Remark on Existence and Uniqueness of Solutions for a Coupled System of Multiterm Nonlinear Fractional Differential Equations

Abstract

The aim of this paper is to extend the work of Sun et al. (2012) to a more general case for a wider range of function classes of 𝑓 and 𝑔. Our results include the case of the previous work, which are essential improvement of the work of Sun et al. (2012), especially.

1. Introduction

Fractional calculus can give a more vivid and accurate description of problems in various fields of sciences than the traditional calculus [1–3]. Recently many complicated dynamic phenomena were modeled by fractional order calculus system and have received more and more attention; see [4–16].

In recent work [12], Sun et al. studied the existence and uniqueness of solutions for a coupled system of multiterm nonlinear fractional differential equations with an initial value condition −𝓓^𝛼𝑥(𝑡)=𝑓𝑡,𝑦(𝑡),𝓓^𝛽1𝑦(𝑡),…,𝓓^𝛽𝑁𝑦(𝑡),𝓓^𝛼−𝑖𝑥(0)=0,𝑖=1,2,…,𝑛₁,−𝓓^𝜎𝑦(𝑡)=𝑔𝑡,𝑥(𝑡),𝓓^𝜌1𝑥(𝑡),…,𝓓^𝜌𝑁𝑥(𝑡),𝓓^𝜎−𝑗𝑦(0)=0,𝑗=1,2,…,𝑛₂, (1) where 𝑡∈(0,1], 𝛼>𝛽₁>𝛽₂>⋯>𝛽_𝑁>0, 𝜎>𝜌₁>𝜌₂>⋯>𝜌_𝑁>0, 𝑛₁=[𝛼]+1, 𝑛₂=[𝜎]+1 for 𝛼,𝜎∉ℕ and 𝑛₁=𝛼, 𝑛₂=𝜎 for 𝛼,𝜎∈ℕ, 𝛽_𝑞,𝜌_𝑞<1 for any 𝑞∈{1,2,…,𝑁}, 𝓓 is the standard Riemann-Liouville derivative, and 𝑓,𝑔∶[0,1]×ℝ^𝑁+1→ℝ are given functions. In order to obtain the existence and uniqueness of solutions of (1), the following growth conditions are introduced in [12].(H1)There exist two nonnegative functions 𝑎(𝑡),𝑏(𝑡)∈𝐿¹[0,1] such that 𝑓𝑡,𝑥₀,𝑥₁,…,𝑥_𝑁≤𝑎(𝑡)+𝑐₀𝑥₀^𝛾0+𝑐₁𝑥₁^𝛾1+⋯+𝑐_𝑁𝑥_𝑁^𝛾𝑁,𝐺𝑡,𝑥₀,𝑥₁,…,𝑥_𝑁≤𝑏(𝑡)+𝑑₀𝑥₀^𝜃0+𝑑₁𝑥₁^𝜃1+⋯+𝑑_𝑁𝑥_𝑁^𝜃𝑁, (2) where 𝑐_𝑖,𝑑_𝑖≥0, 0<𝛾_𝑖, 𝜃_𝑖<1 for 𝑖=0,1,2,…,𝑁.(H2)The functions 𝑓 and 𝑔 satisfy 𝑓𝑡,𝑥₀,𝑥₁,…,𝑥_𝑁≤𝑐₀𝑥₀^𝛾0+𝑐₁𝑥₁^𝛾1+⋯+𝑐_𝑁𝑥_𝑁^𝛾𝑁,𝑔𝑡,𝑥₀,𝑥₁,…,𝑥_𝑁≤𝑑₀𝑥₀^𝜃0+𝑑₁𝑥₁^𝜃1+⋯+𝑑_𝑁𝑥_𝑁^𝜃𝑁, (3) where 𝑐_𝑖,𝑑_𝑖≥0, 𝛾_𝑖,𝜃_𝑖>1 for 𝑖=0,1,2,…,𝑁.

