Abstract
In this work, the aim is to discuss a new class of singular nonlinear higher-order fractional boundary value problems involving multiple Riemann–Liouville fractional derivatives. The boundary conditions are constituted by Riemann–Stieltjes integral boundary conditions. The existence and multiplicity of positive solutions are derived via employing the Guo–Krasnosel’skii fixed point theorem. In addition, the main results are demonstrated by some examples to show their validity.
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1 Introduction
We consider the existence and multiplicity of positive solutions for the following nonlinear singular higher-order fractional differential equations:
where \(D_{0^{+}}^{\alpha}u\), \(D_{0^{+}}^{\alpha_{k}}u\), \(D_{0^{+}}^{\gamma _{k}}u\) (\(k=1,2,\ldots,n-2\)), \(D_{0^{+}}^{\beta_{i}}u \) (\(i=1,2,3\)) are the standard Riemann–Liouville derivatives, and \(n-1<\alpha\leq n\) (\(n\geq3\)), \(k-1<\alpha_{k}\), \(\gamma_{k}\leq k \) (\(k=1,2,\ldots,n-2\)), \(n-j-1<\alpha-\gamma_{j}\leq n-j \) (\(j=1,2,\ldots,n-2\)), \(1<\alpha-\alpha_{n-2}-1\leq2 \), \(\gamma_{n-2}-\alpha_{n-2} \geq0 \), \(\beta_{1}\geq\beta_{2}\), \(\beta_{1}\geq\beta_{3}\), \(\alpha-\beta_{i}\geq1 \), \(\beta_{i}-\alpha_{n-2}-1 \geq0\) (\(i=1,2,3\)), \(\beta_{1}\leq n-1\), \(f:(0,1)\times\mathbb{R}_{+}^{n-1}\rightarrow\mathbb{R}_{+}^{1}=[0,+ \infty) \) is continuous and \(a,h\in C((0,1),\mathbb{R}_{+}^{1}) \), A is a function of bounded variation, \(\int_{0}^{\eta}h(s)D_{0^{+}}^{\beta_{2}}u(s)\,dA(s)\), \(\int_{0}^{1}a(s)D_{0^{+}}^{\beta_{3}}u(s)\,dA(s)\) denote the Riemann–Stieltjes integrals with respect to A.
Fractional differential equations appear naturally in various fields of science and engineering. This is due to the fact that the differential equations of arbitrary order provide an excellent instrument for the description of memory and hereditary properties of various materials and processes and they have numerous applications in multifarious fields of science and engineering including physics, blood flow phenomena, rheology, diffusive transport akin to diffusion, electrical networks, probability, etc. The advantages of fractional derivatives become apparent in modeling mechanical and electrical properties of real materials. For some recent work on this branch of differential equations, see [1–4] and the references therein. Fractional-order differential equations have been addressed by several researchers with the sphere of study ranging from the theoretical aspects to the analytic and numerical methods for finding solutions (see [1–40] and the references therein). The existence of positive solutions for fractional differential equation boundary value problems has attracted much attention, and a great deal of results have been developed for differential and integral boundary value problems. For example, in [5], Ma and Yang studied the following higher-order boundary value problems with a sign-changing nonlinear term:
where \(n\geq2\), \(0<\lambda, \alpha,\beta,\gamma\) and δ are constants satisfying \(\alpha,\gamma>0 \) and \(\beta,\delta\geq0 \), \(f:[0,1]\times\mathbb{R}^{n-1}\rightarrow\mathbb{R}^{1}=(-\infty,+ \infty)\) is continuous. Intervals of λ are determined to ensure the existence of a positive solution of the boundary value problem according to the signs of a and f. By using the Schauder fixed point theorem, the authors obtained the existence of a positive solution.
In [6], by means of the fixed point index theory, Zhang et al. investigated the existence of positive solutions for the fractional differential equation with integral boundary conditions:
where \(0<\beta\leq1<\alpha\leq2\), \(\alpha-\beta>1\), \(D_{t}^{\alpha}x\), \(D_{t}^{\beta}x\) are the standard Riemann–Liouville derivatives, \(\int_{0}^{1}x(s)\,dA(s) \) denotes a Riemann–Stieltjes integral, A is a function of bounded variation, and dA can be a signed measure, \(f:(0,1)\times(0,+\infty)\times(0,+\infty) \rightarrow \mathbb{R}_{+}^{1}\) is continuous, \(f(t,x,y)\) may be singular at both \(t=0,1 \) and \(x=y=0\).
In [7], via employing the topological degree theory, Zhang et al. investigated the existence of positive nontrivial solutions for the following nonlinear fractional differential equations with integral boundary conditions:
where \(3<\alpha\leq4\), \(0<\eta\leq1 \), \(0\leq\frac{\lambda\eta^{\alpha}}{\alpha}<1\), \(D_{0^{+}}^{\alpha}u \) is the Riemann–Liouville fractional derivative, \(h: (0, 1)\rightarrow[0, \infty)\) is continuous, and \(f: [0, 1]\times\mathbb{R}_{+}^{1}\rightarrow\mathbb{R}_{+}^{1}\) is also continuous.
In [8], Graef et al. obtained a new upper estimate for the Green’s function associated with a higher-order factional boundary value problem and applied these properties of \(G(t,s)\) and the well-known Schauder fixed point theorem. Criteria for the existence of positive solutions of the following problem are then established:
where \(n\geq3\), \(n-1< v\leq n\), \(1\leq\alpha\leq n-2 \), \(D_{0^{+}}^{v}\) is the standard Riemann–Liouville fractional derivative of order v, the nonlinearity \(g:[0,1]\times\mathbb{R}_{+}^{1}\rightarrow\mathbb{R}^{1}\) is a continuous function. For more research results, we refer the reader to [18, 19, 22, 24, 25, 27–38].
