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Existence and uniqueness results for the coupled systems of implicit fractional differential equations with periodic boundary conditions
Advances in Difference Equations volume 2018, Article number: 413 (2018)
Abstract
In this paper, we study the periodic boundary value problems for the coupled systems of fractional implicit differential equations. Basing on the coincidence degree theory, we establish the existence and uniqueness theorems. Further, we provide several examples to show our main results.
1 Introduction
In the past two decades, there has been tremendous interest in studying fractional differential equations (FDEs for short) due to their extensive applications in various fields of engineering and scientific disciplines (see [1,2,3,4,5,6,7,8]). For example, in [8], Laskin proposed the following fractional stochastic dynamic model for the considered market:
where \(D_{0+}^{\mu}\) is the standard Riemann–Liouville fractional derivative of order μ, λ and \(F(t)\) respectively denote the expected rate and the random force.
As an important issue for the theory of FDEs, the existence, uniqueness, and multiplicity of solutions for the nonlinear fractional initial value problems (FIVPs for short) and fractional boundary value problems (FBVPs for short) have attracted scholars’ attention. For some recent work on the topic, see papers [9,10,11,12,13,14,15,16,17,18,19], monographs [1, 2, 20, 21], and the references therein. In particular, many researchers focused on studying the FDEs with periodic boundary conditions (PBCs for short) (see [22,23,24,25,26,27,28,29,30]).
In [22], Cabada and Kisela discussed the following FDE with PBC:
where \(\lambda\ne0(\lambda\in\mathbb{R})\), \(D_{0+}^{\alpha}\) is the standard Riemann–Liouville fractional derivative of order α. The existence results were based on the fixed point theorems and monotone iterative technique.
In [23], Staněk dealt with the following FDE with PBC:
where \({}^{c}{D^{(\cdot)}}\) is the Caputo fractional derivative of fractional order. The existence, multiplicity, and uniqueness results were proved by the Schauder fixed point theorem.
Recently, some scholars have considered very interesting aspects of IVPs and BVPs for the implicit FDEs (see [29,30,31,32,33,34,35,36,37,38,39]). For example, Nieto, Ouahab, and Venktesh [32] investigated a class of implicit FIVP:
where \(J=[0,b]\), \(b>0\), \({}^{c}{D^{\alpha}}\) is the Caputo fractional derivative of order α, and \(f:J \times{\mathbb{R}^{2}} \to\mathbb {R}\) is a continuous function. By using fixed point theory and approximation method, the existence and uniqueness results were obtained.
In [29], Benchohra, Bouriah, and Graef studied the following implicit FDE with PBC:
where \(J=[0,T]\), \({}^{c}{D^{\alpha}}\) is the Caputo fractional derivative of order α and \(f:J \times{\mathbb{R}^{2}} \to\mathbb{R}\) is a continuous function. Applying the coincidence degree theory, an existence result was given.
In [33], Ali, Zada, and Shah proved the existence and uniqueness of the solutions for the following implicit FDEs with three-point BCs:
where \({}^{c}{D^{(\cdot)}}\) is the Caputo fractional derivative of fractional order and \(f,g:J \times{\mathbb{R}^{2}} \to\mathbb{R}\) are continuous functions. The results were accomplished by means of the Leray–Schauder fixed point theorem and Banach contraction principle.
Inspired by the above work, in this paper we are mainly concerned with the existence and uniqueness of solutions for the following coupled system of nonlinear implicit FDEs with PBCs:
where \(D_{0 + }^{(\cdot)}\) is the standard Riemann–Liouville fractional derivative of fractional order, \(f,g : [0,1] \times{\mathbb {R}^{2}} \to\mathbb{R}\) are two continuous functions. To state our main results, we assume that the nonlinear terms f and g satisfy the following general conditions:
- \((\mathrm{A}_{1})\) :
-
There exist nonnegative continuous functions \({\gamma_{i}}(t),{\eta_{i}}(t),{\omega_{i}}(t),i = 1,2\), such that, for any \(t \in[0,1]\), \({u_{i}},{v_{i}} \in\mathbb{R}\), \((i = 1,2)\),
