Abstract
By using the coincidence degree theory due to Mawhin and constructing suitable operators, we study the existence of solutions for a third-order functional boundary value problem at resonance with \(\operatorname{dim } \operatorname{Ker}L=1\).
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1 Introduction
A boundary value problem is said to be at resonance if the corresponding homogeneous boundary value problem has a non-trivial solution. Boundary value problems at resonance have been studied by many authors. We refer the readers to [1–9] and the references cited therein. In [10], the authors discussed the second-order differential equation
with functional boundary conditions
where \(\Gamma_{1}, \Gamma_{2}\) are linear functionals on \(C^{1}[0,1]\) satisfying the general resonance condition \(\Gamma_{1}(t)\Gamma_{2}(1) = \Gamma_{1}(1)\Gamma_{2}(t)\). (The authors also studied the non-resonant scenario under condition (\(A_{1}\)): \(\Gamma_{1}(t)\Gamma_{2}(1) \neq\Gamma _{1}(1)\Gamma_{2}(t)\).) To be specific, the following resonant cases received attention:
- (\(A_{2}\)):
-
\(\Gamma_{1}(t), \Gamma_{1}(1), \Gamma_{2}(1) = 0\), \(\Gamma _{2}(t) \neq0\);
- (\(A_{3}\)):
-
\(\Gamma_{1}(t), \Gamma_{1}(1), \Gamma_{2}(t) = 0\), \(\Gamma _{2}(1) \neq0\);
- (\(A_{4}\)):
-
\(\Gamma_{1}(1), \Gamma_{2}(t), \Gamma_{2}(1) = 0\), \(\Gamma _{1}(t) \neq0\);
- (\(A_{5}\)):
-
\(\Gamma_{1}(t), \Gamma_{2}(1), \Gamma_{2}(t) = 0\), \(\Gamma _{1}(1) \neq0\);
- (\(A_{6}\)):
-
\(\Gamma_{1}(1), \Gamma_{1}(t), \Gamma_{2}(1), \Gamma_{2}(t) = 0\).
The cases (\(A_{2}\)) and (\(A_{4}\)) result in \(\operatorname{ker}L = \{c: c \in\mathbb{R}\}\), and (\(A_{3}\)) and (\(A_{5}\)) correspond to \(\operatorname{ker} L = \{ct: c \in\mathbb{R}\}\). The case (\(A_{6}\)) describes a resonance with \(\operatorname{ker}L = \{c_{1}t+c_{2}: c_{1},c_{2} \in\mathbb {R}\}\). In [6], the authors extended the results of [10] as well as [3, 9] in several respects including the study of the case \(\operatorname{ker}L = \{c(at+b): c\in\mathbb{R}\}\), where \(a,b \neq0\).
This paper is a study of third-order functional boundary value problems (FBVPs) at resonance. It improves and generalizes the results of [1, 7] and the results of [2] applicable to third-order problems. We consider
where \(\varphi_{i}:C^{2}[0,1]\rightarrow\mathbb{R}\), \(i=1,2,3\), are bounded linear functionals. To the best of our knowledge, this is the first paper devoted to a third-order FBVP at resonance. We present several generalizations to the existing results and improvements to the method based on Mawhin’s coincidence degree theory.
The framework of this paper is as follows. In Section 2, we present some notations and the fundamentals of coincidence degree theory. In Section 3, we study problem (1.1) under the conditions
respectively. In Section 4, we show the existence of a solution for problem (1.1) under the condition
(Here, if \(\varphi_{2}(t^{j}) =0\) for some \(j\in\{0,1,2\}\), then also \(\varphi_{1}(t^{j}) =0\).)
2 Preliminaries
For convenience, we denote
From the last three determinants we can define and derive the following three relations:
and \(\Delta_{3}(Lx)= -2x(0)\triangle\). Also, \(\Delta_{ij}\), \(i,j=1,2,3\), \(\Delta_{k}(y)_{ij}\), \(i,k=1,2,3\), \(j \in\{1,2,3\} \setminus\{k\}\), are the cofactors of \(\varphi_{i}(t^{3-j})\) in Δ, \(\Delta_{k}(y)\), \(k =1,2,3\), respectively.
We introduce some notations and a theorem. For more details, see [11].
Let X and Y be real Banach spaces and \(L: \operatorname{dom} L\subset X\rightarrow Y\) be a Fredholm operator of index zero, \(P: X\rightarrow X\), \(Q: Y\rightarrow Y\) be projectors such that
It follows that
is invertible. We denote the inverse by \(K_{P}\).
