Skip to main content

Theory and Modern Applications

Existence and uniqueness of nontrivial solutions to a system of fractional differential equations with Riemann–Stieltjes integral conditions

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):

$$ \textstyle\begin{cases} D^{\alpha}_{0+}u(t)+h(t)f(t,v(t))=0,\quad 0< t< 1, \\ D^{\alpha}_{0+}v(t)+h(t)g(t,u(t))=0,\quad 0< t< 1, \\ u(0)=u'(0)=0, \qquad u(1)=\int_{0}^{1}u(\tau)\,d\beta(\tau),\\ v(0)=v'(0)=0, \qquad v(1)=\int_{0}^{1}v(\tau)\,d\beta(\tau), \end{cases} $$
(1.1)

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 [13] 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:

$$ \textstyle\begin{cases} D^{s} u= f(t,v),& 0< t< 1,\\ D^{p} v=g(t,u),& 0< t< 1, \end{cases} $$

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:

$$ \textstyle\begin{cases} D^{\alpha}u(t)=f (t, v(t), D^{\mu}v(t)),\quad 0< t< 1,\\ D^{\beta}v(t)=g (t, u(t), D^{\nu}u(t)),\quad 0< t< 1, \\ u(0)=u(1)=v(0)= v(1)=0, \end{cases} $$

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:

$$ \textstyle\begin{cases} D^{\alpha}u(t)=f (t, v(t), D^{\mu}v(t)),\quad0< t< 1, \\ D^{\beta}v(t)=g (t, u(t), D^{\nu}u(t)),\quad0< t< 1, \\ u(0)=0,\qquad u(1)=\gamma u(\eta),\\ v(0)=0,\qquad v(1)=\gamma v(\eta), \end{cases} $$

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:

$$ \textstyle\begin{cases} D^{\alpha}u(t)+a(t)f (t, v(t))=0,\quad0< t< 1, \\ D^{\beta}v(t)+b(t)g (t, u(t))=0,\quad0< t< 1, \\ u(0)=0,\qquad u(1)=\int_{0}^{1}\phi(t)u(t)\,dt,\\ v(0)=0, \qquad v(1)=\int_{0}^{1}\psi(t)v(t)\,dt, \end{cases} $$

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 [830], 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 [432, 3537].

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

$$\begin{aligned} E^{*}&= \bigl\{ \nu\mid\nu\mbox{ is right continuous on } [0,1) \mbox{ and is bounded variation on } \\ &\quad [0,1] \mbox{ with } \nu(0)=0 \bigr\} . \end{aligned} $$

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

$$I^{\alpha}_{0+}y(t)=\frac{1}{\Gamma(\alpha)} \int_{0}^{t} (t-s)^{\alpha-1}y(s)\,ds, $$

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

$$D^{\alpha}_{0+}y(t)=\frac{1}{\Gamma(n-\alpha)} \biggl(\frac{d}{dt} \biggr)^{n} \int_{0}^{t} \frac{y(s)}{(t-s)^{\alpha-n+1}}\,ds, $$

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

$$D^{\alpha}_{0+}u(t)=0 $$

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

$$I^{\alpha}_{0+}D^{\alpha}_{0+}u(t)=u(t)+c_{1}t^{\alpha-1}+c_{2}t^{\alpha -2}+ \cdots+ c_{N}t^{\alpha-N} $$

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

$$ \textstyle\begin{cases} D^{\alpha}_{0+}u(t)+y(t)=0,\quad0< t< 1,\\ u(0)=u'(0)=0,\quad u(1)=0, \end{cases} $$

is \(u(t)=\int_{0}^{1} G_{0}(t,s)y(s)\,ds\), where

$$G_{0}(t,s)= \textstyle\begin{cases} \frac{[t(1-s)]^{\alpha-1}-(t-s)^{\alpha-1}}{\Gamma(\alpha)},&0\leq s\leq t\leq1,\\ \frac{[t(1-s)]^{\alpha-1}}{\Gamma(\alpha)},&0\leq t\leq s\leq1. \end{cases} $$

Lemma 2.4

([31])

The Green function \(G_{0}(t,s)\) has the following properties:

  1. (1)

    \(\Gamma(\alpha)k(t)q(s)\leq G_{0}(t,s)\leq(\alpha-1)q(s)\) for \(t,s\in[0,1]\),

  2. (2)

    \(\Gamma(\alpha)k(t)q(s)\leq G_{0}(t,s)\leq(\alpha-1)k(t)\) for \(t,s\in[0,1]\),

where

$$k(t)=\frac{t^{\alpha-1}(1-t)}{\Gamma(\alpha)},\qquad q(s)=\frac{s(1-s)^{\alpha-1}}{\Gamma(\alpha)}. $$

By Lemma 2.1, the unique solution of the problem

$$ \textstyle\begin{cases} D^{\alpha}_{0+}u(t)=0,\quad0< t< 1,\\ u(0)=u'(0)=0,\qquad u(1)=1, \end{cases} $$