$$\begin{aligned} & \bigl\vert f\bigl(t,{t^{1 - \beta}} {u_{1}},{v_{1}} \bigr) \bigr\vert \le{\gamma_{1}}(t) \bigl\vert {t^{1 - \beta }} {u_{1}} \bigr\vert + {\eta_{1}}(t) \vert {v_{1}} \vert + {\omega_{1}}(t), \\ & \bigl\vert g\bigl(t,{t^{1 - \alpha}} {u_{2}},{v_{2}} \bigr) \bigr\vert \le{\gamma_{2}}(t) \bigl\vert {t^{1 - \alpha }} {u_{2}} \bigr\vert + {\eta_{2}}(t) \vert {v_{2}} \vert + {\omega_{2}}(t). \end{aligned}$$ - \((\mathrm{A}_{2})\) :
-
There exist nonnegative continuous functions \({p_{i}}(t),{q_{i}}(t),i = 1,2\), such that, for any \(t \in[0,1]\), \({u_{ij}},{v_{ij}} \in\mathbb{R},(i,j = 1,2)\),
$$\begin{aligned} & \bigl\vert f\bigl(t,{t^{1 - \beta}} {u_{11}},{v_{11}} \bigr) - f\bigl(t,{t^{1 - \beta }} {u_{12}},{v_{12}}\bigr) \bigr\vert \leq{p_{1}}(t){t^{1 - \beta}} \vert {u_{11}} - {u_{12}} \vert + {q_{1}}(t) \vert {v_{11}} - {v_{12}} \vert , \\ & \bigl\vert g\bigl(t,{t^{1 - \alpha}} {u_{21}},{v_{21}} \bigr) - g\bigl(t,{t^{1 - \alpha }} {u_{22}},{v_{22}}\bigr) \bigr\vert \leq{p_{2}}(t){t^{1 - \alpha}} \vert {u_{21}} - {u_{22}} \vert + {q_{2}}(t) \vert {v_{21}} - {v_{22}} \vert . \end{aligned}$$ - \((\mathrm{A}_{3})\) :
-
There exist constants \(a,c > 0\), \(b,d \geq0\) such that, for any \(t \in[0,1]\), \({u_{ij}},{v_{ij}} \in\mathbb{R}\), (\(i,j = 1,2\)),
$$\begin{aligned} & \bigl\vert f\bigl(t,{t^{1 - \beta}} {u_{11}},{v_{11}} \bigr) - f\bigl(t,{t^{1 - \beta }} {u_{12}},{v_{12}}\bigr) \bigr\vert \geq a{t^{1 - \beta}} \vert {u_{11}} - {u_{12}} \vert - b \vert {v_{11}} - {v_{12}} \vert , \\ & \bigl\vert g\bigl(t,{t^{1 - \alpha}} {u_{21}},{v_{21}} \bigr) - g\bigl(t,{t^{1 - \alpha }} {u_{22}},{v_{22}}\bigr) \bigr\vert \geq c{t^{1 - \alpha}} \vert {u_{21}} - {u_{22}} \vert - d \vert {v_{21}} - {v_{22}} \vert . \end{aligned}$$
Remark 1.1
Condition \((\mathrm{A}_{2})\) implies condition \((\mathrm{A}_{1})\).
The objective of this paper is twofold. The first one is to study the existence solutions for BVP (1.1), the other is to consider the uniqueness of solution for (1.1). Our work presented in this paper has the following features. Firstly, this article generalizes the results of papers [29, 30] into coupled systems. Secondly, compared with [29, 30], we not only discuss the existence result but also establish the uniqueness result. In addition, the existence results of papers [29, 30] are based on condition \((\mathrm{A}_{2})\), in our paper the existence result can also be obtained under condition \((\mathrm{A}_{1})\). Thirdly, we present two prior estimation ways in using Theorem 2.1 (see Sect. 2) to establish the existence results. It should be pointed out that a number of papers by applying Theorem 2.1 to solve fractional resonance boundary value problems usually used the second way (see Lemma 3.4 in Sect. 3) to estimate the prior bounds. For example [40,41,42,43]. Our results show that the first way is better than the second. We finally remark that our paper investigates the FBVP in the frame of Riemann–Liouville fractional derivative which is more complicated than such a problem involving Caputo fractional derivative, and if \(\alpha=\beta=1\), then BVP (1.1) can be reduced to the implicit first order differential systems with PBCs.
The rest of this paper is built up as follows. We devote Sect. 2 to recalling some preliminary definitions and lemmas. We establish the existence and uniqueness theorems for problem (1.1) in Sect. 3. In order to fully explain our main results, we provide three examples in Sect. 4. Finally, we present some conclusions in Sect. 5.
2 Preliminaries
In this section, we recall some basic definitions, lemmas, and theorems which are used throughout this paper. Firstly, we introduce some definitions and results on fractional calculus [1, 2, 44].
Definition 2.1
The Riemann–Liouville fractional integral of order \(\alpha > 0\) for a function \(x:{\mathbb{R}^{ + }} \to\mathbb{R} \) is given by
provided that the right-hand side integral is pointwise defined on \((0, + \infty)\).
Definition 2.2
The Riemann–Liouville fractional derivative of order \(\alpha > 0\) for a function \(x:{\mathbb{R}^{ + }} \to\mathbb{R}\) is given by
where \(n = [ \alpha]+1\), provided that the right-hand side integral is pointwise defined on \((0, + \infty)\).
