Abstract
In the paper, we establish sufficient conditions for the existence and multiplicity of positive solutions to a class of higher-order delayed nonlinear fractional differential equations with m-point multi-term fractional integral boundary conditions. The results are established by converting the problem into an equivalent integral equation and applying fixed point theorems of the cone expansion and compression of norm type. Our study improves the previous results in the literature. As an application, an example is also provided to illustrate our main results.
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1 Introduction
The fractional differential equation has a significant role to play in many fields such as physics, chemistry, aerodynamics, electrodynamics of a complex medium, polymer rheology, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, and so forth. The fractional differential equation also serves as an excellent tool for the description of hereditary properties of various materials and processes. In consequence, the subject of fractional differential equations is gaining much importance and attention. There are a large number of papers dealing with the existence or multiplicity of solutions or positive solutions of initial or boundary value problems for some differential equations. For details and examples, see [1–23] and the references therein. In [8, 9, 16, 18–23], the authors have discussed the existence of multiple positive solutions for boundary value problems of integer or fractional differential equations.
Integral boundary conditions have various applications in applied fields such as blood flow problems, chemical engineering, thermoelasticity, underground water flow, population dynamics, and so on. For more details of nonlocal and integral boundary conditions, see [5, 10–12, 20] and the references therein. In addition, it is well known that delay arises naturally in practical systems due to the transmission of signal or the mechanical transmission. Though the theory of ordinary differential equations with delays is mature, not much has been done for fractional differential equations with delays; see [6, 10, 13, 14].
In [11], Tariboon et al. are concerned with the existence of at least one, two or three positive solutions for the boundary value problem with three-point multi-term fractional integral boundary conditions,
where \(f:[0,1]\times[0,\infty)\rightarrow [0,\infty)\). \(D^{q}_{0+}\) is the standard Riemann-Liouville derivative of order q. \(I^{p_{i}}_{0+}\) is the Riemann-Liouville fractional integral of order \(p_{i}>0\). \(\alpha_{i}\geq0\) (\(i=1,2,\ldots, m-2\)) are real constants.
To the best of our knowledge, no one has studied the existence of multiple positive solutions with delayed nonlinear fractional differential equations. In this article, motivated by the above-mentioned papers, we study the existence of multiple positive solutions for the following higher-order delayed nonlinear fractional differential equation with m-point multi-term fractional integral boundary conditions:
subject to the following initial conditions:
where \(0\leq\tau_{1}\), \(\tau_{2}<\theta\in(0,1/2)\) are suitably small. \(D_{0^{+}}^{q}\) is the standard Riemann-Liouville derivative of order \(n-1< q\leq n\), \(n\geq3\). \(a, b, c, d\geq0\), \(\eta(t)\in C([-\tau_{1},0], [0,+\infty))\), \(\xi(t)\in C([1,1+\tau_{2}], [0,+\infty))\), \(f\in C([0,1]\times[0,+\infty)\times[0,+\infty),[0,+\infty))\). \(I^{p_{i}}_{0+}\) is the Riemann-Liouville fractional integral of order \(p_{i}\). \(\alpha_{i}\geq0\) (\(i=1,2,\ldots,m-2\)) are real constants such that \(\sum_{i=1}^{m-2}\frac{\alpha_{i}\zeta_{i}^{p_{i}+q-1}\Gamma(q)}{\Gamma (p_{i}+q)}<1\).
Remark 1.1
By (1.2) and (1.3), we have \(\eta(0)=0\) and \(\xi(1)=cu(1)+du'(1)\).
Remark 1.2
When \(1< q\leq2\), \(\tau_{1}=\tau_{2}=0\), \(\varsigma_{i}\equiv\eta\), \(j=0\), then (1.2) and (1.3) degenerate into (1.1). So our models extend (1.1).
2 Preliminaries
For the convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions and properties can be found in the literature.
Definition 2.1
The Riemann-Liouville fractional integral of order \(\alpha>0\) of a function \(u:(0,\infty)\rightarrow \mathbb{R}\) 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 \(u:(0,\infty)\rightarrow \mathbb{R}\) is given by
where \(n-1<{\alpha}\leq n\), provided that the right-hand side is pointwise defined on \((0,\infty)\).
