Nontrivial Solution of Fractional Differential System Involving Riemann-Stieltjes Integral Condition

Abstract

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.

1. Introduction

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: 𝑑𝑥𝑑𝑡=𝑠−𝜇𝑥−𝛽𝑥𝑦𝑑𝑦𝑑𝑡=𝛽𝑥𝑦−𝜐𝑦, (HIV-1) where all parameters and variables are nonnegative. 𝑠 is the assumed constant rate of the production of CD4⁺ T-cells, 𝜇 is their per capita death rate, 𝛽𝑥𝑦 is the rate of infection of CD4⁺ T-cells by virus, and 𝑣𝑦 is the rate of disappearance of infected cells. Recently Arafa1 et al. introduced fractional order into a model of (HIV-1) infection of CD4⁺ T cells. The new system is described by the following set of FODEs of order 𝛼₁,𝛼₂,𝛼₃>0: 𝐷^𝛼1(𝑇)𝐷=𝑠−𝐾𝑉𝑇−𝑑𝑇+𝑏𝐼,_𝛼2(𝐼)()𝐷=𝐾𝑉𝑇−𝑏+𝛿𝐼,_𝛼3(𝐼)=𝑁𝛿𝐼−𝑐𝑉. (HIV-2)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 death rate of infected T cells and includes the possibility of death by the bursting of infected T cells, hence 𝛿𝑑. The parameter 𝑏 is the rate at which infected cells return to uninfected class while 𝑐 is the death rate of virus and 𝑁 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 −𝒟^𝛼_𝐭𝑥(𝑡)=𝜆𝑓𝑡,𝑥(𝑡),𝒟^𝛽_𝐭𝑥(𝑡),𝑦(𝑡),−𝒟^𝛾_𝐭𝑦(𝑡)((𝑡(),𝒟=𝑔𝑡,𝑥)),𝑡∈0,1_𝛽𝐭𝑥(0)=0,𝒟^𝛽_𝐭𝑥(1)=¹₀𝒟^𝛽_𝐭𝑥(𝑠)𝑑𝐴(𝑠),𝑦(0)=0,𝑦(1)=¹₀𝑦(𝑠)𝑑𝐵(𝑠), (1.1) where 𝜆 is a parameter, 1<𝛾<𝛼≤2,1<𝛼−𝛽<𝛾,0<𝛽<1,𝒟^𝛼_𝐭 is the standard Riemann-Liouville derivative. ∫¹₀𝒟^𝛽_𝐭𝑥(𝑠)𝑑𝐴(𝑠) denotes the Riemann-Stieltjes integral, and 𝐴,𝐵∈𝑁𝐵𝑉([0,1]) 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 [9] considered a coupled system of nonlinear fractional differential equations with three-point boundary conditions 𝒟^𝛼_𝐭𝑢(𝑡)=𝑓𝑡,𝑣(𝑡),𝒟^𝑝_𝐭𝑣(𝑡),𝒟^𝛽_𝐭𝑣(𝑡)=𝑓𝑡,𝑢(𝑡),𝒟^𝑞_𝐭𝑢(𝑡)(),𝑢(0)(1)𝜂(0)(1)𝜂,,𝑡∈0,1=0,𝑢=𝛾𝑢,𝑣=0,𝑣=𝛾𝑣 (1.2) where 𝛼,𝛽,𝑝,𝑞,𝛾,𝜂 satisfy certain conditions. Applying the Schauder fixed point theorem, an existence result is proved provided that 𝑓,𝑔∶[0,1]×ℝ×ℝ→ℝ 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 2.1 (see [18–20]). The Riemann-Liouville fractional integral of order 𝛼>0 of a function 𝑥∶(0,+∞)→ℝ is given by 𝐼^𝛼𝑥(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑥(𝑠)𝑑𝑠 (2.1) provided that the right-hand side is pointwise defined on (0,+∞).

