Positive Solutions for System of Nonlinear Fractional Differential Equations in Two Dimensions with Delay

Abstract

We investigate the existence and uniqueness of positive solution for system of nonlinear fractional differential equations in two dimensions with delay. Our analysis relies on a nonlinear alternative of Leray-Schauder type and Krasnoselskii's fixed point theorem in a cone.

1. Introduction

Fractional differential equations have gained considerable importance due to their varied applications [1–5] in viscoelasticity, electroanalytical chemistry, and many physical problems [1]. So far there have been several fundamental works on the fractional derivative and fractional differential equations, written by Oldham and Spanier [3], Miller and Ross [2], Podlubny [1], and others. These works are an introduction to the theory of the fractional derivative and fractional differential equations and provide a systematic understanding of the fractional calculus such as the existence and the uniqueness, some analytic methods for solving fractional differential equations, namely, Green`s function method, the Mellin transform method, and the power series.

Control systems subject to delays have been extensively studied [6] and the delay differential equations are large and important class of dynamic systems. They often arise in either natural or technological control problems. Equations with discontinuity often appear in various control theory models [7, 8]. Time delay, always existing in real systems, usually results in oscillations around the discontinuity surface. Shustin [9] has studied various aspects of oscillations for the system of differential equations with the delay𝑥̇(𝑡)=𝐹_𝑖(𝑥₁(𝑡),…,𝑥_𝑛(𝑡),𝑡)−sign𝑥_𝑖(𝑡−ℎ_𝑖),𝑖=1,…,𝑛, (1.1) in dynamics of oscillations in a multidimension, where 𝐹₁,…,𝐹_𝑛∈𝐶¹(ℝ^𝑛+1) and ℎ₁,…,ℎ_𝑛 are positive constants.

Roy et al. developed a control methodology for linear time-invariant plants that uses multiple delayed observations in feedback [10]. Fractional differential systems have proved to be useful in control processing for the last two decades [11, 12].

As explained in the above text regarding the ordinary differential equations in control theory and others, we can modify the applications of ordinary differential equations to ordinary fractional differential equations.

The existence and uniqueness of solution for the system of fractional differential equation have been studied in the papers [13, 14]. As a pursuit in this paper, we discuss the existence and uniqueness of positive solutions for system of nonlinear fractional differential equations in two-dimensional with the delay𝐷^𝛼[𝑥(𝑡)−𝑥(0)]=𝑓(𝑡,𝑥_𝑡,𝑦_𝑡𝐷),𝑡∈(0,𝑇],_𝛽[𝑦(𝑡)−𝑦(0)]=𝑔(𝑡,𝑥_𝑡,𝑦_𝑡),𝑡∈(0,𝑇],(𝑥(𝑡),𝑦(𝑡))=(𝜙(𝑡),𝜓(𝑡)),𝑡∈[−𝜏,0], (1.2) where (i)0<𝛼,𝛽<1 and 𝐷^𝛼,𝐷^𝛽 are the Riemann-Liouville fractional derivatives,(ii)𝑓,𝑔:(0,𝑇]×𝐶² → [0,+∞) are given continuous functions so that lim_𝑡→0⁺𝑓(𝑡,𝑥_𝑡,𝑦_𝑡) = lim_𝑡→0⁺𝑔(𝑡,𝑥_𝑡,𝑦_𝑡)=+∞ (i.e., 𝑓 and 𝑔 are singular at 𝑡=0) and 𝐶=𝐶([−𝜏,0],ℝ^≥0),(iii)𝑥_𝑡,𝑦_𝑡∈𝐶, where 𝑥_𝑡(𝑠)=𝑥(𝑡+𝑠),𝑦_𝑡(𝑠)=𝑦(𝑡+𝑠),𝑠∈[−𝜏,0],𝑡∈(0,𝑇].

The paper is organized as follows. In Section 2, we give definitions of the fractional derivative and fractional integral with some basic properties. Required topics of functional analysis were also introduced. Section 3 deals with existence of positive solution theorem and gives an illustrative example. The unique positive solution theorem with an example has been discussed in Section 4.

2. Basic Definitions and Preliminaries

We recall some standard definitions and results [1–4, 15, 16]. In the following Π and Σ are real Banach spaces and 𝑊 is an operator (not necessarily linear) with domain in Π and range in Σ.

Definition 2.1. A subset 𝐾 of Π is called a cone if the following conditions hold well:
(i) the set 𝐾 is closed,
(ii) if 𝑓,𝑔∈𝐾,𝛼>0,  𝛽>0 implies that 𝛼𝑓+𝛽𝑔∈𝐾,
(iii) if 𝑓∈𝐾,−𝑓∈𝐾 implies that 𝑓=𝜃 which is the zero element of Π.

Note:
Every cone 𝐾 induces a semiordering in Π, namely, f≺𝑔 if 𝑔−𝑓∈𝐾.

Definition 2.2. For 𝑓,𝑔∈𝐸 the order interval ⟨𝑓,𝑔⟩ is defined as [17] ⟨𝑓,𝑔⟩={ℎ∈𝐸∶𝑓≤ℎ≤𝑔}. (2.1)

Definition 2.3. A subset 𝐸⊂Π is called order bounded if 𝐸 is contained in some order interval.

Definition 2.4. A cone 𝐾 is called normal if there exists a positive constant 𝜇 such that 𝑓,𝑔∈𝑉 and 𝜃≺𝑓≺𝑔 implies that ‖𝑓‖≤𝜇‖𝑔‖.

We state the two fixed point results which will be needed in this paper. Our first result is a nonlinear alternative of Leray-Schauder type in a cone whereas our second is Krasnoselskii`s fixed point theorem.

