Abstract
In this paper, we study existence and nonexistence of positive solutions for a class of Riemann–Stieltjes integral boundary value problems of fractional differential equations with parameters. By using the fixed point index theory, some new sufficient conditions for the existence of at least one, two and the nonexistence of positive solutions are obtained. The results we obtain show the influence of parameter λ and parameter a on the existence of positive solutions. Finally, some examples are given to illustrate our main results.
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1 Introduction
In this paper, we investigate existence and nonexistence of positive solutions for a class of Riemann–Stieltjes integral boundary value problems of fractional differential equations with parameters
where \(D_{0^{+}}^{\alpha }\) and \(D_{0^{+}}^{\beta }\) are the Riemann–Liouville fractional derivatives with \(0<\alpha \leq 1\), \(1<\beta \leq 2\). The parameters \(\lambda >0\), \(a\geq 0\), \(p\in C ([0,1],(0,+\infty ) )\), \(f: [0,1]\times [0,+\infty )\rightarrow [0,+\infty )\) are given functions, and f may be discontinuous but satisfies the \(L^{q}\)-Carathéodory conditions. \(\int _{0}^{1}u(s) \,\mathrm{d}A(s)\) denotes the Riemann–Stieltjes integral with respect to A.
By using the fixed point index theory, some new sufficient conditions for the existence of at least one, two and the nonexistence of positive solutions are obtained. The theorems we obtain show the influence of parameter λ and parameter a on the existence of positive solutions.
In recent decades, with the wide applications of fractional differential equations in physics, engineering, biology, chemistry, and many other fields, researchers have been paying more and more attention to them, see [1,2,3,4,5,6,7,8,9,10,11,12] and the references therein. At the same time, many problems of fluid mechanics, bioengineering, chemical engineering, and so on could be attributed to the integral boundary value problems, which are nonlocal problems. Therefore, a lot of meaningful research results have been obtained, see [13,14,15,16,17,18,19,20] and the references therein. The eigenvalue problem is a relatively active part of the differential equation theory, and there have been many results, see [21,22,23,24,25,26,27,28,29,30] and the references therein. Nowadays, when solving many practical problems, there will inevitably be errors and those errors will often affect the existence of the solution to a large extent. Therefore, it is meaningful to study the boundary value problem of fractional differential equations with disturbance parameters, see [31,32,33,34,35] and the references therein.
As a generalization of classical Riemann integral, Riemann–Stieltjes integral boundary value problem has a stronger applicability, which not only contains the classical Riemann integral boundary value problem, but also includes two-point boundary value and multi-point boundary value. In this paper, we investigate existence and nonexistence of positive solutions for a class of Riemann–Stieltjes integral boundary value problems of fractional differential equations with parameters (1.1).
The paper is organized as follows. In Sect. 2, we present some necessary definitions and lemmas which will be used to prove our main results. We study the properties of integral kernels and obtain inequalities about the integral kernels. We prove the complete continuity of operators. In Sect. 3, we investigate the existence of at least one positive solution for boundary value problem (1.1). In Sect. 4, sufficient conditions for the existence of at least two positive solution of boundary value problem (1.1) and the nonexistence of positive solution of boundary value problem (1.1) are established. In Sect. 5, we give some examples to illustrate our main result.
Throughout this paper, we assume that \(A(t)\) is a monotone increasing function, \(\int _{0}^{1}s^{\beta -2}\,\mathrm{d}A(s)\) exists, and
f satisfies the \(L^{q}\)-Carathéodory conditions, that is,
-
(1)
\(f(\cdot ,u)\) is measurable for all \(u\in [0,+\infty )\);
-
(2)
\(f(t,\cdot )\) is continuous for a.e. \(t\in [0,1]\);
-
(3)
for every \(r>0\), there exists \(\varphi _{r}\in L^{q}[0,1]\) such that
$$ \bigl\vert f\bigl(t,t^{\beta -2}u\bigr) \bigr\vert \leq \varphi _{r}(t) \quad \text{for all } u\in [0,r] \text{ and a.e. }t\in [0,1], $$
where \(q>\frac{1}{\alpha }\) if \(0<\alpha <1\) and \(q=1\) if \(\alpha =1\).
For \(L^{q}[0,1]\), we denote the norm \(\|\varphi \|_{L^{q}}=(\int _{0} ^{1}|\varphi (t)|^{q}\,\mathrm{d}t)^{\frac{1}{q}} \).
2 Preliminaries
The definitions of fractional integral and fractional derivative and the related lemmas can be found in [3, 4].
Lemma 2.1
(See [3], Theorem 2.4 and [4], Lemma 2.5)
Let \(p>0\) and \(n=\lceil {p}\rceil =\min \{z\in \mathbb{Z}:z\geq p\}\). If \(u\in L^{1}[0,1]\) and \(I_{0^{+}}^{n-p}u\in AC^{n}[0,1]\), then the equality
holds a.e. on \([0,1]\).
