Abstract
In this paper, we study the existence and multiplicity of positive solutions for a class of systems of fractional differential equation with parameters. By applying the Krasnosel’skii fixed point theorem for a cone map, we conclude to the existence of at least one and two solutions for our considered system.
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1 Introduction
Fractional differential equations can describe some phenomena in various fields of engineering and scientific, disciplines such as control theory, chemistry, physics, biology, economics, mechanics and electromagnetic. Especially in recent years, a large number of papers dealt with the existence of positive solutions of boundary value problems for nonlinear differential equations of fractional order; for details, see [1–6]. In addition, the existence of positive solutions to fractional differential equations and their systems, especially coupled systems, were well studied by many authors; for details, see [7–10].
In [10], Su studied the existence of solutions for a coupled system of fractional differential equations
where \(1<\alpha, \beta<2\), \(\gamma, \eta>0\), \(\alpha-\eta\geq1\), \(\beta-\gamma\geq1\), \(f, g:[0,1]\times R^{2}\rightarrow R^{2}\) are given functions and \(D_{0^{+}}\) is the standard Riemann-Liouville fractional derivative.
In [11], Dunninger and Wang considered the existence and multiplicity of positive radial solutions for elliptic systems of the form
where \((u,v)\in C^{2}(\overline{\Omega})\times C^{2}(\overline {\Omega})\), with \(\Omega=\{x\in\mathbb{R}^{N}:R_{1}<|x|<R_{2}, R_{1}, R_{2}>0\}\) an annulus with boundary ∂Ω.
Motivated by [10] and [11], in this paper, we consider the system of fractional differential equations with parameters
where \(D^{p}\) is the standard Riemann-Liouville derivative. Moreover, in the rest of this paper we always suppose that the following assumptions hold.
-
(A1)
-
(i)
\(w, h:[0,1]\times[0,+\infty)\times[0,+\infty)\rightarrow [0,+\infty)\) are continuous;
-
(ii)
\(\lambda_{1}\) and \(\lambda_{2}\) are positive parameters;
-
(iii)
\(q_{i}\in(0,1)\), \(p_{i}\in(1,2)\), \(\gamma_{i}\in(2,3)\), \(\eta, \xi \in(0,1)\) (\(i=1,2\)), \(0<\alpha_{1}\eta^{p-1}<1\), \(0<\alpha_{2}\xi^{p-1}<1\).
-
(i)
-
(A2)
\(w(t,x,y), h(t,x,y)>0\) for \(x, y>0\), \(t\in J\).
By applying the Krasnosel’skii fixed point theorem for a cone map, we obtained the existence of at least one and two positive solutions for the system (1).
2 Preliminaries
For the sake of convenience, we introduce following notations:
Theorem A
[12]
Let X be a Banach space, and let \(K \subset X\) be a cone. Assume \(\Omega_{1}\), \(\Omega_{2}\) are two open bounded subsets of X with \(0\in\Omega_{1}\), \(\overline{\Omega}_{1}\subset \Omega_{2}\), and let \(T: K\cap(\overline{\Omega}_{2}\setminus\Omega_{1}) \rightarrow K\) be a completely continuous operator such that
-
(i)
\(\|Tx\|\leq \|x\|\), \(x\in K\cap\partial\Omega_{1}\), and \(\|Tx\|\geq\|x\|\), \(x\in K\cap\partial\Omega_{2}\);
or
-
(ii)
\(\|Tx\|\geq\|x\|\), \(x\in K\cap\partial\Omega_{1}\), and \(\|Tx\| \leq\|x\|\), \(x\in K\cap\partial\Omega_{2}\).
Then T has a fixed point in \(K\cap(\overline{\Omega }_{2}\setminus \Omega_{1})\).
Definition 1
[13]
We call \(D^{p}w(x)=\frac{1}{\Gamma (m-p)}(\frac{d}{dt})^{m}\int_{0}^{t}\frac{w(t)}{(t-s)^{p-m+1}}\,dt\), \(p>0\), \(m=[p]+1\) is the Riemann-Liouville fractional derivative of order p. \([p]\) denotes the integer part of number p.
