1 Introduction

The human immunodeficiency virus (HIV) causes acquired immune deficiency syndrome (AIDS). The viral infection is characterized by a severe impairment of the immune system and related opportunistic infections. HIV is a retrovirus that targets the CD4+ T lymphocytes, which are the most abundant white blood cells of the immune system. Until now, there are several countries, particularly in Africa, with up to 35% of their populations between the ages of 15 and 50 years infected by HIV, and throughout the world, already over 16 million deaths died of AIDS. Mathematical models have been proven valuable in understanding the dynamics of HIV infection [1–4]. Perelson [5, 6] developed a simple model for the primary infection with HIV. In this model, four categories of cells were defined: uninfected CD4+ T-cells, latently infected CD4+ T-cells, productively infected CD4+ T-cells, and the virus population. In [4], two equations were proposed to describe the evolution of the system for HIV-1 population dynamics:

$$\left \{ \textstyle\begin{array}{l} \frac{dx}{dt}=s-\mu x-\beta xy, \\ \frac{dy}{dt}=\beta xy-\alpha y, \end{array}\displaystyle \right . $$

where all parameters and variables are non-negative, and x denotes the number of uninfected CD4+ T-cells and y denotes the number of infected cells, s is the assumed constant rate of production of CD4+ T-cells, μ is their per capita death rate, β is the rate of infection of CD4+ T-cells by virus, and α is the rate of disappearance of infected cells.

Recently, Arafal et al. [7] introduced fractional-order model of infection of CD4+ T-cells which is described by the following set of FODEs of order \(\alpha_{1}, \alpha_{2}, \alpha_{3}>0\):

$$\left \{ \textstyle\begin{array}{l} \mathcal{D} ^{\alpha_{1}}(T)=s-KVT-dT+bI, \\ \mathcal{D}^{\alpha_{2}}(I)=KVT-(b+\delta)I, \\ \mathcal{D}^{\alpha_{3}}(V)=N\delta I-cV, \end{array}\displaystyle \right . $$

where T, I, and V denote the concentration of uninfected CD4+ T-cells, infected CD4+ T-cells, and free HIV virus particles in the blood, respectively. δ represents the death rate of infected T-cells and includes the possibility of death by bursting of infected T-cells, hence \(\delta\geq d\). The parameter b is the rate at which infected cells return to the uninfected class, while c is the death rate of the virus and N is the average number of viral particles produced by an infected cell.

Motivated by the HIV infection model and its application background, in this paper, we consider the existence of solutions for the nonlinear sequential fractional differential system with coupled boundary conditions (BCs) of the type:

$$ \left \{ \textstyle\begin{array}{l} ({}^{\mathrm{c}}\mathcal{D}^{p}+\lambda_{1} {}^{\mathrm{c}}\mathcal {D}^{p-1})u(t)= f_{1}(t,u(t),v(t)),\quad 0< t< 1 , \\ ({}^{\mathrm{c}}\mathcal{D}^{q}+\lambda_{2} {}^{\mathrm{c}}\mathcal {D}^{q-1})v(t)=f_{2}(t,u(t),v(t)),\quad 0< t< 1, \\ u(0)=u'(0)=0, \qquad u(1)=av(\xi), \\ v(0)=v'(0)=0,\qquad v(1)=bu(\eta), \end{array}\displaystyle \right . $$
(1.1)

where \(\lambda_{i}>0\) (\(i=1,2\)) is a parameter, \(2< p, q\leq3\), \({}^{\mathrm{c}}\mathcal {D}^{p}\), \({}^{\mathrm{c}}\mathcal{D}^{q}\) are the Caputo fractional derivatives, ξ, η satisfy \(\xi,\eta\in(0,1)\) and \(ab(\lambda_{1}\eta-1+e^{-\lambda_{1}\eta})(\lambda _{2}\xi-1+e^{-\lambda_{2}\xi}) -(\lambda_{1}-1+e^{-\lambda_{1}})(\lambda_{2}-1+e^{-\lambda_{2}})\triangleq \Lambda\neq0\), the nonlinearities \(f_{1},f_{2}:[0,1]\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}\) are given continuous functions.

In the past decades, fractional calculus has been extensively applied in many fields such as physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, economics, control theory, signal and image processing, biophysics, and blood flow phenomena. Many mathematicians and applied researchers have tried to model real processes using the fractional calculus. In biology, it has been deduced that the membranes of cells of biological organism have fractional-order electrical conductance [8] and thus are classified in groups of non-integer-order models. Fractional derivatives embody essential features of cell rheological behavior and have enjoyed greatest success in the field of rheology [9]. Fractional-order ordinary differential equations are naturally related to systems with long time memory which exists in most biological systems such as HIV infection, hepatitis C virus (HCV) infection, and cancer pervasion. Also, they are closely related to fractals, which are abundant in biological systems. Thus fractional-order differential equations are also regarded as a better tool for the description of hereditary properties of various materials and processes than the corresponding integer-order differential equations. With this advantage, fractional-order models have become more realistic and practical than the corresponding classical integer-order models, moreover, the dynamics behavior of fractional-order models are also as stable as their integer-order counterpart. Since theoretical results can help to get an in-depth understanding for the dynamic behavior in biological process, the study of abstract fractional dynamic models nowadays is quite relevant and important. On the other hand, BCs in (1.1) are referred to as coupled BCs; they arise in the study of reaction-diffusion equations, Sturm-Liouville problems, mathematical biology and so on; see [10–20]. In [16], Leung studied the following reaction-diffusion system for a prey-predator interaction:

$$\left \{ \textstyle\begin{array}{l} u_{t}(t,x)=\sigma_{1}\Delta u+u(a+f(u,v)),\quad t\geq0, x\in\Omega\subset\mathbb{R}^{n}, \\ v_{t}(t,x)=\sigma_{2}\Delta v+v(-r+g(u,v)), \quad t\geq0, x\in\Omega\subset\mathbb{R}^{n}, \end{array}\displaystyle \right . $$

subject to the coupled BCs

$$\frac{\partial u}{\partial\eta}=0, \qquad \frac{\partial v}{\partial \eta}-p(u)-q(v)=0 \quad \mbox{on } \partial\Omega, $$

where the functions \(u(t, x)\), \(v(t, x)\), respectively, represent the density of prey and predator at time \(t\geq0\) and at position \(x=(x_{1},\ldots, x_{n})\). Similar coupled BCs are also studied in [11] for a biochemical system. As far as we know, the nonlinear fractional differential system coupled at equations have been studied extensively. For details, see [21–31] and the references therein. However, there have been a few papers which deal with coupling at boundary conditions for fractional differential systems. Wang et al. [32] obtained the existence and uniqueness of positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations,

$$\left \{ \textstyle\begin{array}{l} \mathcal{D} _{0+}^{\alpha}u(t)= f(t,v(t)),\qquad \mathcal{D}_{0+}^{\beta}v(t)=g(t,u(t)), \quad 0< t< 1, \\ u(0)=v(0)=0, \qquad u(1)=au(\xi),\qquad v(1)=bv(\xi), \end{array}\displaystyle \right . $$

where \(1 < \alpha, \beta< 2\), \(0\leq a, b <1\), \(0 < \xi< 1\), \(f , g: [0, 1]\times[0,+\infty)\to[0,+\infty)\) are continuous functions, and \(\mathcal{D}_{0+}^{\alpha}\), \(\mathcal{D}_{0+}^{\beta}\) are the standard Riemann-Liouville fractional derivative.

The paper is organized as follows. In Section 2, we recall some basic concepts of fractional calculus and present some auxiliary lemmas. The main results are presented in Section 3. We give two results: the first one dealing with the existence of solutions is derived by applying Leray-Schauder’s alternative; the second one concerning the uniqueness of solutions, established by applying Banach’s contraction mapping principle. In Section 4, two examples are given to demonstrate the validity of our main results. Some interesting observations are presented in the conclusions section.