Lemma 2.2
If \(0<\alpha <1\), \(D_{0^{+}}^{\alpha }u\in L^{1}[0,1]\) and \(\lim_{t\rightarrow 0^{+}}t^{1-\alpha }u(t)=c\), where c is a constant, then \(I_{0^{+}}^{1-\alpha }u\in AC^{1}[0,1]\).
Proof
Since \(\lim_{t\rightarrow 0^{+}}t^{1-\alpha }u(t)=c\), then for any \(\varepsilon >0\), there exists a constant \(\delta >0\) such that \(|t^{1-\alpha }u(t)-c|<\frac{\varepsilon }{\varGamma (\alpha )}\) whenever \(0< t<\delta \), and
Hence, we have \(\lim_{t\rightarrow 0^{+}}I_{0^{+}}^{1-\alpha }u(t)=c \varGamma (\alpha )\).
Let \(\phi (t)=D_{0^{+}}^{\alpha }u(t)=\frac{\mathrm{d}}{\mathrm{d}t}I _{0^{+}}^{1-\alpha }u(t)\), then \(\phi \in L^{1}[0,1]\) and
Therefore, \(I_{0^{+}}^{1-\alpha }u\in AC^{1}[0,1]\). □
Let
then E is a Banach space with the norm \(\|u\|=\sup_{t\in [0,1]}t ^{2-\beta }|u(t)|\).
Definition 2.1
A function \(u=u(t)\) is called a solution of fractional boundary value problem (1.1) if \(u\in E\) and satisfies (1.1). Furthermore, \(u=u(t)\) is called a positive solution of fractional boundary value problem (1.1) if \(u(t)>0\), \(t\in (0,1)\).
Lemma 2.3
For any \(y\in L^{q}[0,1]\), the fractional differential initial value problem
has a unique solution
Proof
Suppose that \(v=v(t)\) is a solution of initial value problem (2.1). Since \(y\in L^{q}[0,1]\), then \(D_{0^{+}}^{ \alpha }v\in L^{1}[0,1]\). Because \(\lim_{t\rightarrow 0^{+}}t ^{1-\alpha }v(t)=0\), it follows \(I_{0^{+}}^{1-\alpha }v\in AC^{1}[0,1]\) from Lemma 2.2. Thus, by Lemma 2.1, we have
The initial condition \(\lim_{t\rightarrow 0^{+}}t^{1-\alpha }v(t)=0\) implies that \(c_{1}=0\). Thus,
On the other hand, if \(v=v(t)\) satisfies (2.2), we can easily show that v satisfies the equation of initial value problem (2.1).
Next, we show that \(\lim_{t\rightarrow 0^{+}}t^{1-\alpha }v(t)=0\).
Let
\(F(t)\) is given by the convolution form, that is,
If \(0<\alpha <1\), since \(q>\frac{1}{\alpha }\), we have \(\frac{q( \alpha -1)}{q-1}>-1\) and \(\phi _{1}\in L^{\frac{q}{q-1}}(\mathbb{R})\). Hence,
In view of \(\phi _{2}\in L^{q}(\mathbb{R})\), we can get that
If \(\alpha =1\), it is obvious that \(|F(t+\Delta t)-F(t)|\rightarrow 0\) (\(\Delta t\rightarrow 0\)).
Hence, \(F(t)\) is uniformly continuous on \(\mathbb{R}\), then we can get that
Because
Then we have \(\lim_{t\rightarrow 0^{+}}t^{1-\alpha }v(t)=0\). □
Remark 2.1
For any \(y\in L^{q}[0,1]\), \(v=v(t)\), which satisfies (2.2), is uniformly continuous on \([0,1]\).
For convenience, we denote
Lemma 2.4
For any \(h\in C[0,1]\), the integral boundary value problem of linear fractional differential equation
has a unique solution
where
Proof
Suppose that \(u=u(t)\) is a solution of boundary value problem (2.5). Since \(h\in C[0,1]\), then \(I_{0^{+}}^{2-\beta }u\in AC^{2}[0,1]\). Thus, by Lemma 2.1, we have
The boundary condition \(\lim_{t\rightarrow 0^{+}}t^{2-\beta }u(t)=a\) implies that \(c_{2}=a\). Then
Hence
By the boundary condition \(u(1)=\int _{0}^{1}u(s)\,\mathrm{d}A(s)\), we obtain that
Substituting \(c_{1}\) and \(c_{2}\) into (2.9), we can get that
On the other hand, if u satisfies (2.6), then u satisfies (2.9), too. It follows from (2.9)
which implies that the equation of boundary value problem (2.5) is satisfied.