Lemma 2.1
Let \(\alpha > 0\). If \(x, D_{0 + }^{\alpha}x \in L^{1}(0,1)\), then
where \(n = [\alpha] + 1\), \({c_{i}} \in\mathbb{R}(i = 1,2, \ldots,n)\) are arbitrary constants.
Lemma 2.2
Let \(\alpha> \beta>0\). If \(x \in L^{1}(0,1)\), then
in particular \(D_{0 + }^{\alpha}I_{0 + }^{\alpha}x(t) = x(t)\).
Lemma 2.3
(see [44])
If \(\alpha > 0\), \(\lambda > -1\), \(t > 0\), then
in particular \(D_{0 + }^{\alpha}{t^{\alpha - m}} = 0\), \(m = 1,2, \ldots,n\), where \(n = [\alpha] + 1\).
We recall now the basic knowledge on the coincidence degree theory. For more details, we refer the readers to [45,46,47].
Let \((X,{ \Vert \cdot \Vert _{X}})\) and \((Y,{ \Vert \cdot \Vert _{Y}})\) be two real Banach spaces. Suppose \(L:\operatorname{dom}L \subset X \to Y\) is a Fredholm operator with index zero, then there exist two continuous projectors \(P:X \to X\) and \(Q:Y \to Y\) such that
and \(L|_{\mathrm{dom}}L \cap\operatorname{Ker}P :\operatorname{dom}L \to\operatorname{Im} L\) is invertible. We denote by \({K_{P}} = { ( {L{|_{\operatorname{dom}L \cap\operatorname{Ker}P}}} )^{ - 1}}\). Let Ω be an open bounded subset of X and \(\operatorname{dom}L \cap\bar{\varOmega}\ne\emptyset \). The map \(N:X \to Y\) is called L-compact on Ω̄ if \(QN ( {\bar{\varOmega}} )\) is bounded and \(K_{P} (I - Q)N:\bar{\varOmega}\to X\) is compact.
Theorem 2.1
Let \(L:\operatorname{dom}L \subset X \to Y\) be a Fredholm operator of index zero and \(N:X \to Y\) be L-compact on Ω̄. If the following conditions are satisfied:
-
(i)
\(Lu \ne\lambda Nu\) for any \(u \in(\operatorname{dom}L\backslash\operatorname{Ker}L) \cap\partial\varOmega\), \(\lambda \in(0,1)\);
-
(ii)
\(Nu \notin\operatorname{Im} L\) for any \(u \in\operatorname{Ker}L \cap \partial\varOmega\);
-
(iii)
\(\deg\{ QN{|_{\operatorname{Ker}L}},\varOmega \cap\operatorname {Ker}L,0\} \ne0\);
then the equation \(Lx = Nx\) has at least one solution in \(\operatorname{dom}L \cap\bar{\varOmega}\).
Theorem 2.2
Let \(L:\operatorname{dom}L \subset X \to Y\) be a Fredholm operator of index zero, \(\varOmega\subset X\) be an open bounded set symmetric with \(0\in\varOmega\) and \(N:\bar{\varOmega}\to Y\) is L-compact. If \(Lx - Nx \ne\lambda( - Lx - N( - x))\) for all \((\lambda,x) \in(0,1] \times\operatorname{dom}L \cap\partial\varOmega \), then \(Lx = Nx\) has a solution in \(\operatorname{dom}L \cap\bar{\varOmega}\).
3 Main results
Take
endowed with the norms
respectively, where \(\| \cdot\|_{\infty} = \max_{t \in[0,1]} | \cdot|\). We can easily check that \(({X_{1}},\| \cdot \|_{{{X_{1}}}})\) and \(({X_{2}}, {\| \cdot\|_{{{X_{2}}}}})\) are two Banach spaces. Let \({Z_{1}} = C[0,1]\) with norm \(\|z\|_{{{Z_{1}}}} = \max_{t \in[0,1]} |z(t)|\). According to the basic theory of functional analysis, we have \(X = {X_{1}} \times{X_{2}}\) and \(Z = {Z_{1}} \times{Z_{1}}\) are also Banach spaces, respectively, with the norms
Define the linear operators \({L_{i}}:\operatorname{dom}{L_{i}} \subset{X_{i}} \to {Z_{1}}(i=1,2)\) and the nonlinear operators \({N_{1}}:{X_{2}} \to{Z_{1}}\), \({N_{2}}:{X_{1}} \to{Z_{1}}\) by
where
Define the linear operator \(L:\operatorname{dom}L \subset X \to Z\) and the nonlinear operator \(N:X \to Z\) by
where
Then the coupled system of BVP (1.1) is equivalent to the operator equation \(L(x,y) = N(x,y)\), \((x,y) \in\operatorname{dom}L\).