2 Preliminaries and lemmas

In this section, we will present some preliminaries and lemmas that will be used in the proof of our main results.

Definition 2.1

([33, 34])

The Riemann-Liouville fractional integral of order \(\alpha > 0\) is given by

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

where \(n-1< \alpha< n\), provided that the right-hand side is pointwise defined on \((0,+\infty)\).

Definition 2.2

([33, 34])

For an \((n-1)\)-times absolutely continuous function \(u:[0,\infty)\to\mathbb{R}\), the Caputo derivative of fractional order α is defined as

$${}^{\mathrm{c}}\mathcal{D}^{\alpha}u(t)=\frac{1}{\Gamma(n-\alpha)} \int ^{t}_{0}(t-s)^{n-\alpha-1}u^{(n)}(s) \,ds, \quad n-1< \alpha< n, n=[\alpha]+1, $$

where \([\alpha]\) denotes the integer part of the real number α.

Lemma 2.1

For any \(h,g\in L(0,1)\cap C(0,1)\), the system consisting of the equations

$$ \bigl({}^{\mathrm{c}}\mathcal{D}^{p}+ \lambda_{1} {}^{\mathrm{c}}\mathcal{D}^{p-1}\bigr)u(t)=h(t), \qquad \bigl({}^{\mathrm{c}}\mathcal{D}^{q}+\lambda_{2} {}^{\mathrm{c}}\mathcal{D}^{q-1}\bigr)v(t)=g(t),\quad t\in(0,1) $$
(2.1)

and the BCs

$$ \left \{ \textstyle\begin{array}{l} u(0)=u'(0)=0, \qquad u(1)=av(\xi), \\ v(0)=v'(0)=0,\qquad v(1)=bu(\eta), \end{array}\displaystyle \right . $$
(2.2)

has a unique integral representation

$$\begin{aligned}& u(t)= A_{1}(t) \biggl\{ a\bigl(\lambda_{2} \xi-1+e^{-\lambda_{2}\xi}\bigr) \biggl[ \int_{0}^{1}e^{-\lambda _{2}(1-s)}(Qg) (s)\,ds -b \int_{0}^{\eta}e^{-\lambda_{1}(\eta-s)}(Ph) (s)\,ds \biggr] \\& \hphantom{u(t)={}}{}-\bigl(\lambda_{2}-1+e^{-\lambda_{2}}\bigr) \biggl[a \int_{0}^{\xi}e^{-\lambda_{2}(\xi-s)}(Qg) (s)\,ds - \int_{0}^{1}e^{-\lambda_{1}(1-s)}(Ph) (s)\,ds \biggr] \biggr\} \\& \hphantom{u(t)={}}{}+ \int _{0}^{t}e^{-\lambda_{1}(t-s)}(Ph) (s) \,ds, \end{aligned}$$
(2.3)
$$\begin{aligned}& v(t)= A_{2}(t) \biggl\{ \bigl(\lambda_{1}-1+e^{-\lambda_{1}} \bigr) \biggl[ \int_{0}^{1}e^{-\lambda _{2}(1-s)}(Qg) (s)\,ds -b \int_{0}^{\eta}e^{-\lambda_{1}(\eta-s)}(Ph) (s)\,ds \biggr] \\& \hphantom{v(t)={}}{}-b\bigl(\lambda_{1}\eta-1+e^{-\lambda_{1}\eta}\bigr) \biggl[a \int_{0}^{\xi}e^{-\lambda _{2}(\xi-s)}(Qg) (s)\,ds - \int_{0}^{1}e^{-\lambda_{1}(1-s)}(Ph) (s)\,ds \biggr] \biggr\} \\& \hphantom{v(t)={}}{}+ \int _{0}^{t}e^{-\lambda_{2}(t-s)}(Qg) (s) \,ds, \end{aligned}$$
(2.4)

where

$$\begin{aligned}& A_{1}(t)=\frac{1}{\Lambda}\bigl(\lambda_{1}t-1+e^{-\lambda_{1}t} \bigr),\qquad A_{2}(t)=\frac{1}{\Lambda}\bigl(\lambda_{2}t-1+e^{-\lambda_{2}t} \bigr), \end{aligned}$$
(2.5)
$$\begin{aligned}& (Ph) (s)= \int_{0}^{s}\frac{(s-\tau)^{p-2}}{\Gamma(p-1)}h(\tau)\,d\tau,\qquad (Qg) (s)= \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}g(\tau)\,d \tau. \end{aligned}$$
(2.6)

Proof

Solving (2.1), we obtain

$$\begin{aligned}& u(t)=c_{11}e^{-\lambda_{1}t}+\frac{c_{12}}{\lambda_{1}}\bigl(1-e^{-\lambda _{1}t} \bigr)+\frac{c_{13}}{\lambda_{1}^{2}}\bigl(\lambda_{1}t-1+e^{-\lambda_{1}t}\bigr) + \int_{0}^{t}e^{-\lambda_{1}(t-s)}(Ph) (s) \,ds, \end{aligned}$$
(2.7)
$$\begin{aligned}& v(t)=c_{21}e^{-\lambda_{2}t}+\frac{c_{22}}{\lambda_{2}}\bigl(1-e^{-\lambda _{2}t} \bigr)+\frac{c_{23}}{\lambda_{2}^{2}}\bigl(\lambda_{2}t-1+e^{-\lambda_{2}t}\bigr) + \int_{0}^{t}e^{-\lambda_{2}(t-s)}(Qg) (s) \,ds, \end{aligned}$$
(2.8)

where \(c_{ij}\) (\(1\leq i\leq2\), \(1\leq j\leq3\)) are constants to be determined. In the following, we determine \(c_{ij}\) (\(1\leq i\leq 2\), \(1\leq j\leq3\)), so that \(u(t)\) and \(v(t)\) satisfy (2.2). By BCs (2.2), we obtain

$$ c_{11}= c_{12}=0, \qquad c_{21}= c_{22}=0, $$
(2.9)

and

$$\begin{aligned}& \frac{\lambda_{1}-1+e^{-\lambda_{1}}}{\lambda_{1}^{2}}c_{13}-\frac{a(\lambda _{2}\xi-1+e^{-\lambda_{2}\xi})}{\lambda_{2}^{2}}c_{23} \\& \quad =a \int_{0}^{\xi}e^{-\lambda_{2}(\xi-s)}(Qg) (s)\,ds - \int_{0}^{1}e^{-\lambda_{1}(1-s)}(Ph) (s) \,ds, \end{aligned}$$
(2.10)
$$\begin{aligned}& \frac{b(\lambda_{1}\eta-1+e^{-\lambda_{1}\eta})}{\lambda_{1}^{2}}c_{13}-\frac {\lambda_{2}-1+e^{-\lambda_{2}}}{\lambda_{2}^{2}}c_{23} \\& \quad = \int_{0}^{1}e^{-\lambda_{2}(1-s)}(Qg) (s)\,ds -b \int_{0}^{\eta}e^{-\lambda_{1}(\eta-s)}(Ph) (s) \,ds. \end{aligned}$$
(2.11)

