Case I. \(\varphi_{i}(1)=0\), \(i=1,2,3\).
Clearly, \(\Delta=0\). In this case, we assume that there exists \(j\in\{ 1,2,3\}\) such that \(\Delta_{j3}\neq0\). In what follows, we choose and fix such j.
Lemma 3.1
There exists a function
\(g_{3}\in Y\)
such that
\(\Delta_{3}(g_{3})=1\).
Proof
Suppose the contrary. Then
$$\Delta_{3} \bigl(t^{n} \bigr)= \left \vert \begin{matrix} \varphi_{1} (t^{2} )& \varphi_{1}(t) &\varphi_{1} ( \int_{0}^{t}(t-s)^{2}s^{n} \,ds ) \\ \varphi_{2} (t^{2} )& \varphi_{2}(t)& \varphi_{2} ( \int_{0}^{t}(t-s)^{2}s^{n} \,ds ) \\ \varphi_{3} (t^{2} ) &\varphi_{3}(t)& \varphi_{3} ( \int_{0}^{t}(t-s)^{2}s^{n} \,ds ) \end{matrix} \right \vert =0,\quad n=0,1,\ldots. $$
Hence
$$\left \vert \begin{matrix} \varphi_{1} (t^{2} )& \varphi_{1}(t)& \varphi_{1} (t^{n+3} ) \\ \varphi_{2} (t^{2} ) &\varphi_{2}(t)& \varphi_{2} (t^{n+3} ) \\ \varphi_{3} (t^{2} ) &\varphi_{3}(t)& \varphi_{3} (t^{n+3} ) \end{matrix} \right \vert =0, \quad n=0,1,\ldots. $$
It follows from \(\Delta_{j3}\neq0\) and \(\varphi_{i}(1)=0\), \(i=1,2,3\), that there exist constants a and b such that
$$\varphi_{j} \bigl(t^{i} \bigr)=a\varphi_{k} \bigl(t^{i} \bigr)+b\varphi_{l} \bigl(t^{i} \bigr)=(a \varphi_{k}+b\varphi _{l}) \bigl(t^{i} \bigr),\quad i=0,1,2,\ldots, $$
where \(k,l\in\{1,2,3\}\), \(k,l\neq j\), \(k\neq l\). Hence \(\varphi _{j}(x)=(a\varphi_{k}+b\varphi_{l})(x)\), \(x\in X\). This is a contradiction because \(\varphi_{1},\varphi_{2},\varphi_{3}\) are linearly independent on X. Hence, there exists a function \(h\in Y\) with \(\Delta_{3}(h)\neq 0\) and, as a result, \(g_{3} = {\frac{1}{\Delta_{3}(h)}}h \in Y\) with \(\Delta_{3}(g_{3})=1\). □
Define operators \(L:\operatorname{dom} L\subset X\rightarrow Y\), \(N:X\rightarrow Y\) as follows:
$$Lx(t)=x'''(t),\qquad Nx(t)=f \bigl(t,x(t),x'(t),x''(t) \bigr), $$
where \(\operatorname{dom} L = \{x\in X: x''' \in Y, \varphi_{i}(x)=0, i =1,2,3\}\).
If \(x\in\operatorname{dom} L\) with \(Lx=0\), then \(x=at^{2}+bt+c\), \(a,b,c\in \mathbb{R}\) and \(\varphi_{i}(x)=0\), \(i=1,2,3\), that is,
$$\begin{aligned} &a\varphi_{1} \bigl(t^{2} \bigr)+b\varphi_{1}(t)=0, \\ &a\varphi_{2} \bigl(t^{2} \bigr)+b\varphi_{2}(t)=0, \\ &a\varphi_{3} \bigl(t^{2} \bigr)+b\varphi_{3}(t)=0. \end{aligned}$$
Since \(\Delta_{j3}\neq0\), we have \(a=b=0\). So, \(x\equiv c\), that is, \(\operatorname{Ker} L = \{c:c\in\mathbb{R}\}\).
Lemma 3.2
\(\operatorname{Im} L = \{y\in Y:\Delta_{3}(y)=0\}\).
