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Existence and uniqueness of nontrivial solutions to a system of fractional differential equations with Riemann–Stieltjes integral conditions
Advances in Difference Equations volume 2018, Article number: 306 (2018)
Abstract
This paper studies a system of fractional differential equations with Riemann–Stieltjes integral conditions. The existence and uniqueness of nontrivial solutions to the above system are established under some weaker conditions by the Leray–Schauder topological degree. Two examples are set up to testify the validity of the main results.
1 Introduction
The purpose of this paper is to establish the existence and uniqueness of solutions for the following system of fractional differential equations with Riemann–Stieltjes integral boundary conditions (for short, FBVP):
where \(2<\alpha\leq3\) is a real number, \(D^{\alpha}_{0+}\) is the standard Riemann–Liouville differentiation, β is right continuous on \([0,1)\), left continuous at \(t=1\), and nondecreasing on \([0,1]\) with \(\beta(0)=0\), \(\int_{0}^{1}u(\tau)\,d\beta(\tau)\) denotes the Riemann–Stieltjes integral of u with respect to β. Here the nonlinear terms \(f,g:[0,1]\times(-\infty,+\infty)\to(-\infty,+\infty)\) are continuous sign-changing functions and f, g may be unbounded from below, \(h:(0,1)\to[0,+\infty)\) with \(0<\int_{0}^{1}h(s)\,ds<+\infty\) is continuous and is allowed to be singular at \(t=0,1\).
Fractional differential equations play an important role in many engineering and scientific disciplines such as physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, diffusive transport akin to diffusion, probability, electrical networks, etc. For details, see [1–3] and the references therein. By using a nonlinear alternative of Leray–Schauder theorem and Krasnoselskii’s fixed point theorem in a cone, Bai and Fang in [4] obtained the existence of positive solutions for the following singular coupled system of nonlinear fractional differential equations:
where \(0 < s < 1\), \(0 < p < 1\), \(D^{s}\), \(D^{p}\) are two standard Riemann–Liouville fractional derivatives, \(f , g : (0, 1] \times [0,+ \infty)\to[0,+\infty)\) are two given continuous functions. Su [5] established sufficient conditions for the existence of solutions for the following coupled system of fractional differential equations with two-point boundary conditions:
where \(1 <\alpha,\beta< 2\), \(\mu,\nu> 0\), \(\alpha-\nu\geq1\), \(\beta-\mu\geq1\), \(f , g: [0, 1] \times R\times R \to R\) are given functions, and D is the standard Riemann–Liouville fractional derivative. Ahmad and Nieto [6] extended the results of [5] to a three-point boundary value problem for the following coupled system of fractional differential equations:
where \(1 <\alpha,\beta< 2\), \(\mu,\nu,\gamma> 0\), \(\alpha-\nu\geq1\), \(\beta-\mu\geq1\), \(0<\eta<1\), \(\gamma\eta^{\alpha -1}<1\), \(\gamma\eta^{\beta-1}<1\) \(f , g: [0, 1] \times R\times R \to R\) are given functions, and D is the standard Riemann–Liouville fractional derivative. Yang [7] established sufficient conditions for the existence and nonexistence of positive solutions to boundary values problem for a coupled system of nonlinear fractional differential equations as follows:
where \(1 <\alpha,\beta\leq2\), \(a,b\in C((0,1),[0,+\infty))\), \(\phi, \psi\in L^{1}[0,1]\) are nonnegative and \(f,g\in C([0,1]\times[0,+\infty),[0,+\infty))\), and D is the standard Riemann–Liouville fractional derivative.
Inspired by the above papers and some known results on fractional differential equations with integral boundary conditions [8–30], this paper is to establish the existence and uniqueness of nontrivial solutions to FBVP (1.1) under the conditions that the nonlinear terms f, g of FBVP (1.1) are allowed to be sign-changing and unbounded from below. Finally, it is worth mentioning that the main technique used here is the topological degree theory, the theory of linear operators. As far as we know, there are few works that deal with system of fractional differential equations with Riemann–Stieltjes integral conditions where the nonlinear terms may be unbounded from below. The main results here are different from [4–32, 35–37].