Note that

$$\begin{aligned} \begin{vmatrix} \frac{\lambda_{1}-1+e^{-\lambda_{1}}}{\lambda_{1}^{2}} & - \frac {a(\lambda_{2}\xi-1+e^{-\lambda_{2}\xi})}{\lambda_{2}^{2}} \\ \frac{b(\lambda_{1}\eta-1+e^{-\lambda_{1}\eta})}{\lambda_{1}^{2}} & -\frac{\lambda_{2}-1+e^{-\lambda_{2}}}{\lambda_{2}^{2}} \end{vmatrix} =& \frac{ab(\lambda_{1}\eta-1+e^{-\lambda_{1}\eta})(\lambda_{2}\xi -1+e^{-\lambda_{2}\xi})}{\lambda_{1}^{2}\lambda_{2}^{2}} \\ &{}-\frac{(\lambda_{1}-1+e^{-\lambda_{1}})(\lambda_{2}-1+e^{-\lambda_{2}})}{\lambda _{1}^{2}\lambda_{2}^{2}}=\frac{\Lambda}{\lambda_{1}^{2}\lambda_{2}^{2}}\triangleq \mathfrak{B} \neq0. \end{aligned}$$

Thus, the system (2.10)-(2.11) has a unique solution for \(c_{13}\) and \(c_{23}\). By Cramer’s rule and simple calculations, it follows that

$$\begin{aligned}& c_{13}= \frac{1}{\mathfrak{B}}\frac{a(\lambda_{2}\xi-1+e^{-\lambda_{2}\xi })}{\lambda_{2}^{2}} \biggl[ \int_{0}^{1}e^{-\lambda_{2}(1-s)}(Qg) (s)\,ds -b \int_{0}^{\eta}e^{-\lambda_{1}(\eta-s)}(Ph) (s)\,ds \biggr] \\& \hphantom{c_{13}={}}{}-\frac{1}{\mathfrak{B}}\frac{\lambda_{2}-1+e^{-\lambda_{2}}}{\lambda _{2}^{2}} \biggl[a \int_{0}^{\xi}e^{-\lambda_{2}(\xi-s)}(Qg) (s)\,ds - \int_{0}^{1}e^{-\lambda_{1}(1-s)}(Ph) (s)\,ds \biggr], \end{aligned}$$
(2.12)
$$\begin{aligned}& c_{23}= \frac{1}{\mathfrak{B}}\frac{\lambda_{1}-1+e^{-\lambda_{1}}}{\lambda _{1}^{2}} \biggl[ \int_{0}^{1}e^{-\lambda_{2}(1-s)}(Qg) (s)\,ds -b \int_{0}^{\eta}e^{-\lambda_{1}(\eta-s)}(Ph) (s)\,ds \biggr] \\& \hphantom{c_{23}={}}{}-\frac {1}{\mathfrak{B}}\frac{b(\lambda_{1}\eta-1+e^{-\lambda_{1}\eta})}{\lambda_{1}^{2}} \biggl[a \int_{0}^{\xi}e^{-\lambda_{2}(\xi-s)}(Qg) (s)\,ds - \int_{0}^{1}e^{-\lambda_{1}(1-s)}(Ph) (s)\,ds \biggr]. \end{aligned}$$
(2.13)

Substituting (2.9) and (2.12) in (2.7), one has

$$\begin{aligned} u(t) =&A_{1}(t) \biggl\{ a\bigl(\lambda_{2} \xi-1+e^{-\lambda_{2}\xi}\bigr) \biggl[ \int_{0}^{1}e^{-\lambda _{2}(1-s)}(Qg) (s)\,ds -b \int_{0}^{\eta}e^{-\lambda_{1}(\eta-s)}(Ph) (s)\,ds \biggr] \\ &{}-\bigl(\lambda_{2}-1+e^{-\lambda_{2}}\bigr) \biggl[a \int_{0}^{\xi}e^{-\lambda_{2}(\xi-s)}(Qg) (s)\,ds - \int_{0}^{1}e^{-\lambda_{1}(1-s)}(Ph) (s)\,ds \biggr] \biggr\} \\ &{}+ \int _{0}^{t}e^{-\lambda_{1}(t-s)}(Ph) (s)\,ds. \end{aligned}$$

So (2.3) holds. Similarly, substituting (2.9) and (2.13) in (2.8) we can get (2.4). This completes the proof of the lemma. □

Lemma 2.2

([35])

For any \(h,g\in L(0,1)\cap C(0,1)\), we have

$$\begin{aligned}& \begin{aligned} &\biggl\vert \int_{0}^{\eta}e^{-\lambda_{1}(\eta-s)}(Ph) (s)\,ds\biggr\vert \leq\frac{\eta ^{p}}{\lambda_{1}\Gamma(p)}\bigl(1-e^{-\lambda_{1}\eta}\bigr)\|h\|, \\ &\biggl\vert \int_{0}^{t}e^{-\lambda_{1}(t-s)}(Ph) (s)\,ds\biggr\vert \leq\frac{1}{\lambda _{1}\Gamma(p)}\bigl(1-e^{-\lambda_{1}}\bigr)\|h\|, \end{aligned} \end{aligned}$$
(2.14)
$$\begin{aligned}& \begin{aligned} &\biggl\vert \int_{0}^{\xi}e^{-\lambda_{2}(\xi-s)}(Qg) (s)\,ds\biggr\vert \leq\frac{\xi ^{q}}{\lambda_{2}\Gamma(q)}\bigl(1-e^{-\lambda_{2}\xi}\bigr)\|g\|, \\ &\biggl\vert \int_{0}^{t}e^{-\lambda_{2}(t-s)}(Qg) (s)\,ds\biggr\vert \leq\frac{1}{\lambda _{2}\Gamma(q)}\bigl(1-e^{-\lambda_{2}}\bigr)\|g\|. \end{aligned} \end{aligned}$$
(2.15)

Let \(X=C[0,1]\), then \(X\times X\) is a Banach space with the norm

$$\bigl\Vert (u,v)\bigr\Vert _{1}:=\| u\|+\| v\|,\qquad \| u\| = \max_{0\leq t\leq1}\bigl\vert u(t)\bigr\vert ,\qquad \| v\| = \max _{0\leq t\leq1}\bigl\vert v(t)\bigr\vert , $$

for any \((u,v)\in X\times X\).

In view of Lemma 2.1, we define the operator \(T:X\times X\to X\times X\) by

$$T(u,v)=\bigl(T_{1}(u,v),T_{2}(u,v)\bigr), $$

where operators \(T_{i}:X\times X\to X\) (\(i=1,2\)) are defined by

$$\begin{aligned}& T_{1}(u,v) (t) \\& \quad = A_{1}(t) \biggl\{ a\bigl( \lambda_{2}\xi-1+e^{-\lambda_{2}\xi}\bigr) \biggl[ \int_{0}^{1}e^{-\lambda _{2}(1-s)}Q(u,v) (s)\,ds -b \int_{0}^{\eta}e^{-\lambda_{1}(\eta-s)}P(u,v) (s)\,ds \biggr] \\& \qquad {}-\bigl(\lambda_{2}-1+e^{-\lambda_{2}}\bigr) \biggl[a \int_{0}^{\xi}e^{-\lambda_{2}(\xi -s)}Q(u,v) (s)\,ds - \int_{0}^{1}e^{-\lambda_{1}(1-s)}P(u,v) (s)\,ds \biggr] \biggr\} \\& \qquad {}+ \int _{0}^{t}e^{-\lambda_{1}(t-s)}P(u,v) (s)\,ds \end{aligned}$$
(2.16)

and

$$\begin{aligned}& T_{2}(u,v) (t) \\& \quad = A_{2}(t) \biggl\{ \bigl( \lambda_{1}-1+e^{-\lambda_{1}}\bigr) \biggl[ \int_{0}^{1}e^{-\lambda _{2}(1-s)}Q(u,v) (s)\,ds -b \int_{0}^{\eta}e^{-\lambda_{1}(\eta-s)}P(u,v) (s)\,ds \biggr] \\& \qquad {}-b\bigl(\lambda_{1}\eta-1+e^{-\lambda_{1}\eta}\bigr) \biggl[a \int_{0}^{\xi}e^{-\lambda _{2}(\xi-s)}Q(u,v) (s)\,ds - \int_{0}^{1}e^{-\lambda_{1}(1-s)}P(u,v) (s)\,ds \biggr] \biggr\} \\& \qquad {}+ \int _{0}^{t}e^{-\lambda_{2}(t-s)}Q(u,v) (s)\,ds, \end{aligned}$$
(2.17)

with

$$\begin{aligned}& P(u,v) (s)= \int_{0}^{s}\frac{(s-\tau)^{p-2}}{\Gamma(p-1)}f_{1}\bigl( \tau,u(\tau ),v(\tau)\bigr)\,d\tau, \end{aligned}$$
(2.18)
$$\begin{aligned}& Q(u,v) (s)= \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}f_{2}\bigl( \tau,u(\tau ),v(\tau)\bigr)\,d\tau. \end{aligned}$$
(2.19)