Proof
If \(x\in\operatorname{dom}L\), \(Lx=y\), then there exist constants \(a,b,c\) such that the following equalities hold:
$$\begin{aligned} &x(t)=\frac{1}{2} \int_{0}^{t}(t-s)^{2}y(s) \,ds+at^{2}+bt+c, \\ &\varphi_{1}(x)=\frac{1}{2}\varphi_{1} \biggl( \int_{0}^{t}(t-s)^{2}y(s) \,ds \biggr)+a \varphi_{1} \bigl(t^{2} \bigr)+b\varphi_{1}(t)=0, \\ &\varphi_{2}(x)=\frac{1}{2}\varphi_{2} \biggl( \int_{0}^{t}(t-s)^{2}y(s) \,ds \biggr)+a \varphi_{2} \bigl(t^{2} \bigr)+b\varphi_{2}(t)=0, \\ &\varphi_{3}(x)=\frac{1}{2}\varphi_{3} \biggl( \int_{0}^{t}(t-s)^{2}y(s) \,ds \biggr)+a \varphi_{3} \bigl(t^{2} \bigr)+b\varphi_{3}(t)=0. \end{aligned}$$
So, y satisfies \(\Delta_{3}(y)=0\).
Inversely, if \(y\in Y\) with \(\Delta_{3}(y)=0\), we let
$$x(t)=\frac{1}{2} \int_{0}^{t}(t-s)^{2}y(s) \,ds - \frac{\Delta _{1}(y)_{j3}}{2\Delta_{j3}}t^{2}- \frac{\Delta_{2}(y)_{j3}}{2\Delta_{j3}}t. $$
Obviously, \(x'''(t)=y(t)\). Considering \(\Delta_{1}(y)_{j3}=-\Delta_{3}(y)_{j1}\), \(\Delta_{2}(y)_{j3}=\Delta _{3}(y)_{j2}\), \(\Delta_{j3}=\Delta_{3}(y)_{j3}\) and
$$\varphi_{j}(x)=\frac{1}{2}\varphi_{j} \biggl( \int_{0}^{t}(t-s)^{2}y(s) \,ds \biggr)- \frac{\Delta_{1}(y)_{j3}}{2\Delta_{j3}}\varphi _{j} \bigl(t^{2} \bigr)- \frac{\Delta_{2}(y)_{j3}}{2\Delta_{j3}}\varphi_{j}(t), $$
we have
$$\begin{aligned} \varphi_{j}(x)&=\frac{1}{2\Delta_{j3}} \biggl[ \varphi_{j} \bigl(t^{2} \bigr)\Delta_{3}(y)_{j1}- \varphi_{j}(t)\Delta _{3}(y)_{j2}+ \varphi_{j} \biggl( \int_{0}^{t}(t-s)^{2}y(s) \,ds \biggr) \Delta_{3}(y)_{j3} \biggr] \\ &=\frac{1}{2\Delta_{j3}} \Delta_{3}(y)=0. \end{aligned}$$
Clearly, \(\varphi_{i}(x)=0\), \(i\neq j\), \(i\in\{1,2,3\}\), which implies that \(x\in\operatorname{dom} L\) and, consequently, \(y\in\operatorname{Im} L\). □
Define the operators \(P_{3}:X\rightarrow X\), \(Q_{3}:Y\rightarrow Y\) by
$$P_{3}x = x(0),\qquad Q_{3}y=\Delta_{3}(y)g_{3}. $$
Clearly, \(P_{3}\), \(Q_{3}\) are projectors such that (2.3) hold.
Define the operator \(K_{P_{3}}: Y\rightarrow X\) by
$$K_{P_{3}}y=\frac{1}{2} \int_{0}^{t}(t-s)^{2}y(s) \,ds - \frac{\Delta _{1}(y)_{j3}}{2\Delta_{j3}}t^{2}- \frac{\Delta_{2}(y)_{j3}}{2\Delta_{j3}}t. $$
Lemma 3.3
\(K_{P_{3}}=(L| _{\operatorname{dom}L\cap\operatorname{Ker}P_{3}})^{-1}\).