Theorem 2.5 (Leray-Schauder Theorem). Let 𝐸 be a Banach space with 𝐶⊆𝐸 closed and convex. Assume that 𝑈 is relatively open subset of 𝐶 with 0∈𝑈 and 𝑊∶𝑈→𝐶 is a continuous, compact map. Then either
(i)𝑊 has fixed point in 𝑈 or
(ii) there exist 𝑢∈𝜕𝑈 and 𝛾∈(0,1) with 𝑢=𝛾𝑊𝑢.

Theorem 2.6 (Krasnoselskii`s fixed point theorem [16]). Let 𝐸=(𝐸,‖⋅‖) be a Banach space and let 𝐾⊂𝐸 be a cone in 𝐸. Assume that Ω₁ and Ω₂ are open subsets of 𝐸 with 0∈Ω₁ and Ω₁⊂Ω₂ and let 𝑊∶𝐾∩(Ω₂⧵Ω₁)→𝐾 be continuous and completely continuous. In addition suppose that either
(i)‖𝑊𝑢‖≤‖𝑢‖ for 𝑢∈𝐾∩𝜕Ω₁ and ‖𝑊𝑢‖≥‖𝑢‖ for 𝑢∈𝐾∩Ω₂ or
(ii)‖𝑊𝑢‖≤‖𝑢‖ for 𝑢∈𝐾∩𝜕Ω₁ and ‖𝑊𝑢‖≤‖𝑢‖ for 𝑢∈𝐾∩Ω₂.
Then, 𝑊 has a fixed point in 𝐾∩(Ω₂⧵Ω₁).

In this paper the Beta function 𝐵(𝛼,𝛽) is used also. 𝐵(𝛼,𝛽) is closely related to the Gamma function [1]. If 𝛼,𝛽>0, then it has the integral representation 𝐵(𝛼,𝛽)=¹₀𝑡^𝛼−1(1−𝑡)^𝛽−1𝑑𝑡. (2.2) It may be written in terms of the Gamma function as 𝐵(𝛼,𝛽)=Γ(𝛼)Γ(𝛽)/Γ(𝛼+𝛽).

Definitions of Riemann-Liouville fractional derivative/integral and their properties are given bellow [1–4].

Definition 2.7. Let 𝑥 : [𝑎,𝑏]→ℝ and 𝑥∈𝐿¹[𝑎,𝑏] then the expression 𝐼^𝛼_𝑎+1𝑥(𝑡)=Γ(𝛼)^𝑡_𝑎(𝑡−𝑠)^𝛼−1𝑥(𝑡)𝑑𝑡,𝑥>𝑎 (2.3) is called a left-sided fractional integral of order 𝛼.

Definition 2.8. Let 𝑛 be a positive integer number and 𝛼∈(𝑛−1,𝑛]. The left-sided fractional derivative of a function 𝑥∶[𝑎,𝑏]→ℝ is defined as 𝐷^𝛼_𝑎+𝑥(𝑡)=𝐷^𝑛(𝐼^𝑛−𝛼𝑥(𝑡)),𝑡∈[𝑎,𝑏]. (2.4) We denote 𝐷^𝛼_𝑎+𝑥(𝑡) as 𝐷^𝛼_𝑎𝑥(𝑡) and 𝐼^𝛼_𝑎+𝑥(𝑡) as 𝐼^𝛼_𝑎𝑥(𝑡). Further 𝐷^𝛼₀⁺𝑥(𝑡) and 𝐼^𝛼₀⁺𝑥(𝑡) are referred to as 𝐷^𝛼𝑥(𝑡) and 𝐼^𝛼𝑥(𝑡), respectively.

If the fractional derivative 𝐷^𝛽_𝑎𝑥(𝑡) is integrable then𝐼^𝛼_𝑎(𝐷^𝛽_𝑎𝑥(𝑡))=𝐼_𝑎^𝛼−𝛽𝑥(𝑡)−{𝐼_𝑎^𝑛−𝛽𝑥(𝑡)}_𝑡=𝑎(𝑡−𝑎)^𝛼−𝑛−Γ(𝛼−𝑛+1)_{𝑛−1𝑗=1}{𝐷_𝑎^𝛽−𝑗𝑥(𝑡)}_𝑡=𝑎(𝑡−𝑎)^𝛼−𝑗,Γ(𝛼−𝑗+1) (2.5) where 𝑛−1≤𝛽≤𝛼<𝑛 and 𝑛∈ℕ [1]. Further, if 𝑥∈𝐶[𝑎,𝑏], then {𝐼_𝑎^𝑛−𝛽𝑥(𝑡)}_𝑡=𝑎=0 and then (2.5) implies that𝐼^𝛼_𝑎(𝐷^𝛼_𝑎𝑥(𝑡))=𝑥(𝑡),0<𝛼<1. (2.6)

3. Existence Theorem

In this section we discuss the system of nonlinear fractional differential equation (1.2) which has at least one positive solution.

Lemma 3.1 (see [18]). Let ℎ∶(0,𝑇]→ℝ be a continuous function and lim_𝑡→0⁺ℎ(𝑡)=+∞. If there exits 𝜎∈(0,1) such that 0<𝜎<𝛼 and 𝑡^𝜎ℎ(𝑡) is a continuous function on [0,𝑇], then 𝐻(𝑡)=𝐼^𝛼𝑡^𝜎ℎ(𝑡) is continuous [0,𝑇].

In the following theorem we want to prove that (1.2) is equivalent to a system of integral equations.

