Existence Results for Solutions of Nonlinear Fractional Differential Equations

Abstract

This paper deals with theoretical and constructive existence results for solutions of nonlinear fractional differential equations using the method of upper and lower solutions which generate a closed set. The existence of solutions for nonlinear fractional differential equations involving Riemann-Liouville differential operator in a closed set is obtained by utilizing various types of coupled upper and lower solutions. Furthermore, these results are extended to the finite systems of nonlinear fractional differential equations leading to more general results.

1. Introduction

Fractional derivative, introduced around the 17th century, was developed almost until the 19th century. Although the introduction of the concept of fractional calculus involving fractional differentiation and integral is a few centuries old, it was realized only a few decades ago that these functional operations play an important role in various fields of science and engineering [1–8]. As a reason, since the significance of the fractional calculus has been more clearly perceived, many quality researches have been put forward on this branch of mathematical analysis in the literature (see [9–11] and the references therein), and many physical phenomena, chemical processes, biological systems, and so forth have described with fractional derivatives. In this framework, fractional differential equations have been gaining much interest and attracting the attention of many researchers. Some recent contributions on fractional differential equations can be seen in [9–20] and the references within.

On the other hand, the study for solutions of fractional ordinary and partial differential equations has received great interest by scientists. Especially, in the last decade, there are noteworthy works on the analytical and numerical solutions of fractional partial differential equations (see [21–28] and the references there in).

The attention drawn to basic theoretical concepts like the theory of existence and uniqueness of solutions to nonlinear fractional-order differential equations is obvious. Recently, there have been many paper investigating the existence and uniqueness of fractional-order differential equations [29–34].

An interesting and fruitful technique for providing existence results for nonlinear problems is the method of upper and lower solutions. This technique permits us to establish the existence results in a closed set, namely, the ordered interval, generated by upper and lower solutions. Thus, in this context, we are concerned with the existence of solutions of the following nonlinear fractional-order initial value problem (IVP):𝐷^𝑞𝑥(𝑡)=𝐹(𝑡,𝑥),𝑡∈𝐽=𝑡₀,𝑇,𝑥(𝑡)𝑡−𝑡₀^1−𝑞_𝑡=𝑡0=𝑥⁰, (1.1)

where 𝐹∈𝐶[𝐽×ℝ,ℝ] and 𝐷^𝑞 is Riemann-Liouville (R-L) fractional derivative of order 𝑞, 0<𝑞<1.

The corresponding Volterra fractional integral equation of (1.1) is defined as𝑥(𝑡)=𝑥⁰𝑡−𝑡₀^𝑞−1Γ𝑞+1Γ𝑞^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝐹(𝑠,𝑥(𝑠))𝑑𝑠. (1.2)

In recent years, Lakshmikantham and Vatsala investigated the existence theory and established a Peanos type local existence theorem for (1.1) by using integral inequalities and perturbation techniques [35]. McRae also studied an important existence result utilizing the method of upper and lower solutions [36], by means of which, monotone iterative and quasilinearization techniques are developed to fractional differential equations [37–40].

In this paper, we utilize the technique of upper and lower solutions and establish some existence results in terms of various types of coupled upper and lower solutions. Then we will extend this idea to the finite systems of nonlinear fractional differential equations.

The organization of this paper is as follows. In Section 2, we provide necessary background. In Section 3, we focus on the existence of solutions of nonlinear fractional differential equations in a sector. Finally, in Section 4, we extend our results to the finite systems of fractional differential equations.

2. Preliminaries

Now, we present some basic definitions and theorems which are used throughout the paper.

Definition 2.1. Let 𝑝=1−𝑞, then a function 𝜎(𝑡) is said to be a  𝐶_𝑝 function if 𝜎∈𝐶_𝑝 where 𝐶_𝑝[𝐽,ℝ]=𝑢∈𝐶𝑡₀,𝑇,ℝ∶𝑢(𝑡)𝑡−𝑡₀^𝑝∈𝐶[𝐽,ℝ]. (2.1)
If we replace 𝐹(𝑡,𝑥) in (1.1) by the sum of two functions such that 𝐹=𝑓+𝑔 where 𝑓,𝑔∈𝐶[𝐽×ℝ,ℝ],then problem (1.1) takes the following form:𝐷^𝑞𝑥(𝑡)=𝑓(𝑡,𝑥)+𝑔(𝑡,𝑥),𝑥(𝑡)𝑡−𝑡₀^1−𝑞_𝑡=𝑡0=𝑥⁰. (2.2) We give a variety of possible definitions of upper and lower solutions relative to (2.2).