We can easily show that u satisfies the boundary conditions of boundary value problem (2.5). □
Lemma 2.5
If \(u\in E\), then boundary value problem (1.1) is equivalent to the following integral equation:
where
Proof
Let \(u=u(t)\) be a solution of boundary value problem (1.1) and denote \(v(t)=p(t)D_{0^{+}}^{\beta }u(t)\), \(y(t)= \lambda f(t,u(t))\), \(h(t)=\frac{v(t)}{p(t)}\). By Lemma 2.3 and Lemma 2.4, we have
By exchanging integral order, we can get that
where \(G(t,s)\) is defined by (2.11).
On the other hand, if u satisfies (2.10), the u will also satisfy (2.12). By Lemma 2.3 and Lemma 2.4, u satisfies boundary value problem (1.1). □
Denote constants
and the function
Remark 2.2
Since \(p\in C[0,1]\) and \(p(t)>0\) for \(t\in [0,1]\), we have \(g_{1}(s)>0\) for \(s\in [0,1)\) and
Lemma 2.6
(See [11])
The function \(K(t,s)\), which is defined by (2.8), has the following properties:
-
(1)
\(K(t,s)\) is continuous for any \(t,s\in [0,1]\) and \(K(t,s)>0\) for any \(t,s\in (0,1)\);
-
(2)
$$ \frac{t^{\beta -1}(1-t)s(1-s)^{\beta -1}}{\varGamma (\beta -1)}\leq K(t,s) \leq \frac{t^{\beta -1}(1-t)(1-s)^{\beta -2}}{\varGamma (\beta )}, \quad t,s\in (0,1); $$
-
(3)
$$ K(t,s)\leq \frac{1}{\varGamma (\beta )}t^{\beta -2}s(1-s)^{\beta -1} < \frac{1}{ \varGamma (\beta )}t^{\beta -2}, \quad t,s\in (0,1). $$
Lemma 2.7
The function \(G_{1}(t,s)\), which is defined by (2.7), has the following properties:
-
(1)
\(G_{1}(t,s)\) is continuous for \(t,s\in [0,1]\) and \(G_{1}(t,s)>0\) for \(t,s\in (0,1)\);
-
(2)
$$ \frac{(m_{1}-m_{0})s(1-s)^{\beta -1}t^{\beta -1}}{\varGamma (\beta -1) (1-m _{1})}< G_{1}(t,s)< \frac{(1-m_{1}+m_{2})s(1-s)^{\beta -1} t^{\beta -2}}{ \varGamma (\beta )(1-m_{1})}, \quad t,s\in (0,1), $$
where \(m_{i}\) (\(i=0,1,2\)) are defined by (2.13).
Proof
(1) By the expression of \(G_{1}(t,s)\) and Lemma 2.6, it is easy to check that (1) holds.
(2) For any \(t,s\in (0,1)\), from Lemma 2.6, we have
On the other hand, for any \(t,s\in (0,1)\), \(1<\beta \leq 2\), implies that \(t^{\beta -1}< t^{\beta -2}\), thus, we have
□
Lemma 2.8
The function \(G(t,s)\), which is defined by (2.11), has the following properties:
-
(1)
\(G(t,s)\) is continuous for \(t,s\in [0,1]\) and \(G(t,s)>0\) for \(t,s\in (0,1)\);
-
(2)
$$ (\beta -1) (m_{1}-m_{0})g_{1}(s)t< t^{2-\beta }G(t,s) < (1-m_{1}+m_{2})g _{1}(s), \quad t,s\in (0,1), $$
where \(m_{i}\), \(g_{1}(s)\) are defined by (2.13) and (2.15), respectively.
Proof
(1) By the expression of \(G(t,s)\), we can easily get the results.
(2) According to the definition of \(G(t,s)\) and Lemma 2.7, for any \(t,s\in (0,1)\), we can obtain that
On the other hand, for any \(t,s\in (0,1)\), by Lemma 2.7, we can show that
□
Let
Then P is a cone in E.
Lemma 2.9
If u is a positive solution of boundary value problem (1.1), then \(u\in P\).
Proof
If u is a positive solution of boundary value problem (1.1), then from Definition 2.1, we can get that \(u(t)>0\) for \(t\in (0,1)\) and u satisfies (2.10). It is easy to see \(u\in E\).
For any \(t\in [0,1]\), by Lemma 2.8, we have
On the other hand, we have
Then \(t^{2-\beta }u(t)\geq \gamma _{0}t\|u\|\), which implies \(u\in P\). □
We define \(T:P\rightarrow E\) by
Lemma 2.10
The operator \(T:P\rightarrow P\) is completely continuous.
Proof
By Lemma 2.9, we have \(Tu\in P\) for \(u\in P\), then \(T:P\rightarrow P\).
-
(1)
T is a continuous operator.