If Ω is an open bounded subset of X, \(\operatorname{dom} L \cap \overline{\Omega}\neq\emptyset\), the map \(N:X\rightarrow Y\) is called L-compact on Ω̅ if \(QN(\overline{\Omega})\) is bounded and \(K_{P}(I-Q)N:\overline{\Omega}\rightarrow X\) is compact. We rely on Mawhin’s theorem for coincidences [8].
Theorem 2.1
Let \(L: \operatorname{dom} L\subset X \rightarrow Y\) be a Fredholm operator of index zero and \(N:X\rightarrow Y\) be L-compact on Ω̅. Assume that the following conditions are satisfied:
-
(1)
\(Lx \neq\lambda Nx\) for every \((x,\lambda)\in[(\operatorname {dom} L\setminus\operatorname{Ker} L) \cap\partial\Omega]\times(0,1)\);
-
(2)
\(Nx \notin\operatorname{Im} L\) for every \(x\in\operatorname{Ker} L \cap\partial\Omega\);
-
(3)
\(\operatorname{deg}(JQN| _{\operatorname{Ker} L}, \Omega\cap\operatorname {Ker} L, 0)\neq0\), where \(Q: Y\rightarrow Y\) is a projection such that \(\operatorname{Im} L= \operatorname{Ker} Q\), and \(J: \operatorname{Im} Q \to\operatorname {Ker} L\) is an isomorphism.
Then the equation \(Lx=Nx\) has at least one solution in \(\operatorname{dom} L \cap\overline{\Omega}\).
We work in \(X=C^{2}[0,1]\) with the norm \(\Vert x \Vert = \max\{ \Vert x \Vert _{\infty }, \Vert x' \Vert _{\infty}, \Vert x'' \Vert _{\infty}\}\), where \(\Vert x \Vert _{\infty}=\max_{t\in[0,1]} \vert x(t) \vert \). We define \(Y=L^{1}[0,1]\) with the norm \(\Vert y \Vert _{1}= {\int_{0}^{1} \vert y(t) \vert \,dt}\).
In this paper, we always suppose that the following condition holds:
- \((C)\) :
-
There exist constants \(k_{i}>0\), \(i=1,2,3\), such that \(\vert \varphi_{i}(x) \vert \leq k_{i} \Vert x \Vert \), \(x\in X\) and the function \(f(t,u,v,w)\) satisfies the Carathéodory conditions, that is, \(f(\cdot,u,v,w)\) is measurable for each fixed \((u,v,w)\in \mathbb{R}^{3}\), \(f(t,\cdot,\cdot,\cdot)\) is continuous for a.e. \(t\in[0,1]\).
3 Solvability of (1.1) with condition (1.2)
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
Hence
It follows from \(\Delta_{j3}\neq0\) and \(\varphi_{i}(1)=0\), \(i=1,2,3\), that there exist constants a and b such that
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:
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,
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:
So, y satisfies \(\Delta_{3}(y)=0\).
Inversely, if \(y\in Y\) with \(\Delta_{3}(y)=0\), we let
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
we have
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
Clearly, \(P_{3}\), \(Q_{3}\) are projectors such that (2.3) hold.
Define the operator \(K_{P_{3}}: Y\rightarrow X\) by
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
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
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
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
where, for convenience, we define, using (3.1), the constant
Then
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
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
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,
and
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
So,
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
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
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
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,
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
Obviously, \(P_{2}\) and \(Q_{2}\) are continuous linear projectors satisfying (2.3).
Define the operator \(K_{P_{2}}: Y\rightarrow X\) as
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
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
where \(A_{P_{2}}\) satisfies (3.6). Then the set
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
we have
So,
Thus,
which proves that \(\Omega_{12}\) is bounded. □
Lemma 3.10
Assume that \((H_{6})\) holds. Then the set
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
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,
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,
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
Clearly, \(P_{1}\) and \(Q_{1}\) are continuous linear projectors. Introduce the operator \(K_{P_{1}}:Y\rightarrow X\) by
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
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
where \(A_{P_{1}}\) is given by (3.10). Then the set
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 \(x=P_{1}x+(I-P_{1})x\), \(\Vert (I-P_{1})x \Vert \leq A_{P_{1}} \Vert Lx \Vert _{1} < A_{P_{1}} \Vert Nx \Vert _{1}\),
and \((P_{1}x)''(t_{2})=x''(0)\), we get
Combining the inequalities above, we get
Thus,
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.