Lemma 3.1
The mapping \(L:\operatorname{dom}L \subset X \to Z\) is a Fredholm operator with index zero.
Proof
First, we claim that the operator L satisfies
In fact, by Lemma 2.1, it can easily be checked that (3.1) holds. For any \((u,v) \in\operatorname{Im} L\), there exists \((x,y) \in\operatorname{dom}L\) such that \(D_{0+}^{\alpha}x(t)=u(t)\), \(D_{0+}^{\beta}y(t)=v(t)\). Using Lemma 2.1 and the boundary conditions in (1.1), we find
That is,
Conversely, for any \((u,v) \in Z\) satisfying (3.3), take \(x(t)=I_{0+}^{\alpha}u(t)\) and \(y(t)=I_{0+}^{\beta}v(t)\), then we have
and
which shows \((u,v) \in\operatorname{Im} L\). Therefore, (3.2) holds.
Second, we prove that \(\operatorname{ind}L = \dim\operatorname{Ker}L - co\dim\operatorname{Im} L=0\). Define the linear operators \({Q_{i}}:{Z_{1}} \to{Z_{1}}(i=1,2)\) and \(Q:Z \to Z\) by
Evidently, \({Q_{1}},{Q_{2}},Q\) are continuous operators and \(\operatorname {Im} L=\operatorname{Ker}Q\). For any \((u,v) \in Z\), we have
Thus, Q is a continuous linear projector. For \((u,v) \in Z\), set \(({u_{1}},{v_{1}}) = (u,v) - Q(u,v)\), then \(Q({u_{1}},{v_{1}})=Q(u,v)-{Q^{2}}(u,v)=0\), i.e., \(({u_{1}},{v_{1}}) \in \operatorname{Ker}Q = \operatorname{Im} L\). So, \(Z = \operatorname{Im} L+\operatorname{Im}Q\). Besides, for every \((u,v) \in\operatorname{Im} L \cap\operatorname{Im} Q\), we have \((u,v)=Q(u,v)=(0,0)\). Therefore, \(Z=\operatorname{Im}L \oplus\operatorname{Im}Q\). Furthermore, \(\dim \operatorname{Ker}L = \dim\operatorname{Im} Q = co\dim\operatorname {Im} L = 2\), which means L is a Fredholm operator with index zero. □
Lemma 3.2
Define the linear operator \({K_{P}}:\operatorname{Im} L \to\operatorname{dom}L \cap Y\) by
Then \({K_{P}}\) is the inverse of \(L{|_{\operatorname{dom}L \cap Y}}\) and satisfies
where
Proof
Define the linear operators \({P_{i}}:{X_{i}} \to {X_{i}}\) (\(i = 1,2\)) and \(P:X \to X\) by
We first claim that P is a continuous linear projector operator. In fact, for any \((x,y) \in X\), we have
and
By Lemma 2.3, we also have
Then
Thus, \(P:X \to X\) is a bounded linear projector operator, and it is evident that \(\operatorname{Im} P = \operatorname{Ker}L,Y = \operatorname{Ker}P\).
Next, we show that \({K_{P}} = { ( {L{|_{\operatorname{dom}L \cap \operatorname{Ker}P}}} )^{ - 1}}\). In fact, for any \((u,v) \in \operatorname{Im} L\), by the definition of \({K_{P}}\), we can check that \({K_{P}}(u,v) \in\operatorname{dom}L \cap\operatorname{Ker}P\), that is, \({K_{P}}\) is well defined on ImL. On the one hand, by Lemma 2.2, we have
On the other hand, for every \((x(t),y(t)) \in\operatorname{dom}L \cap \operatorname{Ker}P\), by Lemma 2.1, we get
Because \(({K_{P}}L)(x(t),y(t)) \in\operatorname{Ker}P\) and \(({c_{1}}{t^{\alpha - 1}}, {c_{2}}{t^{\beta - 1}}) \in\operatorname {Ker}L = \operatorname{Im} P\), we can obtain
Thus, \(({K_{P}}L)(x(t),y(t)) = (x(t),y(t))\). Therefore, \({K_{P}} = ( L|_{\operatorname{dom}L \cap\operatorname{Ker}P} )^{ - 1}\). Again by Lemma 2.2, for all \((u,v) \in\operatorname{Im} L\), we have
This completes the proof of Lemma 3.2. □
Lemma 3.3
Assume that \((\mathrm{A}_{2})\) holds, \(\varOmega \subset X\) is an open bounded subset with \(\operatorname{dom}L \cap \bar{\varOmega}\ne\emptyset\). Then N is L-compact on Ω̄.