For the sake of convenience, we set

$$\begin{aligned}& A_{1}=\sup_{t\in[0,1]}\bigl\vert A_{1}(t) \bigr\vert , \qquad A_{2}=\sup_{t\in[0,1]}\bigl\vert A_{2}(t)\bigr\vert , \end{aligned}$$
(2.20)
$$\begin{aligned}& M_{1} = \frac{A_{1}[|ab|(\lambda_{2}\xi-1+e^{-\lambda_{2}\xi})(1-e^{-\lambda_{1}\eta })\eta^{p-1}+(\lambda_{2}-1+e^{-\lambda_{2}})(1-e^{-\lambda_{1}})] +(1-e^{-\lambda_{1}})}{\lambda_{1}\Gamma(p)}, \\& M_{2} =\frac{A_{1}[|a|(\lambda_{2}\xi-1+e^{-\lambda_{2}\xi})(1-e^{-\lambda _{2}})+(\lambda_{2}-1+e^{-\lambda_{2}})|a|(1-e^{-\lambda_{2}\xi})\xi ^{q-1}]}{\lambda_{2}\Gamma(q)}, \\& M_{1}' = \frac{A_{2}[(\lambda_{1}-1+e^{-\lambda_{1}})|b|(1-e^{-\lambda_{1}\eta})\eta ^{p-1}+|b|(\lambda_{1}\eta-1+e^{-\lambda_{1}\eta})(1-e^{-\lambda _{1}})]}{\lambda_{1}\Gamma(p)}, \\& M_{2}' =\frac{A_{2}[(\lambda_{1}-1+e^{-\lambda_{1}})(1-e^{-\lambda _{2}})+|ab|(\lambda_{1}\eta-1+e^{-\lambda_{1}\eta})(1-e^{-\lambda_{2}\xi})\xi^{q-1}] +(1-e^{-\lambda_{2}})}{\lambda_{2}\Gamma(q)}, \end{aligned}$$
(2.21)

and

$$ (P{\mathbf{1}}) (s)= \int_{0}^{s}\frac{(s-\tau)^{p-2}}{\Gamma(p-1)}\,d\tau, \qquad (Q{ \mathbf{1}}) (s)= \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}\,d\tau. $$
(2.22)

Lemma 2.3

The operator T: \(X\times X\rightarrow X\times X\) is a completely continuous.

Proof

By continuity of the functions \(f_{i}\) (\(i=1,2\)), the operator T is continuous.

Let \(\Omega\subset X\times X\) be bounded. Then there exist constants \(L_{i}>0\) (\(i=1,2\)) such that

$$\bigl\vert f_{i}\bigl(t,u(t),v(t)\bigr)\bigr\vert \leq L_{i}, \quad \forall (u,v)\in\Omega, i=1,2. $$

Then for any \((u,v)\in\Omega\), it follows from (2.16), (2.14), (2.15), (2.20), and (2.22) that

$$\begin{aligned}& \bigl\vert T_{1}(u,v) (t)\bigr\vert \\& \quad \leq A_{1} \biggl\{ |a|\bigl(\lambda_{2}\xi-1+e^{-\lambda_{2}\xi} \bigr) \biggl[L_{2} \int _{0}^{1}e^{-\lambda_{2}(1-s)}(Q{\mathbf{1}}) (s) \,ds +|b|L_{1} \int_{0}^{\eta}e^{-\lambda_{1}(\eta-s)}(P{\mathbf{1}}) (s)\,ds \biggr] \\& \qquad {} +\bigl(\lambda_{2}-1+e^{-\lambda_{2}}\bigr) \biggl[|a|L_{2} \int_{0}^{\xi}e^{-\lambda_{2}(\xi -s)}(Q{\mathbf{1}}) (s)\,ds +L_{1} \int_{0}^{1}e^{-\lambda_{1}(1-s)}(P{\mathbf{1}}) (s)\,ds \biggr] \biggr\} \\& \qquad {}+L_{1} \int_{0}^{t}e^{-\lambda_{1}(t-s)}(P{{\mathbf{1}}}) (s) \,ds \\& \quad \leq A_{1} \biggl[|a|\bigl(\lambda_{2} \xi-1+e^{-\lambda_{2}\xi}\bigr) \biggl(\frac{L_{2}(1-e^{-\lambda _{2}})}{\lambda_{2}\Gamma(q)} +\frac{|b|L_{1}(1-e^{-\lambda_{1}\eta})\eta^{p-1}}{\lambda_{1}\Gamma(p)} \biggr) \\& \qquad {} +\bigl(\lambda_{2}-1+e^{-\lambda_{2}}\bigr) \biggl( \frac{|a|L_{2}(1-e^{-\lambda_{2}\xi})\xi ^{q-1}}{\lambda_{2}\Gamma(q)} +\frac{L_{1}(1-e^{-\lambda_{1}})}{\lambda_{1}\Gamma(p)} \biggr) \biggr] \\& \qquad {}+\frac {L_{1}(1-e^{-\lambda_{1}})}{\lambda_{1}\Gamma(p)} \\& \quad \leq L_{1} \frac{A_{1}[|ab|(\lambda_{2}\xi-1+e^{-\lambda_{2}\xi})(1-e^{-\lambda_{1}\eta })\eta^{p-1}+(\lambda_{2}-1+e^{-\lambda_{2}})(1-e^{-\lambda_{1}})] +(1-e^{-\lambda_{1}})}{\lambda_{1}\Gamma(p)} \\& \qquad {}+L_{2}\frac{A_{1}[|a|(\lambda_{2}\xi-1+e^{-\lambda_{2}\xi})(1-e^{-\lambda _{2}})+(\lambda_{2}-1+e^{-\lambda_{2}})|a|(1-e^{-\lambda_{2}\xi})\xi ^{q-1}]}{\lambda_{2}\Gamma(q)}, \end{aligned}$$
(2.23)

which implies that

$$ \bigl\Vert T_{1}(u,v)\bigr\Vert \leq L_{1}M_{1}+L_{2}M_{2}, $$
(2.24)

where \(M_{1}\), \(M_{2}\) are given by (2.21).

By (2.17), (2.14), (2.15), (2.20), (2.22), and proceeding as in (2.23), we can obtain

$$ \bigl\Vert T_{2}(u,v)\bigr\Vert \leq L_{1}M_{1}'+L_{2}M_{2}', $$
(2.25)

where \(M_{1}'\), \(M_{2}'\) are given by (2.21).

Combining (2.24) with(2.25), we obtain

$$\bigl\Vert T(u,v)\bigr\Vert _{1}=\bigl\Vert T_{1}(u,v)\bigr\Vert +\bigl\Vert T_{2}(u,v)\bigr\Vert \leq(L_{1}M_{1}+L_{2}M_{2})+ \bigl(L_{1}M_{1}'+L_{2}M_{2}' \bigr)=M, $$

which implies that the operator T is uniformly bounded.