Definition 2
[13]
We call \(I^{p}w(x)=\frac{1}{\Gamma (\alpha)}\int_{0}^{t}(t-s)^{p-1}w(s)\,ds\), \(t>0\), \(p>0 \) is Riemann-Liouville fractional integral of order p.
Lemma 1
[13]
Let \(p>0\), then, for \(\forall C_{i}\in R\), \(i=0, 1, 2, \ldots, m\), \(m=[p]+1\), we have
Lemma 2
Suppose that \(\varphi\in C(J)\) and (A1) holds, then the unique solution of the linear boundary value problem
is provided by
where \(G_{1}(t,s)\) is the Green’s function defined by
Proof
Let \(x(t)=-I^{p}\varphi (t)+C_{1}t^{p-1}+C_{2}t^{p-2}+C_{3}t^{p-3}+C_{4}t^{p-4}\), then
\(D^{q_{1}}x(0)=0\) implies that \(C_{4}=0\). In fact, if \(t=0\), we see that \(t^{p-q_{1}-1}=0\), \(t^{p-q_{1}-2}=0\), \(t^{p-q_{1}-3}=0\), \(t^{p-q_{1}-4}=\frac{1}{t^{4+q_{1}-p}}\) is not well defined. Similarly, \(D^{p_{1}}x(0)=0\) implies that \(C_{3}=0\) and \(D^{\gamma_{1}}x(0)=0\) implies that \(C_{2}=0\). Thus, \(x(t)=-I^{p}\varphi(t)+C_{1}t^{p-1}\). Now, by using boundary condition \(x(1)=\alpha_{1}x(\eta)\), we get \(C_{1}=\frac{1}{1-\alpha_{1}\eta^{p-1}}(I^{p}\varphi(1)-\alpha _{1}I^{p}\varphi(t))\). Hence, we get the solution as follows:
Then \(G_{1}(t,s)\) can easily be obtained. □
Notation 1
Similarly, we can get
and
In view of Lemma 2, the system (1) is equivalent to the Fredholm integral system of
where \(G_{i}(t,s)\) is the Green’s function defined by Lemma 2.
Define \(X=\{x(t)\mid x\in C(J)\}\), endowed with the norm \(\|x\|=\max_{t\in J}|x(t)|\), further the norm for the product space \(X\times X\), we define as \(\|x+y\|=\|x\|+\|y\|\). Obviously, \((X,\|\cdot\|)\) is a Banach space. We define the cone \(K\subset X\times X\) by
Define an operator \(T:X\times X\rightarrow X\times X\) as
The solutions of the system (1) and the fixed points of operator T coincide with each other.
Lemma 3
The Green’s function \(G_{i}(t,s)\) (\(i=1, 2\)) are continuous on \(J\times J\) and satisfy the following properties:
-
(1)
\(G_{i}(t,s)\in C(J\times J)\) and \(G_{i}(t,s)\geq0\), \(\forall t, s\in J\);
-
(2)
\(\max_{t\in J}G_{i}(t,s)=G_{i}(1,s)\);
-
(3)
\(\frac{\min_{t\in[\delta,1-\delta]}G_{i}(t,s)}{G_{i}(1,s)}\geq \theta_{i}(s)\), \(\delta\in(0,1)\).
Proof
(1) If \(0\leq s\leq t\leq\eta\leq1\), then
Since \(s\leq t\), \(G_{1}(t,s)\geq0\).
If \(0\leq t\leq s\leq\eta\leq1\), then
Hence \(G_{1}(t,s)\geq0\).
If \(0\leq\eta\leq s\leq t\leq1\), then
If \(0\leq\eta\leq t\leq s\leq1\), then
Similarly, we can obtain \(G_{2}(t,s)\geq0\). Thus, \(G_{i}(t,s)\geq0\) for every \(t, s\in J\).
(2) If \(0\leq s\leq t\leq\eta\leq1\), then
and
Therefore, \(G_{1}(t,s)\leq G_{1}(1,s)\).
If \(0\leq t\leq s\leq\eta\leq1\), then
If \(0\leq\eta\leq s\leq t\leq1\), then
and
If \(0\leq\eta\leq t\leq s\leq1\), then
Similarly, we can obtain \(G_{2}(t,s)\leq G_{2}(1,s)\).