Inspired by the works illustrated above, we are committed to establishing the existence and multiplicity of positive solutions for the fractional differential equation boundary value problem (BVP for short) (1.1). The novelty of this article is as follows: Firstly, fractional derivatives are involved in the nonlinear terms and boundary conditions; what is new is that the orders of the fractional derivatives in the nonlinear terms and boundary conditions are different. Moreover, the orders of the fractional derivatives in the boundary conditions can be different, but up to now, there have been few papers dealing with this case where the Riemann–Stieltjes integral boundary conditions contain fractional derivative of different orders. Since fractional derivatives of different orders and high-order fractional derivatives are taken into account in BVP (1.1), it makes the research more complicated. In order to reduce the complexity, we need to use the reduced-order method for fractional differential equation and overcome the difficulties in finding the properties of Green’s function. Secondly, the boundary value conditions involving high-order fractional derivatives of unknown function are more general as they contain multi-point boundary conditions and integral boundary conditions [6, 7, 20, 28–30, 32] as special cases. Thirdly, the given conditions \(f_{0}\), \(f_{\infty}\) and (\(H_{3}\)), (\(H_{5}\)) are quite different from those in other papers such as [18–40] and are weaker and wider.
The remaining part of article is structured as follows. In Sect. 2, we present some preliminaries and lemmas which are required in later considerations. We also develop and prove some properties of Green’s function. In Sect. 3, we discuss the existence and multiplicity of positive solutions for BVP (1.1). In Sect. 4, two examples are presented to illustrate our fundamental results.
2 Preliminaries and lemmas
In this section, some notations and lemmas, which will be used in the proof of our main results, are stated. They can be found in the literature, see [2, 3, 21].
Definition 2.1
The Riemann–Liouville fractional integral of order \(\alpha>0\) of a function \(y : (0, \infty) \rightarrow\mathbb{R}^{1} \) is given by
provided that the right-hand side is pointwise defined on \((0, \infty)\).
Definition 2.2
The Riemann–Liouville fractional derivative of order \(\alpha>0\) of a continuous function \(y : (0, \infty) \rightarrow\mathbb{R}^{1}\) is given by
where \(n = [\alpha]+1\), \([\alpha]\) denotes the integer part of the number α, provided that the right-hand side is pointwise defined on \((0, \infty)\).
Lemma 2.1
([3])
Let \(\alpha>0\). If we assume \(u \in C(0, 1) \cap L^{1}(0, 1)\), then the fractional differential equation
has
for some \(C_{i} \in\mathbb{R}^{1}\) (\(i =1, 2, \ldots, N\)) as the unique solution, where \(N = [\alpha]+1\).
From the definition of the Riemann–Liouville derivative, we can obtain the following lemmas.
Lemma 2.2
([3])
Assume that \(u \in C(0, 1) \cap L^{1}(0, 1)\)with a fractional derivative of order \(\alpha>0\)that belongs to \(C(0, 1) \cap L^{1}(0, 1)\). Then
for some \(C_{i}\in\mathbb{R}^{1}\) (\(i =1, 2, \ldots, N\)), where \(N = [\alpha]+1\).
Lemma 2.3
([3])
If \(x\in L^{1}(0,1)\), \(\alpha>\beta>0 \), then
Lemma 2.4
(Auxiliary lemma)
Let \(l_{1}:=\int_{0}^{\eta}h(t)t^{\alpha-\beta_{2}-1}\,dA(t)>0\), \(l_{2}:=\int_{0}^{1}a(t)t^{\alpha-\beta_{3}-1} \,dA(t)>0\). Given \(y\in C(0, 1) \cap L^{1}(0, 1)\)and the following condition is satisfied:
- (\(H_{1}\)):
-
$$\begin{aligned}& \Delta=\frac{1}{\varGamma(\alpha-\beta_{1})}- \frac{1}{\varGamma(\alpha -\beta_{2})}l_{1}- \frac{1}{\varGamma(\alpha-\beta_{3})}l_{2}>0, \\& \frac{\varGamma(\alpha-\beta_{1})}{\varGamma(\alpha-\beta_{2})} \int_{0}^{ \eta}h(t)t^{\alpha-\beta_{2}-1}\,dA(t) + \frac{\varGamma(\alpha-\beta_{1})}{\varGamma(\alpha-\beta_{3})} \int_{0}^{1}a(t)t^{ \alpha-\beta_{3}-1}\,dA(t)< 1. \end{aligned}$$
The unique solution of
is
where
in which
Proof
We may apply Lemma 2.2 to reduce (2.1) to an equivalent integral equation
for some \(d_{i}\in\mathbb{R}^{1}\) (\(i =1, 2\)). Consequently, the general solution of (2.1) is
By (2.4) and Lemma 2.3, we have
Considering the fact that \(D_{0^{+}}^{\gamma_{n-2}-\alpha_{n-2}}x(0)=0 \), one gets that \(d_{2}=0\) by (2.5). Then we obtain
By (2.6) and Lemma 2.3, we have
So, from (2.7), we have
On the other hand, by
combining with (2.8), we obtain
So, substituting \(d_{1} \) into (2.6), one has the unique solution of problem (2.1)
The proof is complete. □
Using similar arguments as those used in the proof of Lemma 3 from [19], we obtain the following properties of the functions \(L_{i}\) (\(i=1,2,3\)).