Lemma 2.1
(see [17])
Assume that \(u\in C(0,1)\cap L(0,1)\) with a fractional derivative of order \(\alpha>0\) that belongs to \(u\in C(0,1)\cap L(0,1)\). Then
for some \(C_{i} \in\mathbb{R}\), \(i=1,2,\ldots,n\), where n is the smallest integer greater than or equal to α.
Lemma 2.2
(see [26])
Let E be a Banach space, \(P\subseteq E\) is a cone, and \(\Omega_{1}\), \(\Omega_{2}\) are two bounded open balls of E centered at the origin with \(0\in\Omega_{1}\) and \(\overline{\Omega}_{1}\subset\Omega_{2}\). Suppose that \(A: P\cap(\overline{\Omega}_{2}\setminus\Omega _{1})\rightarrow P\) is a completely continuous operator such that either
-
(i)
\(\|Au\| \leq\|u\|\), \(u\in P\cap\partial \Omega_{1}\), and \(\|Au\| \geq\|u\|\), \(u\in P\cap\partial\Omega_{2}\), or
-
(ii)
\(\|Au\| \geq\|u\|\), \(u\in P\cap\partial \Omega_{1}\), and \(\|Au\| \leq\|u\|\), \(u\in P\cap\partial\Omega_{2}\)
holds. Then A has at least one fixed point in \(P\cap(\overline{\Omega}_{2}\setminus\Omega_{1})\).
For simplicity, we put
Now we present the Green’s function for BVP (1.2)-(1.3).
Lemma 2.3
Let \(\sum_{i=1}^{m-2}\frac{\alpha_{i}\zeta_{i}^{p_{i}+q-1}\Gamma(q)}{\Gamma (p_{i}+q)}<1\), \(\alpha_{i}\geq0\), \(p_{i}>0\) (\(i=1,2,\ldots,m-2\)), and \(h\in C([0,1],\mathbb{R})\), \(1\leq n-1< q \leq n\). The unique solution of
is
where \(G(t,s)\) is the Green’s function given by
where
and
Proof
Applying Lemma 2.1, BVP (2.2) can be expressed as an equivalent integral equation
for \(C_{i} \in\mathbb{R}\) (\(i=1,2,\ldots,n\)) \(\in\mathbb{R}\). \(u^{(j)}(0)=0\) (\(j=0,1,2,\ldots,n-2\)) implies that \(C_{2}=C_{3}=\cdots=C_{n}=0\). Taking the Riemann-Liouville fractional integral of order \(p_{i}>0\) for (2.6), we get
\(u(1)=\sum_{i=1}^{m-2}\alpha_{i}(I^{p_{i}}_{0+}u)(\zeta_{i})\) yields
Then we have
Therefore, the unique solution of BVP (2.2) is written as
Hence, by taking into account (2.1), we have
The proof is complete. □
Lemma 2.4
The Green’s function \(G(t,s)\) defined by (2.3)-(2.5) has the following properties:
-
(A1)
\(G(t,s)\in C([0,1]\times[0,1])\) and \(G(t,s)\geq0\), for all \(t, s \in[0,1]\).
-
(A2)
\(G(t,s)\leq\max_{0\leq t\leq1}G(t,s)\leq G(s)\) and \(G(t,s)\geq\min_{0\leq t\leq1}G(t,s)\geq\sigma(t)G(s)\), where
$$ G(s)=\frac{s(1-s)^{q-1}}{\Gamma(q-1)}+\sum_{i=1}^{m-2} \frac {\alpha_{i}}{\Omega\Gamma(p_{i}+q)}g_{i}(\zeta_{i},s),\qquad \sigma(t)= \frac{t^{q-1}(1-t)}{q-1}. $$ -
(A3)
If \(\theta\in(0,1/2)\), then \(\min_{t\in[\theta,1-\theta]}G(t,s)\geq\sigma(\theta)G(s)\).
