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On a sign-changing solution for some fractional differential equations
- Kemei Zhang1Email authorView ORCID ID profile
Received: 4 February 2017
Accepted: 5 April 2017
Published: 20 April 2017
Abstract
In this paper, a kind of αth \((3<\alpha\leq4)\) order differential equation with two-point boundary conditions is considered. The existence result of a sign-changing solution is given by the topological degree theory and the fixed point index theory.
Keywords
sign-changing solution topological degree fixed point indexMSC
34B151 Introduction
Fractional differential equations have gained much importance and attention due to the fact that they have been proved to be valuable tools in the modeling of many phenomena in engineering and sciences such as physics, mechanics, economics and biology; see [1–3]. In recent years, there has been a great deal of research on the existence and/or uniqueness of solution to boundary value problems for fractional-order differential equations. By means of Leray-Schauder degree theory, fixed point theorem on cone, a monotone iterative method combined with lower and upper solutions or by introducing height functions of the nonlinear term on some bounded sets and considering integrations of these height functions, several local existence and multiplicity results of solutions are obtained, for details, please refer to [4–11] and the references therein. In [8–11], the authors studied the following αth (\(3< \alpha\leq4\)) order differential equation:
with different boundary conditions. By using the Leray-Schauder nonlinear alternative theorem, fixed point index theory, the properties of cone and fixed point theorems for a mixed monotone operator, the existence results of positive solutions for singular and nonsingular nonlinear fractional differential equation boundary value problems are obtained.
$$D^{\alpha}_{0^{+}}u(t)=f\bigl(t,u(t)\bigr),\quad t\in(0,1), $$
To the best of our knowledge, there are less papers studying the existence of sign-changing solution for the fractional differential equations. The aim of this paper is to investigate the existence of sign-changing solution for the following Riemann-Liouville fractional differential equations with two-point boundary conditions:
where \(D^{\alpha}_{0^{+}}\) is the standard Riemann-Liouville fractional derivative of order \(3<\alpha\leq4\) defined by
Γ denotes the Euler gamma function, provided that the right side is point wise defined on \((0,+\infty)\). The existence result of a sign-changing solution is given by topological degree theory and fixed point index theory. One can refer for the method and the theoretical knowledge used in this paper to [12–18].
$$ \textstyle\begin{cases} D^{\alpha}_{0^{+}}u(t)=f(t,u(t)), \quad t\in(0,1), \\ u(0)=u(1)=u'(0)=u'(1)=0, \end{cases} $$
(1.1)
$$D^{\alpha}_{0^{+}}u(t)=\frac{1}{\Gamma(4-\alpha)}\biggl(\frac{d}{dt} \biggr)^{4} \int^{t}_{0}\frac{u(s)}{(t-s)^{\alpha-3}}\,ds, $$
2 Preliminaries and some lemmas
Definition 2.1
Let E be a real Banach space and \(A:E\rightarrow E\) be a nonlinear operator. A nonzero solution to the equation \(x=\lambda Ax\) is called an eigenvector of the nonlinear operator A; the corresponding number λ is called a characteristic value of A, and \(\lambda^{-1}\) is called a eigenvalue of A.
Definition 2.2
Let \(E_{1}\), \(E_{2}\) be real Banach spaces and \(D\subset E_{1}\) contain the outside of a ball \(\{x\vert \Vert x\Vert \leq r\}\), \(A:D\rightarrow E_{2}\). The operator A is called asymptotically linear, if there is a continuous linear operator \(B:E_{1}\rightarrow E_{2}\) such that
The operator B involved in the definition of an asymptotically linear operator A is uniquely determined, it is called the derivative of A at infinity and is denoted by \(A'(\infty)\).
$$\lim_{ \Vert x\Vert \rightarrow\infty}\frac{\Vert Ax-Bx\Vert }{\Vert x\Vert }=0. $$
Lemma 2.1
[16]
Let
E
be a Banach space and
P
be a total cone in
E. Suppose
\(T :P\rightarrow P\)
is a bounded linear operator (therefore, T
can be uniquely extended to a bounded linear operator on
\(\overline{P-P}=E\), and the extended operator is denoted by
T
again) with the spectral radius
\(r(T)<1\). If
\(w,w_{0}\in E\)
such that
\(w\leq Tw+w_{0}\), then
where
\((I-T)^{-1}\)
is the inverse operator of the operator
\(I-T\).