Theorem 3.2. Suppose that 𝑓,𝑔 : (0,𝑇]×𝐶²→[0,+∞) are given continuous functions with lim_𝑡→0𝑓(𝑡,𝑥_𝑡,𝑦_𝑡)=+∞=lim_𝑡→0𝑔(𝑡,𝑥_𝑡,𝑦_𝑡)=+∞. If there exist 𝜌,𝜎∈(0,1) such that 0<𝜌,𝜎<𝛼 and 𝑡^𝜌𝑓(𝑡,𝑥_𝑡,𝑦_𝑡),𝑡^𝜎𝑔(𝑡,𝑥_𝑡,𝑦_𝑡) are continuous functions on [0,𝑇], then (1.2) is equivalent to the system of integral equations 𝑥(𝑡)=𝑥(0)+𝐼^𝛼𝑓(𝑡,𝑥_𝑡,𝑦_𝑡),𝑡∈(0,𝑇],𝑦(𝑡)=𝑦(0)+𝐼^𝛽𝑔(𝑡,𝑥_𝑡,𝑦_𝑡),𝑡∈(0,𝑇],𝑥(𝑡)=𝜙(𝑡)≥0,𝑦(𝑡)=𝜓(𝑡)≥0,𝑡∈[−𝜏,0]. (3.1)

Proof. It is sufficient to prove that the first equation of system equation (1.2) is equivalent to the first equation of system equation (3.1). So, 𝐼^𝛼{𝐷^𝛼[𝑥(𝑡)−𝑥(0)]}=𝐼^𝛼𝑓(𝑡,𝑥_𝑡,𝑦_𝑡),𝑡∈(0,𝑇]. (3.2) Using (2.6) we get 𝐼^𝛼{𝐷^𝛼[𝑥(𝑡)−𝑥(0)}=𝑥(𝑡)−𝑥(0). Hence (3.2) implies that 𝑥(𝑡)=𝑥(0)+𝐼^𝛼𝑓(𝑡,𝑥_𝑡,𝑦_𝑡),𝑡∈(0,𝑇].
Conversely, first we note that 𝐼^𝛼𝑓(𝑡,𝑥(𝑡),𝑦(𝑡)) = 𝐼^𝛼{𝑡^𝜌𝑡^−𝜌𝑓(𝑡,𝑥_𝑡,𝑦_𝑡)} exists by Lemma 3.1. 𝐷^𝛼𝐼^𝛼𝑓(𝑡,𝑥_𝑡,𝑦_𝑡)= 𝑓(𝑡,𝑥_𝑡,𝑦_𝑡) as 𝑡^−𝜌𝑓(𝑡,𝑥_𝑡,𝑦_𝑡) is continuous and 𝐼^𝛼𝑓(𝑡,𝑥_𝑡,𝑦_𝑡)∈𝐶[0,𝑇]. Therefore, the system of fractional integral equation (3.1) is a solution of (1.2).
Let 𝜔 : [−𝜏,𝑇]→[0,+∞)×[0,+∞) be a function defined by 𝜔(𝑡)=(𝜙(0),𝜓(0)),𝑡∈[0,𝑇],(𝜙(𝑡),𝜓(𝑡)),𝑡∈[−𝜏,0], (3.3) for each 𝑧(𝑡)=(𝜂(𝑡),𝛾(𝑡)) with 𝑧(0)=(0,0) where 𝜂 and 𝛾 belong to 𝐶([0,𝑇],ℝ). We introduce the notation 𝑧 defined by 𝑧(𝑡)=(𝜂(𝑡),𝛾(𝑡)),𝑡∈[0,𝑇],(0,0),𝑡∈[−𝜏,0]. (3.4) We can decompose (𝑥(⋅),𝑦(⋅)) as (𝑥(𝑡),𝑦(𝑡))=𝑧(𝑡)+𝜔(𝑡),𝑡∈[−𝜏,𝑇] which implies that (𝑥_𝑡,𝑦_𝑡)=𝑧_𝑡+𝜔_𝑡,𝑡∈[0,𝑇]. Hence, by Theorem 3.2, (1.2) is equivalent to the system of integral equations 𝜂(𝑡)=𝐼^𝛼𝑓(𝑡,𝑧_𝑡+𝜔_𝑡),𝛾(𝑡)=𝐼^𝛽𝑔(𝑡,𝑧_𝑡+𝜔_𝑡), (3.5) where 𝑡∈[0,𝑇]. Set Ω={𝑧=(𝜂(𝑡),𝛾(𝑡))∶𝜂,𝛾∈𝐶([0,𝑇],ℝ),𝜂(0)=𝛾(0))=0}. For each 𝑧∈Ω, let ‖𝑧‖_𝑇 be the seminorm in Ω defined by ‖𝑧‖_𝑇=‖𝑧(0)‖+‖𝑧‖=‖𝑧‖=sup{|𝜂(𝑡)|,|𝛾(𝑡)|∶𝑡∈[0,𝑇]}. (3.6)Ω is a Banach space with norm ‖⋅‖_𝑇. Let 𝐾 be a cone of Ω,𝐾𝐾={𝑧=(𝜂(𝑡),𝛾(𝑡))∈Ω∶𝜂(𝑡)≥0,𝛾(𝑡)≥0,𝑡∈[0,𝑇]},_∗={((𝑥,𝑦)∶𝑥,𝑦∈𝐶([−𝜏,𝑇],ℝ⁺)),(𝑥(𝑡),𝑦(𝑡))=(𝜙(𝑡)≥0,𝜓(𝑡)≥0),𝑡∈[−𝜏,0]}. (3.7) For each 𝑡∈[0,𝑇], we define the operator 𝑊 : 𝐾→𝐾 by 𝑊𝑧(𝑡)=(𝐼^𝛼𝑓(𝑡,𝑧_𝑡+𝜔_𝑡),𝐼^𝛽𝑔(𝑡,𝑧_𝑡+𝜔_𝑡))=(𝐹𝜂(𝑡),𝐺𝛾(𝑡)), (3.8) where 𝐹𝜂(𝑡)=𝐼^𝛼𝑓(𝑡,𝑧_𝑡+𝜔_𝑡),𝐺𝜂(𝑡)=𝐼^𝛽𝑔(𝑡,𝑧_𝑡+𝜔_𝑡). (3.9)