Definition 2.2 (see [18–20]). The Riemann-Liouville fractional derivative of order 𝛼>0 of a function 𝑥∶(0,+∞)→ℝ is given by 𝒟^𝛼_𝐭𝑥(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑡₀(𝑡−𝑠)^{𝑛−𝛼−1}𝑥(𝑠)𝑑𝑠, (2.2) where 𝑛=[𝛼]+1, [𝛼] denotes the integer part of number 𝛼, provided that the right-hand side is pointwise defined on (0,+∞).

Lemma 2.3 (see [18–20]). (1) If 𝑥∈𝐿(0,1),𝜈>𝜎>0, then 𝐼^𝜈𝐼^𝜎𝑥(𝑡)=𝐼^𝜈+𝜎𝑥(𝑡),𝒟^𝜎_𝐭𝐼^𝜈𝑥(𝑡)=𝐼^𝜈−𝜎𝑥(𝑡),𝒟^𝜎_𝐭𝐼^𝜎𝑥(𝑡)=𝑥(𝑡). (2.3)
(2) If 𝜈>0,𝜎>0, then 𝒟^𝜈_𝐭𝑡^𝜎−1=Γ(𝜎)Γ()𝑡𝜎−𝜈^{𝜎−𝜈−1}. (2.4)

Lemma 2.4 (see [18–20]). Assume that 𝑥∈𝐿¹(0,1) with a fractional derivative of order 𝛼>0 that belongs to 𝐿¹(0,1). Then 𝐼^𝛼𝒟^𝛼_𝐭𝑥(𝑡)=𝑥(𝑡)+𝑐₁𝑡^𝛼−1+𝑐₂𝑡^𝛼−2+⋯+𝑐_𝑛𝑡^𝛼−𝑛, (2.5) where 𝑐_𝑖∈ℝ(𝑖=1,2,…,𝑛), 𝑛 is the smallest integer greater than or equal to 𝛼.

Let 𝑥(𝑡)=𝐼^𝛽𝑣(𝑡),𝑣(𝑡)∈𝐶[0,1]; by standard discussion, we easily reduce the system (1.1) to the following modified problems: −𝒟_𝐭^𝛼−𝛽𝑣(𝑡)=𝜆𝑓𝑡,𝐼^𝛽𝑣(𝑡),𝑣(𝑡),𝑦(𝑡),−𝒟^𝛾_𝐭𝑦(𝑡)=𝑔𝑡,𝐼^𝛽𝑣(𝑡)(),𝑣(0)(1),𝑡∈0,1=0,𝑣=¹₀𝑣(𝑠)𝑑𝐴(𝑠),𝑦(0)=0,𝑦(1)=¹₀𝑦(𝑠)𝑑𝐵(𝑠), (2.6) and the system (2.6) is equivalent to the system (1.1).

Lemma 2.5 (see [21]). Let ℎ∈𝐿¹(0,1), if 1<𝛼−𝛽,𝛾≤2, then the unique solution of the linear problems −𝒟_𝐭^𝛼−𝛽𝑣(𝑡)(𝑡)(),𝑣(0)(1)=ℎ,𝑡∈0,1=0,𝑣=0,−𝒟^𝛾_𝐭𝑦(𝑡)(𝑡)(),𝑦(0)(1)=ℎ,𝑡∈0,1=0,𝑦=0, (2.7) is 𝑣(𝑡)=¹₀𝐾₁(𝑡,𝑠)ℎ(𝑠)𝑑𝑠,𝑦(𝑡)=¹₀𝐾₂(𝑡,𝑠)ℎ(𝑠)𝑑𝑠, (2.8) respectively, where 𝐾₁()=1𝑡,𝑠Γ[𝑡(𝛼−𝛽1−𝑠)]^{𝛼−𝛽−1}[𝑡(,0≤𝑡≤𝑠≤1,1−𝑠)]^{𝛼−𝛽−1}−(𝑡−𝑠)^{𝛼−𝛽−1}𝐾,0≤𝑠≤𝑡≤1.₂()=1𝑡,𝑠Γ𝛾[𝑡(1−𝑠)]^𝛾−1[𝑡(,0≤𝑡≤𝑠≤1,1−𝑠)]^𝛾−1−(𝑡−𝑠)^𝛾−1,0≤𝑠≤𝑡≤1, (2.9) are the Green functions of the boundary value problems (2.7).