If \(\{u_{n}\}\subset P\), \(u\in P\), and \(\|u_{n}-u\|\rightarrow 0\) as \(n\rightarrow \infty \), there exists a constant \(\gamma >0\) such that \(\|u_{n}\|\leq \gamma \) and \(\|u\|\leq \gamma \), that is, \(\sup_{t\in [0,1]}|t^{2-\beta }u_{n}(t)|\leq \gamma \) and \(\sup_{t\in [0,1]}|t^{2-\beta }u(t)|\leq \gamma \). Then there exists \(\varphi _{\gamma }\in L^{q}[0,1]\), we have
Since f satisfies the \(L^{q}\)-Carathéodory conditions, for a.e. \(t\in [0,1]\), we have
By the Lebesgue dominated convergence theorem, we can get
Hence, \(T:P\rightarrow P\) is continuous.
-
(2)
T is relatively compact.
Let \(\varOmega \subset P\) be any bounded set, then there exists a constant \(r>0\) such that \(\|u\|\leq r\) for each \(u\in \varOmega \), that is, \(\sup_{t\in [0,1]}|t^{2-\beta }u(t)|\leq r\). There exists \(\varphi _{r}\in L^{q}[0,1]\), for any \(u\in \varOmega \), we have
Therefore, by Lemma 2.8, we have
which implies that \(T(\varOmega )\) is uniformly bounded.
In addition, because \(G(t,s)\) is continuous on \([0,1]\times [0,1]\), then it must be uniformly continuous on \([0,1]\times [0,1]\). Thus, for any \(\varepsilon >0\), there exists a constant \(\delta \in (0,\frac{ \varepsilon (1-m_{1})}{2a|1-m_{2}|+1} )\) such that
whenever \(|t_{1}-t_{2}|<\delta \) and \(|s_{1}-s_{2}|<\delta \), where \(t_{1}\), \(t_{2}\), \(s_{1}\), \(s_{2}\in [0,1]\).
Then, for any \(u\in \varOmega \) and \(t_{1}, t_{2}\in [0,1]\) with \(|t_{1}-t_{2}|<\delta \), we have
Thus, we prove that \(T(\varOmega )\) is equicontinuous.
According to the Arzela–Ascoli theorem, T is relatively compact.
Therefore, \(T:P\rightarrow P\) is completely continuous. □
Lemma 2.11
(See [36], Lemma 2.3.1)
Let E be a Banach space and \(P\subseteq E\) be a cone. Assume that Ω is a bounded open subset of E and \(\theta \in \varOmega \) and that \(T:P\cap \bar{\varOmega }\rightarrow P\) is completely continuous. If
then the fixed point index \(i(T,P\cap \varOmega ,P)=1\).
Lemma 2.12
(See [36], Corollary 2.3.1)
Let E be a Banach space and \(P\subseteq E\) be a cone. Assume that Ω is a bounded open subset of E and that \(T:P\cap \bar{ \varOmega }\rightarrow P\) is completely continuous. If there exists \(u_{0}\in P\backslash \{\theta \}\) such that
then the fixed point index \(i(T,P\cap \varOmega ,P)=0\).
Corollary 2.1
Let E be a Banach space and \(P\subseteq E\) be a cone. Assume that Ω is a bounded open subset of E and \(\theta \in \varOmega \) and that \(T:P\cap \bar{\varOmega }\rightarrow P\) is completely continuous.
-
(1)
If \(\|u\|>\|Tu\|\) for \(u\in P\cap \partial \varOmega \), then \(i(T,P\cap \varOmega ,P)=1\);
-
(2)
If \(\|u\|<\|Tu\|\) for \(u\in P\cap \partial \varOmega \), then \(i(T,P\cap \varOmega ,P)=0\).
Proof
(1) If \(\|u\|>\|Tu\|\) for \(u\in P\cap \partial \varOmega \), then we can show that (2.17) holds.
Otherwise, there exist \(u^{*}\in P\cap \partial \varOmega \) and \(\tau ^{*}\geq 1\) such that \(Tu^{*}=\tau ^{*} u^{*}\), then
which contradicts \(\|u\|>\|Tu\|\). In view of Lemma 2.11, we can get \(i(T,P\cap \varOmega ,P)=1\).
(2) If \(\|u\|<\|Tu\|\) for \(u\in P\cap \partial \varOmega \), we can prove that (2.18) holds.
In fact, if for any \(u\in P\backslash \{\theta \}\) there exist \(u^{*}\in P\cap \partial \varOmega \) and \(\tau ^{*}\geq 0\) such that \(u^{*}-Tu^{*}=\tau ^{*}u\), then
Thus, \(\|u^{*}\|\geq \|Tu^{*}\|\), in contradiction with \(\|u\|<\|Tu\|\).
From Lemma 2.12, we can get \(i(T,P\cap \varOmega ,P)=0\). □
3 The existence of at least one positive solution
For convenience, we denote
Let \(B_{r}=\{u\in E:\|u\|< r\}\), \(\partial B_{r}=\{u\in E:\|u\|=r\}\), \(P _{r}=P\cap B_{r}\), \(\partial P_{r}=P\cap \partial B_{r}\).