(3) If \(0\leq s\leq t\leq\eta\leq1\), then
and
Let \(\sigma_{1}\) be a positive number such that \(\min_{t\in[\delta ,1-\delta]}G_{1}(t,s)\geq\sigma_{1}G_{1}(1,s)\). Then we can obtain
If \(0\leq t\leq s\leq\eta\leq1\), then
and
Let \(\sigma_{2}\) be a positive number such that \(\min_{t\in[\delta ,1-\delta]}G_{1}(t,s)\geq\sigma_{2}G_{1}(1,s)\). Then we can obtain \(\sigma_{2}\leq t^{p-1}(1-\alpha_{1}\eta^{p-1})\).
If \(0\leq\eta\leq s\leq t\leq1\), then
and
Let \(\sigma_{3}\) be a positive number such that \(\min_{t\in[\delta ,1-\delta]}G_{1}(t,s)\geq\sigma_{3}G_{1}(1,s)\). Then we can obtain \(\sigma_{3}\leq t^{p-1}\alpha_{1}\eta^{p-1}\).
If \(0\leq\eta\leq t\leq s\leq1\), then
and
Let \(\sigma_{4}\) be a positive number such that \(\min_{t\in[\delta ,1-\delta]}G_{1}(t,s)\geq\sigma_{4}G_{1}(1,s)\). Then we can obtain \(\sigma_{4}\leq t^{p-1}\).
Define \(\theta_{1}=\min\{\sigma_{1}, \sigma_{2}, \sigma_{3}, \sigma_{4}\}\). Then \(\frac{\min_{t\in[\delta,1-\delta ]}G_{1}(t,s)}{G_{1}(1,s)}\geq\theta_{1}(s)\), \(\delta\in(0,1)\). Similarly, we can prove \(\frac{\min_{t\in[\delta,1-\delta ]}G_{2}(t,s)}{G_{2}(1,s)}\geq\theta_{2}(s)\), \(\delta\in(0,1)\). □
Lemma 4
If (A1) holds, then \(T(K)\subset K\) and \(T:K\rightarrow K\) is a completely continuous operator.
Proof
The continuity of T is obvious. To prove \(T(K)\subset K\), let us choose \((x,y)\in K\). Since \(G_{i}(t,s)\leq G_{i}(1,s)\) for \(0\leq s\leq1\), and \(G_{i}(t,s)\geq\theta G_{i}(1,s)\) for \(\delta \leq t\leq1-\delta\), we have
Similarly,
Thus,
Since \(G_{i}(t,s)\geq0\), \(\forall t, s\in J\) and (A1) holds, we conclude that \(T(K)\subset K\). It is not difficult to show that T is uniformly bounded. Combining this with the Arzelà-Ascoli Theorem, we see that \(T:K\rightarrow K\) is a completely continuous operator. □
3 Main results and proofs
Theorem 1
Assume that (A1) holds, then, for all \(\lambda _{i}>0\), \(i=1, 2\), the system (1) has at least one positive solution in the following cases:
-
(a)
\(w_{0}=h_{0}=0\), and either \(w_{\infty}=\infty\) or \(h_{\infty }=\infty\) (superlinear).
-
(b)
\(w_{\infty}=h_{\infty}=0\), and either \(w_{0}=\infty\) or \(h_{0}=\infty\) (sublinear).