Proof
Let \(x\in\operatorname{dom}L\cap\operatorname{Ker}P_{3}\). Then \(\varphi_{i}(x)=0\), \(i=1,2,3\), and \(x(0)=0\). So, we get
$$\begin{aligned} K_{P_{3}}Lx(t) &=\frac{1}{2} \int_{0}^{t}(t-s)^{2}Lx(s) \,ds - \frac {\Delta_{1}(Lx)_{j3}}{2\Delta_{j3}}t^{2}- \frac{\Delta_{2}(Lx)_{j3}}{2\Delta_{j3}}t \\ &= x(t)-\frac{x''(0)}{2}t^{2}-x'(0)t- \frac{\Delta _{1}(Lx)_{j3}}{2\Delta_{j3}}t^{2}- \frac{\Delta_{2}(Lx)_{j3}}{2\Delta_{j3}}t. \end{aligned}$$
It follows from (2.1), (2.2) that \(\Delta _{1}(Lx)_{j3}=-x''(0)\Delta_{j3}\), \(\Delta_{2}(Lx)_{j3}=-2x'(0)\Delta _{j3}\). So, \(K_{P_{3}}Lx=x\).
Inversely, \(y\in\operatorname{Im}L\) results in \(\Delta_{3}(y)=0\). As the proof of Lemma 3.2, \(\varphi_{i}(K_{P_{3}}y)=0\), \(i=1,2,3\). Clearly, \((K_{P_{3}}y)'''=y\). Thus, \(K_{P_{3}}y\in\operatorname{dom}L\) and \(LK_{P_{3}}y=y\), \(y\in\operatorname{Im}L\). □
We introduce the constants \(l_{3} = k_{1} \vert \Delta_{13} \vert +k_{2} \vert \Delta _{23} \vert +k_{3} \vert \Delta_{33} \vert \) and
$$ l=\max\{k_{1}k_{2},k_{1}k_{3},k_{2}k_{3} \}. $$
(3.1)
The latter is frequently used in the remainder of the paper.
The next assumption is fulfilled in the main results by virtue of appropriate assumptions on \(f(t, \cdot,\cdot,\cdot)\):
-
\((H_{1})\)
:
-
For any \(r>0\), there exists a function \(h_{r}\in Y \) such that \(\vert f(t,x(t),x'(t),x''(t)) \vert \leq h_{r}(t)\), \(x\in X\), \(\Vert x \Vert \leq r\).
Lemma 3.4
If
\((H_{1})\)
holds and
\(\Omega\subset X\)
is bounded, then
N
is
L-compact on Ω̅.
Proof
Take \(r\in\mathbb{R}\) large enough such that \(\Vert x \Vert \leq r\), \(x\in\overline{\Omega}\). Then
$$\bigl\vert \Delta_{3}(Nx) \bigr\vert \leq \bigl(k_{1} \vert \Delta_{13} \vert +k_{2} \vert \Delta_{23} \vert +k_{3} \vert \Delta _{33} \vert \bigr) \biggl\Vert \int_{0}^{t}(t-s)^{2}Nx(s) \,ds \biggr\Vert \leq l_{3} \Vert h_{r} \Vert _{1}. $$
So, \(\Vert Q_{3}Nx \Vert _{1}\leq l_{3} \Vert h_{r} \Vert _{1} \Vert g_{3} \Vert _{1}\), which shows that \(Q_{3}N(\overline{\Omega})\) is bounded. For \(y\in Y\), we have
$$\Vert K_{P_{3}}y \Vert \leq \Vert y \Vert _{1}+ \frac{2l}{ \vert \Delta_{j3} \vert }2 \Vert y \Vert _{1}+\frac {4l}{2 \vert \Delta_{j3} \vert }2 \Vert y \Vert _{1}= \biggl(1+\frac{8l}{ \vert \Delta_{j3} \vert } \biggr) \Vert y \Vert _{1}, $$
where, for convenience, we define, using (3.1), the constant
$$ A_{P_{3}} = 1+\frac{8l}{ \vert \Delta_{j3} \vert }. $$
(3.2)
Then
$$\bigl\Vert K_{P_{3}}(I-Q_{3})Nx \bigr\Vert \leq A_{P_{3}} \bigl\Vert (I-Q_{3})Nx \bigr\Vert _{1} \leq A_{P_{3}} \bigl(1+l_{3} \Vert g_{3} \Vert _{1} \bigr) \Vert h_{r} \Vert _{1}. $$
Thus, \(K_{P_{3}}(I-Q_{3})N(\overline{\Omega})\) is bounded.