Lemma 2.5
If condition (\(H_{1}\)) in Lemma 2.4is satisfied, the function \(L_{i}\) (\(i=1,2,3\)) defined by (2.3) has the following properties:
-
(1)
\(L_{1}(t,s)\leq \frac{t^{\alpha-\alpha_{n-2}-1}}{\varGamma(\alpha-\alpha_{n-2})}\)for all \(t,s\in[0,1] \);
-
(2)
\(L_{1}(t,s)\leq\widetilde{h}_{1}(s) \)for all \(t,s\in[0,1] \), where
$$ \widetilde{h}_{1}(s)= \frac{1}{\varGamma(\alpha-\alpha_{n-2})}(1-s)^{ \alpha-\beta_{1}-1} \bigl[1-(1-s)^{\beta_{1}-\alpha_{n-2}} \bigr],\quad s\in[0,1]; $$ -
(3)
\(L_{1}(t,s)\geq t^{\alpha-\alpha_{n-2}-1}\widetilde {h}_{1}(s) \)for all \(t,s\in[0,1] \);
-
(4)
\(L_{2}(t,s)\leq \frac{t^{\alpha-\beta_{2}-1}}{\varGamma(\alpha-\alpha_{n-2})} \)for all \(t,s\in[0,1] \);
-
(5)
\(L_{2}(t,s)\leq \frac{t^{\alpha-\beta_{2}-1}(1-s)^{\alpha-\beta_{1}-1}}{\varGamma (\alpha-\alpha_{n-2})} \)for all \(t,s\in[0,1] \);
-
(6)
\(L_{2}(t,s)\geq t^{\alpha-\beta_{2}-1}\widetilde{h}_{2}(s) \)for all \(t,s\in[0,1] \), where
$$ \widetilde{h}_{2}(s)= \frac{1}{\varGamma(\alpha-\alpha_{n-2})}(1-s)^{ \alpha-\beta_{1}-1} \bigl[1-(1-s)^{\beta_{1}-\beta_{2}} \bigr],\quad s\in[0,1]; $$ -
(7)
\(L_{3}(t,s)\leq \frac{t^{\alpha-\beta_{3}-1}}{\varGamma(\alpha-\alpha_{n-2})}\)for all \(t,s\in[0,1] \);
-
(8)
\(L_{3}(t,s)\leq \frac{t^{\alpha-\beta_{3}-1}(1-s)^{\alpha-\beta_{1}-1}}{\varGamma (\alpha-\alpha_{n-2})} \)for all \(t,s\in[0,1] \);
-
(9)
\(L_{3}(t,s)\geq t^{\alpha-\beta_{3}-1}\widetilde{h}_{3}(s) \)for all \(t,s\in[0,1] \), where
$$ \widetilde{h}_{3}(s)= \frac{1}{\varGamma(\alpha-\alpha_{n-2})}(1-s)^{ \alpha-\beta_{1}-1} \bigl[1-(1-s)^{\beta_{1}-\beta_{3}} \bigr],\quad s\in[0,1]; $$ -
(10)
The functions \(L_{i}\) (\(i=1,2,3\)) are continuous on \([0,1]\times[0,1]\), \(L_{i}(t,s)\geq0 \)for any \(t,s\in[0,1]\), \(L_{i}(t,s)>0 \)for any \(t,s\in(0,1)\) (\(i=1,2,3\)).
Proof
(1) From (2.3), we have
(2) The function \(L_{1}\) is nondecreasing in the first variable. In fact, from (2.3), if \(s\leq t\), we obtain
Then \(L_{1}(t, s) \leq L_{1}(1, s)\) for every \((t, s) \in[0, 1] \times[0, 1]\) with \(s \leq t\). For \(s\geq t \), we acquire
Hence, \(L_{1}(t, s) \leq L_{1}(s, s)\) for every \((t, s) \in[0, 1] \times[0, 1]\) with \(s \geq t\). Therefore, we deduce that \(L_{1}(t, s) \leq\widetilde{h}_{1}(s)\) for every \((t, s) \in[0, 1] \times[0, 1]\), where \(\widetilde{h}_{1}(s)= \frac{1}{\varGamma(\alpha-\alpha_{n-2})}(1-s)^{ \alpha-\beta_{1}-1} [1-(1-s)^{\beta_{1}-\alpha_{n-2}} ]\), \(s\in [0,1] \).
(3) From (2.3), if \(s\leq t\), we obtain
if \(s\leq t\), we obtain
where \(\widetilde{h}_{1}(s)= \frac{1}{\varGamma(\alpha-\alpha_{n-2})}(1-s)^{ \alpha-\beta_{1}-1} [1-(1-s)^{\beta_{1}-\alpha_{n-2}} ]\), \(s\in [0,1] \). Therefore, we conclude \(L_{1}(t,s)\geq t^{\alpha-\alpha_{n-2}-1}\widetilde{h}_{1}(s) \) for all \(t,s\in[0,1] \).
For the proof of properties (4)–(9) is similar to the proof of properties (1)–(3), we omit it here.
(10) This property is evident, it follows from the definitions of \(L_{i}\) (\(i=1,2,3\)) and from properties (3), (6), and (9). The proof is complete. □
Now, from the definitions of the Green’s functions G and the properties of functions \(L_{i}\) (\(i=1,2,3\)), we obtain the following lemma.
Lemma 2.6
If condition (\(H_{1}\)) in Lemma 2.4is satisfied, the functions \(H_{i}\) (\(i=1,2,3\)) defined by (2.3) satisfy the following:
-
(1)
\(H_{1}(t,s)\leq J_{1}(s)\)for any \((t,s)\in[0,1]\times[0,1] \), where
$$ J_{1}(s)=\widetilde{h}_{1}(s)= \frac{1}{\varGamma(\alpha-\alpha _{n-2})}(1-s)^{\alpha-\beta_{1}-1} \bigl[1-(1-s)^{\beta_{1}-\alpha_{n-2}} \bigr],\quad s\in[0,1]; $$ -
(2)
\(H_{1}(t,s)\geq t^{\alpha-\alpha_{n-2}-1}J_{1}(s) \)for all \((t,s)\in[0,1]\times[0,1] \);
-
(3)
\(H_{2}(t,s)\leq J_{2}(s)\)for any \((t,s)\in[0,1]\times[0,1] \), where
$$ J_{2}(s)=\frac{\Delta^{-1}}{\varGamma(\alpha-\beta_{2})} \int_{0}^{ \eta}h(t)L_{2}(t,s)\,dA(t); $$ -
(4)
\(H_{2}(t,s)= t^{\alpha-\alpha_{n-2}-1}J_{2}(s) \)for all \((t,s)\in[0,1]\times[0,1] \);
-
(5)
\(H_{3}(t,s)\leq J_{3}(s)\)for any \((t,s)\in[0,1]\times[0,1] \), where
$$ J_{3}(s)=\frac{\Delta^{-1}}{\varGamma(\alpha-\beta_{3})} \int_{0}^{1}a(t)L_{3}(t,s)\,dA(t); $$ -
(6)
\(H_{3}(t,s)= t^{\alpha-\alpha_{n-2}-1}J_{3}(s) \)for all \((t,s)\in[0,1]\times[0,1] \);
-
(7)
Functions \(H_{i}\) (\(i=1,2,3\)) are continuous on \([0,1]\times[0,1]\), \(H_{i}(t,s)\geq0 \)for any \(t,s\in[0,1]\), \(H_{i}(t,s)>0 \)for all \(t,s\in(0,1)\) (\(i=1,2,3\)).