By Lemma 2.4, the unique solution of the problem 𝒟_𝐭^𝛼−𝛽𝑣(𝑡)𝑣(0)(1)=0,0<𝑡<1,=0,𝑣=1, (2.10) is 𝑡^{𝛼−𝛽−1}. Let 𝒞=¹₀𝑡^{𝛼−𝛽−1}𝑑𝐴(𝑡),ℬ=¹₀𝑡^𝛾−1𝑑𝐵(𝑡), (2.11) and define 𝒢_𝐴(𝑠)=¹₀𝐾₁(𝑡,𝑠)𝑑𝐴(𝑡),𝒢_𝐵(𝑠)=¹₀𝐾₂(𝑡,𝑠)𝑑𝐵(𝑡). (2.12)

As in [14], if 𝒞≠0,ℬ≠0, we can get that the Green function for the following nonlocal system −𝒟_𝐭^𝛼−𝛽𝑣(𝑡)(𝑡)(),𝑣(0)(1)==ℎ,𝑡∈0,1=0,𝑣¹₀𝑣(𝑠)𝑑𝐴(𝑠),−𝒟^𝛾_𝐭𝑦(𝑡)(𝑡)(),𝑦(0)(1)==ℎ,𝑡∈0,1=0,𝑦¹₀𝑦(𝑠)𝑑𝐵(𝑠), (2.13) are given by, respectively, 𝐺()=𝑡𝑡,𝑠^{𝛼−𝛽−1}𝒢1−𝒞^𝐴(𝑠)+𝐾₁()()=𝑡𝑡,𝑠,𝐻𝑡,𝑠^𝛾−1𝒢1−ℬ^𝐵(𝑠)+𝐾₂(𝑡,𝑠). (2.14)

Clearly, 𝐺(𝑡,𝑠),𝐻(𝑡,𝑠) are continuous on [0,1]×[0,1]; thus there exist positive constants 𝑚,𝑛 such that |𝐺(𝑡,𝑠)|≤𝑚,|𝐻(𝑡,𝑠)|≤𝑛. (2.15)

It is well known that (𝑣,𝑦) is a solution of the system (2.6) if and only if (𝑣,𝑦)∈𝐶[0,1]×𝐶[0,1] is a solution of the following nonlinear integral equation system: 𝑣(𝑡)=𝜆¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝐼^𝛽𝑣(𝑠)(𝑠)(𝑠)𝑦(𝑡)=,𝑣,𝑦𝑑𝑠,₁₀𝐻(𝑡,𝑠)𝑔𝑠,𝐼^𝛽𝑣(𝑠)𝑑𝑠. (2.16) Obviously, the system (2.16) is equivalent to the following integral equation: 𝑣(𝑡)=𝜆¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝐼^𝛽𝑣(𝑠),𝑣(𝑠),¹₀𝐻(𝑠,𝜏)𝑔𝜏,𝐼^𝛽𝑣(𝜏)𝑑𝜏𝑑𝑠. (2.17)

Lemma 2.6 (see [22]). Let 𝑋 be a real Banach space and Ω a bounded open subset of 𝑋, where 𝜃∈Ω; 𝑇∶Ω→𝑋 is a completely continuous operator. Then, either there exists 𝑥∈𝜕Ω,𝜆>1 such that 𝑇(𝑥)=𝜆𝑥 or there exists a fixed point 𝑥^∗∈Ω.

3. Main Results

The following definition introduces the Carathèodory conditions imposed on a map 𝑓.

Definition 3.1. Let 𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑛). A map 𝑓∶[0,1]×ℝ^𝑛,(𝑡,𝑧)↦𝑓(𝑡,𝑧) is said to satisfy the Carathéodory conditions if the following conditions hold:(i)for each 𝑧∈ℝ^𝑛, the mapping 𝑡↦𝑓(𝑡,𝑧) is Lebesgue measurable;(ii)for a.e. 𝑡∈[0,1], the mapping 𝑧↦𝑓(𝑡,𝑧) is continuous on ℝ.