Lemma 3.3. Let 𝑓(𝑡,𝑥_𝑡,𝑦_𝑡),𝑔(𝑡,𝑥_𝑡,𝑦_𝑡),and 𝑡∈(0,𝑇] be nonnegative continuous functions, where 𝑥_𝑡,𝑦_𝑡∈𝐶. Then, the operator 𝑊 with the following conditions is maps-bounded set into bounded sets in 𝐾.
(H1)0<lim_𝑡→0⁺𝑓(𝑡,𝑥_𝑡,𝑦_𝑡)=lim_𝑡→0⁺𝑔(𝑡,𝑥_𝑡,𝑦_𝑡)=+∞  (H2) if there exist 𝜌,𝜎∈(0,1)suchthat𝜌,𝜎<𝛼,  𝑡^𝜌𝑓(𝑡,𝑥_𝑡,𝑦_𝑡),and𝑡^𝜎𝑔(𝑡,𝑥_𝑡,𝑦_𝑡) are continuous on [0,𝑇].

Proof. By assumptions of Lemma 3.3 and by using Lemma 3.1, it is clear that 𝑊𝑧(𝑡)=(𝐹𝜂(𝑡),𝛾(𝑡))∈𝐾. There exist positive constants Λ₁,Λ₂ such that |𝑡^𝜌𝑓(𝑡,𝑥_𝑡,𝑦_𝑡)|≤Λ₁ and |𝑡^𝜎𝑔(𝑡,𝑥_𝑡,𝑦_𝑡)|≤Λ₂, as 𝑡^𝜌𝑓(𝑡,𝑥_𝑡,𝑦_𝑡) and 𝑡^𝜎g(𝑡,𝑥_𝑡,𝑦_𝑡) are continuous on [0,𝑇]×𝐶². Hence, 𝐼^𝛼𝑓(𝑡,𝑥_𝑡,𝑦_𝑡)=𝐼^𝛼(𝑡^−𝜌𝑡^𝜌𝑓(𝑡,𝑥_𝑡,𝑦_𝑡))≤Λ₁𝐼^𝛼𝑡^−𝜌=Λ₁Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑠^−𝜌=Λ𝑑𝑠₁𝑡^𝛼−𝜌ΛΓ(𝛼)𝐵(1−𝜌,𝛼)≤¹𝑇^𝛼−𝜌Γ(𝛼)𝐵(1−𝜌,𝛼). (3.10) Similarly, 𝐼^𝛽𝑔(𝑡,𝑥_𝑡,𝑦_𝑡Λ)≤²𝑇^𝛽−𝜎Γ(𝛽)𝐵(1−𝜎,𝛽). (3.11)
Let 𝐻⊂𝐾 be bounded, that is, for each 𝑧=(𝜂,𝛾)∈𝐻 there exist positive constants 𝐿 such that ‖𝑧‖_𝑇≤𝐿. In view of (3.10), (3.11) we have |𝐹_𝜂(𝑡)|≤|𝐼^𝛼𝑓(𝑡,𝑧_𝑡+𝜔_𝑡Λ)|≤¹𝑇^𝛼−𝜌Γ(𝛼)𝐵(1−𝜌,𝛼),|𝐺_𝛾(𝑡)|≤|𝐼^𝛼𝑓(𝑡,𝑧_𝑡+𝜔_𝑡Λ)|≤²𝑇^𝛽−𝜎Γ(𝛼)𝐵(1−𝜎,𝛽). (3.12) Therefore, ‖𝑊𝑧(𝑡)‖≤Λ where ΛΛ=max{¹𝑇^𝛼−𝜌ΛΓ(𝛼)𝐵(1−𝜌,𝛼),²𝑇^𝛽−𝜎Γ(𝛽)𝐵(1−𝜎,𝛽)}. (3.13) Hence, 𝑊(𝐾) is bounded.

Lemma 3.4. Suppose that 𝑓(𝑡,𝑥_𝑡,𝑦_𝑡),𝑔(𝑡,𝑥_𝑡,𝑦_𝑡),𝑥_𝑡,𝑦_𝑡∈𝐶,𝑡∈(0,𝑇] are continuous functions and further two conditions (H1) and (H2) are satisfied. Then, the operator 𝑊 is continuous and completely continuous.