For \(0\leq t_{1}< t_{2}\leq1\), \(x\in\overline{\Omega}\), we have
$$\begin{aligned} \bigl\vert \bigl(K_{P_{3}}(I-Q_{3})Nx \bigr)''(t_{2})- \bigl(K_{P_{3}}(I-Q_{3})Nx \bigr)''(t_{1}) \bigr\vert &= \biggl\vert \int_{t_{1}}^{t_{2}}(I-Q_{3})Nx(s) \,ds \biggr\vert \\ &\leq \int_{t_{1}}^{t_{2}}h_{r}(s) \,ds+ l_{3} \Vert h_{r} \Vert _{1} \int _{t_{1}}^{t_{2}} \bigl\vert g_{3}(s) \bigr\vert \,ds, \end{aligned}$$
that is, \((K_{P_{3}}(I-Q_{3})N)''(\overline{\Omega})\) is equicontinuous on \([0,1]\) as well as \((K_{P_{3}}(I-Q_{3})N)'(\overline{\Omega})\) and \((K_{P_{3}}(I-Q_{3})N)(\overline{\Omega})\) by the mean value theorem. Therefore, by the Arzela-Ascoli theorem, \(K_{P_{3}}(I-Q_{3})N(\overline {\Omega})\) is compact. □
In order to obtain the main results, we impose the following conditions:
-
\((H_{2})\)
:
-
There exist nonnegative functions \(a,b,c,d \in Y\) such that \(\vert f(t,u,v,w) \vert \leq a(t)+b(t) \vert u \vert +c(t) \vert v \vert +d(t) \vert w \vert \), \(t\in[0,1]\), \(u,v,w\in\mathbb{R}\);
-
\((H_{3})\)
:
-
There exists a constant \(M_{03}>0\) such that \(\Delta _{3}(Nx) \neq0\) if \(\vert x(t) \vert >M_{03}\), \(t\in[0,1]\);
-
\((H_{4})\)
:
-
There exists a constant \(M_{13}>0\) such that if \(\vert c \vert > M_{13}\), then one of the following two inequalities holds:
$$ c\Delta_{3}(Nc)>0, $$
(3.3)
or
$$ c\Delta_{3}(Nc)< 0. $$
(3.4)
(Here \(Nc = f(t,c,0,0)\), \(c \in\mathbb{R}\).)
Lemma 3.5
Assume that
\((H_{2})\), \((H_{3})\)
hold and let
$$ A_{P_{3}} \bigl( \Vert b \Vert _{1}+ \Vert c \Vert _{1}+ \Vert d \Vert _{1} \bigr)< \frac{1}{2}, $$
(3.5)
where
\(A_{P_{3}}\)
satisfies (3.2). Then
\(\Omega_{13}=\{x\in \operatorname{dom}L\setminus\operatorname{Ker}L:Lx=\lambda Nx,\lambda\in(0,1)\} \)
is bounded.
Proof
Since \(x\in\Omega_{13}\), then \(\Delta_{3}(Nx) = 0\). By \((H_{3})\), there exists \(t_{0}\in[0,1]\) such that \(\vert x(t_{0}) \vert \leq M_{03}\). Now,
$$\bigl\Vert (I-P_{3})x \bigr\Vert = \bigl\Vert K_{P_{3}}L(I-P_{3})x \bigr\Vert = \Vert K_{P_{3}}Lx \Vert \leq A_{P_{3}} \Vert Lx \Vert _{1} $$
and
$$\bigl\vert P_{3}x(t_{0}) \bigr\vert = \bigl\vert x(t_{0})-(I-P_{3})x(t_{0}) \bigr\vert \leq M_{03}+A_{P_{3}} \Vert Lx \Vert _{1}. $$
Thus, \(\Vert P_{3}x \Vert = \vert P_{3}x(t_{0}) \vert \leq M_{03}+A_{P_{3}} \Vert Lx \Vert _{1}\). It follows from \(x = P_{3}x + (I-P_{3})x\) and \((H_{2})\) that
$$\begin{aligned} \Vert x \Vert \leq M_{03} + 2A_{P_{3}} \Vert Lx \Vert _{1} & < M_{03} + 2A_{P_{3}} \Vert Nx \Vert _{1} \\ &\leq M_{03} + 2A_{P_{3}} \bigl( \Vert a \Vert _{1}+ \bigl( \Vert b \Vert _{1}+ \Vert c \Vert _{1}+ \Vert d \Vert _{1} \bigr) \Vert x \Vert \bigr). \end{aligned}$$
So,
$$\Vert x \Vert \leq\frac{M_{03} + 2A_{P_{3}} \Vert a \Vert _{1}}{1-2A_{P_{3}}( \Vert b \Vert _{1}+ \Vert c \Vert _{1}+ \Vert d \Vert _{1})}. $$
Therefore, \(\Omega_{13}\) is bounded by (3.5). □
Lemma 3.6
Assume that
\((H_{4})\)
holds. Then
\(\Omega_{23}=\{x\in\operatorname{Ker}L: Nx\in\operatorname{Im}L\}\)
is bounded.