However, there are many functions which cannot satisfy conditions (H1) and (H2); for example, 𝑔𝑡,𝑥₀,𝑥₁=𝑡+16.08𝑥25.26⁰+𝑒^𝑥1. (4) Hence the results of [12] are limited only to a small subset of functions which satisfy (H1) and (H2). This paper thus aims to extend the work of Sun et al. [12] to a more general case with more general conditions on 𝑓 and 𝑔. Our major contributions of this paper include three aspects.(1)We extend the function classes to more general case; that is, the power growth assumptions (H1) and (H2) are replaced by a very general assumption where the functions 𝜙(|𝑥₀|,|𝑥₁|,…,|𝑥_𝑁|) and 𝜓(|𝑥₀|,|𝑥₁|,…,|𝑥_𝑁|) are only required to be nondecreasing function classes (see (A1)), which implies that the function classes are extended to more general case and also include the case of [12] as a special case. In mathematics and applied science, this generalization is important and interesting.(2)In [12], the weight functions considered constants 𝑐₀,𝑐₁,…,𝑐_𝑁. But in physics, the influence of weight functions for the whole system is important, so in this work, we improve the weight functions to general Lebesgue integral functions 𝑏(𝑡),𝑑(𝑡)∈𝐿¹[0,1], which is also an essential improvement.(3)In this paper, the nonlinearities 𝑓 and 𝑔 are allowed to be exponential growth. However, in [12], the nonlinearities 𝑓 and 𝑔 are only allowed to be power growth. It is known that in most cases exponential growth is faster than power growth. From this aspect, this is also a major contribution of this paper.

The remaining part of the paper is organized as follows. In Section 2, some preliminary results including definitions, notations, and lemmas are given. Section 3 presents 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

Definition 1 (see [1–3]). The fractional integral of order 𝛼>0 of a function 𝑥∶(𝑎,+∞)→𝑅 is given by 𝐼^𝛼𝑥(𝑡)=1Γ(𝛼)^𝑡_𝑎(𝑡−𝑠)^𝛼−1𝑥(𝑠)𝑑𝑠, (5) provided that the right-hand side is pointwisely defined on (𝑎,+∞).

Definition 2 (see [1–3]). The Riemann-Liouville fractional derivative of order 𝛼>0 of a function 𝑥: (𝑎,+∞)→𝑅 is given by 𝓓^𝛼𝑥(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑡_𝑎(𝑡−𝑠)^{𝑛−𝛼−1}𝑥(𝑠)𝑑𝑠, (6) where 𝑛=[𝛼]+1, [𝛼] denotes the integer part of the number 𝛼, and 𝑡>𝑎, provided that the right-hand side is defined on (𝑎,+∞).

Lemma 3 (see [1]). Assume that 𝑥∈𝐿¹[0,1] with a fractional derivative of order 𝛼>0; then 𝐼^𝛼𝓓^𝛼𝑥(𝑡)=𝑥(𝑡)+𝑐₁𝑡^𝛼−1+𝑐₂𝑡^𝛼−2+⋯+𝑐_𝑛𝑡^𝛼−𝑛, (7) where 𝑐_𝑖∈𝑅, 𝑖=1,2,…,𝑛, 𝑛=[𝛼]+1.

Lemma 4 (see [12]). Suppose that ℎ∈𝐿¹[0,1]. Then the initial value problem 𝓓^𝛼𝑥(𝑡)(𝑡)[],𝓓=ℎ,𝛼>0,𝑡∈𝑎,𝑏_𝛼𝑥(𝑎)=𝑏_𝑘,𝑘=1,2,…𝑛, (8) has a unique solution 𝑥(𝑡)=^𝑛_𝑗=1𝑏_𝑗Γ()𝛼−𝑗+1𝑡−𝑎^𝛼−𝑗+1Γ(𝛼)^𝑡_𝑎ℎ(𝑠)(𝑡−𝑠)^1−𝛼𝑑𝑠, (9) where 𝑛=[𝛼]+1 for 𝛼∉ℕ and 𝛼=𝑛 for 𝛼∈ℕ.