2 Preliminaries and lemmas
Let \(E=C[0,1]\) be a Banach space with the norm \(\|u\|=\max_{0\leq t\leq1}|u(t)|\) for \(u\in E\). Let \(P= \{u\in E\mid u(t)\geq0, t\in[0,1] \}\). Then P is a total cone in E, that is, \(E=\overline{P-P}\). Let \(P^{*}= \{g\in E^{*}\mid g(u)\geq0 \mbox{ for all }u\in P \}\). Then \(P^{*}\) is the dual cone of P. Let \(E^{*}\) denote the dual space of E, then by Riesz representation theorem, \(E^{*}\) is given by
Let \(E^{2}\) be equipped with the norm \(\|(u,v)\|_{1}=\|u\|+\|v\|\). Then \(E^{2}\) is also a real Banach space and \(P^{2}=P\times P\) is a cone in \(E^{2}\). Let \((u_{1},v_{1})\geq(u_{2},v_{2})\) denote \(u_{1}\geq u_{2}\), \(v_{1}\geq v_{2}\) for \((u_{1},v_{1}), (u_{2},v_{2})\in E^{2}\) and \(B_{r}=\{(u,v)\in E ^{2}\mid \|(u,v)\|_{1}< r\}\) for any \(r>0\).
Definition 2.1
The Riemann–Liouville fractional integral of order \(\alpha>0\) of a function \(y : (0,+\infty) \to \mathbb{R}\) is given by
provided the right-hand side is defined on \((0,+\infty)\) pointwisely.
Definition 2.2
The Riemann–Liouville fractional derivative of order \(\alpha>0\) of a function \(y : (0,+\infty) \to\mathbb{R}\) is given by
where \(n=[\alpha]+1\), \([\alpha]\) denotes the integer part of the number α, provided that the right-hand side is defined on \((0,+\infty)\) pointwisely.
Lemma 2.1
Let \(\alpha> 0\). If we assume \(u \in C(0,1)\cap L(0,1)\), then the fractional differential equation
has \(u(t)=c_{1}t^{\alpha-1}+c_{2}t^{\alpha-2}+\cdots+ c_{N}t^{\alpha-N}\), \(c_{i}\in\mathbb{R}\), \(i=1,2,\ldots, N\), as unique solutions, where N is the smallest integer greater than or equal to α.
Lemma 2.2
Assume that \(u \in C(0,1)\cap L(0,1)\) with a fractional derivative of order \(\alpha> 0\) that belongs to \(C(0,1)\cap L(0,1)\). Then
for some \(c_{i}\in\mathbb{R}\), \(i=1,2,\ldots, N\), N is the smallest integer greater than or equal to α.
Lemma 2.3
([31])
Given \(y\in L(0,1)\) and \(2<\alpha\leq3\), then the unique solution of
is \(u(t)=\int_{0}^{1} G_{0}(t,s)y(s)\,ds\), where
Lemma 2.4
([31])
The Green function \(G_{0}(t,s)\) has the following properties:
-
(1)
\(\Gamma(\alpha)k(t)q(s)\leq G_{0}(t,s)\leq(\alpha-1)q(s)\) for \(t,s\in[0,1]\),
-
(2)
\(\Gamma(\alpha)k(t)q(s)\leq G_{0}(t,s)\leq(\alpha-1)k(t)\) for \(t,s\in[0,1]\),
where
By Lemma 2.1, the unique solution of the problem
is \(u(t)=t^{\alpha-1}\). Then it is easy to verify, as a consequence of Lemma 2.3, that FBVP (1.1) is equivalent to the system of perturbed integral equations
Define \(\Gamma=\int_{0}^{1} t^{\alpha-1}\,d\beta(t)\), \(g_{\beta}(s)=\int_{0}^{1}G_{0}(t,s)\,d\beta(t)\). Then we have the following lemma.