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

$$ \textstyle\begin{cases} u(t)=\int_{0}^{1}G_{0}(t,s)h(s)f(s,v(s))\,ds+t^{\alpha-1}\int_{0}^{1} u(\tau)\,d\beta (\tau),\\ v(t)=\int_{0}^{1}G_{0}(t,s)h(s)g(s,u(s))\,ds+t^{\alpha-1}\int_{0}^{1} v(\tau)\,d\beta (\tau). \end{cases} $$
(2.1)

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

$$ \textstyle\begin{cases} D^{\alpha}_{0+}u(t)+y(t)=0,\quad0< t< 1,\\ u(0)=u'(0)=0,\qquad u(1)=\int_{0}^{1}u(\tau)\,d\beta(\tau), \end{cases} $$

has the unique solution

$$u(t)= \int_{0}^{1} G(t,s)y(s)\,ds, $$

where the Green function \(G(t,s)\) is given by

$$ G(t,s)=\frac{t^{\alpha-1}}{1-\Gamma}g_{\beta}(s)+G_{0}(t,s), \quad t,s\in[0,1]. $$
(2.2)

Proof

Multiply (2.1) by \(d\beta(t)\) on both sides and integrate over \([0,1]\) to obtain

$$ \begin{aligned} \int_{0}^{1}u(t)\,d\beta(t)&= \int_{0}^{1} \int_{0}^{1}G_{0}(t,s)y(s)\,ds\,d \beta(t)+ \int _{0}^{1}t^{\alpha-1} \int_{0}^{1}u(\tau)\,d\beta(\tau)\,d\beta(t) \\ &= \int_{0}^{1} \int_{0}^{1}G_{0}(t,s)y(s)\,ds\,d \beta(t)+ \int_{0}^{1}t^{\alpha-1}\,d\beta (t) \int_{0}^{1}u(\tau)\,d\beta(\tau). \end{aligned} $$

Consequently,

$$ \int_{0}^{1}u(t)\,d\beta(t)=\frac{1}{1-\Gamma} \int_{0}^{1} \int _{0}^{1}G_{0}(t,s)y(s)\,ds\,d \beta(t). $$

Replacing \(\int_{0}^{1} u(\tau)\,d\beta(\tau)\) in (2.1) with the above equality, we obtain

$$ \begin{aligned} u(t)&= \int_{0}^{1}G_{0}(t,s)y(s)\,ds+ \frac{t^{\alpha-1}}{1-\Gamma} \int_{0}^{1} \biggl( \int_{0}^{1} G_{0}(t,s)\,d\beta(t) \biggr)y(s)\,ds \\ &= \int_{0}^{1} \biggl(G_{0}(t,s)+ \frac{t^{\alpha-1}}{1-\Gamma}g_{\beta}(s) \biggr)y(s)\,ds \\ &= \int_{0}^{1} G(t,s)y(s)\,ds. \end{aligned} $$

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

$$ \textstyle\begin{cases} u(t)=\int_{0}^{1}G(t,s)h(s)f(s,v(s))\,ds,\\ v(t)=\int_{0}^{1}G(t,s)h(s)g(s,u(s))\,ds. \end{cases} $$

Define

$$\begin{aligned}& (A_{1}v) (t)= \int_{0}^{1}G(t,s)h(s)f \bigl(s,v(s) \bigr)\,ds, \\& (A_{2}u) (t)= \int_{0}^{1}G(t,s)h(s)g \bigl(s,u(s) \bigr)\,ds, \\& A(u,v) (t)= \bigl((A_{1}v) (t),(A_{2}u) (t) \bigr). \end{aligned}$$

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:

$$ (Ku) (t)= \int_{0}^{1}G(t,s)h(s) u(s) \,ds,\quad u\in E. $$
(2.3)

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

$$ \lambda_{1}K\phi=\phi $$
(2.4)

and

$$ \lambda_{1}K^{*}\omega=\omega,\quad\omega(1)=1. $$
(2.5)

Here \(K^{*}:E^{*}\to E^{*}\) is the dual operator of K given by

$$ \bigl(K^{*}v \bigr) (s)= \int_{0}^{s} \int_{0}^{1}G(t,\tau)h(\tau)\,dv(t)\,d\tau, \quad v \in E^{*}. $$

Now we testify that \(K^{*}:E^{*}\to E^{*}\) is the dual operator of K. In fact,

$$\begin{aligned} \bigl\langle K^{*}v(s), u(s)\bigr\rangle &= \int_{0}^{1} u(s)\,dK^{*}v(s)= \int_{0}^{1}u(s) \int_{0}^{1} G(t,s)h(s)\,dv(t)\,ds \\ &= \int_{0}^{1} \biggl( \int_{0}^{1}G(t,s)h(s)u(s)\,ds \biggr)\,dv(t) \\ &= \int_{0}^{1}(Ku) (t)\,dv(t)=\bigl\langle v(t),(Ku) (t)\bigr\rangle . \end{aligned} $$