Proof
If \(x\in\Omega_{23}\), then \(x \equiv c\) and \(Q_{3}(Nc)=0\), that is, \(\Delta_{3}(Nc) = 0\). By \((H_{4})\), it follows that \(\vert c \vert \leq M_{13}\). Thus, \(\Omega_{23}\) is bounded. □
Lemma 3.7
Assume that
\((H_{4})\)
holds. Then
$$\Omega_{33 }= \bigl\{ x:\rho\lambda x+(1-\lambda)\Delta_{3}(Nx) = 0, x\in \operatorname{Ker}L ,\lambda\in[0,1] \bigr\} $$
is bounded, where
\(\rho= \{ \scriptsize{ \begin{array}{l@{\quad}l} 1, &\textit{if } (3.3) \textit{ holds},\\ -1, &\textit{if } (3.4) \textit{ holds}. \end{array}} \)
Proof
Let \(x\in\Omega_{33}\). Then \(x\equiv c\in\mathbb {R}\) and \(\rho\lambda c +(1-\lambda)\Delta_{3}(Nc) =0\). If \(\lambda =0\), then \(\Delta_{3}(Nc) = 0\). By \((H_{4})\), \(\vert c \vert \leq M_{13}\). If \(\lambda=1\), then \(c=0\). If \(\lambda\in(0,1)\), then \(c=- {\frac {1-\lambda}{\lambda\rho}} \Delta_{3}(Nc)\). Hence, \(c^{2}= - {\frac {1-\lambda}{\lambda\rho}} c \Delta_{3}(Nc)\). If \(\vert c \vert >M_{13}\), by \((H_{4})\), we obtain
$$c^{2}=-\frac{1-\lambda}{\lambda\rho} c \Delta_{3}(Nc)< 0, $$
which is a contradiction. Therefore, \(\vert c \vert \leq M_{13}\) and \(\Omega _{33}\) is bounded. □
Theorem 3.8
Assume that
\((H_{2})\)-\((H_{4})\)
and (3.5) hold. Then problem (1.1) has at least one solution.
Proof
Let \(\Omega\supset\overline{\Omega}_{13}\cup \overline{\Omega}_{23}\cup\overline{\Omega}_{33} \) be bounded. It follows from Lemmas 3.5 and 3.6 that \(Lx \neq \lambda Nx\), \(x\in(\operatorname{dom}L\setminus\operatorname{Ker}L)\cap \partial\Omega\), \(\lambda\in(0,1)\) and \(Nx\notin\operatorname{Im}L\), \(x\in\operatorname{Ker}L\cap\partial\Omega\). Let
$$H(x,\lambda)=\lambda\rho x+(1-\lambda)J_{3} Q_{3}Nx, $$
where \(J_{3}: \operatorname{Im}Q_{3} \rightarrow \operatorname{Ker}L\) is an isomorphism defined by \(J_{3}(c g_{3})=c\), \(c\in\mathbb{R}\). By Lemma 3.7, we know \(H(x,\lambda)\neq0\), \(x\in\partial\Omega\cap \operatorname{Ker}L\), \(\lambda\in[0,1]\). Since the degree is invariant under a homotopy,
$$\begin{aligned} \operatorname{deg}(J_{3} Q_{3}N| _{\operatorname{Ker}L},\Omega\cap \operatorname {Ker}L,0)&=\operatorname{deg} \bigl(H(\cdot,0),\Omega\cap \operatorname{Ker}L,0 \bigr) = \operatorname{deg} \bigl(H(\cdot,1),\Omega\cap \operatorname{Ker}L,0 \bigr) \\ & =\operatorname{deg}(\rho\operatorname{I},\Omega\cap\operatorname{Ker}L,0) \neq0. \end{aligned}$$
By Theorem 2.1, \(Lx=Nx\) has a solution in \(\operatorname{dom}L\cap \overline{\Omega}\). □
Case II. \(\varphi_{i}(t)=0\), \(i=1,2,3\).
