Abstract
In this paper, we focus on the existence and asymptotic analysis of positive solutions for a class of singular fractional differential equations subject to nonlocal boundary conditions. By constructing suitable upper and lower solutions and employing Schauder’s fixed point theorem, the conditions for the existence of positive solutions are established and the asymptotic analysis for the obtained solution is carried out. In our work, the nonlinear function involved in the equation not only contains fractional derivatives of unknown functions but also has a stronger singularity at some points of the time and space variables.
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1 Introduction
The purpose of this paper is to establish some new results on existence and asymptotic analysis of positive solutions for the following singular fractional differential equation with nonlocal boundary condition:
where \(2<\alpha \le 3\) with \(0<\gamma \le \mu <\alpha -2\), \(\pmb{\mathscr{D}_{t}}^{\alpha } \) is defined as the Riemann–Liouville derivative, \(\int ^{1}_{0}\pmb{\mathscr{D}_{t}} ^{\mu }x(s)\,d \mathcal{X}(s)\) denotes a linear functional involving the Riemann–Stieltjes integrals, \(\mathcal{X}\) is a function of bounded variation with a changing-sign measure \(d\mathcal{X}\), \(f:(0,1)\times (0, +\infty )\times (0, +\infty )\rightarrow [0,+\infty )\) is continuous, and \(f(t,x_{1},x_{2})\) may be singular at \(t = 0, 1\) and \(x_{1}={x_{2}=0}\).
In describing viscoelasticity, Heymans and Kitagawa [1] pointed out that the accuracy and success of the model are their abilities to describe natural phenomena including memory effects in polymers. However, in many dynamic process, the influence of memory is often persistent, even if the factors affecting the process have disappeared, such as observed in stress relaxation after a nonmonotonous loading program. Thus in order to improve the accuracy of the model, based on the non-locality of fractional order derivative, one can choose a noninteger order differential equation to describe this type of physical phenomena with memory effects. In addition, fractional calculus also has many other applications in various fields of science and engineering, such as a HIV model [2, 3] and a fluid model [4,5,6,7,8]. Recently, Heymans and Podlubny [9] gave some physical interpretation for the fractional spring–pot model, the Zener model, the Maxwell model and the Voigt model. In [10], Abdon introduced a new concept of differentiation and integration combining fractal differentiation and fractional differentiation, which can explain the memory effect of heterogeneity, and elasco-viscosity of the medium and also the fractal geometry of the dynamic system. Using the time-scale fractional calculus, Nadia et al. [11] gave some applications of the fractional derivatives with arbitrary time scales in white noise from signal processing.
In the aspect of mathematical theory and application, to obtain further information of the relative natural phenomena, many authors are interested in the existence and properties of solutions for fractional differential models [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27] and many analytical techniques and methods have been developed to solve various differential equations, such as iterative methods [28,29,30,31,32,33,34,35,36,37], the Mawhin continuation theorem for resonance [38,39,40], the topological degree method [41, 42], the fixed point theorem [43,44,45,46,47,48,49,50,51,52,53,54,55], the variational method [56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73] and the upper and lower solution method [74, 75].
Inspired by the above work, in this paper, we mainly focus on the analytic results for Eq. (1.1). Our strategy is firstly introducing an accurate cone of Banach space and then constructing a couple of suitable upper and lower solutions, and finally establishing some new results on existence and asymptotic behavior of positive solutions for the equation by using the fixed point theorem. It is noteworthy that our approach and technique can solve the singularity of nonlinear term f at the space variables without the need of the complicated supremum and limit condition such as
-
(A)
\(f\in C((0,1)\times (0,+\infty )\times (0,+\infty ),[0,+ \infty ))\) and for any \(0 < r < R < +\infty \),
$$ \lim_{n\to +\infty } \sup_{{{x\in K_{\frac{R}{\varGamma (\beta +1)}}} \atop {y\in \overline{K}_{R}\backslash K_{r} } }} \int _{e(n)}\omega (s)f\bigl(s,x(s),y(s)\bigr)\,ds=0, $$where \(e(n)=[0,\frac{1}{n}] \cup [\frac{n-1}{n},1]\).
This is applied by Zhang et al. [14] for the spectral and singularity analysis for a fractional differential equation with signed measure. The main contributions of this work are as follows:
-
(i)
We present exact cone and suitable growth condition to overcome the difficulty due to the singularity of the nonlinear term f at the space variables.