Next, we show that T is equicontinuous. For any \(t_{1},t_{2}\in[0,1]\) with \(t_{1}\leq t_{2}\), noticing (2.22) then we have

$$\begin{aligned}& \bigl\vert T_{1}(u,v) (t_{2})-T_{1}(u,v) (t_{1})\bigr\vert \\& \quad \leq \biggl\vert \bigl[A_{1}(t_{2})-A_{1}(t_{1}) \bigr] \biggl\{ a\bigl(\lambda_{2}\xi-1+e^{-\lambda_{2}\xi}\bigr) \\& \qquad {}\times \biggl[L_{2} \int _{0}^{1}e^{-\lambda_{2}(1-s)}(Q{\mathbf{1}}) (s) \,ds -bL_{1} \int_{0}^{\eta}e^{-\lambda_{1}(\eta-s)}(P{\mathbf{1}}) (s)\,ds \biggr] \\& \qquad {}-\bigl(\lambda_{2}-1+e^{-\lambda_{2}}\bigr) \biggl[aL_{2} \int_{0}^{\xi}e^{-\lambda_{2}(\xi -s)}(Q{\mathbf{1}}) (s)\,ds -L_{1} \int_{0}^{1}e^{-\lambda_{1}(1-s)}(P{\mathbf{1}}) (s)\,ds \biggr]\biggr\} \biggr\vert \\& \qquad {}+L_{1}\biggl\vert \int_{0}^{t_{1}}\bigl(e^{-\lambda_{1}(t_{2}-s)}-e^{-\lambda _{1}(t_{1}-s)} \bigr) (P{\mathbf{1}}) (s)\,ds + \int_{t_{1}}^{t_{2}}e^{-\lambda_{1}(t_{2}-s)}(P{\mathbf{1}}) (s)\,ds \biggr\vert . \end{aligned}$$

Analogously, we can obtain the following inequalities:

$$\begin{aligned}& \bigl\vert T_{2}(u,v) (t_{2})-T_{2}(u,v) (t_{1})\bigr\vert \\& \quad \leq \biggl\vert \bigl[A_{2}(t_{2})-A_{2}(t_{1}) \bigr] \biggl\{ \bigl(\lambda_{1}-1+e^{-\lambda_{1}}\bigr) \\& \qquad {}\times \biggl[L_{2} \int_{0}^{1}e^{-\lambda _{2}(1-s)}(Q{\mathbf{1}}) (s)\,ds -bL_{1} \int_{0}^{\eta}e^{-\lambda_{1}(\eta-s)}(P{\mathbf{1}}) (s)\,ds \biggr] \\& \qquad {}-b\bigl(\lambda_{1}\eta-1+e^{-\lambda_{1}\eta}\bigr) \biggl[aL_{2} \int_{0}^{\xi }e^{-\lambda_{2}(\xi-s)}(Q{\mathbf{1}}) (s)\,ds -L_{1} \int_{0}^{1}e^{-\lambda_{1}(1-s)}(P{\mathbf{1}}) (s)\,ds \biggr]\biggr\} \biggr\vert \\& \qquad {}+L_{2}\biggl\vert \int_{0}^{t_{1}}\bigl(e^{-\lambda_{2}(t_{2}-s)}-e^{-\lambda _{2}(t_{1}-s)} \bigr) (Q{\mathbf{1}}) (s)\,ds + \int_{t_{1}}^{t_{2}}e^{-\lambda_{2}(t_{2}-s)}(Q{\mathbf{1}}) (s)\,ds \biggr\vert . \end{aligned}$$

Since for any fixed \(s\in[0,1]\), the functions \(e^{-\lambda_{i}(t-s)}\), \(A_{i}(t)\) (\(i=1,2\)) are uniformly continuous on the interval on [0,1], we can conclude that the operator \(T(u,v)\) is equicontinuous. Thus the operator \(T(u,v)\) is completely continuous. The proof is completed. □

Now we state a well-known fixed point theorem, which is needed to prove the existence of solutions for system (1.1).

Lemma 2.4

(Leray-Schauder alternative [36])

Let E be a Banach space. Assume that \(T:E\to E\) be a completely continuous operator. Let

$$V=\{x\in E|x=\mu Tx \textit{ for some } 0< \mu< 1\}. $$

Then either the set V is unbounded, or T has at least one fixed point.

3 Main results

Theorem 3.1

Assume that there exist real constants \(\rho_{i}, \delta_{i}\geq0\) and \(k_{i}>0\) (\(i=1,2\)) such that \(\forall t\in[0,1]\), \(x, y\in\mathbb {R}\),

$$ \bigl\vert f_{i}(t,x,y)\bigr\vert \leq k_{i}+\rho_{i}|x|+\delta_{i}|y|,\quad i=1,2. $$
(3.1)

In addition, assume that

$$\bigl(M_{1}+M_{1}'\bigr)\rho_{1}+ \bigl(M_{2}+M_{2}'\bigr)\rho_{2}< 1 \quad \textit{and} \quad \bigl(M_{1}+M_{1}' \bigr)\delta_{1}+\bigl(M_{2}+M_{2}' \bigr)\delta_{2}< 1, $$

where \(M_{i}\), \(M_{i}'\) (\(i=1,2\)) are defined by (2.21). Then the system (1.1) has at least one solution.

Proof

Let us verify that the set \(V=\{(u,v)\in X\times X: (u,v)=\mu T(u,v), 0\leq\mu\leq1\}\) is bounded. Let \((u,v)\in V\), then \((u,v)=\mu T(u,v)\). For any \(t\in[0,1]\), we have

$$ u(t)=\mu T_{1}(u,v) (t),\qquad v(t)=\mu T_{2}(u,v) (t). $$
(3.2)