Throughout the paper we always assume the following conditions hold. (H0)𝐴,𝐵 are functions of bounded variation such that 𝒞,ℬ≠1, where 𝒞,ℬ are defined by (2.11). (H1)𝑓∶[0,1]×ℝ³→ℝ and 𝑔∶[0,1]×ℝ→ℝ satisfy the Carathéodory condition.

Theorem 3.2. Suppose that (H0)-(H1) hold. If 𝑓(𝑡,0,0,0)≢0, and there exist nonnegative functions 𝑝_𝑖(𝑖=1,2,3),𝑞,𝑟∈𝐿¹[0,1] such that 𝑓𝑡,𝑥₁,𝑥₂,𝑥₃≤𝑝₁(𝑡)𝑥₁+𝑝₂(𝑡)𝑥₂+𝑝₃(𝑡)𝑥₃(𝑡)[],𝑔(𝑡𝑦[].+𝑞,a.e.𝑡∈0,1𝑡,𝑦≤𝑟),a.e.𝑡∈0,1 (3.1) In addition, there exists 𝑡₀∈[0,1] such that 𝑝_𝑖0(𝑡₀)≠0 for some 𝑖₀∈{1,2,3}. Then there exists a constant 𝜆^∗>0, such that, for any 0<𝜆≤𝜆^∗, the system (1.1) has at least one nontrivial solution (𝑥^∗,𝑦^∗)∈𝐶[0,1]×𝐶[0,1].

