t) \bigr\vert \leq\bigl\vert w_{n}'''(t) \bigr\vert , \quad0\leq t\leq1, n=0,1,2,\ldots. \end{aligned}$$
Thus, we see that there exists \(w^{\ast}\in\overline{P_{a}}\) such that \(w_{n}\rightarrow w^{\ast}\). Combining with the continuity of T and \(w_{n+1}=Tw_{n}\), we obtain \(Tw^{\ast}=w^{\ast}\).
On the other hand, another way to approach this is to start off with the zero function. Let \(v_{0}(t)=0\), \(0\leq t\leq1\), then \(v_{0}(t)\in \overline{P_{a}}\). Let \(v_{1}=Tv_{0}\), then \(v_{1}\in \overline{P_{a}}\). We denote \(v_{n+1}=Tv_{n}\), \(n=0,1,2,\ldots\) . Then we have \(v_{n}\subseteq\overline{P_{a}}\), \(n=1,2,\ldots\) . Since T is completely continuous, we assert that \(\{v_{n}\}_{n=1}^{\infty}\) is a sequentially compact set.
In a similar way, since \(v_{1}=Tv_{0}\in\overline{P_{a}}\), we have \(v_{1}(t)=Tv_{0}(t)\geq0\), \(|v_{1}'(t)|=|(Tv_{0})'(t)|\geq0\), \(v_{1}''(t)=(Tv_{0})''(t)\leq0\), \(|v_{1}'''(t)|=|(Tv_{0})'''(t)|\geq0\), for \(0\leq t\leq1\). Then \(v_{2}(t)\geq v_{1}(t)\), \(|v_{2}'(t)|\geq|v_{1}'(t)|\), \(v_{2}''(t)\leq v_{1}''(t)\), \(|v_{2}'''(t)|\geq|v_{1}'''(t)|\), for \(0\leq t\leq1\). By an induction argument similar to the above we easily obtain
$$\begin{aligned}& v_{n+1}\geq v_{n}, \qquad\bigl\vert v_{n+1}'(t) \bigr\vert \geq\bigl\vert v_{n}'(t)\bigr\vert , \qquad v_{n+1}''(t)\leq v_{n}''(t), \\& \bigl\vert v_{n+1}'''(t) \bigr\vert \geq\bigl\vert v_{n}'''(t) \bigr\vert , \quad0\leq t\leq1, n=0,1,2,\ldots. \end{aligned}$$
Hence there exists \(v^{\ast}\in\overline{P_{a}}\) such that \(v_{n}\rightarrow v^{\ast}\). Combining with the continuity of T and \(v_{n+1}=Tv_{n}\), we get \(Tv^{\ast}=v^{\ast}\).
The assumption (H6) indicates that \(f(t,0,0,0,0)\not\equiv0\), \(0\leq t\leq1\), then the zero function is not the solution of (1.1). Thus we have \(v^{\ast}>0\), for \(0< t<1\).
It is well known that each fixed point of T in P is a solution of (1.1). Hence, we assert that the boundary value problem (1.1) has at least one positive symmetric concave solution \(w^{\ast}\) or \(v^{\ast}\).
The proof is completed. □
Remark 3.1
If \(\lim_{n\rightarrow\infty}w_{n}\neq\lim_{n\rightarrow\infty}v_{n}\), then \(w^{\ast}\) and \(v^{\ast}\) are two positive symmetric concave solutions of the problem (1.1). And if \(\lim_{n\rightarrow\infty}w_{n}=\lim_{n\rightarrow\infty}v_{n}\), then \(w^{\ast}=v^{\ast}\) is a positive symmetric concave solution of the problem (1.1). Anyway, the problem (1.1) has at least one positive symmetric concave solution.
The following corollary follows easily.
Corollary 3.1
Assume that (H1)-(H4) and (H6) hold, and there exists
\(a>0\), such that
- (H7)::
-
\(\varliminf\nolimits_{\ell\rightarrow+\infty}\max_{0\leq t\leq1} \frac{f(t,\ell,a,-a,a)}{\ell^{p-1}}\leq\frac{1}{A^{p-1}}\) (particularly, \(\varliminf\nolimits_{\ell\rightarrow+\infty}\max_{0\leq t\leq1} \frac{f(t,\ell,a,-a,a)}{\ell^{p-1}}=0\)).
Then the boundary value problem (1.1) has at least one positive symmetric concave solution
\(w^{\ast}\)
or
\(v^{\ast}\), such that the conclusion of Theorem
3.1
hold.