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

$$ r(K)e(s) = \int_{0}^{1}G(t,s)h(s)e(t)\,dt, \quad \int_{0}^{1} e(t)\,dt=1. $$
(2.6)

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),

$$ G(t,s)=G_{0}(t,s)+\frac{t^{\alpha-1}}{1-\Gamma}g_{\beta}(s)\leq G_{0}(t,s)+\frac{1}{1-\Gamma}g_{\beta}(s)\leq \frac{s(1-s)^{\alpha-1}}{\Gamma(\alpha-1)}+\frac{g_{\beta}(s)}{1-\Gamma} $$

and

$$ \begin{aligned} G(t,s)&=G_{0}(t,s)+\frac{t^{\alpha-1}}{1-\Gamma}g_{\beta}(s)\geq \frac{(1-t)s[t(1-s)]^{\alpha-1}}{(\alpha-1)\Gamma(\alpha-1)}+\frac {(1-t)t^{\alpha-1}g_{\beta}(s)}{(\alpha-1)(1-\Gamma)}\\ & =\frac{(1-t)t^{\alpha-1}}{\alpha-1} \biggl( \frac{s(1-s)^{\alpha-1}}{\Gamma (\alpha-1)}+\frac{g_{\beta}(s)}{1-\Gamma} \biggr). \end{aligned} $$

For any \(u\in P\),

$$ \max_{t\in [0,1]} \bigl\vert (Ku) (t) \bigr\vert =\max _{t\in[0,1]} \biggl\vert \int_{0}^{1}G(t,s)h(s)u(s)\,ds \biggr\vert \leq \int_{0}^{1} \biggl(\frac{s(1-s)^{\alpha-1}}{\Gamma(\alpha-1)}+ \frac{g_{\beta}(s)}{1-\Gamma} \biggr)h(s)u(s)\,ds. $$

Then we have

$$ \begin{aligned} \int_{0}^{1} (Ku) (t)e(t)\,dt&= \int_{0}^{1} \int_{0}^{1}G(t,s)h(s)u(s)\,ds e(t)\,dt \\ &\geq \int_{0}^{1} \int_{0}^{1}\frac{(1-t)t^{\alpha-1}}{\alpha-1} \biggl( \frac {s(1-s)^{\alpha-1}}{\Gamma(\alpha-1)}+\frac{g_{\beta}(s)}{1-\Gamma} \biggr)h(s)u(s)\,ds e(t)\,dt \\ &= \int_{0}^{1}\frac{(1-t)t^{\alpha-1}}{\alpha-1}e(t)\,dt\cdot \int_{0}^{1} \biggl(\frac{s(1-s)^{\alpha-1}}{\Gamma(\alpha-1)}+ \frac{g_{\beta}(s)}{1-\Gamma} \biggr)h(s)u(s)\,ds \\ &\geq \int_{0}^{1}\frac{(1-t)t^{\alpha-1}}{\alpha-1}e(t)\,dt\cdot \Vert Ku \Vert \\ &=\delta \Vert Ku \Vert . \end{aligned} $$

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

$$\operatorname{deg}(I-A,B_{R},0)=0, $$

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

$$ \Vert Hu \Vert + \Vert Hv \Vert < \varepsilon_{0} \bigl( \Vert u \Vert + \Vert v \Vert \bigr). $$
(2.7)

Now we shall show

$$ (u,v)\neq A(u,v)+\mu(\phi,\phi)\quad\mbox{for any } (u,v)\in \partial B_{R} \mbox{ and } \mu\geq0, $$
(2.8)

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

$$ (u,v)=A(u,v)+\mu(\phi,\phi), $$
(2.9)

that is,

$$ (u_{1},v_{1})=(KFv_{1}+ \mu_{1} \phi, KGu_{1}+\mu_{1} \phi). $$
(2.10)