Proof. Let 𝑋=𝐶[0,1] be endowed with the ordering 𝑥≤𝑦 if 𝑥(𝑡)≤𝑦(𝑡) for all 𝑡∈[0,1], and ‖𝑢‖=max_𝑡∈[0,1]|𝑢(𝑡)| is defined as usual by maximum norm. Clearly, it follows that (𝑋,‖⋅‖) is a Banach space. By (2.17) and (2.6), problem (1.1) has a solution (𝑥^∗,𝑦^∗)∈𝐶[0,1]×𝐶[0,1] if and only if 𝑣^∗=𝒟^𝛽_𝑡𝑥^∗ solves the following operator equation: (𝑇𝑣)(𝑡)=𝜆¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝐼^𝛽𝑣(𝑠),𝑣(𝑠),¹₀𝐻(𝑠,𝜏)𝑔𝜏,𝐼^𝛽𝑣(𝜏)𝑑𝜏𝑑𝑠, (3.2) in 𝑋. So we only need to seek a fixed point of 𝑇 in 𝑋. By Ascoli-Arzela Theorem, it is obvious that the operator 𝑇∶𝑋→𝑋 is a completely continuous operator.
Since 𝑓(𝑡,0,0,0)≢0, and 𝑓(𝑡,0,0,0)≤𝑞(𝑡) a.e. 𝑡∈[0,1], we know ∫¹₀𝑞(𝑡)𝑑𝑡>0. On the other hand, by 𝑝_𝑖0(𝑡₀)≠0 for some 𝑖₀∈{1,2,3}, we have ¹₀𝑝₁(𝑡)+𝑝₂(𝑡)+𝑝₃(𝑡)𝑑𝑡>0. (3.3)
Let ∫𝑅=¹⁰𝑞(𝑠)𝑑𝑠1/Γ𝛽+1+𝑛/Γ𝛽∫¹₀𝑟(𝑠)𝑑𝑠∫¹₀𝑝₁(𝑠)+𝑝₂(𝑠)+𝑝₃(𝑠),𝑑𝑠 (3.4) where 𝑛 is defined by (2.15). Define the set Ω={𝑣∈𝐶[0,1]∶||𝑣||≤𝑅}. (3.5) Suppose 𝑣∈𝜕Ω,𝜇>1 such that 𝑇𝑣=𝜇𝑣. Then, for any 𝑣∈𝜕Ω, and noticing that 𝐼^𝛽𝑣(𝜏=1)Γ𝛽¹₀(𝑡−𝑠)^𝛽−1𝑣(𝑠)𝑣𝑑𝑠≤||||Γ𝛽, (3.6) we have ‖𝑇𝑣‖=max_[]𝑡∈0,1|(𝑇𝑣)(𝑡)|≤𝑚𝜆¹₀𝑓𝑠,𝐼^𝛽𝑣(𝑠),𝑣(𝑠),¹₀𝐻(𝑠,𝜏)𝑔𝜏,𝐼^𝛽𝑣(𝜏)𝑑𝜏𝑑𝑠≤𝑚𝜆¹₀𝑝₁(𝑡)𝐼^𝛽𝑣(𝑠)+𝑝₂(𝑡)|𝑣(𝑠)|+𝑝₃(𝑡)¹₀𝐻(𝑠,𝜏)𝑔𝜏,𝐼^𝛽𝑣(𝜏)𝑑𝜏+𝑞(𝑠)𝑑𝑠≤𝑚𝜆¹₀𝑝₁(𝑡)𝐼^𝛽𝑣(𝑠)+𝑝₂(𝑡)|𝑣(𝑠)|+𝑛𝑝₃(𝑡)¹₀𝑟(𝜏)𝐼^𝛽𝑣(𝜏)𝑑𝜏+𝑞(𝑠)𝑑𝑠≤𝑚𝜆¹₀𝑝₁(𝑡)||𝑣||Γ𝛽+𝑝₂(𝑡)||𝑣||+𝑛𝑝₃(𝑡)||𝑣||Γ𝛽¹₀𝑟(𝜏)(𝑠1𝑑𝜏+𝑞)𝑑𝑠≤𝑚𝜆Γ𝛽𝑛+1+Γ𝛽¹₀𝑟(𝑠)𝑑𝑠¹₀𝑝₁(𝑠)+𝑝₂(𝑠)+𝑝₃(𝑠)𝑑𝑠||𝑣||+𝑚𝜆¹₀𝑞(𝑠)𝑑𝑠. (3.7) Thus we take 𝜆^∗=12𝑚1Γ𝛽𝑛+1+Γ𝛽¹₀𝑟(𝑠)𝑑𝑠¹₀𝑝₁(𝑠)+𝑝₂(𝑠)+𝑝₃(𝑠)𝑑𝑠⁻¹. (3.8) Then for any 0<𝜆≤𝜆^∗ and 𝑣∈𝜕Ω, by (3.7), one has 𝜇𝑣1||||≤𝑚𝜆Γ𝛽𝑛+1+Γ𝛽¹₀𝑟(𝑠)𝑑𝑠¹₀𝑝₁(𝑠)+𝑝₂(𝑠)+𝑝₃(𝑠)𝑑𝑠||𝑣||+𝑚𝜆¹₀𝑞(𝑠)≤1𝑑𝑠2𝑣+∫||||¹⁰𝑞(𝑠)𝑑𝑠21/Γ𝛽+1+𝑛/Γ𝛽∫¹₀𝑟(𝑠)𝑑𝑠∫¹₀𝑝₁(𝑠)+𝑝₂(𝑠)+𝑝₃(𝑠).𝑑𝑠 (3.9) Consequently, 1𝜇≤2+12=1. (3.10) This contradicts 𝜇>1, by Lemma 2.6, 𝑇 has a fixed point 𝑣^∗∈Ω. Since 𝑓(𝑡,0,0,0)≢0, then 𝑣^∗≢0, Thus let 𝑥^∗=𝐼^𝛽𝑣^∗,𝑦^∗=¹₀𝐻(𝑡,𝑠)𝑔𝑠,𝑥^∗(𝑠)𝑑𝑠, (3.11) and then the system (1.1) has at least one nontrivial solution (𝑥^∗,𝑦^∗)∈𝐶[0,1]×𝐶[0,1] for any 0<𝜆≤𝜆^∗. This completes the proof of Theorem 3.2.