Proof
From the expression of \(G(t,s)\), it is obvious that \(G(t,s)\in C([0,1]\times[0,1])\). To prove \(G(t,s)\geq0\), we will show that \(g(t,s), g_{i}(\zeta_{i},s)\geq0\), \(i=1,2,\ldots,m-2\), for all \(0\leq s, t \leq1\).
Let \(k_{1}(t,s)=t^{q-1}(1-s)^{q-1}-(t-s)^{q-1}\) for \(0\leq s\leq t\leq1\), then we have
For \(0\leq t\leq s\leq1\), \(k_{2}(t,s)=(t-ts)^{q-1}\geq0\). Therefore, \(g(t,s)\geq0\) for all \(0\leq s ,t \leq1\). Now, let \(k_{3}^{i}(\zeta_{i},s)=\zeta_{i}^{p_{i}+q-1}(1-s)^{q-1}-(\zeta_{i}-s)^{p_{i}+q-1}\) for \(0\leq s\leq\zeta_{i}<1\), then we have
For \(0< \zeta_{i}\leq s\leq1\), \(k_{4}^{i}(\zeta_{i},s)=\zeta_{i}^{p_{i}+q-1}(1-s)^{q-1}\geq0\). Therefore \(g_{i}(\zeta_{i},s)\geq0\), \(i=1,2,\ldots,m-2\), for all \(0\leq s\leq1\).
In the following, we prove (A2). When \(s\leq t\), we have \(1-s\geq1-t\), then
and
When \(t\leq s\), we derive from \(q>2\) that
and
Thus,
From the above analysis, we have for \(0\leq s\leq1\),
and
Now we prove (A3). By (A1) and (A2), we have
and
It is clear that \(\sigma(t)\) is increasing in \(t\in[0,\frac{q-1}{q}]\) and decreasing in \(t\in[\frac{q-1}{q},1]\), respectively. By \(q>2\), we have
thus, \(\sigma(t)\) is increasing in \(t\in[\theta,\frac{q-1}{q}]\), for \(\frac{q-1}{q}\leq1-\theta\) or \(\frac{q-1}{q}\geq1-\theta\), and we have \(\min_{\theta\leq t\leq 1-\theta}\sigma(t)=\min\{\sigma(\theta),\sigma(1-\theta)\}\). For \(0<\theta<1-\theta<1\), \(q>2\), we get
Therefore
This proof is complete. □
3 Existence of multiple positive solutions
In this section, we will consider the existence of multiple positive solutions for the BVP (1.2)-(1.3).
Let \(E= \{u(t):u(t)\in C[-\tau_{1},1+\tau_{2}]\}\) denote a real Banach space with the norm \(\|\cdot\|\) defined by \(\|u\|= \max_{-\tau_{1}\leq t\leq1+\tau_{2}} |u(t) |\). Define the cone \(P\subset E\) by \(P = \{ u\in E:u(t)\geq0 \}\). Let
Suppose that \(u(t)\) is a solution of (1.2)-(1.3); according to Lemma 2.3 and Remark 1.1, it can be written as
where
and
Throughout this paper, we assume that \(v_{0}(t)\) is the solution of (1.2)-(1.3) with \(f\equiv0\). Clearly, \(v_{0}(t)\) can be expressed as follows:
where
and
Obviously, \(v_{0}(t)\geq0\) for each \(t\in[-\tau_{1},1+\tau_{2}]\).
Let \(u(t)\) be a solution of (1.2)-(1.3) and \(v(t)=u(t)-v_{0}(t)\). Noting that \(v(t)\equiv u(t)\) for \(0\leq t\leq1\), we have
where
and
Define an operator \(A:E\rightarrow E\) as follows:
where
and
It is easy to derive that u is a positive solution of BVP (1.2)-(1.3) if \(v=u-v_{0}\) is a nontrivial fixed point of \(A: K \rightarrow K\), where \(v_{0}\) is defined as before.
Lemma 3.1
\(A:K \rightarrow K\) defined by (3.3) is completely continuous.