Proof
(a) Since \(w_{0}=h_{0}=0\), we may choose \(H_{1}>0\) such that \(w(t,x,y)\leq\varepsilon(x+y)\) and \(h(t,x,y)\leq\varepsilon (x+y)\) for \(0< x+y\leq H_{1}\), \(t\in J\), where the constant \(\varepsilon >0\) satisfies
Set \(\Omega_{1}=\{(x,y):(x,y)\in X\times X, \|(x,y)\|< H_{1}\}\). If \((x,y)\in K\cap\partial\Omega_{1}\), \(\|(x,y)\|=H_{1}\), we have
Similarly, \(T_{2}(x,y)(t)\leq\frac{\|(x,y)\|}{2}\). Hence,
for \((x,y)\in K\cap\partial\Omega_{1}\). If we further assume that \(w_{\infty}=\infty\), then there exists \(\widehat{H}>0\) such that \(w(t,x,y)\geq\beta(x+y)\) for \((x+y)\geq\widehat{H}\), \(t\in J\), where \(\beta>0\) is chosen so that \(\lambda_{1}\beta\int _{0}^{1}G_{1}(1,s)\,ds\geq1\). Let \(H_{2}=\max\{2H_{1}, \delta ^{1-p}\widehat{H}\}\) and set \(\Omega_{2}=\{(x,y):(x,y)\in X\times X, \|(x,y)\|< H_{2}\}\). If \((x,y)\in K\cap\partial\Omega_{2}\), we have \(\min_{t\in[\delta,1-\delta]}(x(t)+y(t))\geq\delta^{p-1}\|(x,y)\| \geq\widehat{H}\) and for \(\forall t[\delta,1-\delta]\),
Therefore, \(\|T(x,y)\|=\|T_{1}(x,y), T_{2}(x,y)\|=\|T_{1}(x,y)\|+\| T_{2}(x,y)\|\geq\|(x,y)\|\) for \((x,y)\in K\cap\partial\Omega_{2}\). An analogous estimate holds for \(h_{\infty}=\infty\).
Now by Theorem A, T has a fixed point \((x,y)\in K\cap\partial (\overline{\Omega_{2}}\setminus\Omega_{1})\) such that \(H_{1}\leq\| (x,y)\|\leq H_{2}\) and the system (1) has a positive solution.
(b) If \(w_{0}=\infty\), we choose \(H_{1}>0\) so that \(w(t,x,y)\geq \widetilde{\beta}(x+y)\) for \(0< x+y\leq H_{1}\), \(t\in J\), where β̃ satisfies \(\lambda_{1}\widetilde{\beta}\delta ^{p-1}\int_{0}^{1}G_{1}(1,s)\,ds\geq1\). Let \(\Omega_{1}=\{ (x,y):(x,y)\in X\times X, \|(x,y)\|< H_{1}\}\), if \((x,y)\in K\cap \partial\Omega_{1}\), \(\|(x,y)\|=H_{1}\), and for \(\forall t\in[\delta ,1-\delta]\),
Therefore, \(\|T(x,y)\|=\|T_{1}(x,y), T_{2}(x,y)\|=\|T_{1}(x,y)\|+\| T_{2}(x,y)\|\geq\|(x,y)\|\) for \((x,y)\in K\cap\partial\Omega_{2}\) An analogous estimate holds for \(h_{0}=\infty\).
Set \(w^{\ast}(t)=\max_{0\leq x+y\leq t}w(t,x,y)\) and \(h^{\ast }(t)=\max_{0\leq x+y\leq t}h(t,x,y)\). Then \(w^{\ast}\) and \(h^{\ast}\) are nondecreasing in their respective arguments. Moreover, from \(w_{\infty}=h_{\infty}=0\), we see that \(\lim_{t\rightarrow\infty }\frac{w^{\ast}(t)}{t}=0\), \(\lim_{t\rightarrow\infty}\frac{h^{\ast }(t)}{t}=0\). Therefore, there is an \(H_{2}>2H_{1}\) such that \(w^{\ast }(t)\leq\varepsilon t\), \(h^{\ast}(t)\leq\varepsilon t\) for \(t\geq H_{2}\), where the constant \(\varepsilon>0\) satisfies
Set \(\Omega_{2}=\{(x,y):(x,y)\in X\times X, \|(x,y)\|< H_{2}\}\). If \((x,y)\in K\cap\partial\Omega_{2}\), \(\|(x,y)\|=H_{2}\), we have
Similarly, \(T_{2}(x,y)(t)\leq\frac{\|(x,y)\|}{2}\). Hence,
for \((x,y)\in K\cap\partial\Omega_{2}\).
Applying Theorem A, we conclude to the existence of a positive solution \((x,y)\in K\cap(\overline{\Omega_{2}}\setminus\Omega_{1})\) for the system (1). □
Theorem 2
Assume that (A1) and (A2) hold.
