Complex Boundary Value Problems of Nonlinear Differential Equations: Theory, Computational Methods, and ApplicationsView this Special Issue
Nontrivial Solution of Fractional Differential System Involving Riemann-Stieltjes Integral Condition
We study the existence and uniqueness of nontrivial solutions for a class of fractional differential system involving the Riemann-Stieltjes integral condition, by using the Leray-Schauder nonlinear alternative and the Banach contraction mapping principle, some sufficient conditions of the existence and uniqueness of a nontrivial solution of a system are obtained.
HIV is a retrovirus that targets the CD4+ T lymphocytes, which are the most abundant white blood cells of the immune system. To this day, there have already been over 16 million people who died of AIDS. Although HIV infects other cells also, it wreaks the most havoc on the CD4+ T cells by causing their decline and destruction, thus decreasing the resistance of the immune system [1–3]. Mathematical models have been proven valuable in understanding the dynamics of HIV infection [4–6]. Perelson et al. [7, 8] 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 virus population. And the following two equations describe the evolution of the system: where all parameters and variables are nonnegative. is the assumed constant rate of the production of CD4+ T-cells, is their per capita death rate, is the rate of infection of CD4+ T-cells by virus, and is the rate of disappearance of infected cells. Recently Arafa1 et al. introduced fractional order into a model of infection of CD4+ T cells. The new system is described by the following set of FODEs of order : , and denote the concentration of uninfected CD4+ T cells, infected CD4+ T cells, and free HIV virus particles in the blood, respectively. represents death rate of infected T cells and includes the possibility of death by the bursting of infected T cells, hence . The parameter is the rate at which infected cells return to uninfected class while is the death rate of virus and is the average number of viral particles produced by an infected cell.
Motivated by HIV model, in this paper, we consider the existence of nontrivial solution for fractional differential system where is a parameter, ,, is the standard Riemann-Liouville derivative. denotes the Riemann-Stieltjes integral, and are functions of bounded variation.
In the recent years, there has been a significant development in fractional order differential equations involving fractional derivatives. For example, Ahmad and Nieto  considered a coupled system of nonlinear fractional differential equations with three-point boundary conditions where satisfy certain conditions. Applying the Schauder fixed point theorem, an existence result is proved provided that are given continuous functions and satisfy some growth conditions. For a detailed description of recent work on fractional differential equation, we refer the reader to some recent papers (see [10–17]).
The rest of the paper is organized as follows. Section 2 gives preliminaries and lemmas about fractional calculus. In Section 3, we present the main results and the proof of the results. In addition, an example is given to illustrate the application of the main results.
2. Preliminaries and Lemmas
For the convenience of the reader, we present here some definitions from fractional calculus which are to be used in the later sections.
Definition 2.2 (see [18–20]). The Riemann-Liouville fractional derivative of order of a function is given by where , denotes the integer part of number , provided that the right-hand side is pointwise defined on .
By Lemma 2.4, the unique solution of the problem is . Let and define
As in , if , we can get that the Green function for the following nonlocal system are given by, respectively,
Clearly, are continuous on ; thus there exist positive constants such that
It is well known that is a solution of the system (2.6) if and only if is a solution of the following nonlinear integral equation system: Obviously, the system (2.16) is equivalent to the following integral equation:
Lemma 2.6 (see ). Let be a real Banach space and a bounded open subset of , where ; is a completely continuous operator. Then, either there exists such that or there exists a fixed point .
3. Main Results
The following definition introduces the Carathèodory conditions imposed on a map .
Definition 3.1. Let . A map is said to satisfy the Carathéodory conditions if the following conditions hold:(i)for each , the mapping is Lebesgue measurable;(ii)for a.e. , the mapping is continuous on .
Throughout the paper we always assume the following conditions hold. are functions of bounded variation such that , where are defined by (2.11). and satisfy the Carathéodory condition.
Theorem 3.2. Suppose that hold. If , and there exist nonnegative functions such that In addition, there exists such that for some . Then there exists a constant , such that, for any , the system (1.1) has at least one nontrivial solution .
Proof. Let be endowed with the ordering if for all , and is defined as usual by maximum norm. Clearly, it follows that is a Banach space. By (2.17) and (2.6), problem (1.1) has a solution if and only if solves the following operator equation:
in . So we only need to seek a fixed point of in . By Ascoli-Arzela Theorem, it is obvious that the operator is a completely continuous operator.
Since , and a.e. , we know . On the other hand, by for some , we have
Let where is defined by (2.15). Define the set Suppose such that . Then, for any , and noticing that we have Thus we take Then for any and , by (3.7), one has Consequently, This contradicts , by Lemma 2.6, has a fixed point . Since , then , Thus let and then the system (1.1) has at least one nontrivial solution for any . This completes the proof of Theorem 3.2.
Theorem 3.3. Suppose that hold. If , and there exist nonnegative functions such that In addition, there exists such that for some . Then there exists a constant , such that, for any , the system (1.1) has unique nontrivial solution .
Proof. Let be given in Theorem 3.2; we will show that is a contraction. In fact, If we choose Then by (3.13) and (3.14), we have which implies that is indeed a contraction. Finally, we use the Banach fixed point theorem to deduce the existence of a unique solution to the system (1.1).
Corollary 3.4. Suppose that hold. If , In addition, there exists a nonnegative function such that satisfies Then there exists a constant , such that, for any , the system (1.1) has at least one nontrivial solution .
Proof. We prove satisfies the conditions of Theorem 3.2. Let and choose such that . By (3.16), there exists such that Take , and the measure of is 0. Thus for any , we have From Theorem 3.2 we know the system (1.1) has at least one nontrivial solution.
Example 3.5. Consider the following fractional differential system:
Then the system (3.21) is equivalent to the following 4-point BVP with coefficients of both signs
Clearly, holds. Let
Then also is satisfied.
On the other hand, we have and , which imply all conditions of Theorem 3.2 are satisfied, by Theorem 3.2, there exists a constant , such that for any , the system (3.21) has at least one nontrivial solution .
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