Proof
(1) It follows from \(H_{1}(t,s)=L_{1}(t,s)\), \(J_{1}(s)=\widetilde{h}_{1}(s) \), \(L_{1}(t,s)\leq\widetilde{h}_{1}(s) \) for all \(t,s\in[0,1]\) that \(H_{1}(t,s)\leq J_{1}(s)\) for any \((t,s)\in[0,1]\times[0,1] \).
(2) From property (3) in Lemma 2.5, we have \(H_{1}(t,s)\geq t^{\alpha-\alpha_{n-2}-1}J_{1}(s) \) for all \((t,s)\in[0,1]\times[0,1] \).
(3)–(6) follow from the definitions of \(H_{i}\) (\(i=2,3\)).
(7) This property follows from the definitions of \(L_{i}\), \(H_{i}\) (\(i=2,3\)), and properties (2), (4), (6), and condition (\(H_{1}\)) in Lemma 2.4, property (10) in Lemma 2.5. The proof is complete. □
Let \(u(t)=I_{0^{+}}^{\alpha_{n-2}}x(t)\), \(x(t)\in C[0,1]\), then \(D_{0^{+}}^{\alpha_{n-2}}u(t)=x(t)\), problem (1.1) can turn into the following modified problem of BVP (2.9):
Obviously, the solution of BVP (2.9) is
Lemma 2.7
If \(x\in C([0,1],\mathbb{R}_{+}^{1}) \)is a positive solution of BVP (2.9), let \(u(t)=I_{0^{+}}^{\alpha _{n-2}}x(t)\), then \(u(t)=I_{0^{+}}^{\alpha_{n-2}}x(t)\)is a positive solution of BVP (1.1).
Proof
We assume that \(x\in C([0,1]) \) is a positive solution of BVP (2.9). Let \(u(t)=I_{0^{+}}^{\alpha_{n-2}}x(t)\), we have
which implies that
From \(u(t)=I_{0^{+}}^{\alpha_{n-2}}x(t)\) and the above expression, we have
Hence, we demonstrate that \(u(t)=I_{0^{+}}^{\alpha_{n-2}}x(t)\) is a positive solution of BVP (1.1). The proof is complete. □
Lemma 2.8
If condition (\(H_{1}\)) in Lemma 2.4is satisfied and \(y\in C(0,1)\cap L^{1}(0,1) \)with \(y(t)>0 \)for all \(t\in(0,1)\), then the solution x of problem (2.1) satisfies the inequality \(x(t)>0 \)for all \(t\in(0,1)\). Moreover, we have the inequalities \(x(t)\geq t^{\alpha-\alpha_{n-2}-1}x(t') \)for all \(t,t'\in[0,1] \).
Proof
In view of Lemma 2.6, we acquire
The proof is complete. □
Remark 2.1
Under the assumptions of Lemma 2.8, for \(c\in(0,1/2)\), the solutions of problem (2.1) satisfy the inequality \(\min_{t\in[c,1]}x(t) \geq c^{\alpha-\alpha_{n-2}-1} \|x\| \).
Let \(E=C[0,1] \). Clearly, \((E,\|\cdot\|)\) is a Banach space with supremum norm \(\|x\|=\sup_{t\in[0,1]}|x(t)|\). Let \(P=\{x\in E : x(t)\geq0,0\leq t\leq1\} \). It is easy to see that P is a normal cone of E. We define an operator \(A:E\rightarrow E \) by
Obviously, if x is a fixed point of operator A, then x is a solution of problem (2.9). We present the basic assumptions that we shall use in the sequel.
- (\(H_{2}\)):
-
The function \(f\in C((0,1)\times\mathbb{R}_{+}^{n-1},\mathbb{R}_{+}^{1}) \) and there exist the functions \(p\in C((0,1),\mathbb{R}_{+}^{1}) \) and \(q\in C ([0,1]\times\mathbb{R}_{+}^{n-1},\mathbb{R}_{+}^{1}) \) with \(p\not\equiv0\) and \(\int_{0}^{1}(1-s)^{\alpha-\beta_{1}-1}p(s)\,ds<+\infty\) such that
$$\begin{aligned}& f(t,x_{0},x_{1},\ldots,x_{n-2})\leq p(t)q(t,x_{0},x_{1},\ldots,x_{n-2}), \\& \quad \forall t\in(0,1),x_{i}\in\mathbb{R}_{+}^{1}, i=0,1,\ldots, n-2. \end{aligned}$$
Lemma 2.9
If conditions (\(H_{1}\)) and (\(H_{2}\)) hold, then \(A:P\rightarrow P \)is completely continuous.