Definition 2.2. Let 𝑣,𝑤∈𝐶_𝑝[𝐽,ℝ], 𝑝=1−𝑞,0<𝑞<1 be locally Hölder continuous with exponent 𝜆>𝑞,𝐷^𝑞𝑣,𝐷^𝑞𝑤 exist, and 𝑓,𝑔∈𝐶[𝐽×ℝ,ℝ], then 𝑣 and 𝑤 are said to be as follows:
(i)natural upper and lower solutions of (2.2), respectively, if𝐷^𝑞𝑣≤𝑓(𝑡,𝑣)+𝑔(𝑡,𝑣),𝑣⁰≤𝑥⁰,𝐷^𝑞𝑤≥𝑓(𝑡,𝑤)+𝑔(𝑡,𝑤),𝑤⁰≥𝑥⁰,𝑡∈𝐽, (2.3)(ii)coupled upper and lower solutions of type I of (2.2), respectively, if𝐷^𝑞𝑣≤𝑓(𝑡,𝑣)+𝑔(𝑡,𝑤),𝑣⁰≤𝑥⁰,𝐷^𝑞𝑤≥𝑓(𝑡,𝑤)+𝑔(𝑡,𝑣),𝑤⁰≥𝑥⁰,𝑡∈𝐽, (2.4)(iii)coupled upper and lower solutions of type II of (2.2), respectively, if𝐷^𝑞𝑣≤𝑓(𝑡,𝑤)+𝑔(𝑡,𝑣),𝑣⁰≤𝑥⁰,𝐷^𝑞𝑤≥𝑓(𝑡,𝑣)+𝑔(𝑡,𝑤),𝑤⁰≥𝑥⁰,𝑡∈𝐽, (2.5)(iv)coupled upper and lower solutions of type III of (2.2), respectively, if𝐷^𝑞𝑣≤𝑓(𝑡,𝑤)+𝑔(𝑡,𝑤),𝑣⁰≤𝑥⁰,𝐷^𝑞𝑤≥𝑓(𝑡,𝑣)+𝑔(𝑡,𝑣),𝑤⁰≥𝑥⁰,𝑡∈𝐽, (2.6) where 𝑣⁰=𝑣(𝑡)(𝑡−𝑡₀)^1−𝑞∣_𝑡=𝑡0and 𝑤⁰=𝑤(𝑡)(𝑡−𝑡₀)^1−𝑞|_𝑡=𝑡0.

Lemma 2.3. Let 𝑚∈𝐶_𝑝([𝑡₀,𝑇],𝑅) be locally Hölder continuous with exponent 𝜆>𝑞 and for any 𝑡₁∈(𝑡₀,𝑇], and one has 𝑚𝑡₁=0,𝑚(𝑡)≤0for𝑡₀≤𝑡≤𝑡₁, (2.7) then it follows that 𝐷^𝑞𝑚𝑡₁≥0. (2.8)

Proof. For the proof, see [16].

Remark 2.4. A dual result for Lemma 2.3 is valid.
The explicit solution of the following nonhomogeneous linear fractional differential equation𝐷^𝑞𝑥=𝛼𝑥+𝑓(𝑡),𝑥⁰=𝑥(𝑡)𝑡−𝑡₀^1−𝑞_𝑡=𝑡0, (2.9) involving R-L fractional differential operator of order 𝑞(0<𝑞<1), is necessary for further development of our main results. In (2.9), 𝛼 is a real number and 𝑓∈𝐶_𝑝([𝑡₀,𝑇],ℝ).
When we apply the method of successive approximations [16] to find the solution 𝑥(𝑡)=𝑥(𝑡,𝑡₀,𝑥⁰) explicitly for the given nonhomogeneous IVP (2.9), we obtain𝑥(𝑡)=𝑥⁰𝑡−𝑡₀^𝑞−1𝐸_𝑞,𝑞𝛼𝑡−𝑡₀^𝑞+^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝐸_𝑞,𝑞𝛼(𝑡−𝑠)^𝑞𝑓(𝑠)𝑑𝑠,𝑡∈𝑡₀,𝑇, (2.10) where 𝐸_𝑞,𝑞 denotes the two-parameter Mittag-Leffler function and 𝐸_𝑞,𝑞(𝑡^𝑞)=∑^∞_𝑘=0𝑡^𝑞𝑘/Γ(𝑞(𝑘+1)),𝑞>0.
If 𝑓(𝑡)≡0, we get𝑥(𝑡)=𝑥⁰𝑡−𝑡₀^𝑞−1𝐸_𝑞,𝑞𝛼𝑡−𝑡₀^𝑞,𝑡∈𝑡₀,𝑇, (2.11) as the solution of the corresponding homogeneous IVP for (2.9).
We next give a Peano’s type local existence result and then an existence result in a special closed set generated by upper and lower solutions.