Theorem 3.1
Suppose that there exist constants \(\xi , \eta >0\) such that \(f^{0}<\xi \) and \(f_{\infty }>\eta \). If \(\xi <\frac{\gamma _{0}^{2} \eta \int _{\frac{1}{4}}^{\frac{3}{4}}g_{1}(s)\,\mathrm{d}s}{4\int _{0} ^{1}g_{1}(s)\,\mathrm{d}s}\) and λ satisfies
then there exists a constant \(a_{\lambda }>0\) such that boundary value problem (1.1) with \(0\leq a\leq a_{\lambda }\) has at least one positive solution.
Proof
Since \(f^{0}<\xi \), there exists a constant \(r_{1}>0\) such that
Let
Because λ satisfies (3.1) and \(0\leq a\leq a_{\lambda }\), by Lemma 2.8, for any \(u\in \partial P_{r_{1}}\), we have \(0< t^{2-\beta }u(t)\leq r_{1}\) for \(t\in (0,1]\) and
Hence,
It follows from Corollary 2.1(1) \(i(T,P_{r_{1}},P)=1\).
By \(f_{\infty }>\eta \), there exists a constant \(r_{2}>r_{1}\) such that
For any \(u\in \partial P_{r_{2}}\), we have
and by Lemma 2.8,
Therefore,
It follows from Corollary 2.1(2) \(i(T,P_{r_{2}},P)=0\).
According to the additivity property of the fixed point index, we obtain
Then T has at least one fixed point \(u\in P\cap (P_{r_{2}}\backslash \bar{P}_{r_{1}})\) with \(r_{1}<\|u\|<r_{2}\). Because \(u\in P\), we have \(t^{2-\beta }u(t)\geq \gamma _{0}t\|u\|>0\) for \(t\in (0,1]\), that is, \(u(t)>0\) for \(t\in (0,1)\), \(u=u(t)\) is a positive solution for boundary value problem (1.1) with \(0\leq a\leq a_{\lambda }\). □
Theorem 3.2
If \(f^{0}=0\), \(f_{\infty }=+\infty \), and \(\lambda >0\), then there exists a constant \(a_{\lambda }>0\) such that boundary value problem (1.1) with \(0\leq a\leq a_{\lambda }\) has at least one positive solution.
Proof
Let \(\lambda >0\), \(0<\xi < (\lambda (1-m_{1}+m _{2})\int _{0}^{1} g_{1}(s)\,\mathrm{d}s )^{-1}\) and \(\eta \geq 4 (\lambda (\beta -1)(m_{1}-m_{0})\gamma _{0} \int _{\frac{1}{4}}^{ \frac{3}{4}}g_{1}(s)\,\mathrm{d}s )^{-1}\).
By \(f^{0}=0\), there exists a constant \(r_{1}>0\) such that
and by \(f_{\infty }=+\infty \), there exists a constant \(r_{2}>r_{1}\) such that
Let
For \(0\leq a\leq a_{\lambda }\), similar to the proof of Theorem 3.1, we have
and
According to the additivity property of the fixed point index,
Then T has at least one fixed point \(u\in P\cap (P_{r_{2}}\backslash \bar{P}_{r_{1}})\) with \(r_{1}<\|u\|<r_{2}\), that is, u is a positive solution for boundary value problem (1.1) with \(0\leq a\leq a _{\lambda }\). □
Theorem 3.3
Suppose that there exist constants ξ, \(\eta >0\) such that \(f^{\infty }<\xi \) and \(f_{0}>\eta \). If \(\xi <\frac{\gamma _{0}^{2} \eta \int _{\frac{1}{4}}^{\frac{3}{4}}g_{1}(s)\,\mathrm{d}s}{12\int _{0} ^{1}g_{1}(s)\,\mathrm{d}s}\) and λ satisfies
then boundary value problem (1.1) with \(a\geq 0\) has at least one positive solution.
Proof
By \(f_{0}>\eta \), there exists a constant \(R_{1}>0\) such that
When λ satisfies (3.3) and \(a\geq 0\), similar to the proof of Theorem 3.1, we can obtain
and
On the other hand, by \(f^{\infty }<\xi \), there exists a constant \(M>0\) such that
Since f satisfies the \(L^{q}\)-Carathéodory conditions, for the above \(M>0\), there exists \(\varphi _{M}\in L^{q}[0,1]\) such that
Let
For any \(u\in \partial P_{R_{2}}\), by Lemma 2.8, we have
That is,
From Corollary 2.1(1), we can get \(i(T,P_{R_{2}},P)=1\).
According to the additivity property of the fixed point index, we obtain
Then T has at least one fixed point \(u\in P\cap (P_{R_{2}}\backslash \bar{P}_{R_{1}})\) with \(R_{1}<\|u\|<R_{2}\). That is, u is a positive solution for boundary value problem (1.1) with \(a\geq 0\). □
Theorem 3.4
If \(f_{0}=+\infty \), \(f^{\infty }=0\), \(\lambda >0\), and \(a\geq 0\), then boundary value problem (1.1) has at least one positive solution.