Proof
For \(v\in K\), we find from Lemma 2.4 and the definition of A that \(0\leq(Av)(t)\leq(Av)(0)\), for \(t\in[-\tau_{1},0]\), and \(0\leq(Av)(t)\leq(Av)(1)\) for \(t\in[1,1+\tau_{2}]\). Thus, \(\|Av\|=\|Av\|_{[0,1]}=\max_{0\leq t\leq1}|(Av)(t)|\). It follows from Lemma 2.4 that
Thus, \(A(K)\subset K\). In addition, since f is continuous, it follows that A is continuous.
Let \(Q_{1} \in K\) be bounded, that is, there exists a positive constant \(M_{1}>0\) such that \(\|v\|_{[-\tau_{1},1+\tau_{2}]}\leq M_{1}\) for all \(v \in Q_{1}\). Then \(\|v+v_{0}\|_{[-\tau_{1},1+\tau_{2}]}\leq M_{1}+M_{0}\triangleq M_{2}\) for \(v\in Q_{1}\), where \(v_{0}\) is defined as before. Define a set \(Q_{2}\subset E\) as follows:
Hence, \(\max_{-\tau_{1}\leq t\leq1+\tau_{2}}|(v+v_{0})(t)|\leq \|v+v_{0}\|_{[-\tau_{1},1+\tau_{2}]}\leq \|v\|_{[-\tau_{1},1+\tau_{2}]}+\|v_{0}\|_{[-\tau_{1},1+\tau_{2}]}\leq M_{2}\). Noting that f is continuous on \([0,1]\times[0,M_{2}]\times[0,M_{2}]\), there exists a constant \(M_{3}>0\) such that on \([0,1]\times[0,M_{2}]\times[0,M_{2}]\),
Therefore,
Hence, \(A (Q_{1})\) is bounded.
Finally, we show the operator A is equicontinuous. For \(v\in Q_{1}\), we have
and
In the light of \(f\leq M_{3}\) and
we have \(\|(Av)'\|\leq M\) for some positive constant M. Thus, for \(-\tau_{1}\leq t_{1}< t_{2}\leq1+\tau_{2}\), we have
Therefore, for any \(\epsilon>0\), there exists \(\delta=\delta(\epsilon)>0\) which is independent of \(t_{1}\), \(t_{2}\), and v such that \(\|(Av)(t_{2})-(Av)(t_{1})\|\leq\epsilon\), whenever \(|t_{2}-t_{1}|\leq\delta\). Thus, \(A(Q_{1})\) is equicontinuous. In view of the Ascoli-Arzela theorem, we can easily see that \(A:K \rightarrow K\) is a completely continuous operator. The proof is complete. □
Further we make the following assumptions for \(f(t,x,y)\):
- (H1):
-
There exists a constant \(r_{1}>0\) such that \(0\leq x\leq r_{1}+\|v_{0}\|_{[-\tau_{1},0]}\), \(0\leq y\leq r_{1}+\|v_{0}\|_{[1,1+\tau_{2}]}\), and \(0\leq t\leq1\) implies \(f(t,x,y)<\rho_{1} r_{1}\), where \(\rho_{1}=\frac{1}{\int_{0}^{1}G(s)\,ds}\).
- (H2):
-
There exists a constant \(r_{2}>0\) such that \(\sigma _{1}r_{2}\leq x\leq r_{2}\), \(\sigma_{2}r_{2}\leq y\leq r_{2}\), and \(\theta\leq t\leq1-\theta\) implies \(f(t,x,y)>\rho_{2} r_{2}\), where \(\rho_{2}=\frac{1}{\sigma(\theta)\int_{\theta}^{1-\theta}G(s)\,ds}\), \(\sigma_{1}=\min_{\theta-\tau_{1}\leq t\leq1-\theta-\tau_{1} }|\sigma(t)|\), \(\sigma_{2}=\min_{\theta+\tau_{2}\leq t\leq1-\theta+\tau_{2} }|\sigma(t)|\).
- (H3):
-
\(f_{\infty}=\liminf_{x+y\rightarrow +\infty}\min_{t\in[0,1]}\frac{f(t,x,y)}{x+y}=\infty\).
- (H4):
-
\(f^{\infty}=\limsup_{x+y \rightarrow +\infty}\max_{t\in[0,1]}\frac{f(t,x,y)}{x+y}=0\).