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

$$ \begin{aligned} \int_{0}^{1} u_{1}(t)e(t)\,dt&= \int_{0}^{1}(KFv_{1}) (t)e(t)\,dt+ \mu_{1} \int_{0}^{1}\phi (t)e(t)\,dt \\ &\geq \lambda_{1} (1+\zeta) \int_{0}^{1} \int_{0}^{1}G(t,s)h(s)v_{1}(s)dse(t)\,dt\\ &\quad {}- \int _{0}^{1}(KHv_{1}) (t)e(t)\,dt- \int_{0}^{1}m_{0}(t)e(t)\,dt \\ &=\lambda_{1}(1+\zeta) \int_{0}^{1} \int_{0}^{1}G(t,s)h(s)v_{1}(s)e(t)\,ds \,dt\\ &\quad {}- \int _{0}^{1} \int_{0}^{1}G(t,s)h(s) (Hv_{1}) (s)e(t) \,ds\,dt- \int_{0}^{1}m_{0}(t)e(t)\,dt \\ &=\lambda_{1}(1+\zeta) \int_{0}^{1} \biggl[ \int_{0}^{1}G(t,s)h(s)e(t)\,dt \biggr]v_{1}(s) \,ds\\ &\quad {}- \int_{0}^{1} \biggl[ \int_{0}^{1}G(t,s)h(s)e(t)\,dt \biggr](Hv_{1}) (s)\,ds- \int _{0}^{1}m_{0}(t)e(t)\,dt \\ &=\lambda_{1}(1+\zeta)r(K) \int_{0}^{1}e(s)v_{1}(s)\,ds\\ &\quad {}-r(K) \int _{0}^{1}(Hv_{1}) (s)e(s)\,ds- \int_{0}^{1}m_{0}(t)e(t)\,dt \\ &=(1+\zeta) \int_{0}^{1}v_{1}(t)e(t)\,dt\\ &\quad {}-r(K) \int_{0}^{1}(Hv_{1}) (t)e(t) \,dt- \int_{0}^{1}m_{0}(t)e(t)\,dt, \end{aligned} $$

and

$$ \begin{aligned} \int_{0}^{1}v_{1}(t)e(t)\,dt&= \int_{0}^{1}(KGu_{1}) (t)e(t)\,dt+ \mu_{1} \int _{0}^{1}\phi(t)e(t)\,dt \\ &\geq(1+\zeta) \int_{0}^{1}u_{1}(t)e(t)\,dt-r(K) \int_{0}^{1}(Hu_{1}) (t)e(t)\,dt- \int _{0}^{1}n_{0}(t)e(t)\,dt. \end{aligned} $$

According to the two above inequalities, we obtain

$$ \begin{aligned} \int_{0}^{1} \bigl[u_{1}(t)+v_{1}(t) \bigr]e(t)\,dt&\geq (1+\zeta) \int_{0}^{1} \bigl[u_{1}(t)+v_{1}(t) \bigr]e(t)\,dt\\ &\quad {}-r(K) \int_{0}^{1} \bigl[(Hu_{1}) (t)+(Hv_{1}) (t) \bigr]e(t)\,dt\\ &\quad {} - \int_{0}^{1} \bigl[m_{0}(t)+n_{0}(t) \bigr]e(t)\,dt. \end{aligned} $$

Then we derive

$$ \begin{aligned}[b] \int_{0}^{1} \bigl[u_{1}(t)+v_{1}(t) \bigr]e(t)\,dt&\leq \zeta^{-1} \biggl[ r(K) \int_{0}^{1} \bigl[(Hu_{1}) (t)+(Hv_{1}) (t) \bigr]e(t)\,dt\\ &\quad {} + \int_{0}^{1} \bigl[m_{0}(t)+n_{0}(t) \bigr]e(t)\,dt \biggr]. \end{aligned} $$
(2.11)

By computation, we obtain

$$ \begin{aligned} &\int_{0}^{1}(KHu_{1}) (t)e(t)\,dt=r(K) \int_{0}^{1}(Hu_{1}) (t)e(t)\,dt,\\ &\int_{0}^{1}(KHv_{1}) (t)e(t)\,dt=r(K) \int_{0}^{1}(Hv_{1}) (t)e(t)\,dt. \end{aligned} $$
(2.12)

By (2.6), (2.7), (2.11), and (2.12), we get

$$ \begin{aligned}[b] & \int _{0}^{1} \bigl[u_{1}(t)+v_{1}(t)+(KHv_{1}) (t)+(KHu_{1}) (t)+(Ku_{0}) (t)+(Kv_{0}) (t) \bigr]e(t)\,dt \\ &\quad \leq\zeta^{-1} \biggl[r(K) \int_{0}^{1} \bigl[(Hv_{1}) (t)+(Hu_{1}) (t) \bigr]e(t)\,dt + \int_{0}^{1} \bigl[m_{0}(t)+n_{0}(t) \bigr]e(t)\,dt \biggr] \\ &\qquad {}+r(K) \int_{0}^{1}(Hu_{1}) (t)e(t)\,dt+r(K) \int_{0}^{1}(Hv_{1}) (t)e(t)\,dt\\ &\qquad {} + \int_{0}^{1}(Ku_{0}) (t)e(t)\,dt+ \int_{0}^{1}(Ku_{0}) (t)e(t)\,dt \\ &\quad =\zeta^{-1}(1+\zeta)r(K) \int_{0}^{1} \bigl[(Hu_{1}) (t)+(Hv_{1}) (t) \bigr]e(t)\,dt\\ &\qquad {} +\zeta^{-1} \int_{0}^{1} \bigl[m_{0}(t)+n_{0}(t) \bigr]e(t)\,dt + \int_{0}^{1} \bigl[(Ku_{0}) (t)+(Kv_{0}) (t) \bigr]e(t)\,dt \\ &\quad \leq\zeta^{-1}(1+\zeta)r(K) \bigl( \Vert Hu \Vert + \Vert Hv \Vert \bigr)+\zeta^{-1} \int _{0}^{1} \bigl[m_{0}(t)+n_{0}(t) \bigr]e(t)\,dt\\ &\qquad {} + \int_{0}^{1} \bigl[(Ku_{0}) (t)+(Kv_{0}) (t) \bigr]e(t)\,dt \\ &\quad \leq\zeta^{-1}(1+\zeta)r(K)\varepsilon_{0} \bigl( \Vert u \Vert + \Vert v \Vert \bigr)+l_{1}, \end{aligned} $$
(2.13)