Theorem 2.5. Assume that 𝐹∈𝐶(ℝ₀,ℝ^𝑛) and |𝐹(𝑡,𝑥)|≤𝑀on 𝑅₀ where 𝑅₀={(𝑡,𝑥)∶𝑡₀≤𝑡≤𝑡₀+𝑎 and |𝑥−𝑥⁰(𝑡)|≤𝑏} and 𝑥⁰(𝑡)=𝑥⁰(𝑡−𝑡₀)^𝑞−1/Γ(𝑞), then IVP (1.1) possesses at least one solution 𝑥(𝑡) on [𝑡₀,𝑡₀+𝛼] where 𝛼=min{𝑎,((𝑏/𝑀)Γ(𝑞+1))^1/𝑞}.

For the proof of the theorem, see [35].

If the existence of upper and lower solutions 𝑤,𝑣 such that 𝑣(𝑡)≤𝑤(𝑡),𝑡∈𝐽 for IVP (1.1) is known, the existence of solutions can be proved in the closed set Ω=(𝑡,𝑥)∶𝑣(𝑡)≤𝑥(𝑡)≤𝑤(𝑡),𝑡∈𝑡₀,𝑇. (2.12)

Theorem 2.6. Let 𝑣and 𝑤∈𝐶_𝑝[[𝑡₀,𝑇],ℝ] be natural upper and lower solutions of IVP (1.1), which are locally Hölder continuous with exponent 𝜆>𝑞 such that 𝑣(𝑡)≤𝑤(𝑡) on 𝐽=[𝑡₀,𝑇]and 𝑓∈𝐶(Ω,ℝ), then there exists a solution 𝑥(𝑡) of IVP (1.1) satisfying 𝑣(𝑡)≤𝑥(𝑡)≤𝑤(𝑡),𝑡∈[𝑡₀,𝑇].

For the detailed proof of the above theorem, see [36].

3. Existence Theorems

We are in position to give existence of solutions in the closed setΩ.

Theorem 3.1. Let 𝑣and 𝑤∈𝐶_𝑝[[𝑡₀,𝑇],ℝ] be coupled upper and lower solutions of type I of (2.2) such that 𝑓,𝑔∈𝐶(Ω,ℝ) and 𝑣(𝑡)≤w(𝑡),𝑡∈[𝑡₀,𝑇].Moreover, assume that 𝑔(𝑡,𝑥) is nonincreasing in 𝑥 for each 𝑡, then there exists a solution 𝑥(𝑡) of (2.2) satisfying 𝑣(𝑡)≤𝑥(𝑡)≤𝑤(𝑡) on [𝑡₀,𝑇].