Lemma 2.5
Given \(y(t)\in C(0,1)\cap L(0,1)\) and \(2<\alpha\leq3\), then
has the unique solution
where the Green function \(G(t,s)\) is given by
Proof
Multiply (2.1) by \(d\beta(t)\) on both sides and integrate over \([0,1]\) to obtain
Consequently,
Replacing \(\int_{0}^{1} u(\tau)\,d\beta(\tau)\) in (2.1) with the above equality, we obtain
Reversely, if \(u(t)=\int_{0}^{1}G(t,s)y(s)\,ds\), then \(u(0)=0\) and \(u(1)=\int_{0}^{1} u(\tau)\,d\beta(\tau)\) via (2.1). According to Definition 2.2, Lemma 2.3, and Lemma 2.4, \(D^{\alpha}_{0+}u(t)+y(t)=0\) holds. □
By Lemma 2.5, \((u,v)\in E^{2}\) is a solution of FBVP (1.1) if and only if
Define
It is easy to show that \(A:E^{2}\to E^{2}\) is a completely continuous nonlinear operator, and if \((u,v)\in E^{2}\) is a fixed point of A, then \((u,v)\) is a solution of FBVP (1.1) by Lemma 2.5.
For any \(u\in E\), define \(K:E\to E\) as follows:
Then \(K:E\to E\) is a completely continuous linear operator and \(K(P)\subset P\) holds. Since \(h\in C(0,1)\cap L(0,1)\) with \(\int_{0}^{1} h(t)\,dt>0\), by [32], the spectral radius \(r(K)\) of K is positive. The Krein–Rutman theorem [33] asserts that there exist \(\phi\in P\setminus\{0\}\) and \(\omega\in P^{*}\setminus\{0\}\) corresponding to the number \(\lambda_{1}=1/r(K)\) relative to K such that
and
Here \(K^{*}:E^{*}\to E^{*}\) is the dual operator of K given by
Now we testify that \(K^{*}:E^{*}\to E^{*}\) is the dual operator of K. In fact,
So \(K^{*}:E^{*}\to E^{*}\) is the dual operator of K.
The continuity of G, the integrability of h, and the representation of \(K^{*}\) induce that \(\omega\in C^{1}[0, 1]\). Let \(e(t):= \omega'(t)\). Then \(e\in P \setminus\{0\}\), and (2.5) can be rewritten equivalently as
Lemma 2.6
Let \(0\leq\Gamma=\int_{0}^{1} t^{\alpha-1}\,d\beta(t)<1\) and \(g_{\beta}(s)=\int_{0}^{1}G_{0}(t,s)\,d\beta(t)\geq0\) for \(s\in[0,1]\), then there exists \(\delta>0\) such that \(P_{0}=\{u\in P\mid \int_{0}^{1}u(t)e(t)\,dt\ge\delta\|u\|\}\) is a subcone of P and \(K(P)\subset P_{0}\).
Proof
Let \(\delta=\int_{0}^{1} \frac{(1-t)t^{\alpha-1}}{\alpha-1}e(t)\,dt\). It is obvious that \(G(t,s)>0\) holds for \(t,s\in(0,1)\). By Lemma 2.4 and (2.2),
and
For any \(u\in P\),
Then we have
That means \(K(P)\subset P_{0}\). □
Lemma 2.7
([34])
Let E be a real Banach space and \(\Omega\subset E \) be a bounded open set with \(0\in\Omega\). Suppose that \(A: \bar{\Omega}\to E\) is a completely continuous operator. (1) If there is \(y_{0}\in E\) with \(y_{0} \neq0\) such that \(u\neq Au + \mu y_{0}\), for all \(u\in\partial\Omega\) and \(\mu\geq0\), then \(\operatorname{deg}(I-A,\Omega,0) = 0\). (2) If \(Au\neq\mu u\) for all \(u\in \partial\Omega\) and \(\mu\geq1\), then \(\operatorname{deg}(I-A,\Omega,0) = 1 \). Here deg stands for the Leray–Schauder topological degree in E.