Then, by (3.1), (3.2), (2.14), and (2.15), for any \(t\in[0,1]\), we have

$$\begin{aligned} \bigl\vert u(t)\bigr\vert \leq&\biggl\vert A_{1}(t) \biggl\{ a\bigl(\lambda_{2}\xi-1+e^{-\lambda_{2}\xi} \bigr) \\ &{}\times\biggl[ \int_{0}^{1}e^{-\lambda _{2}(1-s)}Q(u,v) (s)\,ds-b \int_{0}^{\eta}e^{-\lambda_{1}(\eta-s)}P(u,v) (s)\,ds \biggr] \\ &{}-\bigl(\lambda_{2}-1+e^{-\lambda_{2}}\bigr) \biggl[a \int_{0}^{\xi}e^{-\lambda_{2}(\xi -s)}Q(u,v) (s)\,ds - \int_{0}^{1}e^{-\lambda_{1}(1-s)}P(u,v) (s)\,ds \biggr] \biggr\} \\ &{}+ \int _{0}^{t}e^{-\lambda_{1}(t-s)}P(u,v) (s)\,ds\biggr\vert \\ \leq&A_{1} \biggl\{ |a|\bigl(\lambda_{2} \xi-1+e^{-\lambda_{2}\xi}\bigr) \biggl[\bigl(k_{2}+\rho_{2} \Vert u\Vert +\delta_{2}\Vert v\Vert \bigr) \int_{0}^{1}e^{-\lambda_{2}(1-s)}(Q{\mathbf{1}}) (s)\,ds \\ &{}+|b|\bigl(k_{1}+\rho_{1}\Vert u\Vert + \delta_{1}\Vert v\Vert \bigr) \int_{0}^{\eta}e^{-\lambda_{1}(\eta -s)}(P{\mathbf{1}}) (s)\,ds \biggr] \\ &{}+\bigl(\lambda_{2}-1+e^{-\lambda_{2}}\bigr) \biggl[|a| \bigl(k_{2}+\rho_{2}\Vert u\Vert +\delta_{2} \Vert v\Vert \bigr) \int_{0}^{\xi}e^{-\lambda_{2}(\xi-s)}(Q{\mathbf{1}}) (s)\,ds \\ &{}+\bigl(k_{1}+\rho_{1}\Vert u\Vert + \delta_{1}\Vert v\Vert \bigr) \int_{0}^{1}e^{-\lambda_{1}(1-s)}(P{\mathbf{1}}) (s)\,ds \biggr]\biggr\} \\ &{}+\bigl(k_{1}+\rho_{1}\Vert u\Vert + \delta_{1}\Vert v\Vert \bigr) \int_{0}^{t}e^{-\lambda_{1}(t-s)}(P{\mathbf{1}}) (s)\,ds \\ \leq& \frac{A_{1}[|ab|(\lambda_{2}\xi-1+e^{-\lambda_{2}\xi})(1-e^{-\lambda_{1}\eta })\eta^{p-1} +(\lambda_{2}-1+e^{-\lambda_{2}})(1-e^{-\lambda_{1}})]+(1-e^{-\lambda _{1}})}{\lambda_{1}\Gamma(p)} \\ &{} \times\bigl(k_{1}+\rho_{1}\Vert u\Vert + \delta_{1}\Vert v\Vert \bigr)+\bigl(k_{2}+ \rho_{2}\Vert u\Vert +\delta_{2}\Vert v\Vert \bigr) \\ &{}\times \frac{A_{1}[|a|(\lambda_{2}\xi-1+e^{-\lambda_{2}\xi})(1-e^{-\lambda_{2}}) +(\lambda_{2}-1+e^{-\lambda_{2}})|a|(1-e^{-\lambda_{2}\xi})\xi^{q-1}]}{\lambda _{2}\Gamma(q)} \\ \leq& M_{1}\bigl(k_{1}+\rho_{1}\Vert u \Vert +\delta_{1}\Vert v\Vert \bigr)+M_{2} \bigl(k_{2}+\rho_{2}\Vert u\Vert +\delta_{2} \Vert v\Vert \bigr). \end{aligned}$$
(3.3)

Hence we have

$$ \Vert u\Vert \leq M_{1}\bigl(k_{1}+ \rho_{1}\Vert u\Vert +\delta_{1}\Vert v\Vert \bigr)+M_{2}\bigl(k_{2}+\rho_{2}\Vert u\Vert + \delta_{2}\Vert v\Vert \bigr). $$
(3.4)

Similarly, proceeding as in (3.3), we can obtain

$$ \Vert v\Vert \leq M_{1}' \bigl(k_{1}+\rho_{1}\Vert u\Vert +\delta_{1} \Vert v\Vert \bigr)+M_{2}'\bigl(k_{2}+ \rho_{2}\Vert u\Vert +\delta_{2}\Vert v\Vert \bigr). $$
(3.5)

Combining (3.4) with (3.5), we obtain

$$\begin{aligned} \|u\|+\|v\| \leq&\bigl[\bigl(M_{1}+M_{1}' \bigr)k_{1}+\bigl(M_{2}+M_{2}' \bigr)k_{2}\bigr]+\bigl[\bigl(M_{1}+M_{1}' \bigr)\rho_{1}+\bigl(M_{2}+M_{2}' \bigr)\rho_{2}\bigr]\| u\| \\ &{}+\bigl[\bigl(M_{1}+M_{1}'\bigr) \delta_{1}+\bigl(M_{2}+M_{2}'\bigr) \delta_{2}\bigr]\|v\|. \end{aligned}$$

Consequently,

$$\bigl\Vert (u,v)\bigr\Vert _{1}=\|u\|+\|v\|\leq \frac{(M_{1}+M_{1}')k_{1}+(M_{2}+M_{2}')k_{2}}{M_{0}}, $$

where \(M_{0}=\min\{1-[(M_{1}+M_{1}')\rho_{1}+(M_{2}+M_{2}')\rho_{2}],1-[(M_{1}+M_{1}')\delta _{1}+(M_{2}+M_{2}')\delta_{2}]\}\), which proves that the set V is bounded. Thus, by Lemma 2.4, the operator T has at least one fixed point. Hence the system (1.1) has at least one solution. The proof is complete. □

Theorem 3.2

Assume that \(f_{i}:[0,1]\times\mathbb{R}^{2}\to\mathbb{R}\) are continuous functions and there exist constants \(m_{i}, n_{i}\geq0\) (\(i=1,2\)), such that \(\forall t\in[0,1]\), \(x_{i}, y_{i}\in \mathbb{R}\),

$$\bigl\vert f_{i}(t,x_{1},y_{1})-f_{i}(t,x_{2},y_{2}) \bigr\vert \leq m_{i}\vert x_{1}-x_{2}\vert +n_{i}\vert y_{1}-y_{2}\vert , \quad i=1,2. $$

In addition, assume that

$$\bigl(M_{1}+M_{1}'\bigr) (m_{1}+n_{1})+ \bigl(M_{2}+M_{2}'\bigr) (m_{2}+n_{2})< 1, $$

where \(M_{i}\), \(M_{i}'\) (\(i=1,2\)) are defined by (2.21). Then the system (1.1) has a unique solution.

Proof

Define \(\sup_{t\in[0,1]}|f_{i}(t,0,0)|=N_{i}<\infty\) (\(i=1,2\)) such that

$$r\geq\frac {(M_{1}+M_{1}')N_{1}+(M_{2}+M_{2}')N_{2}}{1-(M_{1}+M_{1}')(m_{1}+n_{1})-(M_{2}+M_{2}')(m_{2}+n_{2})}. $$

We show that \(TB_{r}\subset B_{r}\), where \(B_{r}=\{(u,v)\in X\times X: \|(u,v)\|_{1}< r\}\). For any \((u,v)\in B_{r}\), we have