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

$$\begin{aligned}& u_{1}+KHv_{1}+Kv_{0}=KFv_{1}+ \mu_{1}\phi+KHv_{1}+Kv_{0} =K(Fv_{1}+Hv_{1}+v_{0})+ \mu_{1}\phi\in P_{0}, \\& v_{1}+KHu_{1}+Ku_{0}=KGu_{1}+ \mu_{1}\phi+KHu_{1}+Ku_{0} =K(Gv_{1}+Hu_{1}+u_{0})+ \mu_{1}\phi\in P_{0}. \end{aligned}$$

The definition of \(P_{0}\) yields

$$ \begin{aligned} & \int_{0}^{1}(u_{1}+KHv_{1}+Kv_{0}) (t)e(t)\,dt\geq \delta \Vert u_{1}+KHv_{1}+Kv_{0} \Vert \geq \delta \Vert u_{1} \Vert -\delta \Vert KHv_{1} \Vert -\delta \Vert Kv_{0} \Vert , \\ & \int_{0}^{1}(v_{1}+KHu_{1}+Ku_{0}) (t)e(t)\,dt\geq\delta \Vert v_{1}+KHu_{1}+Ku_{0} \Vert \geq \delta \Vert v_{1} \Vert -\delta \Vert KHu_{1} \Vert -\delta \Vert Ku_{0} \Vert , \end{aligned} $$

where δ is given in Lemma 2.6. By adding the above two inequalities, we obtain

$$ \begin{aligned}[b] &\int_{0}^{1}(u_{1}+v_{1}+KHu_{1}+KHv_{1}+Ku_{0}+Kv_{0}) (t)e(t)\,dt\\ &\quad \geq \delta \bigl( \Vert v_{1} \Vert + \Vert u \Vert \bigr)-\delta \bigl( \Vert KHu_{1} \Vert + \Vert KHv_{1} \Vert \bigr)-\delta \bigl( \Vert Ku_{0} \Vert + \Vert Kv_{0} \Vert \bigr). \end{aligned} $$
(2.14)

It follows from (2.7), (2.13), and (2.14) that

$$ \begin{aligned}[b] \Vert u_{1} \Vert + \Vert v_{1} \Vert &\leq\delta^{-1} \int_{0}^{1}(u_{1}+v_{1}+KHu_{1}+KHv_{1}+Ku_{0}+Kv_{0}) (t)e(t)\,dt\\ &\quad {} + \Vert KHu_{1} \Vert + \Vert KHv_{1} \Vert + \Vert Ku_{0} \Vert + \Vert Kv_{0} \Vert \\ &\leq \varepsilon_{0}(\delta\lambda_{1})^{-1} \bigl(1+\zeta^{-1} \bigr) \bigl( \Vert u_{1} \Vert + \Vert v_{1} \Vert \bigr)+l_{1}\delta^{-1}\\ &\quad {} + \varepsilon_{0} \Vert K \Vert \cdot \bigl( \Vert u_{1} \Vert + \Vert v_{1} \Vert \bigr)+ \Vert Ku_{0} \Vert + \Vert Kv_{0} \Vert \\ &=\varepsilon_{0} l_{0} \bigl( \Vert u_{1} \Vert + \Vert v_{1} \Vert \bigr)+l_{2}, \end{aligned} $$
(2.15)

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

$$ \operatorname{deg}(I-A,B_{R},0)=0. $$

 □

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,

$$ \Vert T_{i}w \Vert \leq T_{i} \bigl( \Vert w \Vert \bigr)< \varepsilon \Vert w \Vert , \quad\forall w\in E \mbox{ with } \Vert w \Vert >L, $$

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

$$ (Fv+v_{0}+Hv, Gu+u_{0}+Hu)\in P^{2}\quad \mbox{for all }(u,v)\in E^{2}, $$

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

$$ f(t,x)\geq\lambda_{1}(1+\varepsilon_{1})x,\qquad g(t,y)\geq \lambda_{1}(1+\varepsilon_{1})y,\quad \forall x,y\geq L_{1}. $$
(3.1)