-
(ii)
We establish a sufficient condition for the existence of positive solutions and give the estimation of the positive solution and asymptotic behavior of the derivative of positive solutions at the
-
(iii)
Nonsingular cases for the nonlinear term f at the time and space variables are discussed and some new results are established.
The rest of this paper is organized as follows. In Sect. 2, some preliminaries and lemmas are presented for subsequent developments. The main results are presented in Sect. 3.
2 Preliminaries and lemmas
For the convenience of the reader, we only present here some necessary properties from fractional calculus theory in the sense of Riemann–Liouville, and the corresponding definitions can be found in [76] or [12,13,14,15,16,17,18,19,20,21,22,23,24,25].
Proposition 2.1
([76])
-
(1)
If \(x,y:(0,+\infty )\rightarrow \mathbb{R}\) with order \(\alpha > 0\), then
$$ \pmb{\mathscr{D}_{t}}^{\alpha }\bigl(x(t)+y(t)\bigr)=\pmb{ \mathscr{D}_{t}}^{ \alpha }x(t)+\pmb{\mathscr{D}_{t}}^{\alpha }y(t). $$ -
(2)
If \(x\in L^{1}(0, 1)\), \(\nu >\gamma > 0\) and m is a positive integer, then
$$ \begin{gathered} I^{\nu }I^{\gamma }x(t)=I^{\nu +\gamma }x(t), \qquad \pmb{ \mathscr{D}_{t}}^{\gamma }I^{\nu } x(t)=I^{\nu -\gamma } x(t), \\ \pmb{\mathscr{D}_{t}}^{\gamma }I^{\gamma } x(t)=x(t), \qquad \pmb{\mathscr{D}_{t}}^{m} \bigl(\pmb{ \mathscr{D}_{t}}^{\gamma }x(t) \bigr)= \pmb{\mathscr{D}_{t}}^{\gamma +m}x(t). \end{gathered} $$ -
(3)
If \(\alpha >0\), \(\gamma >0\), then
$$ \pmb{\mathscr{D}_{t}}^{\alpha } t^{\gamma -1}= \frac{\varGamma (\gamma )}{ \varGamma (\gamma -\alpha )}t^{\gamma -\alpha -1}. $$ -
(4)
Suppose \(\gamma > 0\), and \(g(x)\) is integrable, then
$$ I^{\gamma }\pmb{\mathscr{D}_{t}}^{\gamma }g(x)=g(x)+c_{1}x^{\gamma -1}+c _{2}x^{\gamma -2}+\cdots +c_{n}x^{\gamma -n}, $$where \(c_{i}\in \mathbb{R}\) (\(i=1,2,\ldots ,n\)), n is the smallest integer greater than or equal to α.
In the rest of this paper, all discussions are based on the assumption \(2<\alpha -\gamma \le 3\). We first give the following lemma.
Lemma 2.1
Let \(x(t)=I^{\gamma }z(t)\), \(z(t)\in C[0,1]\), then Eq. (1.1) is equivalent to the following boundary value problem:
Proof
Firstly, let \(x(t)=I^{\gamma }z(t)\) and \(z(t)\in C[0,1]\). It follows from Proposition 2.1(2) that
On the other hand, \(2<\alpha \le 3\) and \(0<\gamma \le \mu <\alpha -2\) yield \(\alpha -\gamma ,\alpha -\mu \in (2,3)\). Consequently, by the definition of the Riemann–Liouville derivative and integral and Proposition 2.1(2), one has
It follows from (1.1), (2.2) and (2.3) that \(-\pmb{\mathscr{D}_{t}}^{\alpha -\gamma } z(t)= f(t,I^{\gamma }z(t), z(t))\) with boundary conditions
Thus, Eq. (1.1) is turned into the boundary value problem (2.1).
Conversely, if \(z\in C([0,1],[0,+\infty ))\) is a solution for the problem (2.1). Then letting \(x(t)=I^{\gamma }z(t)\) and using (2.2) and (2.3), we get
with boundary conditions
Consequently, the boundary value problem (2.1) is turned into Eq. (1.1). □
The following lemma is standard according to Proposition 2.1, and we omit the proof.
Lemma 2.2
Given \(h\in L^{1}(0, 1)\), then the boundary value problem
has the unique solution
where \(G(t,s)\) is the Green function of the boundary value problem (2.4) and
On the other hand, by Proposition 2.1, we know that the unique solution of the boundary value problem
is \(\frac{\varGamma (\alpha -\mu )}{\varGamma (\alpha -\gamma )}t^{\alpha - \gamma -1}\). Let
and define
Following the strategy in [24], the Green function for the boundary value problem (2.1) is
Thus we have the following lemma.