- (H5):
-
\(f_{0}=\liminf_{x+y\rightarrow 0}\min_{t\in[0,1]}\frac{f(t,x,y)}{x+y}=\infty\).
- (H6):
-
\(f^{0}=\limsup_{x+y \rightarrow 0}\max_{t\in[0,1]}\frac{f(t,x,y)}{x+y}=0\).
Theorem 3.1
Assume that (H1), (H2), and (H3) are satisfied. If \(r_{1}>r_{2}>0\), then BVP (1.2)-(1.3) has at least two positive solutions \(u_{1}\) and \(u_{2}\) such that
Proof
Let \(A:K\rightarrow K\) be the cone preserving completely continuous that is defined by (3.3). Let \(\Phi_{r_{1}}=\{v\in E:\|v\|< r_{1}\}\), then for any \(v\in K\cap\partial\Phi_{r_{1}}\), we get
Thus, from (A3) and (H1), we have
Therefore,
Let \(\Phi_{r_{2}}=\{v\in E:\|v\|< r_{2}\}\), then for any \(v\in K\cap\partial\Phi_{r_{2}}\), we have
Thus, from (H2) and (A3) of Lemma 2.4, we get
So
Choose \(L>0\) such that
From (H3), there exists \(R_{1}>0\) such that
Choose
Let \(\Phi_{R}=\{v\in E:\|v\|< R,R\geq R_{0}\}\), then for any \(v\in K\cap\partial\Phi_{R}\), we have
Then, from (3.6) and (3.7) we have
Therefore,
Applying Lemma 2.2 to (3.4) and (3.5) yields the result that A has a fixed point \(v_{1}\in K\cap(\overline{\Phi}_{r_{1}}\setminus\Phi_{r_{2}})\) with \(v_{1}(t)\geq \sigma_{1} \|u\|>0\), \(t\in[0,1]\). Similarly, Lemma 2.2 associated with (3.4) and (3.8) shows that A has another fixed point \(v_{2}\in K\cap(\overline{\Phi}_{R}\setminus \Phi_{r_{1}})\) with \(v_{2}(t)\geq\sigma_{2} \|u\|>0\), \(t\in[0,1]\), which means that \(u_{1}(t)=v_{1}(t)+v_{0}(t)\) and \(u_{2}(t)=v_{2}(t)+v_{0}(t)\) are two positive solutions of BVP (1.2)-(1.3). Since
it follows that \(u_{1}(t)\) and \(u_{2}(t)\) satisfy
The proof is complete. □
Theorem 3.2
Assume that (H1), (H3), and (H5) are satisfied. There exist constants \(R>r_{1}>r>0\), then BVP (1.2)-(1.3) has at least two positive solutions \(u_{1}\) and \(u_{2}\) such that
Proof
Choose \(0< r<r_{1}<R\), let \(\Phi_{r}=\{v\in E:\|v\|< r,r<r_{1}\}\). For the same \(L>0\) satisfying (3.6), (H5) implies that
Then for any \(v\in K\cap\partial\Phi_{r}\), we have
Then, from (3.6) and (3.9) we have
Therefore,
Applying Lemma 2.2 to (3.4) and (3.10) yields that A has a fixed point \(v_{1}\in K\cap(\overline{\Phi}_{r_{1}}\setminus\Phi_{r})\) with \(v_{1}(t)\geq \sigma_{1} \|u\|>0\), \(t\in[0,1]\). Similarly, Lemma 2.2 associated with (3.4) and (3.8) yields the result that A has another fixed point \(v_{2}\in K\cap(\overline{\Phi}_{R}\setminus \Phi_{r_{1}})\) with \(v_{2}(t)\geq\sigma_{2} \|u\|>0\), \(t\in[0,1]\). This means that \(u_{1}(t)=v_{1}(t)+v_{0}(t)\) and \(u_{2}(t)=v_{2}(t)+v_{0}(t)\) are two positive solutions of BVP (1.2)-(1.3). Since
it follows that \(u_{1}(t)\) and \(u_{2}(t)\) satisfy
The proof is complete. □
Theorem 3.3
