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The Green’s function of a class of twoterm fractional differential equation boundary value problem and its applications
Advances in Difference Equations volume 2020, Article number: 80 (2020)
Abstract
In this paper, we consider a Riemann–Liouville type twoterm fractional differential equation boundary value problem. Some positive properties of the Green’s function are deduced by using techniques of analysis. As application, we obtain the existence and multiplicity of positive solutions for a fractional boundary value problem under conditions that the nonlinearity \(f(t,x)\) may change sign and may be singular at \(t = 0,1\) and \(x=0\), and we also obtain the uniqueness results of positive solution for a singular problem by means of the monotone iterative technique.
Introduction
In this paper, we study properties of the Green’s function of the following twoterm fractional differential equation boundary value problem (FBVP):
where \(2 < \alpha< 3\), \(a>0\), \(D^{\alpha}_{0+}\) is the standard Riemann–Liouville derivative.
During the past decades, much attention has been paid to the study of fractional differential equations (FDEs) due to the more accurate effect in describing important phenomena in biology, engineering, and so on. It has been proved that a multiterm FDE can be used to describe various types of viscoelastic damping [1, 2]. Most of the model equations proposed can be expressed by the linear form
where \(a_{i}\in\mathbf{R}\), \(i=0,1,\ldots, N1\), equipped with initial conditions (see [3–7] and the references therein). For example, Elshehawey et al. [5] considered the endolymph equation
which can be used to describe the response of the semicircular canals to the angular acceleration.
Recently, many authors have focused on the existence of solutions to nonlinear FBVPs by using the techniques of nonlinear analysis such as fixed point theorems, Leray–Schauder theory, etc. (see [8–32]). Since only positive solutions are meaningful in most practical problems, the existence of positive solutions for FBVPs has particularly attracted a great deal of attention, e.g., the nonlocal FBVPs [10, 22, 25], singular FBVPs [16, 21, 28], semipositone FBVPs [15, 18, 27].
It is known that the cone which usually depends on the positive properties of the Green’s function plays a very important role in discussing positive solutions. When \(1<\alpha< 2\), Jiang and Yuan [14] obtained some properties of the Green’s function for the FDE:
with Dirichlet type boundary value condition. Xu and Fei [30] investigated (2) with threepoint boundary value condition. In [19], we established some new positive properties of the corresponding Green’s function for (2) with multipoint boundary value condition. When \(\alpha> 2\), Zhang et al. [27, 28] obtained triple positive solutions for (2) with conjugate type integral conditions by employing height functions on special bounded sets which were derived from properties of the Green’s function.
While there are a lot of works dealing with multiterm FDEs with initial conditions, the results dealing with boundary value problems of multiterm FDEs are relatively scarce. For some recent literature on Caputo type multiterm FBVPs, we mention the papers [8, 9] and the references therein. In [20], we established some new positive properties of the Green’s function for the Riemann–Liouville type FBVP, in which the linear operator contains two terms:
where \(1 < \alpha< 2\), \(b> 0\). As application, the existence and uniqueness of positive solution are obtained under singular conditions.
Inspired by the above work, in this paper, we aim to deduce some positive properties of the Green’s function for FBVP (1). As application, we investigate the existence and multiplicity of positive solutions for a singular FBVP with changing sign nonlinearity, and we also consider the uniqueness results of positive solution for a singular FBVP. Compared with the existing works, this paper has the following features. Firstly, the fractional derivative discussed in this paper is the standard Riemann–Liouville derivative, which is different from [8, 9], and the linear operator of the FBVP we are considered with contains two terms, which is different from [14, 19, 27, 28, 30]; in other words, we discuss different problem which has been seldom studied before. Secondly, some meaningful properties of the Green’s function for the case that \(2< \alpha<3\) are established; this is different from [20] since Ref. [20] considered the case that \(1< \alpha<2\). Thirdly, we consider a multiplicity of positive solutions under conditions that the nonlinearity \(f(t,x)\) may change sign and possess singularity at \(x=0\); this is different from [15, 18]. It should be noted that there are relatively few results on multiple solutions for FBVPs under this circumstance, not to mention twoterm FBVPs. Finally, we obtain the uniqueness results of positive solution for a singular twoterm FBVP by means of the monotone iterative technique, and the rate of convergence for the iterative sequence is considered.