$$w\leq(I-T)^{-1}w_{0}, $$
Lemma 2.2
Suppose that
E
is a Banach space, \(A:E\rightarrow E\)
is a completely continuous and asymptotically linear operator. If 1 is not the eigenvalue of the linear operator
\(A'(\infty)\), then there exists
\(R_{0}>0\)
such that
for any
\(R\geq R_{0}\), where
\(B_{R}=\{x\in E\vert \Vert x\Vert < R\} \), γ
is the sum of the algebraic multiplicities of the real eigenvalues of
\(A'(\infty)\)
in
\((1,+\infty)\).
$$\operatorname{deg}(I-A, B_{R}, \theta)=(-1)^{\gamma} $$
Lemma 2.3
[16]
Let
E
be a Banach space and Ω be a bounded open set in
E
with
\(\theta\in\Omega\). Suppose that
\(A:\overline{\Omega}\rightarrow E\)
is a completely continuous operator. If
then the topological degree
\(\operatorname{deg}(I-A,\Omega,\theta)=1\).
$$Au\neq\mu u,\quad \forall u\in\partial\Omega, \mu\geq1, $$
Lemma 2.4
[14]
Let
E
be a Banach space, and
P
be a cone in
E, and Ω be a bounded open set in
E. Suppose that
\(A:P\cap\overline{\Omega}\rightarrow P\)
is a completely continuous operator. If
then the fixed point index
\(i(A,P\cap\Omega,P)=1\).
$$Au\neq\mu u,\quad \forall u\in P\cap\partial\Omega, \mu\geq1, $$
Remark 2.1
Let E be a Banach space and Ω be a bounded open set in E with \(\theta\in\Omega\). Suppose that \(A:\overline{\Omega}\rightarrow E\) is a completely continuous operator. If
then the topological degree \(\operatorname{deg}(I-A,\Omega,\theta)=1\). Furthermore, suppose that P is a cone in E and satisfies
then the fixed point index \(i(A,\pm P\cap\Omega,\pm P)=1\).
$$\Vert Au\Vert < \Vert u\Vert ,\quad \forall u\in\partial\Omega, $$
$$\Vert Au\Vert < \Vert u\Vert ,\quad \forall u\in\pm P\cap\partial \Omega, $$
Given \(h\in C[0,1]\), it follows from [8] that the unique solution of the problem
can be expressed uniquely by \(u(t)=\int_{0}^{1}G(t,s)h(s)\,ds\), where
$$\textstyle\begin{cases} D^{\alpha}_{0^{+}}u(t)+h(t)=0,\quad 0< t< 1, \\ u(0)=u(1)=u'(0)=u'(1)=0, \end{cases} $$
$$ G(t,s)=\frac{1}{\Gamma(\alpha)} \textstyle\begin{cases} (t-s)^{\alpha -1}+t^{\alpha-2}(1-s)^{\alpha-2} \bigl[(s-t)+(\alpha-2) (1-t)s \bigr],\\ \quad 0\leq s\leq t\leq1, \\ t^{\alpha-2}(1-s)^{\alpha-2} \bigl[(s-t)+(\alpha-2) (1-t)s \bigr],\\ \quad 0 \leq t\leq s\leq1. \end{cases} $$
(2.1)
It is easy to verify that \(G(t,s)>0\) for \(t,s\in(0,1)\) and
where \(M_{0}=\max\{\alpha-1,(\alpha-2)^{2}\}\).
$$ (\alpha-2)t^{\alpha-2}(1-t)^{2}s^{2}(1-s)^{\alpha-2} \leq\Gamma( \alpha)G(t,s)\leq M_{0}t^{\alpha-2}(1-t)^{2},\quad \forall t,s\in[0,1], $$
(2.2)
In the following, we introduce some notations. Let \(X=C[0,1]\) and
then P is a cone in X. Set \(e(t)=t^{\alpha-2}(1-t)^{2}\),
then \(\Vert x\Vert _{e}\) is called the e-norm of the element \(x\in X_{e}\). It is easy to see that \(X_{e}\) becomes a Banach space under the norm \(\Vert \cdot \Vert _{e}\) and \(P_{e}=P\cap X_{e}\) is a normal solid cone in \(X_{e}\) (\(e\in\stackrel{\circ}{ P_{e}}\)).