Let 𝐼=[0,1] and let 𝐶(𝐼) be the space of all continuous functions defined on 𝐼. We define the space ∣(𝐼)(𝐼),𝓓𝑋×𝑌=𝑥,𝑦𝑥,𝑦∈𝐶×𝐶_𝜌𝑗𝑥(𝑡),𝓓^𝛽𝑗𝑦(𝑡)(𝐼)(𝐼)∈𝐶×𝐶,𝑗=1,2,…,𝑁 (10) endowed with the norm ‖(𝑥,𝑦)‖_𝑋×𝑌=max{‖𝑥‖_𝑋,‖𝑦‖_𝑌}, where ‖𝑥‖_𝑋=max_𝑡∈𝐼|𝑥(𝑡)|+^𝑁_𝑗=1max_𝑡∈𝐼𝓓^𝜌𝑗𝑥(𝑡,𝑦_𝑌)=max_𝑡∈𝐼𝑦(𝑡)+^𝑁_𝑗=1max_𝑡∈𝐼𝓓^𝛽𝑗𝑦(𝑡). (11) Then 𝑋×𝑌 is a Banach space with norm ‖(𝑥,𝑦)‖_𝑋×𝑌.

By Lemma 4, system (1) is equivalent to the following coupled system of integral equations: 𝑥(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑦(𝑠),𝓓^𝛽1𝑦(𝑠),…,𝓓^𝛽𝑁𝑦(𝑠𝑦(𝑡)=1)𝑑𝑠,Γ(𝜎)^𝑡₀(𝑡−𝑠)^𝜎−1𝑔𝑠,𝑥(𝑠),𝓓^𝜌1𝑥(𝑠),…,𝓓^𝜌𝑁𝑥(𝑠)𝑑𝑠. (12) Define an operator 𝑇∶𝑋×𝑌→𝑋×𝑌𝑇(𝑡)=𝑇𝑥,𝑦₁𝑥(𝑡),𝑇₂𝑦(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1×𝑓𝑠,𝑦(𝑠),𝓓^𝛽1𝑦(𝑠),…,𝓓^𝛽𝑁𝑦(𝑠1)𝑑𝑠,Γ(𝜎)^𝑡₀(𝑡−𝑠)^𝜎−1×𝑔𝑠,𝑥(𝑠),𝓓^𝜌1𝑥(𝑠),…,𝓓^𝜌𝑁𝑥(𝑠)𝑑𝑠. (13) It is obvious that a fixed point of operator 𝑇 is the solution of coupled system (1).

3. Main Result

Theorem 5. Let 𝑓,𝑔∶[0,1]×ℝ^𝑁+1→ℝ be continuous. Assume that(A1)there exist nonnegative functions 𝑎,𝑏,𝑐,𝑑∈𝐿¹[0,1] and nonnegative nondecreasing functions 𝜙,𝜓 with respect to each variable 𝑥_𝑖,  𝑖=0,1,2,…,𝑁, such that 𝑓𝑡,𝑥₀,𝑥₁,…,𝑥_𝑁≤𝑎(𝑡)+𝑏(𝑡)𝜙𝑥₀,𝑥₁,…,𝑥_𝑁,𝑔𝑡,𝑥₀,𝑥₁,…,𝑥_𝑁≤𝑐(𝑡)+𝑑(𝑡)𝜓𝑥₀,𝑥₁,…,𝑥_𝑁; (14)(A2)there exists a constant 𝑅₀>max{𝑘₁,𝑙₁} such that 𝜙𝑅₀,𝑅₀,…,𝑅₀≤𝑅₀−𝑘₁𝑘₂,𝜓𝑅₀,𝑅₀,…,𝑅₀≤𝑅₀−𝑙₁𝑙₂, (15) where 𝑘₁=max_𝑡∈𝐼1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1|𝑎(𝑠+)|𝑑𝑠_𝑁𝑗=11Γ𝛼−𝜌_𝑗^𝑡0(𝑡−𝑠)^{𝛼−𝜌𝑗−1}|𝑎(𝑠,𝑘)|𝑑𝑠₂=max_𝑡∈𝐼1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1|𝑏(𝑠+)|𝑑𝑠_𝑁𝑗=11Γ𝛼−𝜌_𝑗^𝑡0(𝑡−𝑠)^{𝛼−𝜌𝑗−1}|𝑏(𝑠,𝑙)|𝑑𝑠₁=max_𝑡∈𝐼1Γ(𝜎)^𝑡₀(𝑡−𝑠)^𝜎−1|𝑐(𝑠+)|𝑑𝑠_𝑁𝑗=11Γ𝜎−𝛽_𝑗^𝑡0(𝑡−𝑠)^{𝜎−𝛽𝑗−1}|𝑐(𝑠,𝑙)|𝑑𝑠₂=max_𝑡∈𝐼1Γ(𝜎)^𝑡₀(𝑡−𝑠)^𝜎−1|𝑑(𝑠+)|𝑑𝑠_𝑁𝑗=11Γ𝜎−𝛽_𝑗^𝑡0(𝑡−𝑠)^{𝜎−𝛽𝑗−1}|𝑑(𝑠)|𝑑𝑠. (16)Then the coupled system (1) has a solution.