Proof
Since \(f,g:[0,1] \times{\mathbb{R}^{2}} \to\mathbb{R}\) are continuous and satisfy \((\mathrm{A}_{2})\), we claim that \(QN(\bar{\varOmega})\) and \((I - Q)N(\bar{\varOmega})\) are uniformly bounded. In fact, for Ω is bounded in X, there exists a constant \(r>0\) such that \(\|(x,y)\|_{X} \le r\), \(\forall(x,y) \in\bar{\varOmega}\), by \((\mathrm{A}_{2})\), we have the following inequalities:
where \(\rho = \sup_{t \in[0,1]} f(t,0,0)\), \({p_{1}} = \sup_{t \in[0,1]} {p_{1}}(t)\), \({q_{1}} = \sup_{t \in[0,1]} {q_{1}}(t)\). In the same way, we have
where \(\varpi = \sup_{t \in[0,1]} g(t,0,0)\), \({p_{2}} = \sup_{t \in[0,1]} {p_{2}}(t)\), \({q_{2}} = \sup_{t \in[0,1]} {q_{2}}(t)\). So we get that
Use of Lemma 3.2 yields
From (3.4), (3.5) it follows that \(QN(\bar{\varOmega})\), \({K_{P}}(I - Q)N(\bar{\varOmega})\) are uniformly bounded. Now, we are going to prove that \({K_{P}}(I - Q)N(x,y)\) is equicontinuous for all \((x,y)\in\bar{\varOmega}\). In fact, take \((x,y) \in\bar{\varOmega}\) and \(0 \le{t_{1}} < {t_{2}} \le1\). Since \({t^{\alpha}},t\) are uniformly continuous on \([{t_{1}},{t_{2}}]\) and \(f(t,u,v)\), \(g(t,u,v)\) are uniformly continuous on \([{t_{1}},{t_{2}}] \times[-r,r] \times[-r,r]\), we have
and
Similarly, it has
and
To summarize, we can conclude that \(\{{K_{P}}(I - Q)N(x,y): (x,y)\in\bar{\varOmega}\}\) is equicontinuous. By the Ascoli–Arzelà theorem, it is immediate that \({K_{P}}(I - Q)N:\bar{\varOmega}\to X\) is compact. Using a similar argument, we can also get N is L-compact if condition \((\mathrm{A}_{1})\) holds. □
In what follows, we shall give several existence results for BVP (1.1). For simplicity of presentation, we let
where \({\gamma_{i}} = \max_{t \in[0,1]} |\gamma (t)|\), \({\eta_{i}} = \max_{t \in[0,1]} |\gamma(t)|\), \({\omega_{i}} = \max_{t \in[0,1]} |\gamma(t)|\), \(i =1,2\). First, we suppose that the following conditions are satisfied:
- \((\mathrm{A}_{4})\) :
-
For \((x,y) \in\operatorname{dom}L\), there exist constants \({B_{i}} > 0\), \(i = 1,2\), such that, for all \(t \in [0,1]\), if either \(|{t^{1 - \alpha}}x(t)| > {B_{1}}\) or \(|{t^{1 - \beta }}y(t)| > {B_{2}}\), then \(QN(x,y) \ne(0,0)\).
- \((\mathrm{A}_{5})\) :
-
For \(({c_{1}}{t^{\alpha - 1}},{c_{2}}{t^{\beta - 1}}) \in\operatorname{Ker}L\), there exist constants \({G_{i}} > 0\), \(i = 1,2\), such that for any \(({c_{1}},{c_{2}}) \in{\mathbb{R}^{2}}\) satisfying either
$$ \textstyle\begin{cases} {c_{1}}{N_{2}}({c_{1}}{t^{\alpha - 1}}) > 0,&\mbox{if } \vert {c_{1}} \vert > {G_{1}},\\ {c_{2}}{N_{1}}({c_{2}}{t^{\beta - 1}}) > 0,&\mbox{if } \vert {c_{2}} \vert > {G_{2}}, \end{cases} $$(3.6)or
$$ \textstyle\begin{cases} {c_{1}}{N_{2}}({c_{1}}{t^{\alpha - 1}}) < 0,&\mbox{if } \vert {c_{1}} \vert > {G_{1}},\\ {c_{2}}{N_{1}}({c_{2}}{t^{\beta - 1}}) < 0,&\mbox{if } \vert {c_{2}} \vert > {G_{2}}. \end{cases} $$(3.7)
Lemma 3.4
Let \((\mathrm{A}_{1})\) and \((\mathrm{A}_{4})\) hold, set
Then \({\varOmega_{1}}\) is bounded provided that
or
Proof
For \((x,y) \in{\varOmega_{1}}\), we have \(N(x,y) \in \operatorname{Im} L = \operatorname{Ker}Q\). Then \(QN(x,y) = (0,0)\). On the one hand, according to hypothesis \((\mathrm{A}_{4})\), it follows that
Since
By substituting (3.10) into (3.11), we obtain
From (3.10), (3.11), and (3.12) we have
On the other hand, by \((\mathrm{A}_{1})\), we have
We now estimate \(\varOmega_{1}\) is bounded under conditions (3.8) and (3.9), respectively.