Combining (3.1) with (A2), the above constants \(b_{1}\), \(b_{2}\) satisfy

$$\begin{aligned}& f(t,x)\geq\lambda_{1}(1+\varepsilon_{1})x-b_{1}-c_{1}B_{1}(x),\\& g(t,y)\geq\lambda_{1}(1+\varepsilon_{1})y-b_{2}-c_{2}B_{1}(y) \quad\mbox{for all }x,y\in\mathbb{R}, \end{aligned}$$

and so

$$ (Fv,Gu)\geq \bigl(\lambda_{1}(1+\varepsilon_{1})v-b_{1}-Hv, \lambda_{1}(1+\varepsilon_{1})u-b_{2}-Hu \bigr) \quad \mbox{for all }(u,v)\in E^{2}. $$
(3.2)

Since K is a positive linear operator, from (3.2), we have

$$ \begin{aligned} \bigl((KFv) (t),(KGu) (t) \bigr)&\geq \bigl(\lambda_{1}(1+ \varepsilon_{1}) (Kv ) (t)-Kb_{1}- (KHv ) (t),\\ &\quad \lambda_{1}(1+\varepsilon_{1}) (Ku ) (t)-Kb_{2}- (KHu ) (t) \bigr)\quad \forall t\in[0,1],(u,v)\in E^{2}. \end{aligned} $$

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

$$ \operatorname{deg}(I-A,B_{R},0)=0. $$
(3.3)

From (A4), it follows that there exist \(0<\varepsilon_{2}<1\) and \(0< r< R\) such that

$$\begin{aligned}& \bigl\vert f(t,x) \bigr\vert \leq(1-\varepsilon_{2}) \lambda_{1} \vert x \vert ,\qquad \bigl\vert g(t,y) \bigr\vert \leq(1- \varepsilon_{2})\lambda_{1} \vert y \vert ,\\& \quad \forall t\in[0,1], x,y\in\mathbb{R} \mbox{ with } \vert x \vert \leq r, \vert y \vert \leq r. \end{aligned}$$

Thus

$$ \begin{aligned}[b] &\bigl\vert (A_{1}u) (t) \bigr\vert \leq(1- \varepsilon_{2})\lambda_{1} \bigl(K \vert v \vert \bigr) (t),\qquad (A_{2}v) (t)|\leq(1-\varepsilon_{2}) \lambda_{1} \bigl(K \vert u \vert \bigr) (t),\\ &\quad \forall t \in[0,1], u,v\in E \mbox{ with } \Vert u \Vert \leq r, \Vert v \Vert \leq r. \end{aligned} $$
(3.4)

Next we will prove that

$$ (u,v)\neq\mu A(u,v)\quad\mbox{ for all }(u,v)\in\partial B_{r} \mbox{ and }\mu\in[0,1]. $$
(3.5)

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,

$$ \begin{aligned}&u_{1}(t)=(A_{1}v_{1}) (t)=\mu_{1} \int_{0}^{1} G(t,s)h(s)f \bigl(s,v_{1}(s) \bigr)\,ds, \\ &v_{1}(t)=(A_{2}u_{1}) (t)=\mu_{1} \int_{0}^{1} G(t,s)h(s)g \bigl(s,u_{1}(s) \bigr)\,ds. \end{aligned} $$

Let \(z(t)=|u_{1}(t)|+|v_{1}(t)|\). Then \(z\in P\) and by (3.4),

$$ \begin{aligned} z(t)&\leq(1-\varepsilon_{2})\lambda_{1} \bigl[ \bigl(K \vert u_{1} \vert \bigr) (t)+ \bigl(K \vert v_{1} \vert \bigr) (t) \bigr] \\ &=(1-\varepsilon_{2})\lambda_{1} \bigl(K \bigl( \vert u_{1} \vert + \vert v_{1} \vert \bigr) \bigr) (t)= (1-\varepsilon_{2})\lambda_{1}(Kz) (t). \end{aligned} $$

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

$$ \operatorname{deg}(I-A,B_{r},0)=1. $$
(3.6)

By (3.3), (3.6), and the additivity of Leray–Schauder degree, we obtain

$$ \operatorname{deg}(I-A,B_{R}\setminus \overline{B}_{r},0)= \operatorname{deg}(I-A,B_{R},0)-\operatorname{deg}(I-A,B_{r},0)=-1. $$