Proof. Define a closed ball of Banach space 𝑋×𝑌𝐵=𝑥,𝑦∈𝑋×𝑌∶𝑥,𝑦_𝑋×𝑌≤𝑅₀. (17) We will prove that 𝑇∶𝐵→𝐵. In fact, for any (𝑥,𝑦)∈𝐵, by (A1), we have 𝑇₁𝑥(𝑡=1)Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑦(𝑠),𝓓^𝛽1𝑦(𝑠),…,𝓓^𝛽𝑁𝑦(𝑠)≤1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1|𝑎(𝑠+𝜙𝑅)|𝑑𝑠₀,𝑅₀,…,𝑅₀Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1|𝑏(𝑠𝓓)|𝑑𝑠,_𝜌𝑗𝑇₁𝑥(𝑡=𝓓_𝜌𝑗)𝐼^𝛼𝑓𝑡,𝑦(𝑡),𝓓^𝛽1𝑦(𝑡),…,𝓓^𝛽𝑁𝑦(𝑡)=𝐼^{𝛼−𝜌𝑗}𝑓𝑡,𝑦(𝑡),𝓓^𝛽1𝑦(𝑡),…,𝓓^𝛽𝑁𝑦(𝑡)=1Γ𝛼−𝜌_𝑗×^𝑡₀(𝑡−𝑠)^{𝛼−𝜌𝑗−1}𝑓𝑠,𝑦(𝑠),𝓓^𝛽1𝑦(𝑠),…,𝓓^𝛽𝑁𝑦(𝑠)≤1𝑑𝑠Γ𝛼−𝜌_𝑗×^𝑡₀(𝑡−𝑠)^{𝛼−𝜌𝑗−1}|𝑎(𝑠+𝜙𝑅)|𝑑𝑠₀,𝑅₀,…,𝑅₀Γ𝛼−𝜌_𝑗^𝑡0(𝑡−𝑠)^{𝛼−𝜌𝑗−1}|𝑏(𝑠)|𝑑𝑠. (18) Thus it follows from (18) and (A2) that 𝑇₁𝑥_𝑋=max_𝑡∈𝐼𝑇₁𝑥(𝑡)+^𝑁_𝑗=1max_𝑡∈𝐼𝓓^𝜌𝑗𝑇₁𝑥(𝑡)≤𝑘₁+𝑘₂𝜙𝑅₀,𝑅₀,…,𝑅₀≤𝑅₀. (19)
In the same way, we also have 𝑇₂𝑦_𝑌=max_𝑡∈𝐼𝑇₂𝑦(𝑡)+^𝑁_𝑗=1max_𝑡∈𝐼𝓓^𝜌𝑗𝑇₂𝑦(𝑡)≤𝑙₁+𝑙₂𝜓𝑅₀,𝑅₀,…,𝑅₀≤𝑅₀. (20) Consequently, ‖𝑇₁𝑥‖_𝑋≤𝑅₀ and ‖𝑇₂𝑦‖_𝑌≤𝑅₀, and then ‖𝑇‖_𝑋×𝑌≤𝑅₀ for any (𝑥,𝑦)∈𝐵; that is, 𝑇∶𝐵→𝐵.
By [12], we know that the operator 𝑇 is completely continuous. Therefore, the Schauder fixed point theorem implies that coupled system (1) has a solution in 𝐵. The proof is completed.

From Theorem 5, we easily obtain the following corollaries.