In this case, assume there exists \(j\in\{1,2,3\}\) such that \(\Delta _{j2}\neq0\). With an adjustment of the method of Lemma 3.1, we can assert the existence of a function \(g_{2}\in Y\) such that \(\Delta _{2}(g_{2})=1\).
Clearly, \(\Delta=0\) and \(\operatorname{Ker}L=\{ct:c\in\mathbb{R}\}\). Similar to the proof of Lemma 3.2, we can show that \(\operatorname {Im}L=\{y\in Y:\Delta_{2}(y)=0\}\).
Define the operators \(P_{2}:X\rightarrow X\), \(Q_{2}:Y\rightarrow Y\) by
$$P_{2}x=x'(0)t,\qquad Q_{2}y= \Delta_{2}(y)g_{2}. $$
Obviously, \(P_{2}\) and \(Q_{2}\) are continuous linear projectors satisfying (2.3).
Define the operator \(K_{P_{2}}: Y\rightarrow X\) as
$$K_{P_{2}}y=\frac{1}{2} \int_{0}^{t}(t-s)^{2}y(s) \,ds- \frac{\Delta _{1}(y)_{j2}}{2\Delta_{j2}}t^{2}-\frac{\Delta_{3}(y)_{j2}}{2\Delta_{j2}}. $$
As above, we can obtain that \(K_{P_{2}}=(L| _{\operatorname{dom}L\cap\operatorname {\operatorname{Ker}}P_{2}})^{-1}\) and \(\Vert K_{P_{2}}y \Vert \leq A_{P_{2}} \Vert y \Vert _{1}\), where
$$ A_{P_{2}}= 1+\frac{8l}{ \vert \Delta_{j2} \vert }. $$
(3.6)
Suppose that the following conditions hold:
-
\((H_{5})\)
:
-
There exists \(M_{02}>0\) such that \(\Delta_{2}(Nx)\neq 0\), if \(\vert x'(t) \vert >M_{02}\), \(t\in[0,1]\);
-
\((H_{6})\)
:
-
There exists \(M_{12}>0\) such that if \(\vert c \vert >M_{12}\), then either
$$ c \Delta_{2} \bigl(N(ct) \bigr)>0, $$
(3.7)
or
$$ c\Delta_{2} \bigl(N(ct) \bigr)< 0. $$
(3.8)
Lemma 3.9
Assume that conditions
\((H_{2})\), \((H_{5})\)
hold and let
$$ A_{P_{2}} \bigl( \Vert b \Vert _{1}+ \Vert c \Vert _{1}+ \Vert d \Vert _{1} \bigr)< \frac{1}{2}, $$
(3.9)
where
\(A_{P_{2}}\)
satisfies (3.6). Then the set
$$\Omega_{12}= \bigl\{ x\in\operatorname{dom}L\backslash \operatorname{Ker}L:Lx=\lambda Nx,\lambda\in(0,1) \bigr\} $$
is bounded.
Proof
If \(x\in\Omega_{12}\), then \(\Delta_{2}(Nx)=0\). By \((H_{5})\), there exists a constant \(t_{1}\in[0,1]\) such that \(\vert x'(t_{1}) \vert \leq M_{02}\). Since \(x(t)=P_{2}x(t)+(I-P_{2})x(t)\), \(x'(t_{1})=x'(0)+((I-P_{2})x)'(t_{1})\) and
$$\bigl\Vert (I-P_{2})x \bigr\Vert = \bigl\Vert K_{P_{2}}L(I-P_{2})x \bigr\Vert = \Vert K_{P_{2}}Lx \Vert \leq A_{P_{2}} \Vert Lx \Vert _{1} < A_{P_{2}} \Vert Nx \Vert _{1}, $$
we have
$$\bigl\vert x'(0) \bigr\vert \leq M_{02}+ \bigl\Vert (I-P_{2})x \bigr\Vert \leq M_{02}+A_{P_{2}} \Vert Nx \Vert _{1}. $$
So,
$$\begin{aligned} \Vert x \Vert \leq \Vert P_{2}x \Vert + \bigl\Vert (I-P_{2})x \bigr\Vert &\leq M_{02}+2A_{P_{2}} \Vert Nx \Vert _{1} \\ &\leq M_{02}+2A_{P_{2}} \bigl( \Vert a \Vert _{1}+ \bigl( \Vert b \Vert _{1}+ \Vert c \Vert _{1}+ \Vert d \Vert _{1} \bigr) \Vert x \Vert \bigr). \end{aligned}$$
Thus,
$$\Vert x \Vert \leq\frac{M_{02}+2A_{P_{2}} \Vert a \Vert _{1}}{1-2A_{P_{2}}( \Vert b \Vert _{1}+ \Vert c \Vert _{1}+ \Vert d \Vert _{1})}, $$
which proves that \(\Omega_{12}\) is bounded. □
Lemma 3.10
Assume that
\((H_{6})\)
holds. Then the set
$$\Omega_{22} = \{x\in\operatorname{Ker}L: Nx\in\operatorname{Im}L\} $$
is bounded.