$$P=\bigl\{ x\in X\vert x(t)\geq0,t\in[0,1]\bigr\} , $$
$$\begin{aligned}& X_{e}=\{x \in X\vert \mbox{ there exists } \lambda>0\mbox{ such that }{-} \lambda e\leq x\leq\lambda e\}, \\& \Vert x\Vert _{e}=\inf\{\lambda>0\vert-\lambda e\leq x\leq \lambda e\},\quad \forall x\in E_{e}, \end{aligned}$$
Before proving the following lemmas, we need the following conditions:
- (H1):
-
\(f:[0,1]\times(-\infty,+\infty)\rightarrow(-\infty,+ \infty)\) is continuous and \(f(t,x)x>0\) for all \(x\in R\setminus \{0 \}\) and \(t\in[0,1]\).
- (H2):
-
\(\lim_{x\rightarrow\infty}\frac{f(t,x)}{x}= \beta_{\infty}(t)\) uniformly with respect to \(t\in[0,1]\).
Evidently, \(\beta_{\infty}(t)\in C[0,1]\) and \(\beta_{\infty}(t) \geq0\) for \(t\in[0,1]\). Define the operators K, F such that
and \(A=KF\).
$$\begin{aligned}& Ku(t)= \int^{1}_{0}G(t,s)u(s)\,ds,\quad u\in X, t\in[0,1], \end{aligned}$$
(2.3)
$$\begin{aligned}& (Fu) (t)=f \bigl(t,u(t) \bigr),\quad t\in[0,1], u\in X, \end{aligned}$$
(2.4)
Lemma 2.5
The operator K defined by (2.3) satisfies \(K: X\rightarrow X_{e}\) and \(K: P\setminus\{\theta\}\rightarrow\stackrel{ \circ}{ P_{e}}\), where \(\stackrel{\circ}{P_{e}}=\{x \in X\vert \textit{ there exist } \tilde{\alpha}>0, \tilde{\beta}>0 \textit{ such that } \tilde{\alpha} e\leq x\leq\tilde{\beta} e\}\).
Proof
For any \(u\in X\), by (2.2), (2.3) we have
where \(\tilde{\lambda}=\frac{M_{0}}{\Gamma(\alpha)}\int_{0}^{1}\vert u(s)\vert \,ds\), i.e., \(K: X\rightarrow X_{e}\).
$$\begin{aligned}& Ku(t)= \int_{0}^{1}G(t,s)u(s)\,ds \leq\frac{M_{0}}{\Gamma(\alpha)} \int_{0}^{1} \bigl\vert u(s) \bigr\vert \,ds\cdot e(t)=\tilde{\lambda}e(t),\quad t\in[0,1], \end{aligned}$$
(2.5)
$$\begin{aligned}& Ku(t)= \int_{0}^{1}G(t,s)u(s)\,ds \geq \int_{0}^{1}G(t,s) \bigl(- \bigl\vert u(s) \bigr\vert \bigr)\,ds \geq-\tilde{\lambda}e(t),\quad t\in[0,1], \end{aligned}$$
(2.6)
Moreover, for \(u\in P\setminus\{\theta\}\), by (2.2), one can get
then it follows from (2.5), (2.7) that \(\tilde{\alpha}e(t)\leq Ku(t) \leq\tilde{\beta}e(t)\), where
i.e., \(Ku\in\mathbin{\stackrel{\circ}{ P_{e}}}\), and so we show that \(K: P_{e}\setminus\{\theta\}\rightarrow\stackrel{\circ}{ P_{e}}\). □
$$ Ku(t)\geq\frac{\alpha-2}{\Gamma(\alpha)} \int_{0}^{1}s^{2}(1-s)^{ \alpha-2}u(s)\,ds \cdot e(t), \quad t\in[0,1], $$
(2.7)
$$\tilde{\alpha}=\frac{\alpha-2}{\Gamma(\alpha)}\int_{0}^{1}s^{2}(1-s)^{ \alpha-2}u(s)\,ds,\qquad \tilde{\beta}=\frac{M_{0}}{\Gamma(\alpha)}\int_{0} ^{1}u(s)\,ds, $$
Lemma 2.6
Suppose that (H2) holds, then the operator
\(A: X_{e}\rightarrow X_{e}\)
is asymptotically linear, and the derivative of
A
at infinity
\(A'(\infty)=B\), where
$$Bu(t)= \int_{0}^{1}G(t,s)\beta_{\infty}(s)u(s)\,ds,\quad u \in X_{e}. $$
Proof
It follows from Lemma 2.5 and the definitions of A, B that \(A,B: X_{e}\rightarrow X_{e}\). By (H2) we know that, for any \(\varepsilon>0\), there exists \(l>0\) such that
for any x with \(\vert x\vert \geq l\) and \(t\in[0,1]\).
