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

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.

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