Proof
Since \(x\in\Omega_{22}\), \(x=ct\), \(c\in\mathbb{R}\) and \(\Delta_{2}(N(ct))=0\). By \((H_{6})\), we have \(\vert c \vert \leq M_{12}\). So, \(\Vert x \Vert = \vert c \vert \leq M_{12}\), that is, \(\Omega_{22}\) is bounded. □
Lemma 3.11
Assume that
\((H_{6})\)
holds. Then the set
$$\Omega_{32} = \bigl\{ x\in\operatorname{Ker}L: \rho\lambda x+(1- \lambda)J_{2} Q_{2}Nx=0,\lambda\in[0,1] \bigr\} $$
is bounded, where
\(J_{2}: \operatorname{Im}Q_{2}\rightarrow\operatorname{Ker}L\), \(J_{2}(cg_{2})(t) = ct\), \(c\in\mathbb{R}\), and
\(\rho= \{ \scriptsize{ \begin{array}{l@{\quad}l} 1, &\textit{if } (3.7) \textit{ holds},\\ -1,& \textit{if } (3.8) \textit{ holds}. \end{array}} \)
Proof
If \(x\in\Omega_{32}\), then \(x=ct\), \(c\in\mathbb {R}\) and \(\lambda\rho c+(1-\lambda)J_{2}Q_{2} (N(ct))=0\). So,
$$\lambda\rho c+(1-\lambda)\Delta_{2} \bigl(N(ct) \bigr)=0. $$
If \(\lambda=0\), then \(\Delta_{2}(N(ct))=0\). By \((H_{6})\), \(\vert c \vert \leq M_{12}\). If \(\lambda=1\), then \(c=0\). If \(\lambda\in(0,1)\), \(c=- \frac{1-\lambda}{\lambda\rho} \Delta_{2}(N(ct))\). So,
$$c^{2}=- \frac{1-\lambda}{\lambda\rho} c \Delta_{2} \bigl(N(ct) \bigr). $$
If \(\vert c \vert >M_{12}\), by \((H_{6})\), we obtain \(c^{2}<0\), a contradiction. So, \(\vert c \vert \leq M_{12}\), that is, \(\Omega_{32}\) is bounded. □
Under assumption \((H_{1})\), N is L-compact on a bounded set Ω̅ as in the proof of Lemma 3.4.
Theorem 3.12
Assume that
\((H_{2})\), \((H_{5})\), \((H_{6})\)
and (3.9) hold. Then FBVP (1.1) has at least one solution.
The proof is similar to that of Theorem 3.8.
Case III. \(\varphi_{i}(t^{2})=0\), \(i=1,2,3\).
In this case, assume that there exists \(j\in\{1,2,3\}\) such that \(\Delta_{j1}\neq0\).
Similarly, there exists a function \(g_{1}\in Y\) such that \(\Delta_{1}(g_{1})=1\).
Obviously, \(\Delta=0\) and \(\operatorname{Ker}L=\{ct^{2}:c\in\mathbb{R}\}\). Similar to the proof of Lemma 3.2, we can obtain \(\operatorname {Im}L=\{y\in Y:\Delta_{1}(y)=0\}\).
Define the operators \(P_{1}:X\rightarrow X\), \(Q_{1}:Y\rightarrow Y\) as
$$P_{1}x=\frac{1}{2}x''(0)t^{2}, \qquad Q_{1}y=\Delta_{1}(y)g_{1}. $$
Clearly, \(P_{1}\) and \(Q_{1}\) are continuous linear projectors. Introduce the operator \(K_{P_{1}}:Y\rightarrow X\) by
$$K_{P_{1}}y=\frac{1}{2} \int_{0}^{t}(t-s)^{2}y(s) \,ds- \frac{\Delta _{2}(y)_{j1}}{2\Delta_{j1}}t-\frac{\Delta_{3}(y)_{j1}}{2\Delta_{j1}}. $$
As above, it is easy to show that \(K_{P_{1}}=(L| _{\operatorname{dom}L\cap \operatorname{Ker}P_{1}})^{-1}\) and \(\Vert K_{P_{1}}y \Vert \leq A_{P_{1}} \Vert y \Vert _{1}\), where
$$ A_{P_{1}} = 1+\frac{4l}{ \vert \Delta_{j1} \vert }. $$
(3.10)
By the same method we used in Lemma 3.4, we can show that N is L-compact on Ω̅.