Lemma 2.8
Assume that the following assumptions are satisfied:
- (C1):
-
There exist \(\phi\in P\setminus\{0\}\), \(\omega\in P^{*}\setminus\{0\}\) such that (2.4), (2.5) hold and K maps P into \(P_{0}\).
- (C2):
-
There exists a continuous operator \(H:E\to P \) such that
$$\lim_{ \Vert u \Vert + \Vert v \Vert \to+\infty}\frac{ \Vert Hu \Vert + \Vert Hv \Vert }{ \Vert u \Vert + \Vert v \Vert }=0. $$ - (C3):
-
There exist two bounded continuous operators \(F,G:E\to E\) and \(u_{0},v_{0}\in E\) such that \((Fv+v_{0}+Hv, Gu+u_{0}+Hu)\in P^{2}\) for all \((u, v)\in E^{2}\).
- (C4):
-
There exist \(m_{0},n_{0}\in E\) and \(\zeta>0\) such that \((KFv,KGu)\geq(\lambda_{1}(1+\zeta)Kv-KHv-m_{0}, \lambda_{1}(1+\zeta)Ku-KHu-n_{0})\) for all \((u, v)\in E^{2}\).
Let \(A_{1}=KF\), \(A_{2}=KG\), \(A(u,v)(t)=((A_{1}v)(t), (A_{2}u)(t))\), then there exists \(R>0\) such that
where \(B_{r}= \{(u, v)\in E^{2}\mid\|(u,v)\|_{1}< r \}\) for any \(r>0\).
Proof
Choose a constant \(l_{0}=(\delta\lambda_{1})^{-1}(1+\zeta^{-1})+\|K\|>0\). By (C2), for \(0<\varepsilon_{0}<l_{0}^{-1}\), there exists \(R_{1}>0\) such that \(\|u\|+\|v\|>R_{1}\) implies
Now we shall show
provided that R is sufficiently large.
In fact, if (2.8) is not true, then there exist \((u_{1},v_{1})\in \partial B_{R}\) and \(\mu_{1}\geq0\) satisfying
that is,
Since \(\phi\in P\setminus\{0\}\), \(e(t)\in P\setminus\{0\}\), \(\int_{0}^{1}\phi(t)e(t)\,dt>0\). Multiply (2.10) by \(e(t)\) on both sides and integrate respectively on \([0,1]\). Then by (C4), (2.6), we get
and
According to the two above inequalities, we obtain
Then we derive
By computation, we obtain
By (2.6), (2.7), (2.11), and (2.12), we get
where \(l_{1}=\zeta^{-1}\int_{0}^{1}[m_{0}(t)+n_{0}(t)]e(t)\,dt +\int_{0}^{1}[(Ku_{0})(t)+(Kv_{0})(t)]e(t)\,dt\) is a constant.
(C3) shows \((Fv_{1}+v_{0}+Hv_{1}, Gu_{1}+u_{0}+Hu_{1})\in P^{2}\) and (C1) implies \(\mu_{1}\phi=\mu_{1}\lambda_{1}K\phi\in P_{0}\). Then (C1), (2.10), and Lemma 2.6 tell us that
The definition of \(P_{0}\) yields
where δ is given in Lemma 2.6. By adding the above two inequalities, we obtain
It follows from (2.7), (2.13), and (2.14) that
where \(l_{2}=l_{1}\delta^{-1}+\|Ku_{0}\|+\|Kv_{0}\|\) is a constant.
Since \(0<\varepsilon_{0} l_{0}<1\), then (2.15) deduces that (2.8) holds provided that R is sufficiently large such that \(R>\max\{l_{2}/(1-\varepsilon_{0} l_{0} ),R_{1}\}\). By (2.15) and Lemma 2.7, we have
□
3 Existence
Theorem 3.1
Assume that the following conditions are satisfied:
- (A1):
-
\(f,g:[0,1]\times\mathbb{R}\to\mathbb{R}\) are continuous.