Basic definitions and preliminaries
Definition 2.1
([33])
The fractional integral of a function \(u:(0,+\infty)\rightarrow R\) is given by
provided that the righthand side is pointwise defined on \((0,+\infty)\).
Definition 2.2
([33])
The Riemann–Liouville fractional derivative of a function \(u:(0,+\infty)\rightarrow R\) is given by
where \(n=[\alpha]+1\), \([\alpha]\)denotes the integer part of number α, provided that the righthand side is pointwise defined on \((0,+\infty)\).
For convenience, we introduce the following notations:
where
is the MittagLeffler function.
It is clear that \(h(x)\) is strictly increasing on \([0,+\infty)\), \(h(0)<0\), and
Therefore, \(h(x)\) has a unique positive root \(a^{\ast}\), that is, \(h(a^{\ast})=0\).
Throughout this paper, we always assume that the following assumption holds:
 \((H_{1})\):
\(a\in(0, a^{\ast}]\) is a constant.
Lemma 2.1
Let\(y\in L^{1}[0,1]\cap C(0,1)\). Then the unique solution of the twoterm FBVP (1) is
where
Proof
It follows from [33] that the general solution of the equation
can be expressed by
By direct calculation, we have
By \(u(0)=u'(0)=0\), there is \(c_{3}=c_{2}=0\). Then we get
It follows from \(u(1)=0\) that
Therefore, the solution of (1) is
□
Remark 2.1
The unique solution given in Lemma 2.1 satisfies \(u\in \mathit{AC}^{1}[0,1]\), where
Proof
In fact, we have
It follows from [32, Lemma 2.1] that \(I^{\alpha}_{0+}y(t)\in \mathit{AC}^{1}[0,1]\) and
Notice that \(g(t)\in \mathit{AC}^{1}[0,1]\), we can get \(u\in \mathit{AC}^{1}[0,1]\). □
Main results
Lemma 3.1
For\(0\leq s\leq t \leq 1\), we have
Proof
For \(t>0\), we have
Therefore, \(g_{1}(t)\) is strictly increasing on \([0,1]\). By direct calculation, we have
which implies \(g_{1}'(t)\) is strictly decreasing on \((0,1]\). Thus
Therefore we can get
that is,
□
Lemma 3.2
Assume that\(s^{\star}\in(0,1)\)satisfies\(s^{\star}=(1s^{\star})^{\alpha2}\), then
Proof
It is clear that \(k(s)\) is strictly decreasing on \([0,1]\). Notice that \(k(0)=1\) and \(k(1)=1\), we know \(k(s)\) has a unique root \(s^{\star}\) on \((0,1)\), that is, \(s^{\star}=(1s^{\star})^{\alpha2}\). Therefore,
Thus
□
Theorem 3.1
The Green’s function\(G(t,s)\)satisfies the following properties:
 \((p_{1})\):
\(G(t,s) > 0, \forall t,s\in(0,1)\);
 \((p_{2})\):
\(G(t,s)=G(1s,1t)\), \(\forall t,s\in [0,1]\);
 \((p_{3})\):
\(G(t,s) \geq M_{1} s (1s)^{\alpha1}(1t)t^{\alpha1}\), \(\forall t,s\in[0,1]\);
 \((p_{4})\):
\(G(t,s) \leq M_{2} s (1s)^{\alpha1}\), \(\forall t,s\in[0,1]\), where
$$M_{1}=\frac{1}{g(1)[\varGamma(\alpha)]^{2}},\qquad M_{2}=\frac{[g'(1)]^{2}}{g(1)s^{\star}}. $$
Proof
Since \((p_{2})\) is trivially true and \((p_{1})\) can be derived from \((p_{3})\), it remains to verify \((p_{3})\) and \((p_{4})\).