Proof
We denote by \(Q_{i}=\int_{0}^{1}J_{i}(s)p(s)\,ds\) (\(i=1,2,3\)), where \(J_{i}\) (\(i=1,2,3\)) are defined in Lemma 2.6. Using (\(H_{2}\)) and Lemma 2.5, we deduce that \(Q_{i}>0\) (\(i=1,2,3\)) and
By Lemma 2.6, we also conclude that \(A:P\rightarrow P \). We prove that A maps bounded sets into relatively compact sets. Suppose that \(D\subset P\) is an arbitrary bounded set of E, then there exists \(M_{1}>0 \) such that \(\|x\|\leq M_{1} \) for all \(x\in D \). By the continuity of q, we obtain
where \(\rho=\sum_{i=0}^{n-2} \frac{M_{1}}{\varGamma(\alpha_{n-2}-\alpha_{i}+1)} \), \(\alpha_{0}=0\). By using Lemma 2.6, for any \(x\in D \) and \(t\in[0,1]\), we obtain
Thus, \(A(D)\) is bounded in E. In what follows, we prove that \(A(D)\) is equicontinuous. By using Lemma 2.4, for any \(x\in D\) and \(t\in[0,1]\), we have
Thus, for any \(t\in(0,1) \), we conclude
We denote
For the integral of the function g, by exchanging the order of integration, we obtain
For the integral of the function μ, we have
We deduce that \(\mu\in L^{1}(0,1) \). Thus, for any \(t_{1},t_{2}\in[0,1]\) with \(t_{1}\leq t_{2} \) and \(x\in D\), by (2.12) and (2.13), we conclude
From (2.13), (2.14) and the absolute continuity of the integral function, we obtain that \(A(D)\) is equicontinuous. By the Ascoli–Arzela theorem, we conclude that \(A(D) \) is a relatively compact set of E. Therefore A is a compact operator. Besides, we can show that A is continuous on P (see the proof of Lemma 1.4.1 in [21]). Hence \(A:P\rightarrow P \) is completely continuous. □
Lemma 2.10
([16])
Let K be a cone of the real Banach space E, \(\varOmega_{1},\varOmega_{2}\subset E \)be bounded open sets of E, \(\theta\in\varOmega_{1}\), \(\overline{\varOmega}_{1}\subset\varOmega_{2} \). Suppose that \(A:K\cap(\overline{\varOmega}_{2}\setminus\varOmega_{1})\rightarrow K \)is a completely continuous mapping such that one of the following two conditions is satisfied:
-
(i)
\(\|Au\|\leq\|u\|\), \(\forall u\in K\cap\partial\varOmega _{1}\); \(\|Au\| \geq\|u\|\), \(\forall u\in K\cap\partial\varOmega_{2}\);
-
(ii)
\(\|Au\|\geq\|u\|\), \(\forall u\in K\cap\partial\varOmega _{1}\); \(\|Au\| \leq\|u\|\), \(\forall u\in K\cap\partial\varOmega_{2}\).
Then A has a fixed point in \(K\cap(\overline{\varOmega}_{2}\setminus\varOmega_{1})\).
For \(c\in(0,\frac{1}{2})\), we define the cone
Under assumptions (\(H_{1}\)), (\(H_{2}\)) and Remark 2.1, we have \(A(P)\subset P_{0}\), and so \(A|_{P_{0}}:P_{0}\rightarrow P_{0} \) (denoted again by A) is also a completely continuous operator.
3 Existence of positive solutions for BVP (1.1)
We first offer some fixed numbers \(k_{i},\mu_{i},\varsigma_{i},\vartheta_{i}\geq0\) (\(i=0,1,2, \ldots,n-2\)) with \(\sum_{i=0}^{n-2}k_{i}>0\), \(\sum_{i=0}^{n-2}\mu_{i}>0\), \(\sum_{i=0}^{n-2}\varsigma_{i}>0\), \(\sum_{i=0}^{n-2}\vartheta_{i}>0\). Now, we list our assumptions:
- (\(H_{3}\)):
-
There exists \(a\geq1\) such that
$$ q_{0}=\limsup_{\sum_{i=0}^{n-2}k_{i}x_{i}\rightarrow0} \max_{t \in[0,1]} \frac{q(t,x_{0},x_{1},\ldots,x_{n-2})}{(k_{0}x_{0}+k_{1}x_{1}+\cdots +k_{n-2}x_{n-2})^{a}}< b_{1}, $$where \(b_{1}=\frac{1}{(Q_{1}+Q_{2}+Q_{3})\varrho_{1}^{a}}>0\), \(\varrho_{1}=\sum_{i=0}^{n-2}k_{i}\varGamma^{-1}(\alpha_{n-2}- \alpha_{i}+1) \), \(Q_{i}=\int_{0}^{1}J_{i}(s)p(s)\,ds\) (\(i=1,2,3\)), in which \(J_{i}\) are defined in Lemma 2.6.
- (\(H_{4}\)):
-
There exists \(c\in(0,\frac{1}{2})\) such that
$$ f_{\infty}=\liminf_{\sum_{i=0}^{n-2}\mu_{i}x_{i}\rightarrow\infty} \min_{t\in[c,1-c]} \frac{f(t,x_{0},x_{1},\ldots,x_{n-2})}{\mu_{0}x_{0} +\mu_{1}x_{1}+\cdots+\mu_{n-2}x_{n-2}}> b_{2}, $$where \(b_{2}= \frac{2}{c^{2(\alpha-\alpha_{n-2}-1)}m_{1}\varrho_{2} c_{1}}\), \(\varrho_{2}=\sum_{i=0}^{n-2}\mu_{i}\varGamma^{-1}(\alpha_{n-2}- \alpha_{i}+1) \), \(c_{1}= \min_{i=0,1,2,\ldots,n-2} \{c^{\alpha_{n-2}-\alpha_{i}}\}\), \(m_{1}=\int_{c}^{1-c}J_{1}(s)\,ds\), in which \(J_{1}\) is defined in Lemma 2.6.
- (\(H_{5}\)):
-
$$ q_{\infty}=\limsup_{\sum_{i=0}^{n-2}\varsigma_{i}x_{i}\rightarrow \infty} \max_{t\in[0,1]} \frac{q(t,x_{0},x_{1},\ldots,x_{n-2})}{\varsigma_{0}x_{0}+\varsigma _{1}x_{1}+\cdots+\varsigma_{n-2}x_{n-2}}< b_{3}, $$
where \(b_{3}=\frac{1}{2\varrho_{3}(Q_{1}+Q_{2}+Q_{3})}\), \(\varrho_{3}=\sum_{i=0}^{n-2}\varsigma_{i}\varGamma^{-1}(\alpha_{n-2}- \alpha_{i}+1)\), \(Q_{i}=\int_{0}^{1}J_{i}(s)p(s)\,ds\) (\(i=1,2,3\)), in which \(J_{i}\) are defined in Lemma 2.6.
- (\(H_{6}\)):
-
There exist \(c\in(0,\frac{1}{2})\), \(\hat{a}\in(0,1]\) such that
$$ f_{0}=\liminf_{\sum_{i=0}^{n-2}\vartheta_{i}x_{i}\rightarrow0} \min _{t\in[c,1-c]} \frac{f(t,x_{0},x_{1},\ldots,x_{n-2})}{(\vartheta_{0}x_{0}+\vartheta _{1}x_{1}+\cdots+\vartheta_{n-2}x_{n-2})^{\hat{a}}}> b_{4}, $$where \(b_{4}= \frac{1}{c^{(1+\hat{a})(\alpha-\alpha_{n-2}-1)}(\varrho_{4} c_{1})^{\hat{a}}m_{1}}\), \(\varrho_{4}=\sum_{i=0}^{n-2}\vartheta_{i}\varGamma^{-1}(\alpha_{n-2}- \alpha_{i}+1)\), \(m_{1}=\int_{c}^{1-c}J_{1}(s)\,ds\), \(c_{1}=\min_{i=0,1,2,\ldots,n-2} \{c^{\alpha_{n-2}-\alpha_{i}}\}\), in which \(J_{1}\) is defined in Lemma 2.6.
