Existence of solutions for a sequential fractional differential system with coupled boundary conditions

This paper is concerned with the existence and uniqueness of solutions for a sequential fractional differential system with coupled boundary conditions. The existence of solutions is derived by applying Leray-Schauder’s alternative, while the uniqueness of the solution is established via Banach’s contraction principle. Two examples are then given to demonstrate the validity of our main results.

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:

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 $\alpha_{1}, \alpha_{2}, \alpha_{3}>0$:

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:

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:

subject to the coupled BCs

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,

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.

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

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

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

and the BCs

has a unique integral representation

where

Proof

Solving (2.1), we obtain

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

and

Note that

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

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

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

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

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

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

and

with

For the sake of convenience, we set

and

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

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

which implies that

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

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

Combining (2.24) with(2.25), we obtain

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

Analogously, we can obtain the following inequalities:

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

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

Theorem 3.1

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

In addition, assume that

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

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

Hence we have

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

Combining (3.4) with (3.5), we obtain

Consequently,

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 $\forall t\in[0,1]$, $x_{i}, y_{i}\in \mathbb{R}$,

In addition, assume that

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

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

Hence

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

Consequently,

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

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

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

It follows from (3.7) and (3.8) that

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. □

Example 4.1

Consider the fractional differential system

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

Let

Notice that

and

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

Noticing

and

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)).

The authors would like to thank the referee for his/her very important comments, which improved the results and the quality of the paper. This work was supported financially by the National Natural Science Foundation of China (11371221), the Natural Science Foundation of Shandong Province of China (ZR2014AL004, ZR2013AQ014) and the Project of Shandong Province Higher Educational Science and Technology Program (J14LI08, J13LI08), Doctoral Scientific Research Foundation of Qufu Normal University and Youth Foundation of Qufu Normal University (BSQD20130140).

Affiliations

1. School of Statistics, Qufu Normal University, Qufu, Shandong, 273165, P.R. China

   • Jiqiang Jiang

2. School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, 273165, P.R. China

   • Jiqiang Jiang
   •  & Lishan Liu

Corresponding author

Correspondence to Jiqiang Jiang.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

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