Proof. Let 𝑝∶[𝑡₀,𝑇]×ℝ→ℝ be defined by 𝑝(𝑡,𝑥)=min[𝑤(𝑡),max(𝑥(𝑡),𝑣(𝑡))]. (3.1)
Then 𝑓(𝑡,𝑝(𝑡,𝑥))+𝑔(𝑡,𝑝(𝑡,𝑥)) defines a continuous extension of 𝑓+𝑔 to [𝑡₀,𝑇]×ℝ which is also bounded since 𝑓+𝑔 is bounded on Ω. Employing Theorem 2.5, we get the following equation:𝐷^𝑞𝑥(𝑡)=𝑓𝑡,𝑝(𝑡,𝑥)+𝑔𝑡,𝑝(𝑡,𝑥),𝑥(𝑡)𝑡−𝑡₀^1−𝑞_𝑡=𝑡0=𝑥⁰, (3.2) having a solution𝑥(𝑡) on [𝑡₀,𝑇].We wish to prove that 𝑣(𝑡)≤𝑥(𝑡)≤𝑤(𝑡) on [𝑡₀,𝑇]. For this purpose, consider the following equations: 𝑤_𝜖(𝑡)=𝑤(𝑡)+𝜖𝛾(𝑡),𝑣_𝜖(𝑡)=𝑣(𝑡)−𝜖𝛾(𝑡), (3.3) where 𝛾(𝑡)=(𝑡−𝑡₀)^𝑞−1𝐸_𝑞,𝑞((𝑡−𝑡₀)^𝑞)and 𝜀>0. This implies that 𝑤_𝜖(𝑡)𝑡−𝑡₀^1−𝑞_𝑡=𝑡0=𝑤⁰_𝜖=𝑤(𝑡)𝑡−𝑡₀^1−𝑞_𝑡=𝑡0+𝜖𝛾(𝑡)𝑡−𝑡₀^1−𝑞_𝑡=𝑡0,𝑣_𝜖(𝑡)𝑡−𝑡₀^1−𝑞_𝑡=𝑡0=𝑣⁰_𝜖=𝑣(𝑡)𝑡−𝑡₀^1−𝑞_𝑡=𝑡0−𝜖𝛾(𝑡)𝑡−𝑡₀^1−𝑞_𝑡=𝑡0, (3.4) which gives 𝑤⁰_𝜖=𝑤⁰+𝜖𝛾⁰,𝑣⁰_𝜖=𝑣⁰−𝜖𝛾⁰ where 𝛾⁰>0. It follows that 𝑣⁰_𝜖<𝑥⁰<𝑤⁰_𝜖in view of the upper and lower definitions of 𝑤(𝑡) and 𝑣(𝑡). It is enough to show that 𝑣_𝜖(𝑡)<𝑥(𝑡)<𝑤_𝜖(𝑡)on𝑡₀,𝑇, (3.5) which proves the claim of the theorem as 𝜖→0. First, suppose that the inequality 𝑥(𝑡)<𝑤_𝜖(𝑡)on [𝑡₀,𝑇] is not true, then there would exist a 𝑡₁∈(𝑡₀,𝑇] such that 𝑥𝑡₁=𝑤_𝜖𝑡₁,𝑥(𝑡)<𝑤_𝜖(𝑡)on𝑡₀,𝑡₁. (3.6) Hence, 𝑥(𝑡₁)>𝑤(𝑡₁)≥𝑣(𝑡₁), therefore 𝑝(𝑡₁,𝑥(𝑡₁))=𝑤(𝑡₁) and 𝑣(𝑡₁)≤𝑝(𝑡₁,𝑥(𝑡₁))≤𝑤(𝑡₁). If we construct 𝑚(𝑡)=𝑥(𝑡)−𝑤_𝜖(𝑡), we get 𝑚(𝑡₁)=0 and 𝑚(𝑡)≤0,𝑡₀≤𝑡≤𝑡₁. Employing Lemma 2.3,we obtain 𝐷^𝑞𝑚(𝑡₁)≥0 which gives a contradiction 𝑓𝑡₁,𝑤𝑡₁+𝑔𝑡₁,𝑤𝑡₁=𝑓𝑡₁,𝑝𝑡₁,𝑥𝑡₁+𝑔𝑡₁,𝑝𝑡₁,𝑥𝑡₁=𝐷^𝑞𝑥𝑡₁≥𝐷^𝑞𝑤_𝜖𝑡₁=𝐷^𝑞𝑤𝑡₁+𝜖𝛾𝑡₁>𝐷^𝑞𝑤𝑡₁≥𝑓𝑡₁,𝑤𝑡₁+𝑔𝑡₁,𝑣𝑡₁≥𝑓𝑡₁,𝑤𝑡₁+𝑔𝑡₁,𝑤𝑡₁. (3.7) Here, we have used the nonincreasing property of 𝑔(𝑡,𝑥) in 𝑥 for each 𝑡 and the fact that 𝛾(𝑡₁)>0.
Similarly, it can be proved that the other side of the inequality (3.5) is valid for 𝑡₀≤𝑡≤𝑇. To do this, we must show that 𝑣_𝜖(𝑡)<𝑥(𝑡)on[𝑡₀,𝑇]. Suppose that it is not true and so there exists a 𝑡₁ such that 𝑣_𝜖(𝑡₁)=𝑥(𝑡₁)and 𝑣_𝜖(𝑡)<𝑥(𝑡) for 𝑡₀≤𝑡<𝑡₁, then 𝑥(𝑡₁)<𝑣(𝑡₁)≤𝑤(𝑡₁) and 𝑝(𝑡₁,𝑥(𝑡₁))=𝑣(𝑡₁). Hence, 𝑣(𝑡₁)≤𝑝(𝑡₁,𝑥(𝑡₁))≤𝑤(𝑡₁). If we put 𝑚(𝑡)=𝑣_𝜖(𝑡)−𝑥(𝑡), we get 𝑚(𝑡₁)=0 and 𝑚(𝑡)≤0,𝑡₀≤𝑡≤𝑡₁. Employing Lemma 2.3, we find 𝐷^𝑞𝑚(𝑡₁)≥0. Since the nonincreasing property of 𝑔(𝑡,𝑥) in 𝑥 for each 𝑡 and the fact that 𝛾(𝑡₁)>0, we arrive at the contradiction𝑓𝑡₁,𝑣𝑡₁+𝑔𝑡₁,𝑣𝑡₁=𝑓𝑡₁,𝑝𝑡₁,𝑥𝑡₁+𝑔𝑡₁,𝑝𝑡₁,𝑥𝑡₁=𝐷^𝑞𝑥𝑡₁≤𝐷^𝑞𝑣_𝜖𝑡₁=𝐷^𝑞𝑣𝑡₁−𝜖𝛾𝑡₁<𝐷^𝑞𝑣𝑡₁≤𝑓𝑡₁,𝑣𝑡₁+𝑔𝑡₁,𝑤𝑡₁≤𝑓𝑡₁,𝑣𝑡₁+𝑔𝑡₁,𝑣𝑡₁. (3.8) Consequently, we have 𝑣_𝜖(𝑡)<𝑥(𝑡)<𝑤_𝜖(𝑡)on [𝑡₀,𝑇], and letting 𝜖→0, we get 𝑣(𝑡)≤𝑥(𝑡)≤𝑤(𝑡) on [𝑡₀,𝑇] proving the theorem.