$$\begin{aligned} \biggl\vert \frac{f(t,x)}{x}-\beta_{\infty}(t) \biggr\vert < \varepsilon \end{aligned}$$
(2.8)
Set \(T_{l}=\{x\in X_{e}\vert \Vert x\Vert _{e}\leq l\}\), and let
For \(u\in X_{e}\), define
Then
where \(l(t)\equiv l\). It follows from (2.9) that
Therefore, by (2.10), (2.11) and (2.12), we have
From (2.8), we get
for any \(u\in X_{e}\), from which one deduces that
and therefore from (2.13), one can get
i.e., the operator A is asymptotically linear, and \(A'(\infty)=B\).
$$\begin{aligned} M=\sup_{ x\in T_{l}} \bigl\{ \Vert Ax\Vert _{e},\Vert Bx\Vert _{e} \bigr\} . \end{aligned}$$
(2.9)
$$\begin{aligned}& \phi_{1}\bigl(u(t)\bigr)=\textstyle\begin{cases} u(t), & \vert u(t)\vert \leq l, \\ l, & \vert u(t)\vert \geq l, \end{cases}\displaystyle \\& \phi_{2}\bigl(u(t)\bigr)= \textstyle\begin{cases} l, & \vert u(t)\vert \leq l, \\ u(t), & \vert u(t)\vert \geq l. \end{cases}\displaystyle \end{aligned}$$
$$\begin{aligned}& Au(t)=A\phi_{1} \bigl(u(t) \bigr)+A\phi_{2} \bigl(u(t) \bigr)-Al(t),\quad t\in[0,1],u\in X_{e}, \end{aligned}$$
(2.10)
$$\begin{aligned}& Bu(t)=B\phi_{1} \bigl(u(t) \bigr)+B\phi_{2} \bigl(u(t) \bigr)-Bl(t),\quad t\in[0,1],u\in X _{e}, \end{aligned}$$
(2.11)
$$\begin{aligned} \bigl\Vert A\phi_{1}(u) \bigr\Vert _{e}\leq M,\qquad \Vert Al\Vert _{e}\leq M,\qquad \bigl\Vert B\phi_{1}(u) \bigr\Vert _{e}\leq M,\qquad \Vert Bl\Vert _{e}\leq M. \end{aligned}$$
(2.12)
$$ \Vert Au-Bu\Vert _{e}\leq 4M+ \bigl\Vert (A \phi_{2})u-(B \phi_{2})u \bigr\Vert _{e}. $$
(2.13)
$$\begin{aligned}& \bigl\vert (A\phi_{2})u(t)-(B\phi_{2})u(t) \bigr\vert \\& \quad = \biggl\vert \int_{0}^{1}G(t,s) \bigl[f \bigl(s, \phi_{2} \bigl(u(s) \bigr) \bigr)-\beta_{\infty}(s) \phi_{2} \bigl(u(s) \bigr) \bigr]\,ds \biggr\vert \\& \quad \leq \int_{0}^{1}G(t,s) \bigl\vert f \bigl(s, \phi_{2} \bigl(u(s) \bigr) \bigr)-\beta_{\infty}(s) \phi_{2} \bigl(u(s) \bigr) \bigr\vert \,ds \\& \quad \leq\frac{M_{0}}{\Gamma(\alpha)}\varepsilon \Vert u\Vert _{e} e(t) \end{aligned}$$
(2.14)
$$\bigl\Vert (A\phi_{2})u-(B\phi_{2})u\bigr\Vert \leq \frac{M_{0}}{\Gamma (\alpha)} \varepsilon \Vert u\Vert _{e} e(t), $$
$$\lim_{ \Vert u\Vert _{e}\rightarrow\infty}\frac{\Vert Au-Bu\Vert _{e}}{\Vert u\Vert _{e}}=0, $$
In this paper, we always denote by \(\Omega_{r}=\{u\in X:\Vert u\Vert _{e}< r\} \) (\(r>0\)) the open ball of radius r and by θ the zero function in \(X_{e}\). For the concepts and properties on the cone and the topological degree, one can refer to [12, 14, 16]. □
3 Main results
Theorem 3.1
Suppose that we have the conditions (H1), (H2) and the following conditions:
- (H3):
-
There is \(p>0\) such that \(\vert u\vert \leq p, t\in [0,1]\) imply that \(\vert f(t,u)\vert <\eta p\), where \(\eta=\frac{\Gamma(\alpha)}{M_{0}}\).