Lemma 2.3
Let \(p\in L^{1}[0,1]\) and \(2<\alpha \le 3\) with \(0<\gamma \le \mu < \alpha -2\). Then the fractional differential equation
has the unique solution
where \(W(t,s)\) is defined by (2.7).
In order to guarantee the nonnegativity of the Green function, the following condition is necessary.
-
(F0)
\(\mathcal{X}\) is a function of bounded variation satisfying \(\mathcal{G}_{\mathcal{X}}(s)\ge 0\), \(s\in [0, 1]\) and \(\mathcal{C}\in [0,1)\).
Lemma 2.4
Assume (F0) is satisfied, then for the Green function in (2.7) one has the following estimation:
-
(1)
\(W(t,s) > 0\) for all \(0< t\), \(s<1\).
-
(2)
$$ \mathcal{B}t^{\alpha -\gamma -1}\mathcal{G}_{\mathcal{X}}(s) \le W(t,s) \le c(s)t^{\alpha -\gamma -1}, $$(2.8)
where
$$ c(s)=\frac{(1-s)^{\alpha -\mu -1}}{\varGamma (\alpha -\gamma )}+ \mathcal{B}\mathcal{G}_{\mathcal{X}}(s). $$
Proof
The conclusion of (1) is clear. In what follows, we prove the conclusion (2). Using (2.5) and (2.7), one gets
and
□
It follows from Lemma 2.3 that we have the following.
Lemma 2.5
If \(z \in C([0,1], \mathbb{R})\) satisfies
and \(\pmb{\mathscr{D}_{t}} ^{\mu -\gamma }z(t)\le 0\) for all \(t\in (0,1)\), then \(z(t)\ge 0\), \(t\in [0,1]\).
3 Singular cases
In this section, we first give the definition of upper and lower solution on the boundary value problem (2.1), and then introduce some theories of function space and give our main results.
Definition 3.1
We call a continuous function \(\psi (t)\) as a lower solution for the boundary value problem (2.1), if
Definition 3.2
We call a continuous function \(\phi (t)\) as a upper solution for the boundary value problem (2.1), if
Let
and define our work space \(E= C[0,1]\) and a subset \(P_{e}\) of E,
Clearly, \(P_{e}\) is nonempty since \(e(t)\in P_{e}\). For any \(z \in P_{e}\), define an operator B by
To overcome the difficulties of the singularity at the space variables, we introduce the following growth conditions for f:
-
(F1)
\(f\in C((0,1)\times (0,\infty )\times (0,\infty ),[0,+ \infty ))\), and \(f(t,x_{1},x_{2})\) is decreasing in \(x_{i}>0\) for \(i=1,2\).
-
(F2)
For any \(\tau >0\), \(f(t,\frac{ \tau }{\varGamma (\gamma +1)} t^{\gamma },\tau )\not \equiv 0\), and
$$ 0< \int _{0}^{1} c(s)f\bigl(s,\tau \kappa (s), \tau e(s)\bigr)\,ds< +\infty . $$
Lemma 3.1
Assume (F0) (F1) and (F2) are satisfied, then \(B (P_{e})\subset P_{e}\) and B is well defined.
Proof
For any \(z\in P_{e}\), it follows from the definition of \(P_{e}\) that there exist two numbers \(0< l_{z}<1<L_{z}\) such that \(l_{z} e(t)\le z(t) \le L_{z} e(t)\) for any \(t\in [0,1]\). Notice that \(\kappa (t)=I^{ \gamma } e(t)\), then by (2.8) and (F1)–(F2), one gets
Take \(\tau =\max_{t\in [0,1]} z(t)\), it follows from (F2) that, for any \(s\in [0,1]\),
Consequently, from the continuity of f, one has
This yields
By (2.8), (3.3) and (3.4), we have
where
On the other hand, in view of (2.8), we also have
where
Thus it follows from (3.3)–(3.6) that \(B (P_{e})\subset P_{e}\) and B is well defined. □
Theorem 3.1
(Existence)
Suppose (F0)–(F2) hold. Then Eq. (1.1) has at least one positive solution.
Proof
Firstly by Lemma 2.3 and (3.2), we have
Next we seek for a couple of lower and upper solutions of the boundary value problem (2.1). To do this, take
Obviously, if \(e(t)=Be(t)\), then \(e(t)\) is a positive solution of Eq. (1.1). If \(e(t)\neq Be(t)\), then we have \(\xi (t), \eta (t) \in P_{e}\) and
Letting
we claim that the functions \(\psi (t)\), \(\phi (t)\) shall be the lower solution and upper solution of the boundary value problem (2.1), respectively.