Corollary 6. Let 𝑓,𝑔∶[0,1]×ℝ^𝑁+1→ℝ be continuous. Assume that(A1)there exist nonnegative functions 𝑐,𝑑∈𝐿¹[0,1] and nonnegative nondecreasing functions 𝜙,𝜓 with respect to each variable 𝑥_𝑖,  𝑖=0,1,2,…,𝑁, such that 𝑓𝑡,𝑥₀,𝑥₁,…,𝑥_𝑁≤𝑏(𝑡)𝜙𝑥₀,𝑥₁,…,𝑥_𝑁,𝑔𝑡,𝑥₀,𝑥₁,…,𝑥_𝑁≤𝑑(𝑡)𝜓𝑥₀,𝑥₁,…,𝑥_𝑁; (21)(A2)there exists a positive constant 𝑅₀ such that 𝜙𝑅₀,𝑅₀,…,𝑅₀≤𝑅₀𝑘₂,𝜓𝑅₀,𝑅₀,…,𝑅₀≤𝑅₀𝑙₂, (22) where 𝑘₂=max_𝑡∈𝐼1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1|𝑏(𝑠+)|𝑑𝑠_𝑁𝑗=11Γ𝛼−𝜌_𝑗^𝑡0(𝑡−𝑠)^{𝛼−𝜌𝑗−1}|𝑏(𝑠,𝑙)|𝑑𝑠₂=max_𝑡∈𝐼1Γ(𝜎)^𝑡₀(𝑡−𝑠)^𝜎−1|𝑑(𝑠+)|𝑑𝑠_𝑁𝑗=11Γ𝜎−𝛽_𝑗^𝑡0(𝑡−𝑠)^{𝜎−𝛽𝑗−1}|𝑑(𝑠)|𝑑𝑠. (23)Then the coupled system (1) has a solution.

Corollary 7. Let 𝑓,𝑔∶[0,1]×ℝ^𝑁+1→ℝ be continuous. Assume that(𝐴^∗1)there exist nonnegative functions 𝑎,𝑐∈𝐿¹[0,1] such that𝑓𝑡,𝑥₀,𝑥₁,…,𝑥_𝑁(𝑡),𝑔≤𝑎𝑡,𝑥₀,𝑥₁,…,𝑥_𝑁≤𝑐(𝑡). (24) Then the coupled system (1) has a solution.

Proof. In fact, let us choose 𝑅₀=max{𝑘₁,𝑙₁}, where 𝑘₁=max_𝑡∈𝐼1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1|𝑎(𝑠+)|𝑑𝑠_𝑁𝑗=11Γ𝛼−𝜌_𝑗^𝑡0(𝑡−𝑠)^{𝛼−𝜌𝑗−1}|𝑎(𝑠,𝑙)|𝑑𝑠₁=max_𝑡∈𝐼1Γ(𝜎)^𝑡₀(𝑡−𝑠)^𝜎−1|𝑐(𝑠+)|𝑑𝑠_𝑁𝑗=11Γ𝜎−𝛽_𝑗^𝑡0(𝑡−𝑠)^{𝜎−𝛽𝑗−1}|𝑐(𝑠)|𝑑𝑠, (25) and construct a closed ball of Banach space 𝑋×𝑌𝐵=𝑥,𝑦∈𝑋×𝑌∶𝑥,𝑦_𝑋×𝑌≤𝑅₀. (26) The rest of proof is similar to Theorem 5.

Remark 8. The condition (A1) is weaker than (H1) and (H2). Clearly, 𝜙(|𝑥₀|,|𝑥₁|,…,|𝑥_𝑁|) and 𝜓(|𝑥₀|,|𝑥₁|,…,|𝑥_𝑁|) include 𝑐₀|𝑥₀|^𝛾0+𝑐₁|𝑥₁|^𝛾1+⋯+𝑐_𝑁|𝑥_𝑁|^𝛾𝑁 and 𝑑₀|𝑥₀|^𝜃0+𝑑₁|𝑥₁|^𝜃1+⋯+𝑑_𝑁|𝑥_𝑁|^𝜃𝑁, 𝜃_𝑖,𝛾_𝑖≠1 as special cases. Moreover (A1) also includes the case 𝜃_𝑖=1 or/and 𝛾_𝑖=1, but (H1) and (H2) do not be allowed.

Remark 9. In Corollary 7, for the special case 𝑎,𝑐∈𝐶[0,1], clearly 𝑓,𝑔∶[0,1]×ℝ^𝑁+1→ℝ are continuous and bounded. This leads to the Corollary 3.1 of [12]. Therefore, Corollary 3.1 of [12] is only a special case of Corollary 7.