- (H4):
-
\(\sup_{t\in[0,1]}\beta_{\infty}(t)<\lambda_{1}\), where \(\lambda_{1}\) is the first characteristic value of K defined by (2.3).
If 1 is not the characteristic value of B and the sum of the algebraic multiplicities γ of the real eigenvalues of B in \((1,+ \infty)\) is odd, then the BVP (1.1) has at least a sign-changing solution.
Proof
Evidently, \(A: X_{e}\rightarrow X_{e}\) is completely continuous. By (H1) and Lemma 2.5, we know that \(A: P_{e}\setminus\{ \theta\}\rightarrow\stackrel{\circ}{ P_{e}}\). From (H3), for any \(u\in\partial\Omega_{p}\), we have
from which one deduces that \(\Vert Au\Vert _{e}<\Vert u\Vert _{e}\), so by Remark 2.1, we get
$$\begin{aligned}& \bigl\vert Au(t) \bigr\vert \\& \quad = \biggl\vert \int^{1}_{0}G(t,s)f \bigl(s,u(s) \bigr)\,ds \biggr\vert \\& \quad \leq \int^{1}_{0}G(t,s) \bigl\vert f \bigl(s,u(s) \bigr) \bigr\vert \,ds \\& \quad < \frac{M_{0}\eta}{\Gamma( \alpha)}t^{\alpha-2}(1-t)^{2}\cdot \Vert u\Vert _{e} \\& \quad =\frac{M_{0}\eta}{ \Gamma(\alpha)}\cdot \Vert u\Vert _{e} \cdot e(t) \\& \quad =\Vert u\Vert _{e}\cdot e(t), \end{aligned}$$
(3.1)
$$\begin{aligned}& \operatorname{deg}(I-A,\Omega_{p},\theta)=1, \end{aligned}$$
(3.2)
$$\begin{aligned}& i(A,\pm P_{e}\cap\Omega_{p},\pm P_{e})=1. \end{aligned}$$
(3.3)
From conditions (H2) and (H4), there exist \(\varepsilon _{1}>0\) small enough and \(R_{1}>p\) large enough such that
Let
then it follows from (3.4) that
$$ f(t,u)< (\lambda_{1}-\varepsilon_{1})u,\quad \mbox{for any }u\geq R_{1} \mbox{ and } t\in[0,1]. $$
(3.4)
$$C_{1}=\max_{0\leq u\leq R_{1},0\leq t\leq1} \bigl\vert f(t,u)-( \lambda_{1}- \varepsilon_{1})u\bigr\vert +1, $$
$$ \bigl\vert f(t,u) \bigr\vert \leq(\lambda_{1}- \varepsilon_{1})\vert u\vert +C_{1}, \quad \forall u\in[0,+ \infty) \mbox{ and } t\in[0,1]. $$
(3.5)
Let \(D=\{u\in P_{e}: Au=\lambda u, \lambda\geq1\}\). In the following, we shall prove that D is bounded. In fact, if \(u\in D\), then by (3.5), there exists \(\lambda\geq1\) such that
where \(\phi(t)=C_{1}\int^{1}_{0}G(t,s)\,ds\in P_{e}\). Since \(r((\lambda_{1}- \varepsilon_{1})K)<1\), from Lemma 2.1 we have \(u(t)\leq(I-( \lambda _{1}-\varepsilon_{1})K)^{-1}\phi(t)\), and then by (2.2), we get
which, together with the definition of \(\Vert \cdot \Vert _{e}\), implies that
and thus
So D is bounded and then there exists a sufficiently large number \(R_{2}>p\) such that, for any \(R\geq R_{2}\), one can get
which together with Lemma 2.4 implies that
$$\begin{aligned} 0 \leq& u(t)\leq\lambda u(t) =Au(t) \\ =& \int^{1}_{0}G(t,s)f \bigl(s,u(s) \bigr)\,ds \\ \leq&(\lambda_{1}-\varepsilon_{1}) \int^{1}_{0}G(t,s)u(s)\,ds+C_{1} \int^{1}_{0}G(t,s)\,ds \\ =&(\lambda_{1}-\varepsilon_{1}) (Ku) (t)+\phi(t), \end{aligned}$$
(3.6)
$$-\frac{C_{1}M_{0}}{\Gamma(\alpha)}\cdot e(t)\leq-\phi(t)\leq\bigl(I-( \lambda_{1}- \varepsilon_{1})K\bigr)u(t)\leq\phi(t)\leq\frac{C_{1}M_{0}}{ \Gamma(\alpha)}\cdot e(t), $$