Proof
Denote
By \(f_{0}=+\infty \), there exists a constant \(R_{1}>0\) such that
On the other hand, by \(f^{\infty }=0\), there exists a constant \(M>0\) such that
Let
Similar to the proof of Theorem 3.3, we can obtain T has at least one fixed point \(u\in P\cap (P_{R_{2}}\backslash \bar{P}_{R _{1}})\) with \(R_{1}<\|u\|<R_{2}\). That is, \(u=u(t)\) is a positive solution for boundary value problem (1.1). □
Theorem 3.5
If \(f_{\infty }=+\infty \), \(r>0\) is a constant and λ satisfies
then there exists a constant \(a_{\lambda }>0\) such that boundary value problem (1.1) with \(0\leq a\leq a_{\lambda }\) has at least one positive solution u with \(\|u\|>r\).
Proof
For any given \(r>0\), when λ satisfies (3.4), let
For \(0\leq a\leq a_{\lambda }\) and any \(u\in \partial P_{r}\), by Lemma 2.8, we have
That is,
By Corollary 2.1(1), we can get \(i(T,P_{r},P)=1\).
Let \(\eta =4 (\lambda (\beta -1)(m_{1}-m_{0})\gamma _{0} \int _{\frac{1}{4}}^{\frac{3}{4}} g_{1}(s)\,\mathrm{d}s )^{-1}\). Since \(f_{\infty }=+\infty \), there exists a constant \(r_{1}>r\) such that
Similar to the proof of Theorem 3.2, we obtain
By Corollary 2.1(2)
According to the additivity property of the fixed point index,
Then T has at least one fixed point \(u\in P\cap (P_{r_{1}}\backslash \bar{P}_{r})\) with \(r<\|u\|<r_{1}\). That is, \(u=u(t)\) is a positive solution for boundary value problem (1.1) with \(0\leq a\leq a _{\lambda }\). □
4 The multiplicity and nonexistence of positive solutions
In this section, we present the existence of at least two positive solutions and nonexistence positive solutions.
Theorem 4.1
Suppose that there exist constants \(\eta _{1},\eta _{2}>0\) such that \(f_{0}>\eta _{1}\) and \(f_{\infty }>\eta _{2}\). Let a constant
If λ satisfies
then there exists a constant \(a_{\lambda }\geq 0\) such that boundary value problem (1.1) with \(0\leq a\leq a_{\lambda }\) has at least two positive solutions \(u_{1}\) and \(u_{2}\).
Proof
Let
by (4.1) and (4.2), we have \(a_{\lambda }\geq 0\).
For \(0\leq a\leq a_{\lambda }\) and any \(u\in \partial P_{r}\), similar to the proof of Theorem 3.5, we have
and
Since \(f_{0}>\eta _{1}\), there exists a constant \(0<\bar{r}_{1}<r\) such that
By \(f_{\infty }>\eta _{2}\), there exists a constant \(\bar{r}_{2}>r\) such that
Similar to the proof of Theorem 3.3 and Theorem 3.1, we can obtain \(\|Tu\|>\|u\|\), \(u\in \partial P_{\bar{r}_{1}}\), and \(\|Tu\|>\|u\|\), \(u\in \partial P_{\bar{r}_{2}} \). Hence, by Corollary 2.1(2), we can get \(i(T,P_{ \bar{r}_{1}},P)=0\) and \(i(T,P_{\bar{r}_{2}},P)=0\).
According to the additivity property of the fixed point index, we can show
and
Then T has at least two fixed points \(u_{1}\in P\cap (P_{r} \backslash \bar{P}_{\bar{r}_{1}})\) with \(\bar{r}_{1}<\|u_{1}\|<r\) and \(u_{2}\in P\cap (\bar{P}_{\bar{r}_{2}}\backslash P_{r})\) with \(r<\|u_{2}\|<\bar{r}_{2}\). That is, \(u_{1}\) and \(u_{2}\) are positive solutions of boundary value problem (1.1) with \(0\leq a\leq a _{\lambda }\). □
Theorem 4.2
Suppose that \(f_{0}=+\infty \), \(f_{\infty }=+\infty \), and a constant \(r>0\) hold. If λ satisfies
then there exists a constant \(a_{\lambda }\geq 0\) such that boundary value problem (1.1) with \(0\leq a\leq a_{\lambda }\) has at least two positive solutions \(u_{1}\) and \(u_{2}\).
Proof
Let
Since λ satisfies (4.3), we have \(a_{\lambda }\geq 0\).
For \(0\leq a\leq a_{\lambda }\) and any \(u\in \partial P_{r}\), similar to the proof of Theorem 3.5, we have
By Corollary 2.1(1), we can get \(i(T,P_{r},P)=1\).