- (A2):
-
There exist nonnegative functions \(b_{i}(t),c_{i}(t)\in C[0,1]\) with \(c_{i}(t)\not\equiv0\) and two continuous even functions \(B_{i}:\mathbb{R}\to\mathbb{R}^{+}\) such that
$$ \begin{aligned} &f(t,x)\geq-b_{1}(t)-c_{1}(t)B_{1}(x) \quad \textit{for all } x\in\mathbb{R}, \\ &g(t,y)\geq-b_{2}(t)-c_{2}(t)B_{2}(y) \quad \textit{for all } y\in\mathbb{R}. \end{aligned} $$Moreover, \(B_{i}\) is nondecreasing on \(\mathbb{R}^{+}\) and satisfies \(\lim_{x\to+\infty}\frac{B_{i}(x)}{x}=0\), (\(i=1,2\)).
- (A3):
-
\(\liminf_{x\to+\infty}\frac{f(t,x)}{x}>\lambda_{1}\), \(\liminf_{x\to+\infty}\frac{g(t,y)}{y}>\lambda_{1}\), uniformly on \(t\in[0,1]\).
- (A4):
-
\(\limsup_{x\to 0} \vert \frac{f(t,x)}{x} \vert <\lambda_{1}\), \(\limsup_{x\to 0} \vert \frac{g(t,y)}{y} \vert <\lambda_{1}\), uniformly on \(t\in[0,1]\).
Here \(\lambda_{1}=1/r(K)\) is a number and the operator K is defined by (2.3).
Then FBVP (1.1) has at least one nontrivial solution.
Proof
We first show that all conditions in Lemma 2.8 are satisfied. By Lemma 2.6, condition (C1) of Lemma 2.8 is satisfied. \((T_{i}u)(t)=B_{i}(u(t))\) (\(i=1,2\)) for any \(u\in E\). Obviously \(T_{1},T_{2}:E\to P\) are continuous operators. By (A2), for any \(\varepsilon>0\), there is \(L>0\) such that when \(z>L\), \(B_{i}(z)<\varepsilon z\) holds. Thus, for \(w\in E \) with \(\|w\|>L\), \(B_{i}(\|w\|)<\varepsilon\|w\|\) holds. The fact that \(B_{i}\) is nondecreasing on \(\mathbb{R}^{+}\) yields \((T_{i}w)(t)\leq T_{i}(\|w\|)\) for any \(w\in P\), \(t\in[0,1]\). Since \(B_{i}:\mathbb{R} \to\mathbb{R}^{+}\) is an even function, \(\|T_{i}w\|\leq T_{i}(\|w\|)\) holds for \(w\in E\). Therefore,
that is, \(\lim_{\|w\| \to+\infty}\frac{\|T_{i}w\|}{\|w\|}=0\).
Define \((Hw)(t)=\max\{C_{1}(T_{1}w)(t),C_{2}(T_{2}w)(t)\}\) for any \(w\in E\), \(t\in[0,1]\), where \(C_{i}=\max_{t\in[0,1]}c_{i}(t)\), \(i=1,2\). By \(\lim_{\|w\| \to+\infty}\frac{\|T_{i}w\|}{\|w\|}=0\), \(\lim_{\|u\|+\|v\| \to +\infty}\frac{\|T_{i}u\|+\|T_{i}v\|}{\|u\|+\|v\|}=0\) holds. Therefore \(\lim_{\|u\|+\|v\| \to+\infty}\frac{\|Hu\|+\|Hv\|}{\|u\|+\|v\|}=0\) holds. Then we obtain that H satisfies condition (C2) in Lemma 2.8.