$$\begin{aligned}& \bigl\vert T_{1}(u,v) (t)\bigr\vert \\& \quad \leq \max _{t\in[0,1]}\biggl\vert A_{1}(t) \biggl\{ a\bigl( \lambda_{2}\xi-1+e^{-\lambda_{2}\xi}\bigr) \\& \qquad {}\times \biggl[ \int_{0}^{1}e^{-\lambda _{2}(1-s)}Q(u,v) (s)\,ds-b \int_{0}^{\eta}e^{-\lambda_{1}(\eta-s)}P(u,v) (s)\,ds \biggr] \\& \qquad {} -\bigl(\lambda_{2}-1+e^{-\lambda_{2}}\bigr) \biggl[a \int_{0}^{\xi}e^{-\lambda_{2}(\xi -s)}Q(u,v) (s)\,ds - \int_{0}^{1}e^{-\lambda_{1}(1-s)}P(u,v) (s)\,ds \biggr] \biggr\} \\& \qquad {}+ \int _{0}^{t}e^{-\lambda_{1}(t-s)}P(u,v) (s)\,ds\biggr\vert \\& \quad \leq A_{1} \biggl\{ |a|\bigl(\lambda_{2} \xi-1+e^{-\lambda_{2}\xi}\bigr) \biggl[\bigl(m_{2}\Vert u\Vert +n_{2}\Vert v\Vert +N_{2}\bigr) \int_{0}^{1}e^{-\lambda_{2}(1-s)}(Q{\mathbf{1}}) (s)\,ds \\& \qquad {}+|b|\bigl(m_{1}\Vert u\Vert +n_{1}\Vert v\Vert +N_{1}\bigr) \int_{0}^{\eta}e^{-\lambda_{1}(\eta-s)}(P{\mathbf{1}}) (s)\,ds \biggr] \\& \qquad {}+\bigl(\lambda_{2}-1+e^{-\lambda_{2}}\bigr) \biggl[|a| \bigl(m_{2}\Vert u\Vert +n_{2}\Vert v\Vert +N_{2}\bigr) \int _{0}^{\xi}e^{-\lambda_{2}(\xi-s)}(Q{\mathbf{1}}) (s) \,ds \\& \qquad {}+\bigl(m_{1}\Vert u\Vert +n_{1}\Vert v\Vert +N_{1}\bigr) \int_{0}^{1}e^{-\lambda_{1}(1-s)}(P{\mathbf{1}}) (s)\,ds \biggr]\biggr\} \\& \qquad {}+\bigl(m_{1}\Vert u\Vert +n_{1}\Vert v\Vert +N_{1}\bigr) \int_{0}^{t}e^{-\lambda_{1}(t-s)}(P{\mathbf{1}}) (s)\,ds \\& \quad \leq \frac{A_{1}[|ab|(\lambda_{2}\xi-1+e^{-\lambda_{2}\xi})(1-e^{-\lambda_{1}\eta })\eta^{p-1} +|(\lambda_{2}-1+e^{-\lambda_{2}})|(1-e^{-\lambda_{1}})]+(1-e^{-\lambda _{1}})}{\lambda_{1}\Gamma(p)} \\& \qquad {}\times\bigl(m_{1}\Vert u\Vert +n_{1}\Vert v \Vert +N_{1}\bigr) +\bigl(m_{2}\Vert u\Vert +n_{2}\Vert v\Vert +N_{2}\bigr) \\& \qquad {}\times\frac{A_{1}[|a|(\lambda_{2}\xi -1+e^{-\lambda_{2}\xi})(1-e^{-\lambda_{2}}) +(\lambda_{2}-1+e^{-\lambda_{2}})|a|(1-e^{-\lambda_{2}\xi})\xi^{q-1}]}{\lambda _{2}\Gamma(q)} \\& \quad \leq M_{1}\bigl[(m_{1}+n_{1})r+N_{1} \bigr]+M_{2}\bigl[(m_{2}+n_{2})r+N_{2} \bigr]. \end{aligned}$$
(3.6)

Hence

$$\bigl\Vert T_{1}(u,v)\bigr\Vert \leq M_{1} \bigl[(m_{1}+n_{1})r+N_{1}\bigr]+M_{2} \bigl[(m_{2}+n_{2})r+N_{2}\bigr]. $$

Similarly, for any \((u,v)\in B_{r}\), proceeding as in (3.6), we can get

$$\bigl\Vert T_{2}(u,v)\bigr\Vert \leq M_{1}' \bigl[(m_{1}+n_{1})r+N_{1}\bigr]+M_{2}' \bigl[(m_{2}+n_{2})r+N_{2}\bigr]. $$

Consequently,

$$\begin{aligned} \bigl\Vert T(u,v)\bigr\Vert _{1} =&\bigl\Vert T_{1}(u,v)\bigr\Vert +\bigl\Vert T_{2}(u,v)\bigr\Vert \\ \leq& M_{1}\bigl[(m_{1}+n_{1})r+N_{1} \bigr]+M_{2}\bigl[(m_{2}+n_{2})r+N_{2} \bigr] \\ &{}+ M_{1}'\bigl[(m_{1}+n_{1})r+N_{1} \bigr]+M_{2}'\bigl[(m_{2}+n_{2})r+N_{2} \bigr] \\ \leq&r. \end{aligned}$$

Now for \((u_{1},v_{1}), (u_{2},v_{2})\in X\times X\), and for any \(t\in[0,1]\), we have

$$\begin{aligned}& \bigl\vert T_{1}(u_{2},v_{2}) (t)-T_{1}(u_{1},v_{1}) (t)\bigr\vert \\& \quad \leq \biggl\vert A_{1}(t) \biggl\{ a\bigl(\lambda_{2} \xi-1+e^{-\lambda_{2}\xi}\bigr) \biggl[ \int_{0}^{1}e^{-\lambda _{2}(1-s)}\bigl[Q(u_{2},v_{2}) (s)-Q(u_{1},v_{1}) (s)\bigr]\,ds \\& \qquad {}-b \int_{0}^{\eta}e^{-\lambda_{1}(\eta -s)}\bigl[P(u_{2},v_{2}) (s)-P(u_{1},v_{1}) (s)\bigr]\,ds \biggr] \\& \qquad {}-\bigl(\lambda_{2}-1+e^{-\lambda_{2}}\bigr) \biggl[a \int_{0}^{\xi}e^{-\lambda_{2}(\xi -s)}\bigl[Q(u_{2},v_{2}) (s)-Q(u_{1},v_{1}) (s)\bigr]\,ds \\& \qquad {}- \int_{0}^{1}e^{-\lambda_{1}(1-s)}\bigl[P(u_{2},v_{2}) (s)-P(u_{1},v_{1}) (s)\bigr]\,ds \biggr]\biggr\} \\& \qquad {}+ \int_{0}^{t}e^{-\lambda_{1}(t-s)}\bigl[P(u_{2},v_{2}) (s)-P(u_{1},v_{1}) (s)\bigr]\,ds\biggr\vert \\& \quad \leq A_{1} \biggl\{ |a|\bigl(\lambda_{2} \xi-1+e^{-\lambda_{2}\xi}\bigr) \biggl[\bigl(m_{2}\Vert u_{2}-u_{1}\Vert +n_{2}\Vert v_{2}-v_{1}\Vert \bigr) \int_{0}^{1}e^{-\lambda_{2}(1-s)}(Q{\mathbf{1}}) (s)\,ds \\& \qquad {}+|b|\bigl(m_{1}\Vert u_{2}-u_{1}\Vert +n_{1}\Vert v_{2}-v_{1}\Vert \bigr) \int_{0}^{\eta}e^{-\lambda_{1}(\eta -s)}(P{\mathbf{1}}) (s)\,ds \biggr] \\& \qquad {}+\bigl(\lambda_{2}-1+e^{-\lambda_{2}}\bigr) \biggl[|a| \bigl(m_{2}\Vert u_{2}-u_{1}\Vert +n_{2}\Vert v_{2}-v_{1}\Vert \bigr) \int_{0}^{\xi}e^{-\lambda_{2}(\xi-s)}(Q{\mathbf{1}}) (s)\,ds \\& \qquad {}+\bigl(m_{1}\Vert u_{2}-u_{1}\Vert +n_{1}\Vert v_{2}-v_{1}\Vert \bigr) \int_{0}^{1}e^{-\lambda_{1}(1-s)}(P{\mathbf{1}}) (s)\,ds \biggr]\biggr\} \\& \qquad {}+\bigl(m_{1}\Vert u_{2}-u_{1}\Vert +n_{1}\Vert v_{2}-v_{1}\Vert \bigr) \int_{0}^{t}e^{-\lambda_{1}(t-s)}(P{\mathbf{1}}) (s)\,ds \\& \quad \leq \frac{A_{1}[|ab|(\lambda_{2}\xi-1+e^{-\lambda_{2}\xi})(1-e^{-\lambda_{1}\eta })\eta^{p-1} +(\lambda_{2}-1+e^{-\lambda_{2}})(1-e^{-\lambda_{1}})]+(1-e^{-\lambda _{1}})}{\lambda_{1}\Gamma(p)} \\& \qquad {}\times\bigl(m_{1}\Vert u_{2}-u_{1}\Vert +n_{1}\Vert v_{2}-v_{1}\Vert \bigr)+ \bigl(m_{2}\Vert u_{2}-u_{1}\Vert +n_{2}\Vert v_{2}-v_{1}\Vert \bigr) \\& \qquad {}\times\frac{A_{1}[|a|(\lambda_{2}\xi-1+e^{-\lambda_{2}\xi})(1-e^{-\lambda_{2}}) +(\lambda_{2}-1+e^{-\lambda_{2}})|a|(1-e^{-\lambda_{2}\xi})\xi^{q-1}]}{\lambda _{2}\Gamma(q)} \\& \quad \leq M_{1}\bigl(m_{1}\Vert u_{2}-u_{1} \Vert +n_{1}\Vert v_{2}-v_{1}\Vert \bigr)+M_{2}\bigl(m_{2}\Vert u_{2}-u_{1} \Vert +n_{2}\Vert v_{2}-v_{1}\Vert \bigr) \\& \quad \leq \bigl[M_{1}(m_{1}+n_{1})+M_{2}(m_{2}+n_{2}) \bigr]\bigl(\Vert u_{2}-u_{1}\Vert +\Vert v_{2}-v_{1}\Vert \bigr). \end{aligned}$$