In fact, it follows from (F1) that B is nonincreasing relative to z. By (3.8)–(3.9), we have
and \(\psi (t), \phi (t)\in P_{e}\). Thus (3.7) and (3.10) yield
and
Thus (3.10)–(3.12) show that \(\phi (t)\) and \(\psi (t)\) are the lower and upper solutions of the boundary value problem (2.1), respectively, and \(\psi (t), \phi (t)\in P_{e}\).
Now define the function
then from (3.13), \(F[0,1]\times [0,+\infty )\to [0,+\infty )\) is a continuous function.
Next let us consider the following auxiliary boundary value problem:
Define an operator A in E by
Obviously, from Lemma 2.3, a fixed point of A is a solution of the boundary value problem (3.14).
For all \(z\in E\), as \(\psi \in P_{e}\), there exists a constant \(0< l_{\psi }<1\) such that \(\psi (t)\ge l_{\psi }e(t)\), \(t\in [0,1]\). Thus by Lemma 2.4, we have
So A is bounded. In addition, according to the continuity of F and K, we find that \(A:E \to E\) is continuous.
Let Ω be a bounded subset of E, then we have \(\Vert z\Vert\le N\) for some positive constant \(N>0\) and all \(z\in \varOmega \). Let \(L = \max_{0\le t\le 1, 0\le z\le N}|F(t,z)| + 1\). It follows from the uniform continuity of \(W(t,s)\) that, for any \(\epsilon > 0\) and \(s\in [0,1]\), there exists \(\sigma > 0\) such that
for \(\vert t_{1}-t_{2}\vert < \sigma \). Then
This implies that \(A(\varOmega )\) is equicontinuous.
Thus according to the Arzelà–Ascoli theorem, \(A:E\to E\) is a completely continuous operator. Consequently it follows from the Schauder fixed point theorem that A has a fixed point w such that \(w=Aw\).
In order to show that w is also a fixed point of the operator B, we only need to prove
We firstly verify that \(w(t)\le \phi (t)\). Let \(z(t)=\phi (t)-w(t)\), \(t\in [0,1]\). Noticing that w is a fixed point of A and (3.12), we have
On the other hand, it follows from (3.11) and (F1) that
Thus by the definition of F and (3.16), one gets
It follows from (3.7) and (3.17) that
Thus Lemma 2.5, (3.15) and (3.18) imply that
Similarly, we also have \(w(t)-\psi (t)\ge 0 \) on \([0, 1]\). Thus the following estimation is valid:
which also implies \(F(t,w(t))=f(t,I^{\gamma }w(t),w(t))\), \(t\in [0,1]\).
Combined with the above facts, we get that the fixed point of A is also the fixed point of B. So \(w(t)\) is a positive solution of the boundary value problem (2.1), and consequently \(x(t)=I^{\gamma }w(t)\) is a positive solution of Eq. (1.1). □
Theorem 3.2
(Estimation and asymptotic behavior)
Assume (F0)–(F2) are satisfied. Then there exist two positive constants m, n such that the solution \(x(t)\) to Eq. (1.1) satisfies
Proof
It follows from \(\psi \in P_{e}\) and (3.19) that there exists \(0< l_{\psi }<1\) such that
Thus, from (3.20) and (2.8), we have
Consequently, one gets
We have
By (3.22), we have
In the end, by the l’Hospital rule,
that is, \(\pmb{\mathscr{D}_{t}} ^{\gamma }x(t)=o(t^{\alpha -2})\). □
4 Nonsingular cases
In this section, we are interested in some nonsingular cases of the nonlinear term f at time and space variables.
Case 1: f may be singular at \(t=0\) and (or) \(t=1\), but f is nonsingular at \(x_{1}=x_{2}=0\):
Theorem 4.1
Suppose (F0) and the following assumptions are satisfied:
-
(B1)
\(f\in C((0,1)\times [0,\infty )\times [0,\infty ),[0,+ \infty ))\), and \(f(t,x_{1},x_{2})\) is decreasing in \(x_{i}>0\) for \(i=1,2\).