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

$$\begin{aligned}& \begin{aligned} \Vert u_{1}-u_{2} \Vert & =\max _{t\in[0,1]} \bigl\vert (A_{1}v_{1}) (t)-(A_{1}v_{2}) (t) \bigr\vert \\ &\leq \max _{t\in[0,1]} \int_{0}^{1}G(t,s)h(s) \bigl\vert f \bigl(s,v_{1}(s) \bigr)-f \bigl(s,v_{2}(s) \bigr) \bigr\vert \,ds \\ &\leq k \Vert v_{1}-v_{2} \Vert \max _{t\in[0,1]} \int_{0}^{1}G(t,s)h(s)\,ds \\ &\leq k \Vert v_{1}-v_{2} \Vert \int_{0}^{1} \biggl(\frac{s(1-s)^{\alpha-1}}{\Gamma(\alpha -1)}+ \frac{g_{\beta}(s)}{1-\Gamma} \biggr)h(s)\,ds \\ &< \Vert v_{1}-v_{2} \Vert , \end{aligned} \\& \begin{aligned} \Vert v_{1}-v_{2} \Vert & =\max _{t\in[0,1]} \bigl\vert (A_{2}u_{1}) (t)-(A_{2}u_{2}) (t) \bigr\vert \\ &\leq \max _{t\in[0,1]} \int_{0}^{1}G(t,s)h(s) \bigl\vert f \bigl(s,u_{1}(s) \bigr)-f \bigl(s,u_{2}(s) \bigr) \bigr\vert \,ds \\ &\leq k \Vert u_{1}-u_{2} \Vert \max _{t\in[0,1]} \int_{0}^{1}G(t,s)h(s)\,ds \\ &\leq k \Vert u_{1}-u_{2} \Vert \int_{0}^{1} \biggl(\frac{s(1-s)^{\alpha-1}}{\Gamma(\alpha -1)}+ \frac{g_{\beta}(s)}{1-\Gamma} \biggr)h(s)\,ds \\ &< \Vert u_{1}-u_{2} \Vert . \end{aligned} \end{aligned}$$

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

$$\beta(t)= \textstyle\begin{cases} 0,&[0,\frac{1}{3}),\\ \frac{1}{8},&[\frac{1}{3},\frac{2}{3}),\\ \frac{1}{2},&[\frac{2}{3},1], \end{cases} $$

\(h(t)=\frac{1}{\sqrt{t(1-t)}}\) and

$$\begin{aligned}& f(t,x)= \textstyle\begin{cases} \sum_{i=1}^{n}(-1)^{i}a_{i}-(1+t^{2}) \vert x \vert ^{\frac{1}{3}}\ln( \vert x \vert +1)+(1+t^{2})\ln2,& x\in(-\infty,-1),\\ \sum_{i=1}^{n}a_{i}x^{i}, &x\in[-1,+\infty), \end{cases}\displaystyle \\& g(t,y)= \textstyle\begin{cases} \sum_{i=1}^{n}(-1)^{i}a_{i}-(1+t^{4}) \vert y \vert ^{\frac{1}{3}}\ln( \vert y \vert ^{\frac {1}{3}}+1)+(1+t^{4})\ln2,& y\in(-\infty,-1),\\ \sum_{i=1}^{n}a_{i}y^{i}, &y\in[-1,+\infty), \end{cases}\displaystyle \end{aligned}$$

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

$$\begin{aligned}& \beta(t)= \textstyle\begin{cases} 0,&[0,\frac{1}{3}),\\ \frac{1}{3},&[\frac{1}{3},1], \end{cases}\displaystyle \\& f(t,x)=g(t,x)= \textstyle\begin{cases} -a_{1}-(1+t^{2})\ln( \vert x \vert +1)+(1+t^{2})\ln2,& x\in(-\infty,-1),\\ a_{1}x, &x\in[-1,1),\\ a_{2}+a_{2}\ln(x+1)+a_{1}-a_{2}-a_{2}\ln2, &x\in[1,+\infty). \end{cases}\displaystyle \end{aligned}$$

\(h(t)=\frac{\Gamma(\alpha-1)}{2(5a_{2}+2)\sqrt{t}(1-t)^{\alpha-1}}\), 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.

References

  1. Podlubny, I.: Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198. Academic Press, New York (1999)

    MATH  Google Scholar 

  2. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  3. Diethelm, K.: The Analysis of Fractional Differential Equation. Springer, Heidelberg (2010)

    Book  MATH  Google Scholar 

  4. Bai, C., Fang, J.: The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations. Appl. Math. Comput. 150, 611–621 (2004)

    MathSciNet  MATH  Google Scholar 

  5. Su, X.: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 22, 64–69 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ahmad, B., Nieto, J.: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58, 1838–1843 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Wang, W.: Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions. Comput. Math. Appl. 63, 288–297 (2012)

    Article  MathSciNet  Google Scholar 

  8. Li, Y., Li, F.: Sign-changing solutions to second-order integral boundary value problems. Nonlinear Anal. 69, 1179–1187 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Yang, Z.: Existence of nontrivial solutions for a nonlinear Sturm–Liouville problem with integral boundary value conditions. Nonlinear Anal. 68, 216–225 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Boucherif, A.: Second-order boundary value problems with integral boundary conditions. Nonlinear Anal. 70, 364–371 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Liu, L., Liu, B., Wu, H.: Nontrivial solutions of m-point boundary value problems for singular second-order differential equations with a sign-changing nonlinear term. J. Comput. Appl. Math. 224, 373–382 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jiang, J., Liu, L., Wu, Y.: Second-order nonlinear singular Sturm–Liouville problems with integral boundary conditions. Appl. Math. Comput. 215, 1573–1582 (2009)