4 Solvability of (1.1) with condition (1.3)
We define, for convenience,
By the same method as we used in the proof of Lemma 3.1 (see also [6]), there exists \(g \in Y\) such that
It is easy to see that
Lemma 4.1
Proof
In fact, if \(x\in\operatorname{dom}L\), \(Lx=y\), then
and \(\varphi_{i}(x)=0, i=1,2,3\). So, we have
In view of (4.1),
On the other hand, if \(y\in Y\) satisfies the identity on the right-hand side of (4.2), we choose
Obviously, \(Lx=y\). If \(\Delta_{11}\neq0\), then
Considering \(\Delta_{11}=\Delta_{1}(y)_{11}\), \(\Delta_{2}(y)_{11}=\Delta _{1}(y)_{12}\), \(\Delta_{3}(y)_{11}=-\Delta_{1}(y)_{13}\) and \(\Delta _{1}(y)=0\), we get
Similarly, \(\varphi_{1}(x)= -\frac{1}{2\Delta_{12}}\Delta_{2}(y)=0\), if \(\Delta_{12}\neq0\) and \(\varphi_{1}(x)= \frac{1}{2\Delta_{13}}\Delta _{3}(y)=0\), if \(\Delta_{13}\neq0\). It is easy to check \(\varphi _{2}(x)=\varphi_{3}(x)=0\).
Thus, \(x\in\operatorname{dom}L\), that is, \(y\in\operatorname{Im}L\). So, (4.2) holds. □
Define operators \(P:X\rightarrow X\), \(Q:Y\rightarrow Y\) by
where g is introduced at the beginning of the section. Moreover, \(J: \operatorname{Im}Q \to\operatorname{Ker}L\) is defined by
We define \(K_{P}:Y\rightarrow X\) as follows:
Lemma 4.2
\(K_{P}=(L| _{\operatorname{dom}L\cap\operatorname{Ker}P})^{-1}\) and
Proof
If \(\Delta_{11}\neq0\), for \(x\in\operatorname{dom}L\cap \operatorname{Ker}P\), considering \(\Delta_{2}(Lx)_{11}=-x''(0)\Delta_{12}-2x'(0)\Delta_{11}\), \(\Delta _{3}(Lx)_{11}=x''(0)\Delta_{13}-2x(0)\Delta_{11}\), \(\Delta _{11}x''(0)-\Delta_{12}x'(0)+\Delta_{13}x(0)=0\), we have
Inversely, if \(y\in\operatorname{Im}L\), then \(\Delta_{i}(y)=0\), \(i=1,2,3\). This, together with \(\Delta_{2}(y)_{11}=\Delta_{1}(y)_{12}\), \(\Delta _{3}(y)_{11}=-\Delta_{1}(y)_{13}\), \(\Delta_{11}=\Delta_{1}(y)_{11}\), \(\Delta=0\), implies that
Obviously, \(\varphi_{2}(K_{P}y)=\varphi_{3}(K_{P}y)=0\). So, \(K_{P}y\in\operatorname {dom}L\). By a simple calculation, we can obtain \(K_{P}y\in\operatorname {Ker}P\) and \(LK_{P}y=y\). Therefore, \(K_{P}=(L| _{\operatorname{dom}L\cap\operatorname {Ker}P})^{-1}\). If \(\Delta_{11}=0\), \(\Delta_{12}\neq0\) or \(\Delta _{11}=\Delta_{12}=0\), \(\Delta_{13}\neq0\), we can similarly get the result.
If \(\Delta_{11}\neq0\), then
If \(\Delta_{11}=0\), \(\Delta_{12}\neq0\), then
Similarly, if \(\Delta_{11}=\Delta_{12}=0\), \(\Delta_{13}\neq0\), then
□
Lemma 4.3
Assume that \((H_{1})\) holds and \(\Omega\subset X\) is bounded. Then N is L-compact on Ω.
Proof
For \(x\in\overline{\Omega}\), there exists a constant \(r>0\) such that \(\Vert x \Vert \leq r\). By \((H_{1})\), we get
that is, \(QN(\overline{\Omega})\) is bounded. Hence,
So, \(K_{P}(I-Q)N(\overline{\Omega})\) is bounded.