First. We show that \(\varOmega_{1}\) is bounded if condition (3.8) holds. In fact, (3.14) and (3.15) imply that
If we plug (3.16) back into (3.13), we get
So that
It follows from (3.17) and (3.18) that
Substituting (3.19) into (3.16), we obtain
Thus, \({\varOmega_{1}}\) is bounded.
Second. We prove that \({\varOmega_{1}}\) is bounded under condition (3.9). In such a case, by Lemma 3.2, one has
Therefore, from (3.13)–(3.15), we can derive that
Next, we separate the proof into four cases.
Case 1. \(\|(x,y)\|_{X} \le{B_{1}} + \frac {2}{{\varGamma(\alpha + 1)}}\|D_{0 + }^{\alpha}x\|_{\infty} + \Delta \|D_{0 + }^{\beta}y\|_{\infty}\) because
By (3.14) and (3.15), one gets
that is,
Case 2. \(\|(x,y)\|_{X} \le{B_{1}} + ( {\frac {2}{{\varGamma(\alpha + 1)}} + \Delta} )\|D_{0 + }^{\alpha}x\|_{\infty}\). By (3.14) and (3.20), we have
which implies that
Case 3. \(\|(x,y)\|_{X} \le{B_{2}} + \frac {2}{{\varGamma(\beta + 1)}}\|D_{0 + }^{\beta}y\|_{\infty} + \Delta \|D_{0 + }^{\alpha}x\|_{\infty}\). Using a similar proof as that in Case 1, we can get
Case 4. \(\|(x,y)\|_{X} \le{B_{2}} + ( {\frac {2}{{\varGamma(\beta + 1)}} + \Delta} )\|D_{0 + }^{\beta}y\|_{\infty}\). By applying a method similar to Case 2, we can obtain
To summarize, \({\varOmega_{1}}\) is bounded and the proof is completed. □
Remark 3.1
If \(\alpha=\beta\), then condition (3.8) can be derived by (3.9).
In fact, from
we can obtain
On the other hand, by
we have
Using the fact \(\alpha=\beta\), we also have
Then it follows that
where \(\zeta= \max\{ {\gamma_{1}},{\eta_{1}},{\gamma_{2}},{\eta_{2}}\}\). From (3.21) and (3.22), we have
According to the above inequalities, it follows (3.8) holds.
Lemma 3.5
Let \((\mathrm{A}_{4})\) hold, set
Then \({\varOmega_{2}}\) is bounded.
Proof
For \((x,y) \in\operatorname{Ker}L\), then we can write \(x = {c_{1}}{t^{\alpha - 1}}\), \(y = {c_{2}}{t^{\beta - 1}}\), \(({c_{1}},{c_{2}}) \in{\mathbb{R}^{2}}\), and \(N({c_{1}}{t^{\alpha - 1}},{c_{2}}{t^{\beta - 1}}) \in\operatorname{Im} L = \operatorname {Ker}Q\), that is, \(QN({c_{1}}{t^{\alpha - 1}},{c_{2}}{t^{\beta - 1}}) = (0,0)\). By \((\mathrm{A}_{4})\), there exist \({t_{3}},{t_{4}} \in[0,1]\) such that \(|t_{3}^{1 - \alpha}x({t_{3}})| = |{c_{1}}| \le{B_{1}}\), \(|t_{4}^{1 - \beta}y({t_{4}})| = |{c_{2}}| \le{B_{2}}\). Therefore,
The proof is completed. □
Lemma 3.6
Let \((\mathrm{A}_{5})\) hold, set
Then \({\varOmega_{3}}\) is bounded, where
and \(J:\operatorname{Ker}L \to\operatorname{Im}Q\) is the linear isomorphism given by
Proof
Without loss of generality, we suppose that (3.7) holds, then for \((x,y) \in{\varOmega_{3}}\), we have
By the preceding lemma, it suffices to show that \(|{c_{1}}|\), \(|{c_{2}}|\) are bounded. In fact, if \(\lambda = 1\), then \({c_{1}} = {c_{2}} = 0\). Otherwise, for \(\lambda \in[0,1)\), we get
If \(|{c_{1}}| > {G_{1}}\) or \(|{c_{2}}| > {G_{2}}\), by (3.7), it is easy to verify that at least one of the above equations is not true. Therefore, \(|{c_{1}}|\), \(|{c_{2}}|\) are bounded, which completes the proof of Lemma 3.6. □
Lemma 3.7
Let \((\mathrm{A}_{2})\) hold, set
Then \(\varOmega_{4}\) is bounded provided that
where \({p_{i}} = \sup_{t \in[0,1]} {p_{i}}(t)\), \({q_{i}} = \sup_{t \in[0,1]} {q_{i}}(t)\), \(i=1,2\).