Theorem 3.2. Let 𝑣and 𝑤∈𝐶_𝑝[[𝑡₀,𝑇],ℝ] be coupled upper and lower solutions of type II of (2.2) such that 𝑓,𝑔∈𝐶(Ω,ℝ) and 𝑣(𝑡)≤𝑤(𝑡),𝑡∈[𝑡₀,𝑇].Moreover, assume that 𝑓(𝑡,𝑥) is nonincreasing in 𝑥 for each 𝑡, then there exists a solution 𝑥(𝑡) of (2.2) satisfying 𝑣(𝑡)≤𝑥(𝑡)≤𝑤(𝑡) on [𝑡₀,𝑇].

Proof. Let 𝑝∶[𝑡₀,𝑇]×ℝ→ℝ be defined by 𝑝(𝑡,𝑥)=min[𝑤(𝑡),max(𝑥(𝑡),𝑣(𝑡))]. (3.9)
Then 𝑓(𝑡,𝑝(𝑡,𝑥))+𝑔(𝑡,𝑝(𝑡,𝑥)) defines a continuous extension of 𝑓+𝑔 to [𝑡₀,𝑇]×ℝ which is also bounded since 𝑓+𝑔 is bounded on Ω. Therefore, by Theorem 2.5,𝐷^𝑞𝑥(𝑡)=𝑓𝑡,𝑝(𝑡,𝑥)+𝑔𝑡,𝑝(𝑡,𝑥),𝑥(𝑡)𝑡−𝑡₀^1−𝑞_𝑡=𝑡0=𝑥⁰ (3.10) has a solution𝑥(𝑡) on [𝑡₀,𝑇].We intend to show that 𝑣(𝑡)≤𝑥(𝑡)≤𝑤(𝑡)on [𝑡₀,𝑇]. For this purpose, consider the following equations: 𝑤_𝜖(𝑡)=𝑤(𝑡)+𝜖𝛾(𝑡),𝑣_𝜖(𝑡)=𝑣(𝑡)−𝜖𝛾(𝑡), (3.11) where 𝛾(𝑡) and 𝜀 are defined as before. It follows that 𝑣⁰_𝜖<𝑥⁰<𝑤⁰_𝜖. It is enough to show that 𝑣_𝜖(𝑡)<𝑥(𝑡)<𝑤_𝜖(𝑡)on𝑡₀,𝑇. (3.12) Suppose that it is not true. Thus, there would exist a 𝑡₁∈(𝑡₀,𝑇] such that 𝑥𝑡₁=𝑤_𝜖𝑡₁,𝑣_𝜖(𝑡)<𝑥(𝑡)<𝑤_𝜖(𝑡)on𝑡₀,𝑡₁. (3.13) Hence, 𝑥(𝑡₁)>𝑤(𝑡₁)≥𝑣(𝑡₁); therefore, we get 𝑝(𝑡₁,𝑥(𝑡₁))=𝑤(𝑡₁) and 𝑣(𝑡₁)≤𝑝(𝑡₁,𝑥(𝑡₁))≤𝑤(𝑡₁). Setting 𝑚(𝑡)=𝑥(𝑡)−𝑤_𝜖(𝑡), we have 𝑚(𝑡₁)=0 and 𝑚(𝑡)≤0, 𝑡₀≤𝑡≤𝑡₁. Employing Lemma 2.3, we obtain 𝐷^𝑞𝑚(𝑡₁)≥0 which yields a contradiction 𝑓𝑡₁,𝑤𝑡₁+𝑔𝑡₁,𝑤𝑡₁=𝑓𝑡₁,𝑝𝑡₁,𝑥𝑡₁+𝑔𝑡₁,𝑝𝑡₁,𝑥𝑡₁=𝐷^𝑞𝑥𝑡₁≥𝐷^𝑞𝑤_𝜖𝑡₁=𝐷^𝑞𝑤𝑡₁+𝜖𝛾𝑡₁>𝐷^𝑞𝑤𝑡₁≥𝑓𝑡₁,𝑣𝑡₁+𝑔𝑡₁,𝑤𝑡₁≥𝑓𝑡₁,𝑤𝑡₁+𝑔𝑡₁,𝑤𝑡₁. (3.14)
Here, we have used the nonincreasing property of 𝑓(𝑡,𝑥) in 𝑥 for each 𝑡 and the fact that 𝛾(𝑡₁)>0. Thus, we get 𝑥(𝑡)<𝑤_𝜖(𝑡)on [𝑡₀,𝑇].
After utilizing the similar procedure, the other case can be proved easily. Consequently, we have 𝑣_𝜖(𝑡)<𝑥(𝑡)<𝑤_𝜖(𝑡)on [𝑡₀,𝑇], and letting 𝜖→0, we get 𝑣(𝑡)≤𝑥(𝑡)≤𝑤(𝑡) on [𝑡₀,𝑇] which proves the theorem.