Proof. In the following we want to prove that 𝑊 is continuous. It is sufficient to prove that 𝐹 and 𝐺 are continuous. Let 𝑧₀=(𝑢₀,𝑣₀)∈𝐾 with ‖𝑢₀‖=𝑐₀ and ‖𝑣₀‖=𝑑₀. If 𝑧=(𝑢,𝑣)∈𝐾 with ‖𝑢−𝑢₀‖<𝑙₁ and ‖𝑣−𝑣₀‖<𝑙₂, then ‖𝑢‖<𝑙₁+𝑐₀ and ‖𝑣‖<𝑙₂+𝑑₀. By the continuity of 𝑡^𝜌𝑓(𝑡,𝑧_𝑡+𝜔_𝑡) and 𝑡^𝜎𝑔(𝑡,𝑧_𝑡+𝜔_𝑡) on [0,𝑇], we conclude that 𝑡^𝜌𝑓(𝑡,𝑧_𝑡+𝜔_𝑡) and 𝑡^𝜎𝑔(𝑡,𝑧_𝑡+𝜔_𝑡) are uniformly continuous on [0,𝑇]×[0,𝑐₀]×[0,𝑑_𝑜]. Therefore, |𝐹𝑢(𝑡)−𝐹𝑢₀(𝑡)|,|𝐺𝑣(𝑡)−𝐺𝑣₀(𝑡)|,𝑡∈[0,𝑇] are continuous and hence, ‖𝑊𝑧(𝑡)−𝑊𝑧₀(𝑡)‖=max{|𝐹𝑢(𝑡)−𝐹𝑢₀(𝑡)|,|𝐺𝑣(𝑡)−𝐺𝑣₀(𝑡)|}. (3.14) is continuous.
Finally, we want to prove that the operator 𝑊 is equicontinuous. Let 𝐻⊂𝐾 be bounded. Suppose that, 𝑧=(𝑢,𝑣)∈𝐾 and 𝑡,𝑟∈[0,𝑇] such that 𝑡<𝑟. For given 𝜖>0, there exists 𝛿₁>0, so that if |𝑡−𝑟|<𝛿₁, then |𝑡^𝜌𝑓(𝑡,𝑧_𝑡+𝜔_𝑡)−𝑡^𝜌𝑓(𝑡,𝑧_𝑟+𝜔_𝑟)|<𝜖:|𝐹𝑢(𝑡)−𝐹𝑢(𝑟)|=|𝐼^𝛼{𝑓(𝑡,𝑧_𝑡+𝜔_𝑡)−𝑓(𝑟,𝑧_𝑟+𝜔_𝑟≤1)}||Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑠^−𝜌𝑠^𝜌𝑓(𝑠,𝑧_𝑠+𝜔_𝑠)𝑑𝑠−^𝑟₀(𝑟−𝑠)^𝛼−1𝑠^−𝜌𝑠^𝜌𝑓(𝑠,𝑧_𝑠+𝜔_𝑠≤1)𝑑𝑠||Γ(𝛼)^𝑟₀(𝑡−𝑠)^𝛼−1𝑠^−𝜌𝑠^𝜌𝑓(𝑠,𝑧_𝑠+𝜔_𝑠)−^𝑟₀(𝑟−𝑠)^𝛼−1𝑠^−𝜌𝑠^𝜌𝑓(𝑠,𝑧_𝑠+𝜔_𝑠+1)𝑑𝑠||Γ(𝛼)^𝑟_𝑡(𝑡−𝑠)^𝛼−1𝑠^−𝜌𝑠^𝜌𝑓(𝑠,𝑧_𝑠+𝜔_𝑠≤1)𝑑𝑠|Γ(𝛼)^𝑟₀|(𝑡−𝑠)^𝛼−1𝑠^−𝜌−(𝑟−𝑠)^𝛼−1𝑠^−𝜌|×|𝑠^𝜌𝑓(𝑠,𝑧_𝑠+𝜔_𝑠)−𝑠^𝜌𝑓(𝑠,𝑧_𝑠+𝜔_𝑠+1)|𝑑𝑠Γ(𝛼)^𝑟_𝑡|(𝑡−𝑠)^𝛼−1𝑠^−𝜌||𝑠^𝜌𝑓(𝑠,𝑧_𝑠+𝜔_𝑠<𝜖)|𝑑𝑠Γ(𝛼)^𝑟₀{(𝑟−𝑠)^𝛼−1𝑠^−𝜌+(𝑟−𝑠)^𝛼−1𝑠^−𝜌Λ}𝑑𝑠+¹Γ(𝛼)^𝑟_𝑡(𝑠−𝑡)^𝛼−1𝑠^−𝜌=𝑑𝑠2𝜖ΛΓ(𝛼)𝐵(1−𝜌,𝛼)+¹Γ(𝛼)^𝑟_𝑡(𝑠−𝑡)^𝛼−1𝑠^−𝜌𝑑𝑠. (3.15) Similarly, for given 𝜖>0, there exists 𝛿₂>0, so that if |𝑡−𝑟|<𝛿₂, then |𝐺𝑢(𝑡)−𝐺𝑢(𝑟)|≤2𝜖ΛΓ(𝛽)𝐵(1−𝜎,𝛽)+²Γ(𝛽)^𝑟_𝑡(𝑠−𝑡)^𝛽−1𝑠^−𝜎𝑑𝑠 (3.16)

Case 1. If 𝑡=0 and 0<𝑟<𝛿_𝑖,𝑖=1,2, then ^𝑟_𝑡(𝑠−𝑡)^𝛼−1𝑠^−𝜌𝑑𝑠≤max_{0≤𝑟≤𝑇}^𝑟₀𝑠^{𝛼−𝜌−1}𝑇𝑑𝑠≤^𝛼−𝜌.𝛼−𝜌 (3.17) Set 𝜆₁1=max{2Γ(𝛼)𝐵(1−𝜌,𝛼),2Λ₁𝑇𝜖(𝛼−𝜌)^𝛼−𝜌𝜆},₂1=max{2Γ(𝛽)𝐵(1−𝜎,𝛽),2Λ₂𝑇𝜖(𝛽−𝜎)^𝛽−𝜎}. (3.18) Hence, |𝐹𝑢(𝑡)−𝐹𝑢(𝑟)|<𝜆₁𝜖 and |𝐺𝑣(𝑡)−𝐺𝑣(𝑟)|<𝜆₂𝜖.