$$\bigl\Vert \bigl(I-(\lambda_{1}-\varepsilon_{1})K\bigr)u \bigr\Vert _{e}\leq\frac{C_{1}M_{0}}{ \Gamma(\alpha)}, $$
$$\begin{aligned} \Vert u\Vert _{e} =& \bigl\Vert \bigl(I-( \lambda_{1}-\varepsilon_{1})B \bigr)^{-1} \bigl[ \bigl(I-(\lambda_{1}-\varepsilon_{1})B \bigr)u \bigr] \bigr\Vert _{e} \\ \leq& \bigl\Vert \bigl(I-(\lambda_{1}-\varepsilon_{1})B \bigr)^{-1} \bigr\Vert \cdot\frac{C_{1}M _{0}}{\Gamma(\alpha)}. \end{aligned}$$
(3.7)
$$ Au\neq\lambda u, \quad \forall u\in P_{e}\cap\partial \Omega_{R} \mbox{ and } \lambda\geq1, $$
(3.8)
$$ i(A,P_{e}\cap\Omega_{R},P_{e})=1,\quad R\geq R_{2}. $$
(3.9)
Similar to the proof of (3.9), there exists \(R_{3}>p\) such that, for \(R\geq R_{3}\), we have
For any \(R_{4}>\{R_{2},R_{3}\}\), it follows from (3.3), (3.9) and (3.10) and the excision property of fixed point index that
Since A has no fixed point on \(\partial(\pm P_{e}\setminus\{ \theta\})\), (3.11) and the permanence of the topological degree imply that
Let \(R_{4}\) be large enough, then by Lemma 2.2, (3.2) and (3.12), one can obtain
which implies that A has at least one fixed point \(x^{*}\in \Omega_{R_{4}}\setminus[\overline{\Omega}_{p}\cup((\pm P_{e}) \cap(\overline{\Omega_{R_{4}}}\setminus\Omega_{p}))]\). i.e., BVP (1.1) has at least a sign-changing solution. The proof is completed. □
$$ i \bigl(A,(-P_{e})\cap\Omega_{R},-P_{e} \bigr)=1. $$
(3.10)
$$ i \bigl(A,(\pm P_{e})\cap(\Omega_{R_{4}}\setminus\overline{ \Omega_{p}}), \pm P_{e} \bigr)=1-1=0. $$
(3.11)
$$ \operatorname{deg}\bigl(I-A,\mathbin{\mathrel{\pm} \stackrel{\circ}{ P_{e}}}\cap( \Omega_{R_{4}} \setminus\overline{ \Omega_{p}}),\theta \bigr)=i \bigl(A,( \pm P_{e})\cap(\Omega_{R_{4}}\setminus\overline{ \Omega_{p}}), \pm P_{e} \bigr)=0. $$
(3.12)
$$\begin{aligned}& \operatorname{deg}\bigl(I-A,\Omega_{R_{4}}\setminus \bigl[\overline{ \Omega}_{p}\cup \bigl((\pm P _{e})\cap(\overline{ \Omega_{R_{4}}}\setminus\Omega_{p}) \bigr) \bigr],\theta \bigr) \\& \quad =\operatorname{deg}(I-A,\Omega_{R_{4}},\theta)-\operatorname{deg}(I-A,\Omega_{p},\theta)-\operatorname{deg}\bigl(I-A, \mathbin{\mathrel{\pm}\stackrel{\circ}{ P_{e}}}\cap( \Omega_{R_{4}}\setminus\overline{ \Omega_{p}}),\theta \bigr) \\& \quad =(-1)^{\gamma}-1-0=-2\neq0, \end{aligned}$$
(3.13)
Declarations
Acknowledgements
This work was supported by National Natural Science Foundation of China (No. 11571197), and by Shandong Provincial Natural Science Foundation of China (No. 2016ZRB01076).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
(1)
School of Mathematics Sciences, Qufu Normal University
References
- Podlubny, I: Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198. Academic Press, New York (1999) MATHGoogle Scholar
- Kilbas, AA, Srivastava, HM, Nieto, JJ: Theory and Applicational Differential Equations. Elsevier, Amsterdam (2006) Google Scholar