Theorem 3.3. Let 𝑣and 𝑤∈𝐶_𝑝[[𝑡₀,𝑇],ℝ] be coupled upper and lower solutions of type III of (2.2) such that 𝑓,𝑔∈𝐶(Ω,ℝ) and 𝑣(𝑡)≤𝑤(𝑡),𝑡∈[𝑡₀,𝑇].Moreover, assume that both 𝑓(𝑡,𝑥) and 𝑔(𝑡,𝑥) are nonincreasing in 𝑥 for each 𝑡, then there exists a solution 𝑥(𝑡) of (2.2) satisfying 𝑣(𝑡)≤𝑥(𝑡)≤𝑤(𝑡) on [𝑡₀,𝑇].

Proof. In a similar manner in previous theorems, the existence of the solution can be proved. Thus, we omit the details.

4. Extensions to the Systems of Differential Equations

We can generalize the result of Theorem 2.6 to finite systems of fractional differential equations. Consider the following fractional differential system:𝐷^𝑞𝑥(𝑡)=𝐹(𝑡,𝑥),𝑥(𝑡)𝑡−𝑡₀^1−𝑞_𝑡=𝑡0=𝑥⁰, (4.1)

where 𝐹∈𝐶[[𝑡₀,𝑇]×ℝ^𝑛,ℝ^𝑛], and 𝐷^𝑞𝑥 is the fractional derivative of 𝑥∈ℝ^𝑛 and 0<𝑞<1,𝑝=1−𝑞.

At this point, we shall need an important property, known as quasimonotone nondecreasing relative to systems of inequalities.

Definition 4.1. A function 𝐹∈𝐶[𝐽×ℝ^𝑛,ℝ^𝑛] is said to possess quasimonotone nondecreasing property if 𝑢≤𝑣, 𝑢_𝑖=𝑣_𝑖for some 𝑖,1≤𝑖≤𝑛,then 𝐹_𝑖(𝑡,𝑢)≤𝐹_𝑖(𝑡,𝑣).
Here, we shall be using vectorial inequalities, which are understood to mean the same inequalities hold between their corresponding components.
Next, we give the following existence result for systems of differential equations.

Theorem 4.2. Let 𝑣 and 𝑤∈𝐶_𝑝[[𝑡₀,𝑇],ℝ^𝑛] be upper and lower solutions of (4.1), respectively, such that 𝐹∈𝐶(Ω,ℝ^𝑛) and 𝑣(𝑡)≤𝑤(𝑡),𝑡∈[𝑡₀,𝑇] where Ω=[(𝑡,𝑥)∶𝑣(𝑡)≤𝑥(𝑡)≤𝑤(𝑡),𝑡∈[𝑡₀,𝑇]]. Moreover, assume that 𝐹(𝑡,𝑥) is quasimonotone nondecreasing in 𝑥 for each 𝑡, then there exists a solution 𝑥(𝑡) of (4.1) satisfying 𝑣(𝑡)≤𝑥(𝑡)≤𝑤(𝑡) on [𝑡₀,𝑇].