Proof
For \((x,y) \in\varOmega_{4}\), we have
that is,
From (3.24) it follows that, for any \(t \in[0,1]\),
and
Taking account of (3.26) and (3.27), we derive
Similarly, by (3.25), it can be shown that, for any \(t \in[0,1]\),
According to (3.28)–(3.31), we get
Now, by using (3.32) and (3.33), we obtain
So we get that
This completes the proof of the lemma. □
Theorem 3.1
Assume that \((\mathrm{A}_{1})\), \((\mathrm{A}_{4})\), \((\mathrm{A}_{5})\), and (3.8) hold or \((\mathrm{A}_{1})\), \((\mathrm{A}_{4})\), \((\mathrm{A}_{5})\), and (3.9) hold. Then BVP (1.1) has at least one solution in X.
Proof
Set Ω be a bounded open set of X such that \(\bigcup_{i = 1}^{3}{\bar{\varOmega}_{i}} \subset\varOmega\). By Lemma 3.3, N is L-compact on Ω̄. Lemmas 3.4 and 3.5 imply that (i) and (ii) of Theorem 2.1 are satisfied. In order to achieve the thesis, we have to prove that condition (iii) of Theorem 2.1 holds. Define the homotopy mapping as follows:
By Lemma 3.6, we get \(H((x,y),\lambda) \ne(0,0)\) for all \((x,y) \in \operatorname{Ker}L \cap\partial\varOmega\). Using the homotopy invariance of the topological degree,
Then, by Theorem 2.1, BVP (1.1) has at least one solution in X. Thus the theorem is proved. □
Theorem 3.2
If \((\mathrm{A}_{2})\) and (3.23) hold, then BVP (1.1) has at least one solution in X.
Proof
Set \(\varOmega = \{ (x,y) \in X:\|(x,y)\|_{X} < m + 1\}\). Obviously, Ω is symmetric with \((0,0) \in\varOmega\) and \(X \cap\bar{\varOmega}\ne\emptyset\). By Lemma 3.7, we get, for every \((x,y) \in\partial\varOmega\) and \(\lambda \in(0,1]\),
which together with Theorem 2.2 yields that problem (1.1) has at least one solution in X. □
Theorem 3.3
If \((\mathrm{A}_{2})\), \((\mathrm{A}_{3})\), and (3.23) hold, then BVP (1.1) has exactly one solution in X provided that
where
Proof
By Theorem 3.2, we obtain that BVP (1.1) has at least one solution in X. Now, we prove the uniqueness result. Suppose that BVP (1.1) has two solutions \(({x_{1}},{y_{1}}),({x_{2}},{y_{2}}) \in \operatorname{dom}L\). Then, for \(i = 1,2\), we have
and
Let \(x = {x_{1}} - {x_{2}}\), \(y = {y_{1}} - {y_{2}}\). Then x, y satisfy the equations
Noting that \(\operatorname{Im} L = \operatorname{Ker}Q\), we have
which imply there exist \({t_{5}},{t_{6}} \in[0,1]\) such that
Basing on condition \((\mathrm{A}_{3})\), we conclude that
Considering that
thus,
Therefore, we can draw a fact
On the other hand, using hypothesis \((\mathrm{A}_{2})\) and (3.35)–(3.36), we find that
Consequently, we infer that
By Lemma 3.2, we obtain
Proceeding as in the proof of Lemma 3.4, we divide the proof in four cases.
Case 1. \(\|(x,y)\|_{X} \le\kappa\|D_{0 + }^{\alpha}x\|_{\infty} + \Delta\|D_{0 + }^{\beta}y\|_{\infty}\). By (3.20) and (3.39), we get
and
Again, by (3.20), we obtain
In view of condition (3.34), we have
As a result, we get \({x_{1}} = {x_{2}}\), \({y_{1}} = {y_{2}}\).
Case 2. \(\|(x,y)\|_{X} \le ( \kappa+ \Delta )\|D_{0 + }^{\alpha}x\|_{\infty}\). Then (3.39) and (3.20) imply
and
Using (3.20), we derive
According to assumption (3.34), we obtain
Consequently, \({x_{1}} = {x_{2}}\), \({y_{1}} = {y_{2}}\).