Theorem 3.3. Suppose that (H0)-(H1) hold. If 𝑓(𝑡,0,0,0)≢0, and there exist nonnegative functions 𝑝_𝑖(𝑖=1,2,3),𝑟∈𝐿¹[0,1] such that 𝑓𝑡,𝑥₁,𝑥₂,𝑥₃−𝑓𝑡,𝑦₁,𝑦₂,𝑦₃≤𝑝₁(𝑡)𝑥₁−𝑦₁+𝑝₂(𝑡)𝑥₂−𝑦₂+𝑝₃(𝑡)𝑥₃−𝑦₃[],𝑔,𝑎.𝑒.𝑡∈0,1𝑡,𝑦₁−𝑔𝑡,𝑦₂≤𝑟(𝑡)𝑦₁−𝑦₂,a.e.𝑡∈[0,1]. (3.12) In addition, there exists 𝑡₀∈[0,1] such that 𝑝_𝑖0(𝑡₀)≠0 for some 𝑖₀∈{1,2,3}. Then there exists a constant 𝜆^∗>0, such that, for any 0<𝜆≤𝜆^∗, the system (1.1) has unique nontrivial solution (𝑥^∗,𝑦^∗)∈𝐶[0,1]×𝐶[0,1].

Proof. Let 𝑇 be given in Theorem 3.2; we will show that 𝑇 is a contraction. In fact, 𝑇𝑣₁−𝑇𝑣₂≤𝑚𝜆¹₀𝑓𝑠,𝐼^𝛽𝑣₁(𝑠),𝑣₁(𝑠),¹₀𝐻(𝑠,𝜏)𝑔𝜏,𝐼^𝛽𝑣₁(𝜏)𝑑𝜏−𝑓𝑠,𝐼^𝛽𝑣₂(𝑠),𝑣₂(𝑠),¹₀𝐻(𝑠,𝜏)𝑔𝜏,𝐼^𝛽𝑣₂(𝜏)𝑑𝜏𝑑𝑠≤𝑚𝜆¹₀𝑝₁(𝑡)𝐼^𝛽𝑣₁(𝑠)−𝑣₂(𝑠)+𝑝₂(𝑡)𝑣₁(𝑠)−𝑣₂(𝑠)+𝑛𝑝₃(𝑡)¹₀𝑟(𝜏)𝐼^𝛽𝑣₁(𝜏)−𝑣₂(𝜏)𝑑𝜏𝑑𝑠≤𝑚𝜆¹₀𝑝₁(𝑡)𝑣₁−𝑣₂Γ𝛽+𝑝₂(𝑡)𝑣₁−𝑣₂+𝑛𝑝₃(𝑡)Γ𝛽¹₀𝑟(𝜏)𝑑𝜏𝑣₁−𝑣₂1𝑑𝑠≤𝑚𝜆Γ𝛽𝑛+1+Γ𝛽¹₀𝑟(𝑠)𝑑𝑠¹₀𝑝₁(𝑠)+𝑝₂(𝑠)+𝑝₃(𝑠)𝑑𝑠𝑣₁−𝑣₂. (3.13) If we choose 𝜆^∗=12𝑚1Γ𝛽𝑛+1+Γ𝛽¹₀𝑟(𝑠)𝑑𝑠¹₀𝑝₁(𝑠)+𝑝₂(𝑠)+𝑝₃(𝑠)𝑑𝑠⁻¹. (3.14) Then by (3.13) and (3.14), we have 𝑇𝑣₁−𝑇𝑣₂≤12𝑣₁−𝑣₂, (3.15) which implies that 𝑇 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 (𝐻0)-(𝐻1) hold. If 𝑓(𝑡,0,0,0)≢0, limsup_{𝑥1+𝑥2+𝑥3→+∞}max_𝑡∈[0,1]𝑓𝑡,𝑥₁,𝑥₂,𝑥₃𝑥₁+𝑥₂+𝑥₃<+∞. (3.16) In addition, there exists a nonnegative function 𝑟∈𝐿¹[0,1] such that 𝑔 satisfies 𝑔𝑡,𝑦≤𝑟(𝑡)𝑦,a.e.𝑡∈[0,1]. (3.17) Then there exists a constant 𝜆^∗>0, such that, for any 0<𝜆≤𝜆^∗, the system (1.1) has at least one nontrivial solution (𝑥^∗,𝑦^∗)∈𝐶[0,1]×𝐶[0,1].