Case 2. If 0<𝑡<𝑟<𝛿_𝑖≤𝑇,𝑖=1,2, then 𝑚₁=Λ₁Γ(𝛼)^𝑟_𝑡(𝑠−𝑡)^𝛼−1𝑠^−𝜌𝑑𝑠,𝑚₂=Λ₂Γ(𝛽)^𝑟_𝑡(𝑠−𝑡)^𝛽−1𝑠^−𝜎𝑑𝑠 (3.19) are finite. Set 𝜇₁1=max{2Γ(𝛼)𝐵(1−𝜌,𝛼)},𝜇₂1=max{2Γ(𝛽)𝐵(1−𝜎,𝛽)}. (3.20) Hence, |𝐹𝑢(𝑡)−𝐹𝑢(𝑟)|<𝜇₁𝜖 and |𝐺𝑣(𝑡)−𝐺𝑣(𝑟)|<𝜇₂𝜖. Thus, for any given 𝜖>0, for all 𝑧=(𝑢,𝑣)∈𝐾 and for all 𝑡,𝑟∈[0,𝑇] with |𝑟−𝑡|<𝛿 where 𝛿=min{𝛿₁,𝛿₂}, for the Euclidian distance 𝑑 on ℝ² we have 𝑑(𝑊𝑧(𝑡),𝑊𝑧(𝑟))={[𝐹𝑢(𝑟)−𝐹𝑢(𝑡)]²}^1/2+{[𝐺𝑣(𝑟)−𝐺𝑣(𝑡)]²}^1/2<𝜂√2𝜖, (3.21) where 𝜂=max{𝜇₁,𝜇₂,𝜆₁,𝜆₂}. Therefore, 𝑊(𝐻) is equicontinuous and the Ascoli-Arzela theorem implies that 𝑊(𝐻) is compact and hence 𝑊∶𝐾→𝐾 is completely continuous.

Theorem 3.5. If 𝑓(𝑡,𝑥_𝑡,𝑦_𝑡) and 𝑔(𝑡,𝑥_𝑡,y_𝑡) on (0,𝑇]×𝐶² are nonnegative continuous functions, then the system of nonlinear fractional differential equation (1.2) with the conditions (H1) and (H2) has at least one positive solution 𝑥^∗∈𝐾^∗, satisfying ‖𝑥^∗‖≤max{‖𝜙‖,‖𝜓‖,ℎ}, where ℎ is observed in the proof of the theorem.

Proof. By Lemma 3.3, the operator 𝑊∶𝐾→𝐾 is continuous and completely continuous. We prove that there exits an open set 𝑈⊆𝐾, with 𝑧≠𝛾𝑊(𝑧) for 𝛾∈(0,1) and 𝑧∈𝜕𝑈. Let 𝑧=(𝑢,𝑣)∈𝐾 be any solution of 𝑧=𝛾𝑊(𝑧),𝛾∈(0,1). In view of Theorem 3.5, we have 𝑧(𝑡)=𝛾𝑊𝑧(𝑡)=𝛾(𝐹𝑢(𝑡),𝐺𝑣(𝑡)),𝑡∈[0,𝑇]. (3.22) Hence, 𝑧(𝑡)=𝛾𝑊𝑧(𝑡)=(𝛾𝐼^𝛼𝑡^−𝜌𝑡^𝜌𝑓(𝑡,𝑧_𝑡+𝜔_𝑡),𝛾𝐼^𝛽𝑡^−𝜎𝑡^𝜎𝑔(𝑡,𝑧_𝑡+𝜔_𝑡)). (3.23) There exist positive constants 𝑁₁,𝑁₂ such that |𝑡^𝜌𝑓(𝑡,𝑧_𝑡+𝜔_𝑡)|≤𝑁₁ and |𝑡^𝜌𝑔(𝑡,𝑧_𝑡+𝜔_𝑡)|≤𝑁₂, as 𝑡^𝜌𝑓(𝑡,𝑧_𝑡+𝜔_𝑡) and 𝑡^𝜎𝑔(𝑡,𝑧_𝑡+𝜔_𝑡) are continuous on [0,𝑇]×(𝐶[−𝜏,𝑇])². Hence, 𝑁|𝑢(𝑡)|≤¹Γ(𝛼)max_{0≤𝑡≤𝑇}𝑡^𝛼−𝜌𝑁𝐵(1−𝜌,𝛼)≤¹𝑇^𝛼−𝜌𝑁Γ(𝛼)𝐵(1−𝜌,𝛼),|𝑣(𝑡)|≤²Γ(𝛽)max_{0≤𝑡≤𝑇}𝑡^𝛽−𝜎𝑁𝐵(1−𝜎,𝛽)≤²𝑇^𝛽−𝜎Γ(𝛽)𝐵(1−𝜎,𝛽). (3.24) If we consider 𝑁ℎ=1+max{¹𝑇^𝛼−𝜌𝑁Γ(𝛼)𝐵(1−𝛼,𝛼),²𝑇^𝛽−𝜎Γ(𝛽)𝐵(1−𝛽,𝛽)}, (3.25) then any solution 𝑧=𝛾𝑊(𝑧) satisfies ‖𝑧‖≠ℎ. Set 𝑈={𝑧∈𝐾∶‖𝑧‖<ℎ}. Theorem 2.5 guarantees that 𝑊 has fixed point 𝑧∈𝑈. Theorem 3.2 gives that (1.2) has a positive solution 𝑥^∗∈𝐾^∗ satisfying ‖𝑥^∗‖≤max{‖𝜙‖,‖𝜓‖,ℎ}.