The proof of this theorem is a special case of the following theorem in which we choose 𝐹not to be quasimonotone nondecreasing in 𝑥 provided that we strengthen the notion of upper and lower solutions of (4.1) as follows:

For each 𝑖,1≤𝑖≤𝑛,𝐷^𝑞𝑣_𝑖(𝑡)≤𝐹_𝑖𝑡,𝜌∀𝜌suchthat𝑣_𝑖(𝑡)=𝜌_𝑖(𝑡),𝑣(𝑡)≤𝑥(𝑡)≤𝑤(𝑡)on𝑡₀,𝐷,𝑇_𝑞𝑤_𝑖(𝑡)≤𝐹_𝑖𝑡,𝜌∀𝜌suchthat𝑤_𝑖(𝑡)=𝜌_𝑖(𝑡),𝑣(𝑡)≤𝑥(𝑡)≤𝑤(𝑡)on𝑡₀,𝑇. (4.2) We state and prove the following existence result relative to the definition of upper and lower solutions in (4.2).

Theorem 4.3. Let 𝑣and 𝑤∈𝐶_𝑝[[𝑡₀,𝑇],ℝ^𝑛] be upper and lower solutions of (4.1), respectively, satisfying the relations given in (4.2), which are also locally Hölder continuous with exponent 𝜆>𝑞 such that 𝑣(𝑡)≤𝑤(𝑡) and 𝐹∈𝐶(Ω,ℝ^𝑛), then there exists a solution 𝑥(𝑡) of (4.1) satisfying 𝑣(𝑡)≤𝑥(𝑡)≤𝑤(𝑡) on [𝑡₀,𝑇].

Proof. Let 𝑝∶[𝑡₀,𝑇]×ℝ^𝑛→ℝ^𝑛 given by 𝑝(𝑡,𝑥)=min[𝑤(𝑡),max(𝑥(𝑡),𝑣(𝑡))]foreach𝑖, (4.3) then 𝐹(𝑡,𝑝(𝑡,𝑥)) defines a continuous extension of 𝐹 to [𝑡₀,𝑇]×ℝ^𝑛 which is also bounded since 𝑓+𝑔 is bounded on Ω. Therefore, by Theorem 2.5, 𝐷^𝑞𝑥(𝑡)=𝐹𝑡,𝑝(𝑡,𝑥),𝑥(𝑡)𝑡−𝑡₀^1−𝑞_𝑡=𝑡0=𝑥⁰ (4.4) has a solution𝑥(𝑡) on [𝑡₀,𝑇]. For 𝜖>0 and 𝑒=(1,1,…,1),consider 𝑤_𝜖(𝑡)=𝑤(𝑡)+𝜖𝛾(𝑡)𝑒 and 𝑣_𝜖(𝑡)=𝑣(𝑡)−𝜖𝛾(𝑡)𝑒 where 𝛾(𝑡)=(𝑡−𝑡₀)^𝑞−1𝐸_𝑞,𝑞((𝑡−𝑡₀)^𝑞). It is clear that 𝑣⁰_𝜖<𝑥⁰<𝑤⁰_𝜖. We wish to show that 𝑣_𝜖(𝑡)<𝑥(𝑡)<𝑤_𝜖(𝑡)on [𝑡₀,𝑇]. Suppose that it is not true, then there exists an index 𝑗,1≤𝑗≤𝑛 and a 𝑡₁∈(𝑡₀,𝑇] such that 𝑥_𝑗𝑡₁=𝑤_𝜖𝑗𝑡₁,𝑥(𝑡)≤𝑤_𝜖(𝑡),𝑡₀≤𝑡≤𝑡₁,𝑥_𝑖𝑡₁≤𝑤_𝜖𝑖𝑡₁for𝑖≠𝑗. (4.5) Thus, we have 𝑣(𝑡₁)≤𝑝(𝑡₁,𝑥(𝑡₁))≤𝑤(𝑡₁) and 𝑝_𝑗(𝑡₁,𝑥(𝑡₁))=𝑤_𝑗(𝑡₁). Setting 𝑚_𝑗(𝑡)=𝑣_𝑗(𝑡)−𝑤_𝑗(𝑡), it follows that 𝑚_𝑗𝑡₁=0,𝑚_𝑗(𝑡)≤0,𝑡∈𝑡₀,𝑡₁;𝑚_𝑖𝑡₁≤0,𝑖≠𝑗. (4.6)
Applying Lemma 2.3 to the component 𝑚_𝑗(𝑡), we get 𝐷^𝑞𝑚_𝑗(𝑡₁)≥0 or 𝐷^𝑞𝑥_𝑗(𝑡₁)≥𝐷^𝑞𝑤_𝜖𝑗(𝑡₁) which yields a contradiction𝐹_𝑗𝑡₁,𝑤𝑡₁=𝐹_𝑗𝑡₁,𝑝𝑡₁,𝑥𝑡₁=𝐷^𝑞𝑥_𝑗𝑡₁≥𝐷^𝑞𝑤_𝜖𝑗𝑡₁=𝐷^𝑞𝑤_𝑗𝑡₁+𝜖𝛾𝑡₁>𝐷^𝑞𝑤_𝑗𝑡₁≥𝐹_𝑗𝑡₁,𝑤𝑡₁. (4.7) Now, letting 𝜖→0, we arrive at 𝑣(𝑡)≤𝑥(𝑡)≤𝑤(𝑡) on [𝑡₀,𝑇] which proves the conclusion of the theorem.
Sometimes, we can have arbitrary coupling relative to upper and lower solutions. Let 𝑝_𝑖 and 𝑞_𝑖be nonnegative integers for each  𝑖,1≤𝑖≤𝑛, so that we can split the vector 𝑥 into (𝑥_𝑖,[𝑥]_𝑝𝑖,[𝑥]_𝑞𝑖). Then the system (4.1) can be written as𝐷^𝑞𝑥_𝑖(𝑡)=𝐹_𝑖𝑡,𝑥_𝑖,[𝑥]_𝑝𝑖,[𝑥]_𝑞𝑖,𝑥(𝑡)𝑡−𝑡₀^1−𝑞_𝑡=𝑡0=𝑥⁰, (4.8) where 𝐹∈𝐶[[𝑡₀,𝑇]×ℝ^𝑛,ℝ^𝑛].