In the following, we focus on the uniqueness of the solution of the system (1).

Theorem 10. Let 𝑓,𝑔∶[0,1]×ℝ^𝑁+1→ℝ be continuous. Assume that(B1)there exist nonnegative functions 𝑎,𝑐∈𝐿¹[0,1] and nonnegative nondecreasing functions 𝜙,𝜓 with respect to each variable 𝑥_𝑖,  𝑖=0,1,2,…,𝑁, such that 𝑓𝑡,𝑢₀,𝑢₁,…,𝑢_𝑁−𝑓𝑡,𝑣₀,𝑣₁,…,𝑣_𝑁≤𝑎(𝑡)𝜙𝑢₀−𝑣₀,𝑢₁−𝑣₁,…,𝑢_𝑁−𝑣_𝑁,𝑔𝑡,𝑢₀,𝑢₁,…,𝑢_𝑁−𝑔𝑡,𝑣₀,𝑣₁,…,𝑣_𝑁≤𝑏(𝑡)𝜓𝑢₀−𝑣₀,𝑢₁−𝑣₁,…,𝑢_𝑁−𝑣_𝑁; (27)(B2)for any 𝑠>0, 𝜙(𝑠,𝑠,…,𝑠)≤𝑠,𝜓(𝑠,𝑠,…,𝑠)≤𝑠, (28) and max{𝑘²₁,𝑙²₁}<1, where 𝑘₁=max_𝑡∈𝐼1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑎(𝑠)+𝑑𝑠_𝑁𝑗=11Γ𝛼−𝜌_𝑗^𝑡0(𝑡−𝑠)^{𝛼−𝜌𝑗−1}𝑎(𝑠),𝑙𝑑𝑠₁=max_𝑡∈𝐼1Γ(𝜎)^𝑡₀(𝑡−𝑠)^𝜎−1𝑐(𝑠)+𝑑𝑠_𝑁𝑗=11Γ𝜎−𝛽_𝑗^𝑡0(𝑡−𝑠)^{𝜎−𝛽𝑗−1}𝑐(𝑠)𝑑𝑠. (29)Then coupled system (1) has a unique solution.