For \(t\in[0,1]\), it is easy to check that
and
Combining the notations of g and \(g'\) with
one has
Case (I): \(0\leq t\leq s\leq1\).
By (6), one has
By (7), one has
Case (II): \(0< s< t< 1\).
It is obvious that
Therefore, it follows from Lemma 3.1 and (7) that
By the monotonicity of \(g'(t)\), we have
By Lagrange’s mean value theorem, there exist \(\zeta\in(1s,1)\) and \(\eta\in(t,1)\) such that
On the other hand, it follows from (8) that
Combining (13) and (14) with (6), we have
It follows from (10) and (12) that \((p_{3})\) holds. On the other hand, (11) and (15) yield \((p_{4})\) holds. □
Corollary 3.1
It follows from \((p_{2})\) and \((p_{4})\) of Theorem 3.1 that
Applications
Semipositone problem
In this section, we consider the existence and multiplicity of positive solutions to the semipositone FBVP:
For convenience, we list here the hypotheses to be used in this section:
 \((H_{2})\):
\(f\in C( (0,1)\times(0,+\infty),(\infty,+\infty))\) and satisfies
$$f(t,x)\geqe(t), \quad(t,x)\in(0,1)\times (0,+\infty), $$where \(e\in L^{1}[0,1]\cap C(0,1)\) is nonnegative and \(\int_{0}^{1}e(s)\,ds>0\).
 \((H_{3})\):
For any \(R\geq r>0\), there exists \(\varPsi_{r,R}\in L^{1}[0,1]\cap C(0,1)\) such that
$$f(t,x)+e(t)\leq\varPsi_{r,R}(t),\quad\forall t \in(0,1), x\in\bigl[ r(1t)t^{\alpha1},R\bigr]. $$ \((H_{4})\):
There exists \([c_{1},d_{1}]\subset(0,1)\) such that
$$\liminf_{x\rightarrow0^{+}}\min_{t\in[c_{1},d_{1}]}f(t,x)=+\infty. $$ \((H_{5})\):
There exists \([c_{2},d_{2}]\subset(0,1)\) such that
$$\liminf_{x\rightarrow+\infty}\min_{t\in [c_{2},d_{2}]}\frac{f(t,x)}{x}=+ \infty. $$
Remark 4.1
Condition \((H_{4})\) implies that \(f(t,x)\) is singular at \(x = 0\).
Let \(E=C[0,1]\) be endowed with the maximum norm \(\ u\ = \max_{0\leq t\leq1} u(t)\). Define a cone
Denote \(B_{r}=\{u(t)\in E : \ u(t)\< r\}\) and
Lemma 4.1
The unique solution of the FBVP
is
with
Proof
The lemma can be deduced from Lemma 2.1 and Corollary 3.1, so we omit it. □
Next we consider the auxiliary FBVP:
where \([u(t)\lambda\omega(t)]^{+}=\max\{u(t)\lambda\omega(t),0 \}\).
Let
Lemma 4.2
Suppose that\((H_{2})\)and\((H_{3})\)hold. Then, for any\(\lambda>0\)and
\(A:P\setminus P_{r}\rightarrow P\)is completely continuous.
Proof
For any \(u \in P\) with \(\u\\geq r\), one has
The rest of the proof is similar to Lemma 2.6 in [21], we omit it here. □
By the extension theorem of completely continuous operator (see [34]), there exists an extension operator \(\widetilde{A}:P \rightarrow P\), which is still completely continuous. Without loss of generality, we still write it as A.
Lemma 4.3
([34])
LetEbe a real Banach space, \(P\subset E\)be a cone. Assume that\(\varOmega_{1}\)and\(\varOmega_{2}\)are two bounded open subsets ofEwith\(\theta\in\varOmega_{1}\), \(\overline{\varOmega}_{1}\subset\varOmega_{2} \), \(A: P\cap(\overline{\varOmega}_{2}\setminus \varOmega_{1})\rightarrow P\)is a completely continuous operator such that either
 (1)
\(\Au\\leq \u\\), \(u\in P\cap\partial\varOmega_{1}\)and\(\Au\\geq\u\\), \(u\in P\cap\partial\varOmega_{2}\); or
 (2)
\(\Au\\geq\u\\), \(u\in P\cap\partial\varOmega_{1}\)and\(\Au\\leq\u\\), \(u\in P\cap\partial\varOmega_{2}\).