Assume that (H2), (H4), and (H6) are satisfied. There exist constants \(R>r_{2}>r>0\), then BVP (1.2)-(1.3) has at least two positive solutions \(u_{1}\) and \(u_{2}\) such that
Proof
Choose \(0< r<r_{2}<R\). By (H4), for any \(0<\varepsilon<\frac{1}{2\int_{0}^{1}G(s)\,ds}\), there exists \(R'>0\) such that
Putting
then
Choose
Let \(\Phi_{R}=\{v\in E: \|v\|< R, R\geq\max\{r_{2},R_{0}\}\}\). Then, for any \(v\in K\cap\partial\Phi_{R}\), we have
Therefore,
Let \(\Phi_{r}=\{v\in E:\|v\|< r,r<r_{2}\}\). By (H6), for any \(0<\varepsilon<\frac{1}{2\int_{0}^{1}G(s)\,ds}\), there exists \(0< r'<r\) such that
Putting
then
Choose
Then, for any \(v\in K\cap\partial\Phi_{r}\), we have
So,
Applying Lemma 2.2 to (3.5) and (3.12) yields the result that A has a fixed point \(v_{1}\in K\cap(\overline{\Phi}_{r_{1}}\setminus\Phi_{r})\) with \(v_{1}(t)\geq \sigma_{1} \|u\|>0\), \(t\in[0,1]\). Similarly, from Lemma 2.2 associated with (3.5) and (3.11) one derives that A has another fixed point \(v_{2}\in K\cap(\overline{\Phi}_{R}\setminus \Phi_{r_{1}})\) with \(v_{2}(t)\geq\sigma_{2} \|u\|>0\), \(t\in[0,1]\). This means that \(u_{1}(t)=v_{1}(t)+v_{0}(t)\) and \(u_{2}(t)=v_{2}(t)+v_{0}(t)\) are two positive solutions of BVP (1.2)-(1.3). Since
it follows that \(u_{1}(t)\) and \(u_{2}(t)\) satisfy
This completes the proof. □
We account for the control functions
where \(r_{0}=\|v_{0}\|_{[-\tau_{1},0]}\), \(r'_{0}=\|v_{0}\|_{[1,1+\tau_{2}]}\).
Theorem 3.4
Suppose that there exist two positive numbers \(\xi_{2}<\xi_{1}\) such that one of the following conditions is satisfied:
- (B1):
-
\(\varphi(\xi_{1})< \rho_{1}\xi_{1}\), \(\psi(\xi _{2})>\rho_{2}\xi_{2}\).
- (B2):
-
\(\psi(\xi_{1})>\rho_{2}\xi_{1}\), \(\varphi(\xi_{2})<\rho _{1}\xi_{2}\).
Then BVP (1.2)-(1.3) has at least one positive solution \(u\in K\) such that
Proof
Because of the similarity of the proof, we prove only this theorem under condition (B1). By assumption (B1), we have
which are the assumptions (H1) and (H2). By Theorem 3.1, we find that A has a fixed point \(v\in K\cap(\overline{\Phi}_{\xi_{1}}\setminus\Phi_{\xi_{2}})\), which means that (1.2)-(1.3) has at least one positive solution u and \(\xi_{2}<\|u\|_{[0,1]}<\xi_{1}\). This completes the proof. □
Similarly, we can obtain the existence of multiple positive solutions for BVP (1.2)-(1.3).
Theorem 3.5
Suppose that there exist three positive numbers \(\xi_{3}<\xi_{2}<\xi_{1}\) such that one of the following conditions is satisfied:
- (B3):
-
\(\varphi(\xi_{1})<\rho_{1}\xi_{1}\), \(\psi(\xi_{2})>\rho _{2}\xi_{2}\), \(\varphi(\xi_{3})<\rho_{1}\xi_{3} \).
- (B4):
-
\(\psi(\xi_{1})>\rho_{2}\xi_{1}\), \(\varphi(\xi_{2})<\rho _{1}\xi_{2}\), \(\psi(\xi_{3})>\rho_{2}\xi_{3} \).