Proof. We prove 𝑓 satisfies the conditions of Theorem 3.2. Let 𝔚=limsup_{𝑥1+𝑥2+𝑥3→+∞}max_[]𝑡∈0,1𝑓𝑡,𝑥₁,𝑥₂,𝑥₃𝑥₁+𝑥₂+𝑥₃, (3.18) and choose 𝜀>0 such that 𝔚+1−𝜀>0. By (3.16), there exists 𝑁>0 such that 𝑓𝑡,𝑥₁,𝑥₂,𝑥₃≤(𝔚+1−𝜀)𝑥₁+𝑥₂+𝑥₃,𝑥₁+𝑥₂+𝑥₃>𝑁,𝑡∈[0,1]. (3.19) Take 𝕄=max_{𝑡∈[0,1]/Ω,|𝑥1|+|𝑥2|+|𝑥3|≤𝑁},|𝑓(𝑡,𝑥₁,𝑥₂,𝑥₃)|, and the measure of Ω is 0. Thus for any (𝑡,𝑥_,𝑥₂,𝑥₃)∈[0,1]×ℝ³, we have 𝑓𝑡,𝑥₁,𝑥₂,𝑥₃≤(𝔚+1−𝜀)𝑥₁+𝑥₂+𝑥₃+𝕄,a.e.𝑡∈[0,1]. (3.20) 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: −𝒟_𝐭^3/2𝑥=𝜆𝑡²+sin𝑡𝑥−𝑡^1/2cos²𝒟_𝐭^3/8𝑥+₃1√𝑡1sin𝑦+√,1−𝑡+2−𝒟_𝐭^5/4((𝒟𝑦=𝑡cos𝑡+ln𝑡+1))𝑥,_𝐭3/8𝑥(0)=0,𝒟_𝐭^3/8𝑥(1)=¹₀𝒟_𝐭^3/8𝑥(𝑡)𝑑𝐴(𝑡),𝑦(0)=0,𝑦(1)=¹₀𝑦(𝑡)𝑑𝐵(𝑡), (3.21) where 𝐴(𝑡)=10,𝑡∈0,2,321,𝑡∈2,34,31,𝑡∈4,𝐵(𝑡)=1,10,𝑡∈0,4,11,𝑡∈4,34,32,𝑡∈4,1. (3.22) Then the system (3.21) is equivalent to the following 4-point BVP with coefficients of both signs −𝒟_𝐭^3/2𝑥=𝜆𝑡²+sin𝑡𝑥−𝑡^1/2cos²𝒟_𝐭^3/8𝑥+₃1√𝑡1sin𝑦+√,1−𝑡+2−𝒟_𝐭^5/4((𝒟𝑦=𝑡cos𝑡+ln𝑡+1))𝑥,_𝐭3/8𝑥(0)=0,𝒟_𝐭^3/8𝑥(1)=32𝒟_𝐭^3/812−12𝒟_𝐭^3/834,𝑦(0)(1)1=0,𝑦=𝑦43+𝑦4. (3.23) Clearly, (H0) holds. Let 𝑓𝑡,𝑥₁,𝑥₂,𝑥₃=𝑡²+sin𝑡𝑥₁−𝑡^1/2cos²𝑥₂+₃1√𝑡sin𝑥₃+1√=((1−𝑡+2,𝑔𝑡,𝑦𝑡cos𝑡+ln𝑡+1))𝑦. (3.24) Then (H1) also is satisfied.
On the other hand, we have 𝑓𝑡,𝑥₁,𝑥₂,𝑥₃≤𝑡²+sin𝑡𝑥₁+𝑡^1/2𝑥₂+₃1√𝑡𝑥₃+1√𝑔≤((𝑦,1−𝑡+2,𝑡,𝑦𝑡cos𝑡+ln𝑡+1)) (3.25) and 𝑓(𝑡,0,0,0)=1/√1−𝑡+2≢0, which imply all conditions of Theorem 3.2 are satisfied, by Theorem 3.2, there exists a constant 𝜆^∗>0, such that for any 0<𝜆≤𝜆^∗, the system (3.21) has at least one nontrivial solution (𝑥^∗,𝑦^∗)∈𝐶[0,1]×𝐶[0,1].