To prove the main result, we need the following hypotheses:
-
\((H_{7})\)
:
-
There exists \(M_{01}>0\) such that \(\Delta_{1}(Nx)\neq0\) if \(\vert x''(t) \vert >M_{01}\), \(t\in[0,1]\);
-
\((H_{8})\)
:
-
There exists \(M_{11}\) such that if \(\vert c \vert >M_{11}\), then either \(c \Delta_{1}(N(ct^{2}))>0\) or \(c \Delta_{1}(N(ct^{2}))<0\).
Lemma 3.13
Assume that
\((H_{2})\), \((H_{7})\)
hold. In addition, assume that
$$ A_{P_{1}} \bigl( \Vert b \Vert _{1}+ \Vert c \Vert _{1}+ \Vert d \Vert _{1} \bigr)< \frac{1}{2}, $$
(3.11)
where
\(A_{P_{1}}\)
is given by (3.10). Then the set
$$\Omega_{11}= \bigl\{ x\in\operatorname{dom}L\backslash \operatorname{Ker}L:Lx=\lambda Nx,\lambda\in(0,1) \bigr\} $$
is bounded.
Proof
For \(x\in\Omega_{11}\), we have \(\Delta_{1}(Nx)=0\). By \((H_{7})\), there exists \(t_{2}\in[0,1]\) such that \(\vert x''(t_{2}) \vert \leq M_{01}\). Since \(x=P_{1}x+(I-P_{1})x\), \(\Vert (I-P_{1})x \Vert \leq A_{P_{1}} \Vert Lx \Vert _{1} < A_{P_{1}} \Vert Nx \Vert _{1}\),
$$\bigl\vert (P_{1}x)''(t_{2}) \bigr\vert = \bigl\vert x''(t_{2})- \bigl((I-P_{1})x \bigr)''(t_{2}) \bigr\vert \leq M_{01}+ \bigl\Vert (I-P_{1})x \bigr\Vert $$
and \((P_{1}x)''(t_{2})=x''(0)\), we get
$$\bigl\vert x''(0) \bigr\vert = \bigl\vert (P_{1}x)''(t_{2}) \bigr\vert \leq M_{01}+ \bigl\Vert (I-P_{1})x \bigr\Vert \leq M_{01}+A_{P_{1}} \Vert Lx \Vert _{1}. $$
Combining the inequalities above, we get
$$\Vert x \Vert < M_{01}+2A_{P_{1}} \bigl( \Vert a \Vert _{1}+ \bigl( \Vert b \Vert _{1}+ \Vert c \Vert _{1}+ \Vert d \Vert _{1} \bigr) \Vert x \Vert \bigr). $$
Thus,
$$\Vert x \Vert \leq\frac{M_{01}+2A_{P_{1}} \Vert a \Vert _{1}}{1-2A_{P_{1}}( \Vert b \Vert _{1}+ \Vert c \Vert _{1}+ \Vert d \Vert _{1})}. $$
In view of (3.11), \(\Omega_{11}\) is bounded. □
Similarly, if \((H_{7})\) and \((H_{8})\) hold, we can prove that \(\Omega _{21} = \{x\in\operatorname{Ker}L: Nx\in\operatorname{Im}L\}\) and \(\Omega_{31} = \{x\in\operatorname{Ker}L: \rho\lambda x+(1-\lambda)J_{1} Q_{1}Nx=0, \lambda\in[0,1]\}\), with an isomorphism \(J_{1}: \operatorname{Im}Q \to \operatorname{Ker}L\), \(J_{1}(cg_{1})(t) =ct^{2}\), \(c\in\mathbb{R}\), are bounded.
Theorem 3.14
Assume that
\((H_{2})\), \((H_{7})\), \((H_{8})\)
and (3.11) hold. Then FBVP (1.1) has at least one solution.