Proof. We prove that the operator 𝑇∶𝑋×𝑌→𝑋×𝑌 is contraction. To do this, let (𝑥₁,𝑦₁), (𝑥₂,𝑦₂)∈𝑋×𝑌; we have 𝑇₁𝑥₂(𝑡)−𝑇₁𝑥₁(𝑡=1)Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1×𝑓𝑠,𝑦₂(𝑠),𝓓^𝛽1𝑦₂(𝑠),…,𝓓^𝛽𝑁𝑦₂(𝑠)−1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1×𝑓𝑠,𝑦₁(𝑠),𝓓^𝛽1𝑦₁(𝑠),…,𝓓^𝛽𝑁𝑦₁(𝑠)≤1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑎(𝑠)×𝜙𝑦₂(𝑠)−𝑦₁(𝑠),𝓓^𝛽1𝑦₂(𝑠)−𝓓^𝛽1𝑦₁(𝑠𝓓),…,_𝛽𝑁𝑦₂(𝑠)−𝓓^𝛽𝑁𝑦₁(𝑠≤1)𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑎(𝑠)×𝜙𝑦₂−𝑦₁,𝓓^𝛽1𝑦₂−𝓓^𝛽1𝑦₁𝓓,…,_𝛽𝑁𝑦₂−𝓓^𝛽𝑁𝑦₁≤1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑎(𝑠)×𝜙𝑦₂−𝑦_1𝑌,𝑦₂−𝑦_1𝑌,…,𝑦₂−𝑦_1𝑌≤1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑎(𝑠)𝑑𝑠𝑦₂−𝑦_1𝑌,𝓓^𝜌𝑗𝑇₁𝑥₂(𝑡)−𝓓^𝜌𝑗𝑇₁𝑥₁(𝑡=𝐼_{𝛼−𝜌𝑗})𝑓𝑡,𝑦₂(𝑡),𝓓^𝛽1𝑦₂(𝑡),…,𝓓^𝛽𝑁𝑦₂(𝑡)−𝑓𝑡,𝑦₁(𝑡),𝓓^𝛽1𝑦₁(𝑡),…,𝓓^𝛽𝑁𝑦₁(𝑡)≤1Γ𝛼−𝜌_𝑗×^𝑡₀(𝑡−𝑠)^{𝛼−𝜌𝑗−1}×𝑓𝑡,𝑦₂(𝑡),𝓓^𝛽1𝑦₂(𝑡),…,𝓓^𝛽𝑁𝑦₂(𝑡)−𝑓𝑡,𝑦₁(𝑡),𝓓^𝛽1𝑦₁(𝑡),…,𝓓^𝛽𝑁𝑦₁(𝑡)≤1𝑑𝑠Γ𝛼−𝜌_𝑗×^𝑡₀(𝑡−𝑠)^{𝛼−𝜌𝑗−1}𝑎(𝑠)×𝜙𝑦₂−𝑦₁,𝓓^𝛽1𝑦₂−𝓓^𝛽1𝑦₁𝓓,…,_𝛽𝑁𝑦₂−𝓓^𝛽𝑁𝑦₁≤1𝑑𝑠Γ𝛼−𝜌_𝑗×^𝑡₀(𝑡−𝑠)^{𝛼−𝜌𝑗−1}𝑎(𝑠)×𝜙𝑦₂−𝑦_1𝑌,𝑦₂−𝑦_1𝑌,…,𝑦₂−𝑦_1𝑌≤1𝑑𝑠Γ𝛼−𝜌_𝑗^𝑡0(𝑡−𝑠)^{𝛼−𝜌𝑗−1}𝑎(𝑠)𝑑𝑠𝑦₂−𝑦_1𝑌. (30) Thus it follows from (30) and (B2) that 𝑇₁𝑥₂−𝑇₁𝑥_1𝑋=max_𝑡∈𝐼𝑇₁𝑥₂(𝑡)−𝑇₁𝑥₁(𝑡+_𝑁𝑗=1)max_𝑡∈𝐼𝓓^𝜌𝑗𝑇₁𝑥₂(𝑡)−𝑇₁𝑥₁(𝑡)≤1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑎(𝑠)+𝑑𝑠_𝑁𝑗=11Γ𝛼−𝜌_𝑗^𝑡0(𝑡−𝑠)^{𝛼−𝜌𝑗−1}𝑎(𝑠)𝑑𝑠𝑦₂−𝑦_1𝑌≤𝑘₁𝑦₂−𝑦_1𝑌. (31) Similarly, we can get 𝑇₁𝑦₂(𝑡)−𝑇₁𝑦₁(𝑡≤1)Γ(𝜎)^𝑡₀(𝑡−𝑠)^𝜎−1𝑐(𝑠)𝑑𝑠𝑥₂−𝑥_1𝑋,𝓓^𝛽𝑗𝑇₁𝑦₂(𝑡)−𝓓^𝛽𝑗𝑇₁𝑦₁(𝑡≤1)Γ𝜎−𝛽_𝑗^𝑡0(𝑡−𝑠)^{𝜎−𝛽𝑗−1}𝑐(𝑠)𝑑𝑠𝑥₂−𝑥_1𝑋,𝑇₁𝑦₂−𝑇₁𝑦_1𝑌=max_𝑡∈𝐼𝑇₁𝑦₂(𝑡)−𝑇₁𝑦₁(𝑡+_𝑁𝑗=1)max_𝑡∈𝐼𝒟^𝜌𝑗𝑇₁𝑦₂(𝑡)−𝑇₁𝑦₁(𝑡≤1)Γ(𝜎)^𝑡₀(𝑡−𝑠)^𝜎−1𝑐(𝑠)+𝑑𝑠_𝑁𝑗=11Γ𝜎−𝛽_𝑗^𝑡0(𝑡−𝑠)^{𝜎−𝛽𝑗−1}𝑐(𝑠)𝑑𝑠𝑥₂−𝑥_1𝑋≤𝑙₁𝑥₂−𝑥_1𝑋. (32)
Hence, for the Euclidean distance 𝑑 on ℝ², we get 𝑑𝑇𝑥₂,𝑦₂,𝑇𝑥₁,𝑦₁=𝑇₁𝑥₂−𝑇₁𝑥₁²_𝑋+𝑇₁𝑦₂−𝑇₁𝑦₁²_𝑌≤𝑘²₁𝑥₂−𝑥₁²_𝑋+𝑙²₁𝑦₂−𝑇₁²_𝑌≤max𝑘²₁,𝑙²₁𝑥₂−𝑥₁²_𝑋+𝑦₂−𝑇₁²_𝑌=max𝑘²₁,𝑙²₁𝑑𝑥₂,𝑦₂,𝑥₁,𝑦₁. (33) Thus 𝑇 is a contraction since max{𝑘²₁,𝑙²₁}<1.
By Banach contraction principle, 𝑇 has a unique fixed point, which is a solution of the coupled system (1). The proof is completed.