Case 3. \(\|(x,y)\|_{X} \le\mu\|D_{0 + }^{\beta}y\|_{\infty} + \Delta\|D_{0 + }^{\alpha}x\|_{\infty}\). By a method similar to that used in Case 1, we can conclude that
Case 4. \(\|(x,y)\|_{X} \le(\mu + \Delta)\|D_{0 + }^{\beta}y\|_{\infty}\). Similar to the analysis in Case 2, we can deduce that
From (3.34), (3.42), and (3.43), we also obtain that
that is, \({x_{1}} = {x_{2}},{y_{1}} = {y_{2}}\). In summary, BVP (1.1) has a unique continuous solution in X. □
4 Example
Example 4.1
Consider the boundary value problem
Corresponding to problem (1.1), here
Let
Then \((\mathrm{A}_{1})\) holds and
Accordingly,
Consequently, (3.8) holds. Since
and
So if we put \({B_{1}} = {G_{1}} = 1\), \({B_{2}} = {G_{2}} = 3\), then we have
Therefore, \((\mathrm{A}_{4})\) and \((\mathrm{A}_{5})\) hold. By Theorem 3.1, we can conclude that BVP (4.1) has at least one solution.
Remark 4.1
Obviously, for BVP (4.1), condition \((\mathrm{A}_{2})\) is not valid and (3.9) does not hold.
In fact, we can obtain that
So that (3.9) is not true.
Example 4.2
Consider the boundary value problem
Corresponding to problem (1.1), here
Let
then \({p_{1}} = \frac{1}{5}\), \({q_{1}} = \frac{4}{{15}}\), \({p_{2}} = \frac{1}{7}\), \({q_{2}} = \frac{2}{7}\). We can easily check that \((\mathrm {A}_{2})\) holds and
By Theorem 3.2, BVP (4.2) has at least one solution. If we let
then \((\mathrm{A}_{1})\) holds and
Therefore,
that is, (3.8) holds. Since
Then, for any \({B_{i}} > 0\) and \({G_{i}} > 0\), \((i = 1,2)\), we have \((\mathrm{A}_{4})\) and \((\mathrm{A}_{5})\) hold. By Theorem 3.1, we can also obtain that BVP (4.2) has at least one solution.
Remark 4.2
The existence result of BVP (4.2) cannot be obtained by verifying conditions \((\mathrm{A}_{1})\), \((\mathrm{A}_{4})\), \((\mathrm{A}_{5})\), and (3.9) of Theorem 3.1.
In fact, we can check that
This implies that (3.9) does not hold.
Example 4.3
Consider the following fractional boundary value problem:
Corresponding to problem (1.1), here
Let
Then
Choose \(a = \frac{1}{{10}}\), \(b = \frac{1}{{20}}\), \(c = \frac {1}{{14}}\), \(d = \frac{1}{{28}}\). It is easy to show that \((\mathrm {A}_{2})\) and \((\mathrm{A}_{3})\) hold. Since
By Theorem 3.2, BVP (4.3) has at least one solution. We also can check that
By Theorem 3.3, BVP (4.3) has a unique solution. If we let
then \((\mathrm{A}_{1})\) holds and
Thus,
and
That is, both (3.8) and (3.9) hold because
Therefore, for any \({B_{i}} > 0\) and \({G_{i}} > 0\), (\(i = 1,2\)) \((\mathrm {A}_{4})\) and \((\mathrm{A}_{5})\) hold, which means the existence result of BVP (4.3) can be obtained by Theorem 3.1.
5 Conclusion
In the present paper, we investigate the existence and uniqueness of solutions for the coupled systems of nonlinear implicit fractional periodic boundary value problems in the frame of Riemann–Liouville fractional derivative. By using Theorems 2.1 and 2.2, the new existence and uniqueness results are established. The results in papers [29, 30] are improved and extended in this paper. First, we extend the results of [29, 30] to coupled systems; second, in [29, 30], the authors only studied the existence results based on Lemma 2.1 and established existence theorems under condition \((\mathrm{A}_{2})\). Our results show that the existence results can also be obtained under condition \((\mathrm{A}_{1})\). Besides, compared with [40,41,42,43], we used a different technique to prove that \(\varOmega_{1}\) is bounded (see Lemma 3.4, the first way). By Remark 3.1, we show that the first way is superior to the second way used by [40,41,42,43]. Finally, our main results are well illustrated with the aid of several interesting examples.
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Zhang, W., Liu, W. & Xue, T. Existence and uniqueness results for the coupled systems of implicit fractional differential equations with periodic boundary conditions. Adv Differ Equ 2018, 413 (2018). https://doi.org/10.1186/s13662-018-1867-5
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DOI: https://doi.org/10.1186/s13662-018-1867-5