Example 3.6. Consider the system of nonlinear fractional differential equation 𝐷^4/5𝑒[𝑥(𝑡)−𝑥(0)]=^−𝑡√𝑡(𝑥_𝑡+𝑦_𝑡𝐷),𝑡∈(0,1],_3/4𝑒[𝑦(𝑡)−𝑦(0)]=^−𝑡₄√𝑡(𝑥_𝑡+𝑦_𝑡),𝑡∈(0,1],(𝑥(𝑡),𝑦(𝑡))=(−sin𝜋𝑡,−sin(𝜋𝑡2)),𝑡∈[−1,0], (3.26) where 𝑥_𝑡=^−𝑡₋₁𝑑𝑠1+𝑥²=(𝑡+𝑠)⁰_𝑡−1𝑑𝑠1+𝑥²𝑦(𝑠),𝑡∈[−1,0],_𝑡=^−𝑡₋₁𝑦²(𝑡+𝑠)1+𝑦²(𝑡+𝑠)𝑑𝑠=⁰_𝑡−1𝑦²(𝑠)1+𝑦²(𝑠)𝑑𝑠,𝑡∈[−1,0]. (3.27) Here, 𝛼=4/5,𝛽=3/4,𝑓(𝑡,𝑥_𝑡,𝑦_𝑡)=𝑒^−𝑡𝑡^−1/2(𝑥_𝑡+𝑦_𝑡), and 𝑔(𝑡,𝑥_𝑡,𝑦_𝑡)=𝑒^−𝑡𝑡^−1/4(𝑥_𝑡+𝑦_𝑡) with lim_𝑡→0⁺𝑓(𝑡,⋅,⋅)=+∞=lim_𝑡→0⁺𝑔(𝑡,⋅,⋅)=+∞. If we select 𝜌=3/4 and 𝜎=1/2, then 𝑡^𝜌𝑓(𝑡,⋅,⋅) and 𝑡^𝜎𝑔(𝑡,⋅,⋅) are continuous on [0,1].
By Theorem 3.2, the system of nonlinear fractional differential equation (3.26) is equivalent to the system of integral equation 𝑥(𝑡)=𝑥(0)+𝐼^4/5(𝑒^−𝑡𝑡^−1/2(𝑥_𝑡+𝑦_𝑡)),𝑡∈(0,1],𝑦(𝑡)=𝑦(0)+𝐼^3/4(𝑒^−𝑡𝑡^−1/4(𝑥_𝑡+𝑦_𝑡𝜋)),𝑡∈(0,1],(𝑥(𝑡),𝑦(𝑡))=(−sin𝜋𝑡,−sin2𝑡),𝑡∈[−1,0]. (3.28) Therefore, by Theorem 3.5, (3.26) has at least one positive solution 𝑥^∗∈𝐾^∗ satisfying ‖𝑥^∗‖≤max{‖𝜙‖,𝜓,ℎ} where ‖𝜙‖=max{max_{−1≤𝑡≤0}|−sin𝜋𝑡|,max_{−1≤𝑡≤0}𝜋|−sin2𝑁𝑡|}=1,ℎ=1+max{¹1Γ(4/5)𝐵(4,45𝑁),²1Γ(3/4)𝐵(2,34𝑁)},₁=max_0≤𝑡≤1{𝑡^𝜌(𝑒^−𝑡𝑡^−1/2(𝑥_𝑡+𝑦_𝑡))}=max_0≤𝑡≤1{𝑒^−𝑡𝑡^1/4(𝑥_𝑡+𝑦_𝑡4)}≤₄√,𝑁4𝑒₂=max_0≤𝑡≤1{𝑡^𝜎(𝑒^−𝑡𝑡^−1/4(𝑥_𝑡+𝑦_𝑡))}=max_0≤𝑡≤1{𝑒^−𝑡𝑡^1/2(𝑥_𝑡+𝑦_𝑡6)}≤√.2𝑒 (3.29) Note that, for each 𝑡∈[0,1], we have 0≤𝑥_𝑡,𝑦_𝑡≤2 and max_0≤𝑡≤1{𝑡^1/4𝑒^−𝑡1}=₄√4𝑒,max_0≤𝑡≤1{𝑡^1/2𝑒^−𝑡1}=√.2𝑒 (3.30)

Theorem 3.7. Suppose that 𝑓,𝑔 : (0,𝑇]×𝐶²→[0,∞) are nonnegative continuous with lim_𝑡→+∞𝑓(𝑡,𝑥_𝑡,𝑦_𝑡)=lim_𝑡→+∞𝑔(𝑡,𝑥_𝑡,𝑦_𝑡)=+∞. If there exist 𝜌,𝜎∈(0,1) such that 0<𝜌<𝛼,0<𝜎<𝛽 and 𝑡^𝜌𝑓(𝑡,𝑥_𝑡,𝑦_𝑡) and 𝑡^𝜎𝑔(𝑡,𝑥_𝑡,𝑦_𝑡) are continuous on [0,𝑇]×𝐶², then (1.2) has at least one positive solution with the following conditions:
(H3)  𝑡^𝜌𝑓(𝑡,𝑥_𝑡,𝑦_𝑡)≤𝜌₁𝑇^𝜌−𝛼(Γ(1−𝜌+𝛼)/Γ(1−𝜌)),𝑡^𝜎𝑔(𝑡,𝑥_𝑡,𝑦_𝑡)≤𝜌₂𝑇^𝜎−𝛽(Γ(1−𝜎+𝛽)/Γ(1−𝜎)), 
 (H4) 𝑡^𝜌𝑓(𝑡,𝑥_𝑡,𝑦_𝑡)≥𝜇₁𝑇^𝜌−𝛼(Γ(1−𝜌+𝛼)/Γ(1−𝜌)),𝑡^𝜎𝑔(𝑡,𝑥_𝑡,𝑦_𝑡)≥𝜇₂𝑇^𝜎−𝛽(Γ(1−𝜎+𝛽)/Γ(1−𝜎)),
where (𝑡,𝑥_𝑡,𝑦_𝑡)∈[0,𝑇]×[0,𝜌]²,(𝑡,𝑥_𝑡,𝑦_𝑡)∈[0,𝑇]×[0,𝜇]²,𝜌=max{𝜌₁,𝜌₂}, 𝜇=min{𝜇₁,𝜇₂}, and 0<𝜇<𝜌.