Then BVP (1.2)-(1.3) has at least two positive solutions \(u_{1}, u_{2}\in K\) such that
Theorem 3.6
Suppose that there exist four positive numbers \(\xi_{4}<\xi_{3}<\xi_{2}<\xi_{1}\) such that one of the following conditions is satisfied:
- (B5):
-
\(\varphi(\xi_{1})<\rho_{1}\xi_{1}\), \(\psi(\xi_{2})>\rho _{2}\xi_{2}\), \(\varphi(\xi_{3})<\rho_{1}\xi_{3}\), \(\psi(\xi_{4})>\rho_{2}\xi _{4} \).
- (B6):
-
\(\psi(\xi_{1})>\rho_{2}\xi_{1}\), \(\varphi(\xi_{2})<\rho _{1}\xi_{2}\), \(\psi(\xi_{3})>\rho_{2}\xi_{3}\), \(\varphi(\xi_{4})<\rho _{1}\xi_{4}\).
Then BVP (1.2)-(1.3) has at least three positive solutions \(u_{1}, u_{2}, u_{3}\in K\) such that
Theorem 3.7
Suppose that there exist \(n+1\) positive numbers \(\xi_{n+1}<\xi_{n}<\cdots<\xi_{2}<\xi_{1}\) such that one of the following conditions is satisfied:
- (B7):
-
\(\varphi(\xi_{2k-1})< \rho_{1}\xi_{2k-1}\), \(\psi (\xi_{2k})>\rho_{2}\xi_{2k}\), \(k=1,2,\ldots, [\frac {n+2}{2} ]\).
- (B8):
-
\(\psi(\xi_{2k-1})>\rho_{2}\xi_{2k-1}\), \(\varphi (\xi_{2k})<\rho_{1}\xi_{2k}\), \(k=1,2,\ldots, [\frac {n+2}{2} ] \).
Then BVP (1.2)-(1.3) has at least n positive solutions \(u_{i}\in K\) (\(i=1,2,\ldots,n\)) such that
4 Example
Consider the following four-point BVP of delayed nonlinear fractional differential equations:
where \(f(t,x,y)=\frac{4(t^{2}+t+1)}{3}+\frac{x^{2}+y^{2}}{8}\), \(0\leq t\leq1\), \(x, y\geq0\), \(q=5/2\), \(\tau_{1}=\frac{1}{6}\), \(\tau_{2}=\frac{1}{9}\), \(\alpha_{1}=8\), \(\alpha_{2}=5\), \(p_{1}=\frac{1}{2}\), \(p_{2}=\frac{3}{2}\), \(\zeta_{1}=\frac{1}{4}\), \(\zeta_{2}=\frac{1}{2}\), \(m=4\). Choosing \(\theta=\frac{1}{4}\), \(r_{1}=2\), \(r_{2}=\frac{1}{50}\), \(r_{0}=r'_{0}=\frac{1}{5}\).
By a simple calculation, we get
By Lemmas 2.3 and 2.4 and the aid of a computer, we obtain
and
Therefore, for \((t,x,y)\in[\frac{1}{4}, \frac{3}{4}]\times[\frac{11\sqrt{3}}{64\text{,}800}, \frac{1}{50}]\times[\frac{155\sqrt{31}}{583\text{,}200}, \frac {1}{50}]\), we have
For \((t,x,y)\in[0,1]\times[0,\frac{11}{5}]\times[0,\frac{11}{5}]\), we obtain
and
Thus (H1)-(H3) hold. With the use of Theorem 3.1, BVP (4.1) has at least two positive solutions \(u_{1}\) and \(u_{2}\) such that \(0<\frac{1}{50}<\|u_{1}\|_{[0,1]}<2<\|u_{2}\|_{[0,1]}\).
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Acknowledgements
The authors would like to thank the anonymous referees for their useful and valuable suggestions. This work was supported by the National Natural Sciences Foundation of Peoples Republic of China under Grant (No. 11161025), Yunnan Province natural scientific research fund project (No. 2011FZ058).
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Zhao, K., Gong, P. Positive solutions of m-point multi-term fractional integral BVP involving time-delay for fractional differential equations. Bound Value Probl 2015, 19 (2015). https://doi.org/10.1186/s13661-014-0280-6
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DOI: https://doi.org/10.1186/s13661-014-0280-6