Consequently, for \((u_{1},v_{1}), (u_{2},v_{2})\in X\times X\), we obtain

$$\begin{aligned}& \bigl\Vert T_{1}(u_{2},v_{2})-T_{1}(u_{1},v_{1}) \bigr\Vert \\& \quad \leq \bigl[M_{1}(m_{1}+n_{1})+M_{2}(m_{2}+n_{2}) \bigr]\bigl(\Vert u_{2}-u_{1}\Vert +\Vert v_{2}-v_{1}\Vert \bigr). \end{aligned}$$
(3.7)

Similarly, for \((u_{1},v_{1}), (u_{2},v_{2})\in X\times X\), we can obtain

$$\begin{aligned}& \bigl\Vert T_{2}(u_{2},v_{2})-T_{2}(u_{1},v_{1}) \bigr\Vert \\& \quad \leq \bigl[M_{1}'(m_{1}+n_{1})+M_{2}'(m_{2}+n_{2}) \bigr]\bigl(\Vert u_{2}-u_{1}\Vert +\Vert v_{2}-v_{1}\Vert \bigr). \end{aligned}$$
(3.8)

It follows from (3.7) and (3.8) that

$$\begin{aligned}& \bigl\Vert T(u_{2},v_{2})-T(u_{1},v_{1}) \bigr\Vert _{1} \\& \quad \leq \bigl[\bigl(M_{1}+M_{1}' \bigr) (m_{1}+n_{1})+\bigl(M_{2}+M_{2}' \bigr) (m_{2}+n_{2})\bigr]\bigl(\Vert u_{2}-u_{1} \Vert +\Vert v_{2}-v_{1}\Vert \bigr). \end{aligned}$$

Since \((M_{1}+M_{1}')(m_{1}+n_{1})+(M_{2}+M_{2}')(m_{2}+n_{2})<1\), T is a contraction operator. So, by the fixed point theorem of the contraction mapping principle, the operator T has a unique fixed point, which is the unique solution of the system (1.1). The proof is complete. □

4 Applications

Example 4.1

Consider the fractional differential system

$$ \left \{ \textstyle\begin{array}{l} {}^{\mathrm{c}}D^{\frac{5}{2}}(D+2)u(t)=f_{1}(t,u(t),v(t)), \quad 0< t< 1 , \\ {}^{\mathrm{c}}D^{\frac{9}{4}}(D+3)v(t)=f_{2}(t,u(t),v(t)), \quad 0< t< 1, \\ u(0)=u'(0)=0, \qquad u(1)=a v(\xi), \\ v(0)=v'(0)=0,\qquad v(1)=b u(\eta), \end{array}\displaystyle \right . $$
(4.1)

where \(\lambda_{1}=2\), \(\lambda_{2}=3\) are two positive parameters. We take \(a=\frac{2}{3}\), \(b=3e\), \(\xi=\frac{1}{3}\), \(\eta=\frac{1}{2}\).

By direct calculation, we have

$$ \begin{aligned} &\Lambda \approx-1.5914367\neq0, \\ &A_{1} \approx0.71340273,\qquad A_{2}\approx1.28801043, \\ &M_{1} \approx0.92074193,\qquad M_{2}\approx0.09483186, \\ &M_{1}' \approx2.25908866,\qquad M_{2}' \approx0.80969458. \end{aligned} $$
(4.2)

Let

$$\begin{aligned}& f_{1}(t,x,y)=\frac{t}{t^{2}+1} \biggl(2+\frac{1}{10}\sin x+ \frac{1}{8}\cos^{2} y \biggr), \quad t\in[0,1], x,y\in\mathbb{R}, \\& f_{2}(t,x,y)=\frac{1}{(t+2)^{2}}(2+x+2\sin y), \quad t\in[0,1], x,y\in \mathbb{R}. \end{aligned}$$

Notice that

$$\begin{aligned}& \bigl\vert f_{1}(t,x,y)\bigr\vert =\biggl\vert \frac{t}{t^{2}+1}\biggl(2+\frac{1}{10}\sin x+\frac{1}{8} \cos^{2} y\biggr)\biggr\vert \leq2+\frac{1}{10}\vert x\vert +\frac{1}{8}\vert y\vert , \\& \bigl\vert f_{2}(t,x,y)\bigr\vert =\biggl\vert \frac{1}{(t+2)^{2}}(2+x+2\sin y)\biggr\vert \leq\frac{1}{2}+ \frac{1}{4}\vert x\vert +\frac{1}{2}\vert y\vert , \end{aligned}$$

and

$$\begin{aligned}& \bigl(M_{1}+M_{1}'\bigr)\rho_{1}+ \bigl(M_{2}+M_{2}'\bigr)\rho_{2} \approx0.54411467< 1, \\& \bigl(M_{1}+M_{1}'\bigr)\delta_{1}+ \bigl(M_{2}+M_{2}'\bigr)\delta_{2} \approx0.84974204< 1. \end{aligned}$$

Therefore, all conditions of Theorem 3.1 are satisfied, and hence by Theorem 3.1 the system (4.1) has at least one solution.

Example 4.2

In Example 4.1, we only change \(f_{1}\), \(f_{2}\), and keep the other conditions unchanged. Λ, \(A_{1}\), \(A_{2}\), \(M_{1}\), \(M_{2}\), \(M_{1}'\), \(M_{2}'\) are as in (4.2). Let

$$ \begin{aligned} & f_{1}(t,x,y)=\frac{t}{18(t+1)^{2}} x+ \frac{1}{9}\arctan y, \quad t\in[0,1], x,y\in\mathbb{R}, \\ & f_{2}(t,x,y)=-\frac{1}{8}\cos x+\frac{1}{12}\sin y, \quad t\in[0,1], x,y\in\mathbb{R}. \end{aligned} $$
(4.3)

Noticing

$$\begin{aligned}& \bigl\vert f_{1}(t,x_{1},y_{1})-f_{1}(t,x_{2},y_{2}) \bigr\vert \leq\frac{1}{18}\vert x_{1}-x_{2}\vert +\frac {1}{9}\vert y_{1}-y_{2}\vert , \\& \bigl\vert f_{2}(t,x_{1},y_{1})-f_{2}(t,x_{2},y_{2}) \bigr\vert \leq\frac{1}{8}\vert x_{1}-x_{2}\vert +\frac{1}{12}\vert y_{1}-y_{2}\vert , \end{aligned}$$

and

$$\bigl(M_{1}+M_{1}'\bigr) (m_{1}+n_{1})+ \bigl(M_{2}+M_{2}'\bigr) (m_{2}+n_{2}) \approx0.71841477< 1. $$

Thus all conditions of Theorem 3.2 are satisfied and, consequently, the system (4.1) has a unique solution (with \(f_{1}\), \(f_{2}\) as in (4.3)).