Take \(v_{0}(t)\equiv b_{1}=\max_{t\in[0,1]}b_{1}(t)>0\), \(u_{0}(t)\equiv b_{2}=\max_{t\in[0,1]}b_{2}(t)>0\), and \((Fv)(t)=f(t,v(t))\), \((Gu)(t)=g(t,u(t))\) for \(t\in[0,1]\), \((u,v)\in E^{2}\), then it follows from (A1) that
which shows that condition (C3) in Lemma 2.8 holds.
By (A3), there exist \(\varepsilon_{1}>0\) and a sufficiently large number \(L_{1}>0\) such that
Combining (3.1) with (A2), the above constants \(b_{1}\), \(b_{2}\) satisfy
and so
Since K is a positive linear operator, from (3.2), we have
Let \(m_{0}(t)=(Kb_{1})(t)\), \(n_{0}(t)=(Kb_{2})(t)\). Then condition (C4) in Lemma 2.8 is satisfied.
According to Lemma 2.8, we derive that there exists a sufficiently large number \(R>0\) such that
From (A4), it follows that there exist \(0<\varepsilon_{2}<1\) and \(0< r< R\) such that
Thus
Next we will prove that
If there exist \((u_{1},v_{1})\in\partial B_{r}\) and \(\mu_{1}\in[0,1]\) such that \((u_{1},v_{1})\neq\mu A(u_{1},v_{1})\), that is,
Let \(z(t)=|u_{1}(t)|+|v_{1}(t)|\). Then \(z\in P\) and by (3.4),
The nth iteration of this inequality shows that \(z(t)\leq(1-\varepsilon_{2})^{n}\lambda_{1}^{n} (K^{n}z)(t)\) (\(n=1,2,\ldots\)), so \(\|z\|\leq (1-\varepsilon_{2})^{n}\lambda_{1}^{n}\|K^{n}\|\cdot\|z\|\), that is, \(1\leq (1-\varepsilon_{2})^{n}\lambda_{1}^{n}\|K^{n}\|\). This yields \(1-\varepsilon_{2}=(1-\varepsilon_{2})\lambda_{1}r(K)=(1-\varepsilon_{2})\lambda _{1}\lim_{n\to \infty}\sqrt[n]{\|K^{n}\|}\geq1\), which is a contradictory inequality. Hence, (3.5) holds.
It follows from (3.5) and Lemma 2.7 that
By (3.3), (3.6), and the additivity of Leray–Schauder degree, we obtain
So A has at least one fixed point on \(B_{R}\setminus\overline{B}_{r}\), namely FBVP (1.1) has at least one nontrivial solution. □
4 Uniqueness
Theorem 4.1
Assume that (A1)–(A4) are satisfied. Moreover, the following conditions are satisfied:
- (A5):
-
\(0<\int_{0}^{1} g_{\beta}(s)h(s)\,ds<+\infty\) and there exists a constant \(k<[\int_{0}^{1}(\frac{s(1-s)^{\alpha-1}}{\Gamma(\alpha-1)}+\frac{g_{\beta}(s)}{1-\Gamma} )h(s)\,ds]^{-1}\) such that
$$\bigl\vert f(t,x)-f(t,y) \bigr\vert \leq k \vert x-y \vert ,\qquad \bigl\vert g(t,x)-g(t,y) \bigr\vert \leq k \vert x-y \vert \quad \textit{for any } x,y\in\mathbb{R}. $$
Then FBVP (1.1) has a unique solution.