- (\(H_{7}\)):
-
\(M_{q}>0\) satisfies
$$ M_{q} \biggl[\max_{t\in[0,1]} \int_{0}^{1}G(t,s)p(s)\,ds \biggr]< 1, $$where \(M_{q}=\max\{q(t,x_{0},x_{1},\ldots,x_{n-2}): 0\leq t\leq1, x_{i} \in[0,\widetilde{\rho}], i=0,1,\ldots,n-2\}\), \(\widetilde{\rho}=\sum_{i=0}^{n-2} \frac{1}{\varGamma(\alpha_{n-2}-\alpha_{i}+1)} \), \(\alpha_{0}=0\).
Theorem 3.1
Assume that (\(H_{1}\)), (\(H_{2}\)), (\(H_{3}\)), and (\(H_{4}\)) hold, then fractional BVP (1.1) has at least one positive solution.
Proof
From (\(H_{3}\)), for fixed \(b_{1}>\epsilon_{1}>0 \), there exists \(r\in(0,1)\) such that
for all \(t\in[0,1]\), \(x_{i}\geq0\), \(\sum_{i=0}^{n-2}k_{i}x_{i}\leq r \). Taking \(r_{1}<\min\{\frac{r}{\varrho_{1}}, r\}\). By Definition 2.1, for any \(t\in[0,1] \), we can obtain that
where \(\alpha_{0}=0\). For any \(x\in\partial B_{r_{1}}\cap P_{0} \) and \(t\in[0,1]\), by (3.2), we know that
Thus, by (3.1), Lemma 2.6, and (3.2), for any \(x\in\partial B_{r_{1}}\cap P_{0} \) and \(t\in[0,1]\), we obtain
Therefore
From (\(H_{4}\)), for fixed \(\epsilon_{2}>0\), there exists \(c_{2}>0 \) such that
for any \(t\in[c,1-c]\), \(x_{i}\geq0\) (\(i=0,1,2,\ldots,n-2\)). By Definition 2.1, for any \(t\in[0,1] \), we can obtain that
where \(\alpha_{0}=0\). Then, by using (3.4) and (3.5), for any \(x\in P_{0}\) and \(t\in[c,1]\), we have
where \(\widetilde{c_{2}}=c^{\alpha-\alpha_{n-2}-1}c_{2}m_{1}\). We can choose \(R\geq\max\{\widetilde{c_{2}},1\} \), then we deduce
By (3.3), (3.6), and the Guo–Krasnosel’skii fixed point theorem, we conclude that A has a fixed point \(x\in(\overline{B}_{R}\setminus B_{r_{1}})\cap P_{0}\), that is, \(r_{1}\leq\|x\|\leq R \). By Lemma 2.8, we obtain that \(x(t)>0 \) for all \(t\in(0,1)\). By Lemma 2.7, we obtain that BVP (1.1) has at least one positive solution. Therefore, the proof of Theorem 3.1 is completed. □
Theorem 3.2
Assume that (\(H_{1}\)), (\(H_{2}\)), (\(H_{5}\)), and (\(H_{6}\)) hold, then fractional BVP (1.1) has at least one positive solution.
Proof
From (\(H_{5}\)), for fixed \(b_{3}>\epsilon_{3}>0\), there exists \(c_{3}>0\) such that
By using of (3.7) and (\(H_{2}\)), for any \(x\in P_{0}\), \(t\in[0,1]\), we conclude
where \(\widetilde{c_{3}}=(Q_{1}+Q_{2}+Q_{3})c_{3}\). We can choose large \(\widetilde{R}>\max\{2\widetilde{c_{3}},1\}\) such that
From (\(H_{6}\)), for small enough \(\epsilon_{4}>0\), there exists \(\widetilde{r}\in(0,1]\) such that
for all \(t\in[c,1-c]\), \(x_{i}\geq0\), \(\sum_{i=0}^{n-2}\vartheta_{i}x_{i}\leq \widetilde{r}\). Taking \(\widetilde{r_{1}}<\min\{\frac{\widetilde{r}}{\varrho_{4}}, \widetilde{r}\}\). For any \(x\in\partial B_{\widetilde{r_{1}}}\cap P_{0} \) and \(t\in[0,1]\), from (3.2), we know that
Thus, by using of (3.9), for any \(x\in\partial B_{\widetilde{r_{1}}}\cap P_{0}\) and \(t\in[c,1-c]\), we have
Therefore
By (3.8), (3.10) and the Guo-Krasnosel’skii fixed point theorem, we deduce that A has at least one point \(x\in(\overline{B}_{\widetilde{R}}\setminus B_{\widetilde{r_{1}}}) \cap P_{0}\), that is, \(\widetilde{r_{1}}\leq\|x\|\leq\widetilde{R} \). By Lemma 2.7, we obtain that BVP (1.1) has at least one positive solution. Therefore, the proof of Theorem 3.2 is completed. □
Theorem 3.3
Assume that (\(H_{1}\)), (\(H_{2}\)), (\(H_{4}\)), (\(H_{6}\)), and (\(H_{7}\)) are satisfied, then BVP (1.1) has at least two positive solutions.