-
(a)
If \(w_{0}=h_{0}=w_{\infty}=h_{\infty}=0\), then there is a positive constant \(\sigma_{1}\) such that (1) has at least two positive solutions for all \(\lambda_{1}, \lambda_{2}\geq\sigma_{1}\).
-
(b)
If \(w_{0}=\infty\) or \(h_{0}=\infty\), and either \(w_{\infty}=0\) or \(h_{\infty}=0\), then there is a positive constant \(\sigma_{2}\) such that the system (1) has at least two positive solutions for all \(\lambda_{1}, \lambda_{2}\geq\sigma_{2}\).
Proof
(a) For \((x,y)\in K\) and \(\|(x,y)\|=l\), let
By assumption \(m(l)>0\) for \(l>0\). Choose two numbers \(0< H_{3}< H_{4}\), and let
Then, for \(\lambda_{1}, \lambda_{2}\geq\sigma_{1}\), \((x,y)\in K\cap \partial\Omega_{i}\) (\(i=3, 4\)), and \(\|(x,y)\|=H_{i}\), we have
Similarly, \(\min_{t\in[\delta,1-\delta]}T_{2}(x,y)(t)\geq\frac {H_{i}}{2}\) (\(i=3, 4\)). This implies that \(\|T(x,y)\|\geq H_{i}=\|(x,y)\|\) for \((x,y)\in K\cap\partial\Omega_{i}\) (\(i=3, 4\)). Since \(w_{0}=h_{0}=w_{\infty}=h_{\infty}=0\), it follows from the proof of Theorem 1(a) and (b), respectively, we can choose \(H_{1}<\frac {H_{3}}{2}\) and \(H_{2}>2H_{4}\) such that \(\|T(x,y)\|\leq\|(x,y)\|\) for \((x,y)\in K\cap\partial\Omega_{i}\) (\(i=1, 2\)), where \(\Omega_{i}=\{ (x,y):(x,y)\in X\times X, \|(x,y)\|< H_{i}\}\) (\(i=1, 2\)).
Applying Theorem A to \(\Omega_{1}\), \(\Omega_{3}\) and \(\Omega_{2}\), \(\Omega_{4}\) we get a positive solution \((x_{1},y_{1})\) such that \(H_{1}\leq\|(x_{1},y_{1})\|\leq H_{3}\) and another positive solution \((x_{2},y_{2})\) such that \(H_{4}\leq\|(x_{1},y_{1})\|\leq H_{2}\). Since \(H_{3}< H_{4}\), these two solutions are distinct.
(b) For \((x,y)\in K\) and \(\|(x,y)\|=L\), let
Then \(M(L)>0\) for \(L>0\). Choose two numbers \(0< H_{3}< H_{4}\), let \(\sigma_{2}=\min\{\frac{H_{3}}{2M(H_{3})}, \frac{H_{4}}{2M(H_{4})}\} \) and set \(\Omega_{i}=\{(x,y):(x,y)\in X\times X, \|(x,y)\|< H_{i}\}\) (\(i=3, 4\)). Then, for \(\lambda_{1}, \lambda_{2}\leq\sigma_{2}\) and \((x,y)\in K\cap\partial\Omega_{i}\) (\(i=3, 4\)), \(\|(x,y)\|=H_{i}\), we have \(T_{1}(x,y)(t)\leq\lambda_{1}M(H_{i})\leq\frac{H_{i}}{2}\) (\(i=3, 4\)), and, \(T_{2}(x,y)(t)\leq\frac{H_{i}}{2}\) (\(i=3,4\)), which implies \(\| T(x,y)\|\leq H_{i}=\|(x,y)\|\) for \((x,y)\in K\cap\partial\Omega _{i}\) (\(i=3, 4\)). Since either \(w_{0}=\infty\) or \(h_{0}=\infty\), and either \(w_{\infty}=\infty\) or \(h_{\infty}=\infty\), it follows from the proof of Theorem 1(a) and (b), we can choose \(H_{1}<\frac {H_{3}}{2}\) and \(H_{2}>2H_{4}\) such that \(\|T(x,y)\|\geq\|(x,y)\|\) for \((x,y)\in K\cap\partial\Omega_{i}\) (\(i=1,2\)), where \(\Omega_{i}=\{ (x,y):(x,y)\in X\times X, \|(x,y)\|< H_{i}\}\) (\(i=1, 2\)).