Denote
By \(f_{0}=+\infty \), there exists a constant \(0<\bar{r}_{1}<r\) such that
And by \(f_{\infty }=+\infty \), there exists a constant \(\bar{r}_{2}>r\) such that
Similar to the proof of Theorem 3.3 and Theorem 3.1, we obtain
By Corollary 2.1(2), we can get \(i(T,P_{\bar{r}_{1}},P)=0\) and \(i(T,P_{\bar{r}_{2}},P)=0\).
According to the additivity property of the fixed point index, we obtain
and
Then T has at least two fixed points \(u_{1}\in P\cap (P_{r} \backslash \bar{P}_{\bar{r}_{1}})\) with \(\bar{r}_{1}<\|u_{1}\|<r\), and \(u_{2}\in P\cap (\bar{P}_{\bar{r}_{2}}\backslash P_{r})\) with \(r<\|u_{2}\|<\bar{r}_{2}\). That is, \(u_{1}\) and \(u_{2}\) are positive solutions for boundary value problem (1.1) with \(0\leq a\leq a _{\lambda }\). □
Theorem 4.3
Suppose that \(0<\liminf_{u\rightarrow +\infty }\inf_{t\in [\frac{1}{4},\frac{3}{4}]} f(t,t^{\beta -2}u)<+\infty \), \(f^{0}=0\), and \(f^{\infty }=0\) hold. Then there exist constants \(\lambda ^{*}>0\) and \(a_{0}>0\) such that boundary value problem (1.1) has at least two positive solutions with \(\lambda \geq \lambda ^{*}\) and \(0\leq a\leq a_{0}\).
Proof
By \(0<\liminf_{u\rightarrow +\infty }\inf_{t\in [\frac{1}{4},\frac{3}{4}]} f(t,t^{\beta -2}u)<+\infty \), there exist constants \(L>0\) and \(R_{2}>0\) such that
Let
We denote \(\xi = (3\lambda (1-m_{1}+m_{2})\int _{0}^{1}g_{1}(s) \,\mathrm{d}s) )^{-1}\) for \(\lambda \geq \lambda ^{*}\).
By \(f^{\infty }=0\), there exists a constant \(M>R_{2}\) such that
By \(f^{0}=0\), there exists a constant \(0< R_{1}< R_{2}\) such that
Let
for \(0< R_{1}< R_{2}< R_{3}\), we define
It is easy to see that \(\varOmega _{1}\), \(\varOmega _{2}\), and \(\varOmega _{3}\) are nonempty bounded convex open sets in E, and \(\varOmega _{1}\subset \varOmega _{3}\), \(\varOmega _{2}\subset \varOmega _{3}\), and \(\varOmega _{1}\cap \varOmega _{2}=\varnothing \). Let
Then, when \(0\leq a\leq a_{0}\), for any \(u\in P\cap \partial \varOmega _{1}\), similar to Theorem 3.1, we obtain
and by Corollary 2.1(1), we can get that
For any \(u\in P\cap \partial \varOmega _{3}\), by Lemma 2.8, we have
that is,
it follows \(i(T,P\cap \varOmega _{3},P)=1\) from Corollary 2.1(1).
Similarly, for any \(u\in P\cap \partial \varOmega _{2}\), we have
then \(\|Tu\|< R_{3}\), and
so, \(Tu\in P\cap \varOmega _{2}\).
Let \(u_{0}\equiv \frac{1}{2}(4^{2-\beta }R_{2}+R_{3})\), and
Because \(u_{0}\in E\), \(\|u_{0}\|=\frac{1}{2}(4^{2-\beta }R_{2}+R _{3})\), \(\gamma _{0}<1\), then \(t^{2-\beta }u_{0}\geq \gamma _{0}t\|u _{0}\|\), that is, \(u_{0}\in P\). Since \(R_{3}>4^{2-\beta }R_{2}\), we can see that \(\|u_{0}\|< R_{3}\), \(\min_{t\in [\frac{1}{4},\frac{3}{4}]}t^{2-\beta }u_{0}>R_{2}\), which implies \(u_{0}\in P\cap \varOmega _{2}\). So, we have
Therefore, we have
By the complete continuity of the operator T and the definition of H, we can know that \(H:[0,1]\times (P\cap \varOmega _{2})\rightarrow P\) is completely continuous.
According to the homotopy invariance and normality of fixed point index,
Thus, T has one fixed point \(u_{1}\) in \(P\cap \varOmega _{2}\).
By the additivity of the fixed point index, we obtain
Thus, T has one fixed point \(u_{2}\) in \(P\cap (\varOmega _{3}\backslash (\bar{\varOmega }_{1}\cup \bar{\varOmega }_{2}))\).