Proof
Firstly, when (\(H_{1}\)), (\(H_{2}\)), (\(H_{4}\)) hold, by the proof of Theorem 3.1, we know that there exists \(R>1 \) such that
Again, when (\(H_{1}\)), (\(H_{2}\)), (\(H_{6}\)) hold, by the proof of Theorem 3.2, we know that there exists \(\widetilde{r_{1}}<1 \) such that
On the other hand, let \(\varOmega=\{x\in P:\|x\|=1\}\). By (\(H_{7}\)), for any \(x\in\partial\varOmega\cap P_{0}\) and \(t\in[0,1]\), we have
Consequently,
Therefore, from (3.12) and (3.13) and Lemma 2.9, it follows that BVP (2.10) has one positive solution \(x_{1}^{*}\) with \(\widetilde{r_{1}}\leq\|x_{1}^{*}\|<1 \). In the same way, from (3.11) and (3.13) and Lemma 2.10, it follows that BVP (2.9) has another positive solution \(x_{2}^{*}\) with \(1\leq\|x_{2}^{*}\|\leq R \). By Lemma 2.7, we obtain that BVP (1.1) has at least two positive solutions. Therefore, the proof of Theorem 3.3 is completed. □
4 Examples
Example 4.1
We consider the following fractional boundary value problem:
where \(\alpha=\frac{7}{2}\), \(\alpha_{1}=\frac{1}{2}\), \(\alpha_{2}= \frac{11}{10}\), \(\gamma_{1}=\frac{3}{4}\), \(\gamma_{2}=\frac{8}{5}\), \(\beta_{1}=\frac{5}{2}\), \(\beta_{2}=\frac{49}{20}\), \(\beta_{3}= \frac{12}{5} \), \(h(s)=a(s)=1\), \(s\in[0, 1]\), \(\eta=\frac{2}{3}\), and
with \(a_{0}>1\) and \(\tau_{1}\in(0,1)\). Let \(u(t)=I_{0^{+}}^{\frac{11}{10}}x(t)\), the equation can be changed to the following fractional boundary value problem:
Here, \(f(t,x,y,z)=p(t)q(t,x,y,z)\), where \(p(t)= \frac{1}{(1-t)^{\tau_{1}}}\) for all \(t\in(0,1) \) and \(q(t,x,y,z)=(x+y+z)^{a_{0}}\) for all \(t\in(0,1) \), \(x,y,z\geq0\), we have \(\int_{0}^{1}p(s)\,ds<+\infty\). By direct calculation, \(l_{1}=\int_{0}^{\frac{2}{3}}t^{\frac{1}{20}}\,dA(t)\approx 0.00965936>0\), \(l_{2}=\int_{0}^{1}t^{\frac{1}{10}}\,dA(t)\approx 0.0093303299>0\),
In (\(H_{3}\)) for \(a=1\), \(k_{0}=3\), \(k_{1}=4\), \(k_{2}=5\), we obtain \(q_{0}=0\). In (\(H_{4}\)), for \(c\in(0,\frac{1}{2})\), \(\mu_{0}=\mu_{1}=\mu_{2}=\frac{1}{2}\), we have \(f_{\infty}=+\infty\). Then, by Theorem 3.1, we deduce that BVP (1.1) has at least one positive solution.
Example 4.2
We consider the following fractional boundary value problem:
where \(\alpha=\frac{7}{2}\), \(\alpha_{1}=\frac{1}{2}\), \(\alpha_{2}= \frac{11}{10}\), \(\gamma_{1}=\frac{3}{4}\), \(\gamma_{2}=\frac{8}{5}\), \(\beta_{1}=\frac{5}{2}\), \(\beta_{2}=\frac{49}{20}\), \(\beta_{3}= \frac{12}{5} \), \(h(s)=a(s)=1\), \(s\in[0, 1]\), \(\eta=\frac{2}{3}\), and
with \(\tau_{1}\in(0,1)\). Let \(u(t)=I_{0^{+}}^{\frac{11}{10}}x(t)\), the equation can be changed to the following fractional boundary value problem:
Here, \(f(t,x,y,z)=p(t)q(t,x,y,z)\), where \(p(t)= \frac{1}{(1-t)^{\tau_{1}}}\) for all \(t\in(0,1) \) and \(q(t,x,y,z)=e^{x+y+z}\) for all \(t\in(0,1) \), \(x,y,z\geq0\), we have \(\int_{0}^{1}p(s)\,ds<+\infty\). By direct calculation, \(l_{1}=\int_{0}^{\frac{2}{3}}t^{\frac{1}{20}}\,dA(t)\approx0.00965936>0\), \(l_{2}=\int_{0}^{1}t^{\frac{1}{10}}\,dA(t)\approx0.0093303299>0\),
In (\(H_{4}\)), for \(\mu_{0}=\mu_{1}=\mu_{2}=5\), we obtain
In (\(H_{6}\)), for \(c\in(0,\frac{1}{2})\), \(\vartheta_{0}=\vartheta_{1}=\vartheta_{2}=4\), \(\hat{a}=1\), we obtain
Choose \(0< M_{q}< \frac{1}{\max_{t\in[0,1]}\int_{0}^{1}G(t,s)\frac{1}{(1-s)^{\tau _{1}}}\,ds}\), we see that all the conditions of Theorem 3.3 are satisfied. Thus, by Theorem 3.3, we deduce that BVP (1.1) has at least two positive solutions.