An Example. Consider the existence of solutions for the following coupled system of multiterm nonlinear fractional differential equations: −𝓓^3.5𝑥(𝑡)=𝑡+16.08𝑦(𝑡)25.26+𝑒^{(𝓓0.8𝑦(𝑡))},𝓓^3.5𝑥(0)=0,𝑖=1,2,…,4,−𝓓^4.2𝑦(𝑡)=10000𝑡5501^−1/2𝑥^0.2(𝑡)+𝑡²𝓓^0.5𝑥(𝑡)^0.5,𝓓^4.2−𝑗𝑦(0)=0,𝑗=1,2,…,5, (34) where 𝑡∈(0,1].

Let 𝑓𝑡,𝑥₀,𝑥₁=𝑡+16.08𝑥25.26⁰+𝑒^𝑥1,𝑔𝑡,𝑥₀,𝑥₁=𝑡^−1/2𝑥₀^0.2+𝑡²𝑥₁^0.5, (35) and choose 𝑎(𝑡)=𝑡(𝑡)=16.08,𝑏,𝜙𝑥25.26₀,𝑥₁=𝑥₀+𝑒^𝑥1(𝑡)𝑑(𝑡)=,𝑐=0,10000𝑡5501^−1/2+𝑡²,𝜓𝑥₀,𝑥₁=𝑥₀^0.2+𝑥₁^0.5. (36) Then 𝑓𝑡,𝑥₀,𝑥₁≤𝑎(𝑡)+𝑏(𝑡)𝜙𝑥₀,𝑥₁,𝑔𝑡,𝑥₀,𝑥₁≤𝑐(𝑡)+𝑑(𝑡)𝜓𝑥₀,𝑥₁; (37) consequently, (A1) holds.

In the following, we check the condition (A1). Since 𝑘₁1=maxΓ(3.5)^𝑡₀(𝑡−𝑠)^2.5𝑠16.08𝑑𝑠+Γ(3)^𝑡₀(𝑡−𝑠)²𝑠𝑘6.08𝑑𝑠=0.01,₂1=maxΓ(3.5)^𝑡₀(𝑡−𝑠)^2.5125.26𝑑𝑠+Γ(3)^𝑡₀(𝑡−𝑠)²𝑙25.26𝑑𝑠=0.01,₁𝑙=0,₂=1000015501×maxΓ(4.2)^𝑡₀(𝑡−𝑠)^3.2𝑠^−1/2+𝑠²+1𝑑𝑠Γ(3.4)^𝑡₀(𝑡−𝑠)^2.4𝑠^−1/2+𝑠²𝑑𝑠=1, (38) take 𝑅₀=5; we have 𝜙𝑅₀,𝑅₀=𝑅₀+𝑒^𝑅0=5+𝑒⁵𝑅=153.44<⁰−𝑘₁𝑘₂=5−0.01𝜓𝑅0.01=499,₀,𝑅₀=𝑅₀^0.2+𝑅₀^0.5=5^0.2+5^0.5𝑅=3.6158<⁰−𝑙₁𝑙₂=5, (39) which implies that (A2) is satisfied. Hence, by Theorem 5, the coupled system of fractional differential equation (34) has a solution.

Remark 11. In the coupled system of fractional differential equation (34), the nonlinear function 𝑓 involves exponential growth, but the results of [12] are only allowed to be power growth; that is, (34) cannot be solved by using the results of [12]. So the results obtained in this paper give a significant improvement of the previous work in [12].

Acknowledgments

The authors would like to express their sincere gratitude to the anonymous reviewers and academic editor for a number of valuable comments and suggestions. The authors were supported financially by the Natural Science Foundation of Shandong Province of China (ZR2010AM022).