Definition 4.4. A function 𝐹∈𝐶[[𝑡₀,𝑇]×ℝ^𝑛,ℝ^𝑛] is said to possess a mixed quasimonotone property if for each 𝑖, 𝐹_𝑖(𝑡,𝑥_𝑖,[𝑥]_𝑝𝑖,[𝑥]_𝑞𝑖) is monotone nondecreasing in [𝑥]_𝑝𝑖 and monotone nonincreasing in [𝑥]_𝑞𝑖.

Definition 4.5. The functions 𝑣 and 𝑤∈𝐶_𝑝[[𝑡₀,𝑇],ℝ^𝑛] are said to be coupled upper and lower quasisolutions of (4.8) if they satisfy 𝐷^𝑞𝑣_𝑖≤𝐹_𝑖𝑡,𝑣_𝑖,[𝑣]_𝑝𝑖,[𝑣]_𝑞𝑖,𝑣⁰≤𝑥⁰,𝐷^𝑞𝑤_𝑖≥𝐹_𝑖𝑡,𝑤_𝑖,[𝑤]_𝑝𝑖,[𝑤]_𝑞𝑖,𝑤⁰≥𝑥⁰, (4.9) for each  𝑖,1≤𝑖≤𝑛.
Next, we give an existence result which is also a special case of Theorem 4.3.

Theorem 4.6. Let 𝑣,𝑤∈𝐶_𝑝[[𝑡₀,𝑇],ℝ^𝑛] be coupled upper and lower quasisolutions of (4.8) and 𝐹∈𝐶[[𝑡₀,𝑇]×ℝ^𝑛,ℝ^𝑛]. If 𝐹(𝑡,𝑥) possesses a mixed quasimonotone property, then there exists a solution 𝑥(𝑡) of (4.8) such that 𝑣(𝑡)≤𝑥(𝑡)≤𝑤(𝑡) on [𝑡₀,𝑇].

It should be noted that if 𝐹 satisfies a mixed quasimonotone property, then (4.2) holds for coupled upper and lower quasisolutions given by (4.9). Therefore, Theorem 4.3 includes Theorem 4.6 as a special case.

5. Conclusion

In this work, some existence theorems have been established for nonlinear fractional-order differential equations relative to coupled upper and lower solutions. The differential operator is taken in the Riemann-Liouville sense. For the further developments in applications of dynamical systems, we have generalized these results to the finite systems of nonlinear fractional differential equations. Being defined by a suitable differential operator, the process of finding a solution between upper and lower solutions generating a closed set could be applied to various types of linear and nonlinear fractional partial differential equations as a future work.