References
Delbosco, D.: Fractional calculus and function spaces. J. Fract. Calc. 6, 45–53 (1994)
Podlubny, I.: Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198. Academic Press, New York (1999)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Lazarevic, M.P., Spasic, A.M.: Finite-time stability analysis of fractional order time-delay systems: Gronwall’s approach. Math. Comput. Model. 49, 475–481 (2009)
Ma, D., Yang, X.: On eigenvalue intervals of higher-order boundary value problems with a sign-changing nonlinear term. Appl. Math. Comput. 235, 2314–2324 (2011)
Zhang, X., Liu, L., Wu, Y., Wiwatanapataphee, B.: The spectral analysis for a singular fractional differential equation with a signed measure. Appl. Math. Lett. 257, 252–263 (2015)
Zhang, X., Wang, L., Sun, Q.: Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter. Appl. Math. Comput. 226, 708–718 (2014)
Graef, J.R., Kong, L., Yang, B.: Positive solutions for a fractional boundary value problem. Appl. Math. Lett. 56, 49–55 (2016)
Cabada, A., Wang, G.: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389, 403–411 (2012)
Guo, L., Liu, L., Wu, Y.: Uniqueness of iterative positive solutions for the singular fractional differential equations with integral boundary conditions. Bound. Value Probl. 2016, 147 (2016)
Ahmad, B., Ntouyas, S., Alsaedi, A.: On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions. Chaos Solitons Fractals 83, 234–241 (2016)
Qarout, D., Ahmad, B., Alsaedi, A.: Existence theorems for semi-linear Caputo fractional differential equations with nonlocal discrete and integral boundary conditions. Fract. Calc. Appl. Anal. 19, 463–479 (2016)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integral and Derivative: Theory and Applications. Gordon & Breach, Yverdon (1993)
Cao, Z., Jiang, D., Yuan, C.: Existence and uniqueness of solutions for singular integral equation. Positivity 12, 725–732 (2008)
Guo, D., Cho, Y.J., Zhu, J.: Partial Ordering Methods in Nonlinear Problems. Nova Science Publishers, New York (2004)
Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, San Diego (1988)
Liu, L., Sun, F., Zhang, X., Wu, Y.: Bifurcation analysis for a singular differential system with two parameters via to topological degree theory. Nonlinear Anal., Model. Control 22(1), 31–50 (2017)
Henderson, J., Luca, R.: Existence of nonnegative solutions for a fractional integro-differential equation. Results Math. 72, 747–763 (2017)
Henderson, J., Luca, R.: Existence of positive solutions for a singular fractional boundary value problem. Nonlinear Anal., Model. Control 22, 99–114 (2016)
Henderson, J., Luca, R.: Systems of Riemann–Liouville fractional equations with multi-point boundary conditions. Appl. Math. Comput. 309, 303–323 (2017)
Henderson, J., Luca, R.: Boundary Value Problems for Systems of Differential, Difference and Fractional Equations. Positive Solutions. Elsevier, Amsterdam (2016)
Liu, L., Kang, P., Wu, Y., Wiwatanapataphee, B.: Positive solutions of singular boundary value problems for systems of nonlinear fourth order differential equations. Nonlinear Anal. 68, 485–498 (2008)
Liu, L., Zhang, X., Jiang, J., Wu, Y.: The unique solution of a class of sum mixed monotone operator equations and its application to fractional boundary value problems. J. Nonlinear Sci. Appl. 9, 2943–2958 (2016)
Liu, L., Li, H., Liu, C., Wu, Y.: Existence and uniqueness of positive solutions for singular fractional differential systems with coupled integral boundary value problems. J. Nonlinear Sci. Appl. 10, 243–262 (2017)
Xu, J., Wei, Z.: Positive solutions for a class of fractional boundary value problems. Nonlinear Anal., Model. Control 21, 1–17 (2016)
Zhu, B., Liu, L., Wu, Y.: Existence and uniqueness of global mild solutions for a class of nonlinear fractional reaction–diffusion equations with delay. Comput. Math. Appl. 78(6), 1811–1818 (2019)
Wang, Y., Liu, L.: Uniqueness and existence of positive solutions for the fractional integro-differential equation. Bound. Value Probl. 2017, 12 (2017)
Wang, Y., Liu, L.: Positive solutions for a class of fractional 3-point boundary value problems at resonance. Adv. Differ. Equ. 2017, 7 (2017)
Guo, L., Liu, L., Wu, Y.: Existence of positive solutions for singular fractional differential equations with infinite-point boundary conditions. Nonlinear Anal., Model. Control 21, 635–650 (2016)
Guo, L., Liu, L., Wu, Y.: Existence of positive solutions for singular higher-order fractional differential equations with infinite-points boundary conditions. Bound. Value Probl. 2016, 114 (2016)
Wang, F., Liu, L., Kong, D., Wu, Y.: Existence and uniqueness of positive solutions for a class of nonlinear fractional differential equations with mixed-type boundary value conditions. Nonlinear Anal., Model. Control 24(1), 73–94 (2019)
Guo, L., Liu, L.: Unique iterative positive solutions for singular p-Laplacian fractional differential equation system with infinite-point boundary conditions. Bound. Value Probl. 2019, 113 (2019)
Wang, F., Liu, L., Wu, Y.: Iterative unique positive solutions for a new class of nonlinear singular higher order fractional differential equations with mixed-type boundary value conditions. J. Inequal. Appl. 2019, 210 (2019)
Wang, F., Liu, L., Wu, Y.: Iterative analysis of the unique positive solution for a class of singular nonlinear boundary value problems involving two types of fractional derivative with p-Laplacian operator. Complexity 2019, Article ID 2319062 (2019). https://doi.org/10.1155/2019/2319062
Min, D., Liu, L., Wu, Y.: Uniqueness of positive solution for the singular fractional differential equations involving integral boundary value conditions. Bound. Value Probl. 2018, 23 (2018)
Liu, X., Liu, L., Wu, Y.: Existence of positive solutions for a singular nonlinear fractional differential equation with integral boundary conditions involving fractional derivatives. Bound. Value Probl. 2018, 24 (2018)
Guo, L., Liu, L., Wu, Y.: Iterative unique positive solutions for singular p-Laplacian fractional differential equation system with several parameters. Nonlinear Anal., Model. Control 23(2), 182–203 (2018)
Guo, L., Liu, L., Wu, Y.: Maximal and minimal iterative positive solutions for singular infinite-point p-Laplacian fractional differential equations. Nonlinear Anal., Model. Control 23(6), 851–865 (2018)
Liu, L., Sun, F., Wu, Y.: Blow-up of solutions for a nonlinear Petrovsky type equation with initial data at arbitrary high energy level. Bound. Value Probl. 2019, 15 (2019)
Wang, F., Liu, L., Wu, Y.: A numerical algorithm for a class of fractional BVPs p-Laplacian operator and singularity—the convergence and dependence analysis. Appl. Math. Comput. 382, 125339 (2020). https://doi.org/10.1016/j.amc.2020.125339
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Liu, L., Min, D. & Wu, Y. Existence and multiplicity of positive solutions for a new class of singular higher-order fractional differential equations with Riemann–Stieltjes integral boundary value conditions. Adv Differ Equ 2020, 442 (2020). https://doi.org/10.1186/s13662-020-02892-7
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DOI: https://doi.org/10.1186/s13662-020-02892-7