Proof
It follows from \(|f(t,x)-f(t,y)|\leq k|x-y|\), \(|g(t,x)-g(t,y)|\leq k|x-y|\) for any \(x,y\in\mathbb{R}\) that (A1) holds. Then by Theorem 3.1, FBVP (1.1) has at least one nontrivial solution. Suppose that FBVP (1.1) has two different solutions \((u_{1}(t),v_{1}(t))\) and \((u_{2}(t),v_{2}(t))\). By Lemma 2.6, \(G(t,s)\leq \frac{s(1-s)^{\alpha-1}}{\Gamma(\alpha-1)}+\frac{g_{\beta}(s)}{1-\Gamma}\). Then from (A5) it follows that
By adding the above two inequalities, we obtain \(\|v_{1}-v_{2}\|+\|u_{1}-u_{2}\|<\|v_{1}-v_{2}\|+\|u_{1}-u_{2}\|\), which is a contradictory inequality. Therefore \((u_{1}(t),v_{1}(t))=(u_{2}(t),v_{2}(t))\) and FBVP (1.1) has a unique solution. □
5 Examples
Example 5.1
Consider FBVP (1.1) with
\(h(t)=\frac{1}{\sqrt{t(1-t)}}\) and
where \(0< a_{1}<\lambda_{1}\), \(a_{n}>0\). Obviously, \(\Gamma=\int_{0}^{1}t^{\alpha-1}\,d\beta(t)=\frac{1}{8}(\frac{1}{3})^{\alpha-1} +\frac{3}{8}(\frac{2}{3})^{\alpha-1}<\frac{7}{24}<1\). Then h is singular at \(t=0,1\) and f, g are unbounded from below. Take \(c_{1}(t)=1+t^{2}\), \(c_{2}(t)=1+t^{4}\), \(b_{1}(t)=\sum_{i=1}^{n}a_{i}+(1+t^{2})\ln2\), \(b_{2}(t)=\sum_{i=1}^{n}a_{i}+(1+t^{4})\ln2\), \(B_{1}(x)=|x|^{\frac{1}{3}}\ln(|x|+1)\), \(B_{2}(x)=|x|^{\frac{1}{3}}\ln(|x|^{\frac{1}{3}}+1)\). Then all the conditions in Theorem 3.1 are satisfied. Therefore, FBVP (1.1) with the above \(\beta(t)\), \(h(t)\), \(f(t,x)\), \(g(t,y)\) has at least one nontrivial solution.
Example 5.2
Consider FBVP (1.1) with
\(h(t)=\frac{\Gamma(\alpha-1)}{2(5a_{2}+2)\sqrt{t}(1-t)^{\alpha-1}}\), where \(0< a_{1}<\lambda_{1}<a_{2}\).
Take \(c_{i}(t)=1+t^{2}\), \(b_{i}(t)=a_{1}+(1+t^{2})\ln2\), \(B_{i}(x)=\ln(|x|+1)\), \(i=1,2\). Then (A2) is satisfied. The choice of \(a_{1}\), \(a_{2}\) guarantees that (A3) and (A4) are satisfied. By some simple computation, we obtain that \(\Gamma=\int_{0}^{1}t^{\alpha-1}\,d\beta(t) =\frac{1}{3^{\alpha}}<1\), \(|f(t,x)-f(t,y)|\leq (a_{1}+\frac{3}{2}a_{2}+1)|x-y|\), \(|g(t,x)-g(t,y)|\leq (a_{1}+\frac{3}{2}a_{2}+1)|x-y|\) for any \(x,y\in\mathbb{R}\) and \(\int_{0}^{1}(\frac{s(1-s)^{\alpha-1}}{\Gamma(\alpha-1)}+\frac{g_{\beta}(s)}{1-\Gamma} )h(s)\,ds<\frac{4}{3(5a_{2}+2)}\). Hence (A5) holds. So FBVP (1.1) with the above \(\beta(t)\), \(h(t)\), \(f(t,x)\), \(g(t,y)\) has a unique solution.
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This paper is supported by the National Natural Science Foundation of China (11701252) and the Natural Science Foundation of Shangdong Province of China (ZR2017MA036).
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Liu, B., Li, J., Liu, L. et al. Existence and uniqueness of nontrivial solutions to a system of fractional differential equations with Riemann–Stieltjes integral conditions. Adv Differ Equ 2018, 306 (2018). https://doi.org/10.1186/s13662-018-1762-0
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DOI: https://doi.org/10.1186/s13662-018-1762-0
MSC
- 34B15
- 47H10
- 47H11
Keywords
- Fractional differential equations
- Riemann–Stieltjes integral
- Topological degree
- Uniqueness