Consequently, \(u_{1}\) and \(u_{2}\) are positive solutions of boundary value problem (1.1) with \(\lambda \geq \lambda ^{*}\) and \(0\leq a\leq a_{0}\). □
Remark 4.1
If \(f(t,u)\neq 0\), by (4.4), we can get that boundary value problem (1.1) has at least one positive solution in \(P\cap \varOmega _{1}\). Then boundary value problem (1.1) with \(\lambda \geq \lambda ^{*}\) and \(0\leq a\leq a_{0}\) has at least three positive solutions in \(P\cap \varOmega _{3}\).
Similar to the proof of Theorem 4.3, we can prove the following theorem.
Theorem 4.4
Suppose that \(\lambda >0\), \(\liminf_{u\rightarrow +\infty } \inf_{t\in [\frac{1}{4},\frac{3}{4}]} f(t,t^{\beta -2}u)=+ \infty \), \(f^{0}=0\), and \(f^{\infty }=0\) hold. Then there exists a constant \(a_{0}>0\) such that boundary value problem (1.1) has at least two positive solutions with \(0\leq a\leq a_{0}\).
Theorem 4.5
If \(f_{\infty }>0\), then there exist constants \(\lambda ^{*}>0\) and \(a_{0}>0\) such that boundary value problem with \(\lambda \geq \lambda ^{*}\) and \(a\geq a_{0}\) (1.1) has no positive solution.
Proof
Since \(f_{\infty }>0\), there exist constants \(\eta >0\) and \(r_{1}>0\) such that
Let
If u is a positive solution of boundary value problem (1.1) with \(\lambda \geq \lambda ^{*}\) and \(a\geq a_{0}\), we will show that this leads to a contradiction.
In fact, since \(Tu=u\), we have
Hence, \(\|u\|>r_{1}\).
Because \(u\in P\), by (4.5), we can get that
and
That is, \(\|u\|>\|u\|+r_{1}\), which is a contradiction. Therefore, boundary value problem (1.1) with \(\lambda \geq \lambda ^{*}\) and \(a\geq a_{0}\) has no positive solution. □
5 Illustration
To illustrate our main results, we present the following examples.
Example 5.1
Consider the boundary value problem
where \(\alpha =\frac{1}{4}\), \(\beta =\frac{7}{4}\), \(f(t,u)=(t^{2}+1) (160t^{\frac{1}{4}}u-\frac{51{,}199}{320}\sin (t^{\frac{1}{4}}u) )\), \(p(t)=\frac{1}{t+1}>0\), and
Hence,
We can obtain the following results.
-
(1)
It is easy to check that all the conditions of Theorem 3.1 are satisfied. By Theorem 3.1, for each λ with \(2.75654\leq \lambda \leq 62.0449\), let a constant \(r_{1}=0.01\), then for each a satisfying \(0\leq a\leq 0.00632303-0.00010191 \lambda \), boundary value problem (5.1) has at least one positive solution.
-
(2)
It is easy to see that all the conditions of Theorem 4.5 are satisfied. By Theorem 4.5, let \(r_{1}=381\), for all \(\lambda \geq 5.51308\) and \(a\geq 295.175\), boundary value problem (5.1) has no positive solution.
Example 5.2
Consider the boundary value problem
where \(\alpha =\frac{1}{2}\), \(\beta =\frac{3}{2}\), \(f(t,u)= ( \mathrm{e}^{tu^{2}} +t\sin (t^{\frac{1}{2}}u) )\), \(p(t)=t+1>0\) for any \(t\in [0,1]\), and
Then
All the conditions of Theorem 4.2 are satisfied. By Theorem 4.2, for given \(r>1\), \(0<\lambda \leq 1.1124\) and each a satisfying \(0\leq a\leq 1-0.402948\lambda \), boundary value problem (5.2) has at least two positive solutions \(u_{1}\), \(u_{2}\).
Example 5.3
Consider the boundary value problem
where \(\alpha =\frac{1}{2}\), \(\beta =\frac{3}{2}\), \(p(t)=t+1\), \(A(t)=\frac{1}{2}t^{2}\), and
Then \(\liminf_{u\rightarrow +\infty }\inf_{t\in [\frac{1}{4},\frac{3}{4}]} f(t,t^{\beta -2}u)=+\infty \), \(f^{0}=0\), and \(f^{\infty }=0\),
All the conditions of Theorem 4.4 are satisfied. By Theorem 4.4, there exists a constant \(a_{0}>0\) such that boundary value problem (5.3) with \(0\leq a\leq a_{0}\) has at least two positive solutions for \(\lambda >0\).
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This work is supported by the National Natural Science Foundation of China (No. 11171220, 11571207).
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This work is supported by the National Natural Science Foundation of China (No. 11171220 and 11571207).
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Jia, M., Li, L., Liu, X. et al. A class of nonlocal problems of fractional differential equations with composition of derivative and parameters. Adv Differ Equ 2019, 280 (2019). https://doi.org/10.1186/s13662-019-2181-6
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DOI: https://doi.org/10.1186/s13662-019-2181-6