    MathSciNet  MATH  Google Scholar 

  13. Kong, L.: Second order singular boundary value problems with integral boundary conditions. Nonlinear Anal. 72, 2628–2638 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Graef, J.R., Kong, L.: Periodic solutions for functional differential equations with sign-changing nonlinearities. Proc. R. Soc. Edinb. 140A, 597–616 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chamberlain, J., Kong, L., Kong, Q.: Nodal solutions of boundary value problems with boundary conditions involving Riemann–Stieltjes integrals. Nonlinear Anal. 74, 2380–2387 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, H., Liu, Y.: On sign-changing solutions for a second-order integral boundary value problem. Comput. Math. Appl. 62, 651–656 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Feng, M.: Existence of symmetric positive solutions for a boundary value problem with integral boundary conditions. Appl. Math. Lett. 24, 1419–1427 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, Y., Liu, L., Wu, Y.: Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity. Nonlinear Anal. 74, 6434–6441 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, Y., Liu, L., Wu, Y.: Positive solutions for a nonlocal fractional differential equation. Nonlinear Anal. 74, 3599–3605 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chen, T., Liu, W., Hu, Z.: A boundary value problem for fractional differential equation with p-Laplacian operator at resonance. Nonlinear Anal. 75, 3210–3217 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, B., Li, J., Liu, L.: Nontrivial solutions for a boundary value problem with integral boundary conditions. Bound. Value Probl. 2014, 15 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, H., Liu, L., Wu, Y.: Positive solutions for singular nonlinear fractional differential equation with integral boundary conditions. Bound. Value Probl. 2015, 232 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wang, Y., Liu, L.: Necessary and sufficient condition for the existence of positive solution to singular fractional differential equations. Adv. Differ. Equ. 2015, 207 (2015)

    Article  MathSciNet  Google Scholar 

  24. Jiang, J., Liu, L.: Existence of solutions for a sequential fractional differential system with coupled boundary conditions. Bound. Value Probl. 2016, 159 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hao, X., Wang, H., Liu, L., Cu, Y.: Positive solutions for a system of nonlinear fractional nonlocal boundary value problems with parameters and p-Laplacian operator. Bound. Value Probl. 2017, 18 (2017)

    Article  MathSciNet  Google Scholar 

  26. Bai, Z., Lü, H.: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495–505 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Bai, Z., Chen, Y., Lian, H., Sun, S.: On the existence of blow up solutions for a class of fractional differential equations. Fract. Calc. Appl. Anal. 17, 1175–1187 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Bai, Z.: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 72, 916–924 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Bai, Z., Zhang, S., Sun, S., Yin, C.: Monotone iterative method for fractional differential equations. Electron. J. Differ. Equ. 2016, 6 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Song, Q., Dong, X., Bai, Z., Chen, B.: Existence for fractional Dirichlet boundary value problem under barrier strip conditions. J. Nonlinear Sci. Appl. 10, 3592–3598 (2017)

    Article  MathSciNet  Google Scholar 

  31. Yuan, C.: Multiple positive solutions for \((n-1, 1)\)-type semipositone conjugate boundary value problems of nonlinear fractional differential equations. Electron. J. Qual. Theory Differ. Equ. 2010, 36 (2010)

    MathSciNet  MATH  Google Scholar 

  32. Webb, J.R.L., Lan, K.: Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type. Topol. Methods Nonlinear Anal. 27, 91–115 (2006)

    MathSciNet  MATH  Google Scholar 

  33. Krein, M.G., Rutman, M.A.: Linear operators leaving invariant a cone in a Banach space. Transl. Am. Math. Soc. 26, 199–325 (1950)

    Google Scholar 

  34. Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, Orlando (1988)

    MATH  Google Scholar 

  35. Ahmad, B., Luca, R.: Existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions. Fract. Calc. Appl. Anal. 21, 423–441 (2018)

    Article  MathSciNet  Google Scholar 

  36. Agarwal, R.P., Ahmad, B., Garout, D.: Existence results for coupled nonlinear fractional differential equations equipped with nonlocal coupled flux and multi-point boundary conditions. Chaos Solitons Fractals 102, 149–161 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ahmad, B., Alsaedi, A., Aljoudi, S.: On a coupled system of sequential fractional differential equations with variable coefficients and coupled integral boundary conditions. Bull. Math. Soc. Sci. Math. Roum. 60, 3–18 (2017)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to editors and reviewers for their valuable suggestions.

Funding

This paper is supported by the National Natural Science Foundation of China (11701252) and the Natural Science Foundation of Shangdong Province of China (ZR2017MA036).

Author information

Authors and Affiliations

Authors

Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Bingmei Liu.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13662-018-1762-0

MSC

Keywords