-
(B2)
\(f(t,0,0)\not \equiv 0\) for any \(t\in (0,1)\), and
$$ 0< \int _{0}^{1} c(s)f(s,0,0)\,ds< +\infty . $$
Then Eq. (1.1) has at least one positive solution \(x(t)\) satisfying
for some constant \(\mathcal{M}^{*}> 0\). Moreover, the positive solution \(x(t)\) has boundary asymptotic behavior
Proof
In fact, we only replace the set \(P_{e}\) in Theorem 3.1 by using
Let
and set
Then we have \(\phi (t), \psi (t)\in P_{1} \) and
On the other hand, we also have
and
Thus from (4.1)–(4.3), \(\phi (t)\) and \(\psi (t)\) are still the lower and upper solutions of the boundary value problem (2.1), respectively.
Finally, it follows from Lemma 2.3 that
Thus according to the proofs of Theorems 3.1–3.2, the conclusion of Theorem 4.1 is true. □
Case 2: \(f(t,x_{1},x_{2})\) is nonsingular at both \(t=0,1\) and \(x_{i} =0\), \(i=1,2\). Then, by Theorem 4.1, the following conclusion is valid.
Theorem 4.2
Assume that \(f(t,x_{1},x_{2}): [0,1]\times [0,\infty )\times [0, \infty ) \to [0,+\infty )\) is a continuous and decreasing function in \(x_{i}\), \(i=1,2\) with \(f(t,0,0)\not \equiv 0\) for any \(t\in [0,1]\). If (F0) holds, then Eq. (1.1) has at least one positive solution \(x(t)\) with the estimation
for some constant \(\mathcal{M}^{*}> 0\) and boundary asymptotic behavior \(\pmb{\mathscr{D}_{t}} ^{\gamma }x(t)=o(t^{\alpha -2})\).
Proof
In fact, if \(f(t,x_{1},x_{2}): [0,1]\times [0,\infty )\times [0, \infty ) \to [0,+\infty )\) is continuous and \(f(t,0,0)\not \equiv 0\), then the condition (B2) holds naturally. □
5 Numerical examples
Example 1
Consider the existence of positive solutions for the following singular fractional differential equation with nonlocal boundary condition:
where \(\mathcal{X}\) is a function of bounded variation such that
By simple calculation, Eq. (5.1) can be transformed to the following 4-point boundary value problem with coefficients of both signs in the boundary condition:
Conclusion: The BVP (5.1) has at least one positive solution \(x(t)\), and there exist two positive constants m, n such that
with boundary asymptotic behavior \(\pmb{\mathscr{D}_{t}} ^{ \frac{1}{4}}x(t)=o(t^{\frac{1}{2}})\).
Proof
Let \(\alpha =\frac{5}{2}\), \(\gamma =\frac{1}{4}\), \(\mu =\frac{1}{3}\), \(f(t,x_{1},x_{2})=10 t^{-\frac{1}{4}} [x_{1}^{-\frac{1}{3}}+x_{2} ^{-\frac{1}{8}} ]\). Then \(2<\alpha \le 3\) satisfying \(0<\gamma \le \mu <\alpha -2\) and f is singular at \(t=0\) and \(x_{1}=x_{2}=0\).
Clearly,
and
Thus
Consequently, (F0) and (F1) hold.
Since
for any \(\tau >0\) and \(t\in (0,1)\), we have \(f(t,\frac{ \tau }{ \varGamma (\gamma +1)} t^{\gamma },\tau )= 10 t^{-\frac{1}{4}} [ (\frac{ \tau }{ \varGamma (\frac{5}{4})} )^{-\frac{1}{3}} t^{-\frac{1}{12}}+ \tau ^{-\frac{1}{8}} ]\not \equiv 0\) and
Thus (F2) holds.
It follows from Theorem 3.1 that Eq. (5.1) has at least a positive solution \(x(t)\) satisfying the estimation
for some positive constants m, n and boundary asymptotic behavior \(\pmb{\mathscr{D}_{t}} ^{\frac{1}{4}}x(t)=o(t^{\frac{1}{2}})\). □
Remark 5.1
In [14], Zhang et al. use the condition (A) to overcome the singularity of the equation. Obviously, Example 1 indicates that (F1) and (F2) are easier to check than (A), thus the growth condition in this paper is more popular in handling a singularity in the space variable.
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He, J., Zhang, X., Liu, L. et al. Existence and asymptotic analysis of positive solutions for a singular fractional differential equation with nonlocal boundary conditions. Bound Value Probl 2018, 189 (2018). https://doi.org/10.1186/s13661-018-1109-5
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DOI: https://doi.org/10.1186/s13661-018-1109-5