- Lakshmikantham, V, Vatsala, AS: General uniqueness and monotone iterative technique for fractional differential equations. Appl. Math. Lett. 21, 828-834 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Jankowski, T: Boundary problems for fractional differential equations. Appl. Math. Lett. 28, 14-19 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Zhang, XG, Liu, LS, Wu, YH, Wiwatanapataphee, B: The spectral analysis for a singular fractional differential equation with a signed measure. Appl. Math. Comput. 257, 252-263 (2015) MathSciNetMATHGoogle Scholar
- Zhang, XG, Liu, LS, Wu, YH: The eigenvalue for a class of singular p-Laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition. Appl. Math. Comput. 235, 412-422 (2014) MathSciNetMATHGoogle Scholar
- Zhang, XQ: Positive solutions for a class of singular fractional differential equation with infinite-point boundary value conditions. Appl. Math. Lett. 39, 22-27 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Xu, XJ, Jiang, DQ, Yuan, CJ: Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation. Nonlinear Anal. 71, 4676-4685 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Xu, XJ, Hu, WW: A new existence results of positive solution for a class of nonlinear fractional differential equation boundary value problems. J. Syst. Sci. Math. Sci. 32(5), 580-590 (2012) MathSciNetMATHGoogle Scholar
- Zhang, XQ, Wang, LQ, Wang, S: Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions. Appl. Math. Comput. 226, 708-718 (2014) MathSciNetMATHGoogle Scholar
- Zhai, CB, Hao, MR: Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems. Nonlinear Anal. TMA 75, 2542-2555 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Liu, YS: Multiple positive solutions of nonlinear singular boundary value problem for fourth-order equations. Appl. Math. Lett. 17, 747-757 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Krasnoselskii, MA, Zabreiko, PP: Geometrical Methods of Nonlinear Analysis. Springer, New York (1984) View ArticleGoogle Scholar
- Guo, DJ, Laksmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, Boston (1988) Google Scholar
- Zhang, KM: Non-trivial solutions of fourth-order singular boundary value problems with sign-changing nonlinear terms. Topol. Methods Nonlinear Anal. 40, 53-70 (2012) MathSciNetMATHGoogle Scholar
- Deimling, K: Nonlinear Functional Analysis. Springer, Berlin (1985) View ArticleMATHGoogle Scholar
- Zhang, KM, Xie, XJ: Existence of sign-changing solutions for some asymptotically linear three-point boundary value problems. Nonlinear Anal. TMA 70(7), 2796-2805 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Cui, YJ: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51, 48-54 (2016) MathSciNetView ArticleMATHGoogle Scholar
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