By the same method as used in the proof of Lemma 3.3, we can demonstrate that \(K_{P}(I-Q)N(\overline{\Omega})\) is compact. Thus N is L-compact. □
When \(\Delta_{11}\neq0\), we assume that the following conditions hold:
- \((H_{9})\) :
-
There exists a constant \(M_{0}>0\) such that if \(\vert x''(t) \vert > M_{0}\), then
$$(\varphi_{1}-k\varphi_{2}) \biggl( \int_{0}^{t}(t-s)^{2}Nx(s) \,ds \biggr) \neq0; $$ - \((H_{10})\) :
-
There exists a constant \(M_{1}>0\) such that if \(\vert c \vert > M_{1}\), either
$$ c(\varphi_{1}-k\varphi_{2}) \biggl( \int_{0}^{t}(t-s)^{2}N \bigl(c \bigl( \Delta _{11}s^{2}-\Delta_{12}s+\Delta_{13} \bigr) \bigr) \,ds \biggr) > 0, $$(4.5)or
$$ c(\varphi_{1}-k\varphi_{2}) \biggl( \int_{0}^{t}(t-s)^{2}N \bigl(c \bigl( \Delta _{11}s^{2}-\Delta_{12}s+\Delta_{13} \bigr) \bigr) \,ds \biggr) < 0. $$(4.6)
Lemma 4.4
Assume that \(\Delta_{11}\neq0\), \((H_{2}), (H_{9})\) and
hold. Then the set
is bounded.
Proof
Since \(x\in\Omega_{1}\), then \(QNx=0\). By \((H_{9})\), there exists \(t_{0}\in[0,1]\) such that \(\vert x''(t_{0}) \vert \leq M_{0}\). Since \(x= Px +(I-P)x\),
and \(x''(t)=(Px)''(t)+((I-P)x)''(t)\), it follows that
Considering
we have
This, together with (4.8) and \((H_{2})\), means
So,
Therefore, \(\overline{\Omega}_{1}\) is bounded due to (4.7). □
Lemma 4.5
Assume that \(\Delta_{11}\neq0\) and \((H_{10})\) holds. Then the set \(\Omega_{2}=\{ x\in\operatorname{Ker}L:Nx\in\operatorname{Im}L\}\) is bounded.
Proof
For \(x\in\Omega_{2}\), we have \(x=c(\Delta _{11}t^{2}-\Delta_{12}t+\Delta_{13})\) and \(QNx=0\). By \((H_{10})\), we get that \(\vert c \vert \leq M_{1}\). So, \(\Omega_{2}\) is bounded. □
Let \(\rho= \{ \scriptsize{ \begin{array}{l@{\quad}l} 1, &\mbox{if } (4.5) \mbox{ holds},\\ -1, &\mbox{if } (4.6) \mbox{ holds}. \end{array}} \)
Lemma 4.6
Assume that \(\Delta_{11}\neq0\) and \((H_{10})\) holds. The set
is bounded, where \(J: \operatorname{Im}Q \rightarrow \operatorname{Ker}L\) is defined by (4.3).
Proof
If \(x\in\Omega_{3}\), then \(\lambda\rho x +(1-\lambda )JQNx=0\), \(x=c(\Delta_{11}t^{2}-\Delta_{12}t+\Delta_{13})\).
If \(\lambda=0\), then \(QNx=0\). It follows from \((H_{10})\) that \(\Vert x \Vert \leq M_{1}(2 \vert \Delta_{11} \vert + \vert \Delta_{12} \vert + \vert \Delta_{13} \vert )\). If \(\lambda=1\), then \(x \equiv0\). For \(\lambda\in(0,1)\), we have
That is,
By \((H_{10})\), we know that \(\vert c \vert \leq M_{1}\). So, \(\Omega_{3}\) is bounded. □
Theorem 4.7
Assume that \(\Delta_{11}\neq0\) and \((H_{2})\), \((H_{9})\), \((H_{10})\) hold and
where \(A_{p}\) is given by (4.4). Then FBVP (1.1) has at least one solution.
The proof is similar to that of Theorem 3.8.
The example below illustrates Theorem 4.7.