Then A has a fixed point in\(P\cap(\overline{\varOmega}_{2}\setminus \varOmega_{1})\).
Theorem 4.1
Assume that\((H_{2})\)–\((H_{5})\)hold. Then there exists\(\lambda^{\ast}> 0 \)such that FBVP (16) has at least two positive solutions for any\(\lambda\in (0,\lambda^{\ast})\).
Proof
For
\((H_{4})\) guarantees there exists \(X_{1}\in(0,1)\) such that
Let
For any \(\lambda\in(0,\lambda^{\ast})\), let
It is clear that \(r_{1} < X_{1} <1\).
\(\forall u \in \partial P_{r_{1}}\), one has
and
Then
which implies
Let \(r_{2}=1+M_{2}\). For any \(u \in \partial P_{r_{2}}\), one has
and
This and \((H_{3})\) yield
Then
Therefore
For
\((H_{5})\) guarantees there exists \(X_{2} > r_{2}\) such that
Let
where
It is easy to see that
For any \(u \in \partial P_{r_{3}}\), one has
Hence
Thus
Then
that is,
Combining (18)–(20) with Lemma 4.3, we get A has at least two fixed points \(u_{1}\), \(u_{2}\) with \(r_{1}< \u_{1}\<r_{2} < \u_{2}\< r_{3}\), that is, \(u_{1}\) and \(u_{2}\) are solutions of the auxiliary FBVP (17). It is clear that \(u_{i}(t)\lambda\omega(t)> 0\) on \((0,1)\), \(i=1,2\). Let \(\bar{u}_{i}(t)=u_{i}(t)\lambda\omega(t)\), \(i=1,2\). Then \(\bar{u}_{1}(t)\) and \(\bar{u}_{2}(t)\) are two positive solutions of the semipositone FBVP (16). □
Corollary 4.1
Suppose that either\((H_{2})\)–\((H_{4})\)or\((H_{2})\), \((H_{3})\), and\((H_{5})\)hold. Then FBVP (16) has at least one positive solution providedλis small enough.
Example 4.1
Consider the following problem:
with
It is clear that \(f(t,x)\) is singular at \(t = 0,1\), and \(x = 0\). For \(x\in[0,+\infty)\), notice that \(\varGamma(\cdot)\) is strictly increasing on \([2,+\infty)\), we have
By direct calculation, we have
that is, \(h(\frac{1}{4})<0\). This yields \(a^{\ast}>\frac{1}{4}\), so \((H_{1})\) holds.
Let
It is easy to check that \((H_{2})\)–\((H_{5})\) hold. Therefore Theorem 4.1 ensures that FBVP (21) has at least two positive solutions provided λ is small enough.
Uniqueness results
In this section, we consider the uniqueness results of positive solution to the singular FBVP:
For convenience, we assume that the following assumptions hold in the rest of this paper:
 \((H_{6})\):
\(f\in C((0,1)\times[0,+\infty)\times (0,+\infty)\rightarrow[0,+\infty))\), \(f(t,x,y)\) is nondecreasing on x, nonincreasing on y, and there exists \(\mu\in(0,1)\) such that
$$\begin{aligned} f \biggl(t,rx,\frac{y}{r} \biggr)\geq r^{\mu}f(t,x,y),\quad \forall x,y> 0, r \in(0,1). \end{aligned}$$(23) \((H_{7})\):
\(0<\int_{0}^{1}f(s,(1s)s^{\alpha1},(1s)s^{\alpha1})\,ds<+\infty\).
Remark 4.2
Inequality (23) is equivalent to
Define a cone Q by
Define a mixed monotone operator T by
Set \(Q_{1}=Q\setminus \{\theta\}\), where θ is the zero element of E.
Lemma 4.4
\(T:Q_{1}\times Q_{1}\rightarrow Q_{1}\).