Once again, we conclude to the existence of two distinct positive solutions. □
Theorem 3
Assume (A1) and (A2) hold.
-
(a)
If \(w_{0}=h_{0}=0\) or \(w_{\infty}=h_{\infty}=0\), then there is a positive constant \(\sigma_{3}\) such that (1) has at least two positive solutions for all \(\lambda_{1}, \lambda_{2}\geq\sigma_{3}\).
-
(b)
If \(w_{0}=\infty\) or \(h_{0}=\infty\), or if \(w_{\infty}=\infty\) or \(h_{\infty}=\infty\), then there is a positive constant \(\sigma _{4}\) such that the system (1) has at least two positive solutions for all \(\lambda_{1}, \lambda_{2}\leq\sigma_{4}\).
Example 1
Consider the system of fractional differential equation provided by
where for \(\forall x, y>0\), \(w(t,x(t),y(t))=[x(t)+y(t)]^{4}>0\), \(h(t,x(t),y(t))=[x(t)+y(t)]^{6}>0\), \(q_{1}=\frac{1}{2}\), \(q_{2}=\frac {2}{3}\in(0,1)\); \(p_{1}=\frac{4}{3}\), \(p_{2}=\frac{3}{2}\in(1,2)\); \(\gamma_{1}=\frac{9}{4}\), \(\gamma_{2}=\frac{5}{2}\in(2,3)\); \(\lambda _{1}=1\), \(\lambda_{2}=2\), \(\alpha_{1}=\frac{1}{2}\), \(\alpha_{2}=\frac {1}{3}\), \(\eta=\frac{1}{2}\), \(\xi=\frac{1}{3}\), and \(\alpha_{1}\eta ^{p-1}=\frac{1}{2^{p}}\), \(\alpha_{2}\xi^{p-1}=\frac{1}{3^{p}}\in (0,1)\). By direct calculation we obtain \(w_{0}=h_{0}=0\) and \(w_{\infty }=h_{\infty}=\infty\). Then, by Theorem 1(a), the system (4) has at least one positive solution.
Example 2
Consider the system of fractional differential equation provided by
where \(\forall x, y>0\), \(w(t,x(t),y(t))=\sqrt[4]{x(t)+y(t)}>0\), \(h(t,x(t),y(t))=\sqrt[3]{x(t)+y(t)}>0\), \(q_{1}=q_{1}=\frac{1}{3}\in (0,1)\); \(p_{1}=p_{2}=\frac{3}{2}\in(1,2)\); \(\gamma_{1}=\gamma_{2}=\frac {9}{4}\in(2,3)\); \(\lambda_{1}=\lambda_{2}=1\), \(\alpha_{1}=\alpha _{2}=1\), \(\eta=\xi=\frac{1}{2}\), and \(\alpha_{1}\eta^{p-1}=\alpha _{2}\xi^{p-1}=\frac{1}{2^{p-1}}\in(0,1)\). By direct calculation we see that \(w_{0}=h_{0}=\infty\) and \(w_{\infty}=h_{\infty}=0\). Then, by Theorem 1(b), the system (5) has at least one positive solution.
4 Conclusions
In this research, by using the Krasnosel’skii fixed point theorem for a cone map, we studied the existence and multiplicity of positive solutions for a class of systems of fractional differential equations with parameters, and we obtained the existence of at least one and two solutions for our considered system.
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Acknowledgements
The authors would like to thank very much the referees for their expert and constructive comments, which have further enriched our article. This work is supported by the National Natural Science Foundation of China (Nos. 11561063, 11401479, 71561024).
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Han, X., Yang, X. Existence and multiplicity of positive solutions for a system of fractional differential equation with parameters. Bound Value Probl 2017, 78 (2017). https://doi.org/10.1186/s13661-017-0808-7
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DOI: https://doi.org/10.1186/s13661-017-0808-7