Consider
where \(A =1/113{,}414\), along with the functional conditions
In this case \(\phi_{1}(t^{2}) = \phi_{2}(t^{2}) =8\), \(\phi_{3}(t^{2})=1\), \(\phi _{1}(t) = \phi_{2}(t) =10\), \(\phi_{3}(t)=1\), and \(\phi_{1}(1) = \phi_{2}(1) =19\), \(\phi_{3}(1)=2\), so that \(k=1\). Also, \(\triangle= 0\) and \(\triangle_{11}=1\), \(\triangle_{12} = -3\), \(\triangle_{13} = -2\), and
Subsequently,
and
We can easily check \((\phi_{1}-k\phi_{2})(-t^{5})=1\).
The following estimates hold for \(x''(t) < -M_{0}\):
and hence
provided \(M_{0} > 2 + \frac{128}{35A}\).
If \(x''(t) > M_{0}\), then
and hence
provided \(M_{0} > 2\).
Therefore, if we choose \(M_{0} > 2 + \frac{64}{35A}\), then (\(H_{9}\)) holds.
Let now \(c\in\mathbb{R}\) and \(x_{c}(t)= c(t^{2}+3t-2)\). Then
and
Then, repeating the computation leading to the choice of \(M_{0}\), we obtain that \(\vert c \vert > M_{1} = M_{0}/2\) results in
that is, (\(H_{10}\)) holds.
Finally, note that \(k_{1} = 25\), \(k_{2}=21\), \(k_{3} = 25/2\), so that \(l= 525\). Hence \(A_{P} = 1 +16l/ \vert \triangle_{11} \vert =8{,}401\). Then
All the conditions of Theorem 4.7 are verified.
The following corollaries are stated without proof.
Corollary 4.8
Let \(\Delta_{11}=0\), \(\Delta_{12}\neq0\) and assume that (\(H_{2}\)) and the following conditions hold:
- \((H_{11})\) :
-
There exists a constant \(M'_{0}>0\) such that if \(\vert x'(t) \vert \geq M'_{0}\), then
$$(\varphi_{1}-k\varphi_{2}) \biggl( \int_{0}^{t}(t-s)^{2}Nx(s) \,ds \biggr) \neq0; $$ - \((H_{12})\) :
-
There exists a constant \(M'_{1} > 0\) such that if \(\vert c \vert > M'_{1}\), then either
$$c(\varphi_{1}-k\varphi_{2}) \biggl( \int_{0}^{t}(t-s)^{2}N \bigl(c(- \triangle _{12}s+\triangle_{13}) \bigr) \,ds \biggr) > 0, $$or
$$c(\varphi_{1}-k\varphi_{2}) \biggl( \int_{0}^{t}(t-s)^{2}N \bigl(c(- \triangle _{12}s+\triangle_{13}) \bigr) \,ds \biggr) < 0. $$
Then FBVP (1.1) has at least one solution.
Corollary 4.9
Let \(\Delta_{11}=\Delta_{12}=0\), \(\Delta _{13}\neq0\) and assume that (\(H_{2}\)) and the following conditions hold:
- \((H_{13})\) :
-
There exists a constant \(M''_{0}>0\) such that if \(\vert x(t) \vert \geq M''_{0}\), then
$$(\varphi_{1}-k\varphi_{2}) \biggl( \int_{0}^{t}(t-s)^{2}Nx(s) \,ds \biggr) \neq0; $$ - \((H_{14})\) :
-
There exists a constant \(M''_{1} > 0\) such that if \(\vert c \vert > M''_{1}\), then either
$$c(\varphi_{1}-k\varphi_{2}) \biggl( \int_{0}^{t}(t-s)^{2}N(c \triangle _{13}) \,ds \biggr) > 0, $$or
$$c(\varphi_{1}-k\varphi_{2}) \biggl( \int_{0}^{t}(t-s)^{2}N(c \triangle _{13}) \,ds \biggr) < 0. $$
Then FBVP (1.1) has at least one solution.
5 Conclusion
This paper is a study of third-order functional boundary value problems at resonance; it improves and generalizes some of the existent results. We present several generalizations to the existing results and improvements to the method based on Mawhin’s coincidence degree theory.
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Acknowledgements
The authors are grateful to anonymous reviewers for carefully reading this paper and for their comments and suggestions which have improved the paper. The work WJ is supported by the Natural Science Foundation of Hebei Province (A2013208108).
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Jiang, W., Kosmatov, N. Solvability of a third-order differential equation with functional boundary conditions at resonance. Bound Value Probl 2017, 81 (2017). https://doi.org/10.1186/s13661-017-0811-z
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DOI: https://doi.org/10.1186/s13661-017-0811-z