Proof
For \(u,v\in Q_{1}\), \(\exists l_{u},l_{v}> 0\) such that
Denote
It follows from Corollary 3.1 and Remark 4.2 that
By \((p_{3})\) and \((p_{4})\) of Theorem 3.1, we have
and
which implies
This and (25) yield \(T:Q_{1}\times Q_{1}\rightarrow Q_{1}\) is well defined. □
Theorem 4.2
The singular FBVP (22) has a unique positive solution.
Proof
Let \(w\in Q_{1}\), it follows from Lemma 4.4 that \(T(w,w)\in Q_{1}\). Then we can select \(r_{0}\in(0,1)\) such that
Set
where
It is easy to see that \(u_{i}, v_{i} \in Q_{1}\), \(i=0,1,\dots \), and
It follows from (23) and (24) that
Then we have
By induction, we can get
Therefore, (28) and (29) yield
Then \(\{u_{n}\}\) is a Cauchy sequence. Similarly, we can get \(\{v_{n}\}\) is a Cauchy sequence. It follows from (28) that there exist \(u^{\ast}, v^{\ast}\in Q_{1}\) such that \(\{u_{n}\}\) and \(\{v_{n}\}\) converge to \(u^{\ast}\) and \(v^{\ast}\) respectively. Moreover,
This and (29) imply that
Hence
By (30), we have
Let \(n\rightarrow +\infty\), we get
Then we have \(u^{\ast}= T(u^{\ast},u^{\ast})\), that is, \(u^{\ast}\) is a positive fixed point of T.
Next, we will show that the positive fixed point of T is unique. In fact, if \(u\neq u^{\ast}\) is a positive fixed point of T, by Lemma 4.4, we have \(u\in Q_{1}\). Denote
It is clear that \(r_{1}\in(0,1)\) and
Then
and
Therefore,
This contradicts with the definition of \(r_{1}\) since \(r_{1}^{\mu}>r_{1}\). Consequently, the positive fixed point of T is unique, that is, FBVP (22) has a unique positive solution. □
Remark 4.3
The iterative sequence \(\{u_{n}\}\) defined by (27) converges uniformly to the unique positive solution \(u^{\ast}\). Moreover, we have the error estimation
with the rate of convergence
Example 4.2
Consider the following problem:
with
It follows from Example 4.1 that \(h(\frac{1}{4})<0\), that is, \((H_{1})\) holds. Clearly, \((H_{5})\) and \((H_{6})\) hold. Then Theorem 4.2 ensures that FBVP (31) has a unique positive solution \(u^{\ast}\).
By direct calculation, we have
Therefore,
Let
By Theorem 3.1 and Corollary 3.1, we have
Set
Then (26) holds and \(\v_{0}\=\frac{54}{25\sqrt{5}}\approx 0.966\). Moreover,
Then we have the rate of convergence
and the error estimation
Conclusions
In this paper, we establish some positive properties of the Green’s function for a class of FBVPs. The interesting point is that the linear operator of the FBVPs contains two terms. As application of the main results, we investigate the existence and multiplicity results of positive solutions for an FBVP under conditions that the nonlinearity may change sign and possess singularity, and we also consider the uniqueness results of positive solution for a singular FBVP.
Abbreviations
 FBVP:

Fractional differential equations boundary value problem
 FDEs:

fractional differential equations
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Acknowledgements
The author would like to thank the referees for their pertinent comments and valuable suggestions.
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This work was supported by the Natural Science Foundation of Shandong Province of China (ZR2017MA036); the National Natural Science Foundation of China (11871302); a Project of Shandong Province Higher Educational Science and Technology Program (J18KA217), and the International Cooperation Program of Key Professors by Qufu Normal University.
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Wang, Y. The Green’s function of a class of twoterm fractional differential equation boundary value problem and its applications. Adv Differ Equ 2020, 80 (2020). https://doi.org/10.1186/s13662020025495
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Keywords
 Twoterm fractional differential equation
 Boundary value problems
 Green’s function
 Positive solution
 Singularity