Proof. For proving this theorem we provide the conditions required in Theorem 2.6. Set Ω₁Ω={𝑧=(𝑢,𝑣)∈𝐾∶‖𝑧‖<𝜇},₂={𝑧=(𝑢,𝑣)∈𝐾∶‖𝑧‖<𝜌}. (3.31) For each 𝑧=(𝑢,𝑣)∈𝐾∩𝜕Ω₂ and 𝑡∈[0,𝑇] we have 0≤𝑢(𝑡)≤𝜌 and 0≤𝑣(𝑡)≤𝜌. Condition (H3) implies that 1𝐹𝑢(𝑡)=Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑠^−𝜌𝑠^𝜌𝑓(𝑠,𝑧_𝑠+𝜔_𝑠)𝑑𝑠≤𝜌_𝑖𝑇^𝜌1−𝛼Γ(1−𝜌+𝛼)1Γ(1−𝜌)Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑠^−𝜌𝑑𝑠=𝜌₁𝑇^𝜌−𝛼𝑡^𝛼−𝜌Γ(1−𝜌+𝛼)Γ(1−𝜌)Γ(1−𝜌)Γ(1−𝜌+𝛼)=𝜌₁𝑇^𝜌−𝛼𝑡^𝛼−𝜌. (3.32) Hence ‖𝐹𝑢‖=max_{0≤𝑡≤𝑇}|𝐹𝑢(𝑡)|≤𝜌. Similarly ‖𝐺𝑣‖=max_{0≤𝑡≤𝑇}|𝐺𝑣(𝑡)|≤𝜌. Thus, ‖𝑊𝑧‖≤𝜌=‖𝑧‖ for 𝑧=(𝑢,𝑣)∈𝐾∩𝜕Ω₂. By using condition (H4) and the above-mentioned proof, we have ‖𝑊𝑧‖≥𝜇=‖𝑧‖ for 𝑧∈𝐾∩𝜕Ω₁. Therefore, by Theorem 2.6, (ii), and Theorem 3.2, the proof is completed.

Example 3.8. Consider the system of nonlinear fractional differential equation 𝐷^1/41[𝑥(𝑡)−𝑥(0)]=₅√𝑡𝑧_𝑡𝐷,𝑡∈(0,1],_1/21[𝑦(𝑡)−𝑦(0)]=₄√𝑡𝑥_𝑡𝐷,𝑡∈(0,1],_3/41[𝑧(𝑡)−𝑧(0)]=√𝑡𝑦_𝑡,𝑡∈(0,1],𝑥(𝑡)=(−sin𝜋𝑡,−sin(𝜋𝑡4),−sin(𝜋𝑡2)),𝑡∈[−1,0], (3.33) where 𝑥_𝑡,𝑦_𝑡,𝑧_𝑡 are continuous functions from [−1,0] to [0,+∞) so that 𝑥_𝑡=Γ(5/4)Γ(3/4){1+^−𝑡₋₁2𝑥²(𝑡+𝑠)+11+𝑥²𝑦(𝑡+𝑠)𝑑𝑠},_𝑡=Γ(5/4)Γ(1/4){2+^−𝑡₋₁3𝑦²(𝑡+𝑠)+21+𝑦²𝑧(𝑡+𝑠)𝑑𝑠},_𝑡=Γ(21/20)Γ(4/5){3+^−𝑡₋₁4𝑧²(𝑡+𝑠)+31+𝑧²(𝑡+𝑠)𝑑𝑠}. (3.34) Here 𝛼 = 1/4,𝛽 = 1/2,𝛾 = 3/4,𝑓(𝑡,𝑥_𝑡,𝑦_𝑡,𝑧_𝑡) = 𝑡^−1/5𝑧_𝑡, 𝑔(𝑡,𝑥_𝑡,𝑦_𝑡,𝑧_𝑡) = 𝑡^−1/4𝑥_𝑡,ℎ(𝑡,𝑥_𝑡,𝑦_𝑡,𝑧_𝑡) = 𝑡^−1/2𝑦_𝑡  such that lim_𝑡→0⁺𝑓(𝑡,⋅,⋅) = lim_𝑡→0⁺𝑔(𝑡,⋅,⋅) = lim_𝑡→0⁺ℎ(𝑡,⋅,⋅) = +∞. Also 𝑡^𝜌𝑓(𝑡,⋅,⋅),𝑡^𝜎𝑔(𝑡,⋅,⋅), and 𝑡^𝜐ℎ(𝑡,⋅,⋅) are continuous, where 𝜌 = 1/5,𝜎 = 1/4, and 𝜐 = 1/2. In the following calculations, we review conditions (H3) and (H4) by using (3.34): 𝑡^1/5𝑓(𝑡,𝑥_𝑡,𝑦_𝑡,𝑧_𝑡)=Γ(21/20)Γ(4/5){3+^−𝑡₋₁2𝑧²(𝑡+𝑠)+11+𝑧²=(𝑡+𝑠)𝑑𝑠}Γ(21/20){Γ(4/5)⁰_𝑡−12𝑧²(𝑠)+11+𝑧(𝑠)²𝑑𝑠}≤Γ(21/20)Γ(4/5){3+4⁰_𝑡−1𝑑𝑠}≤7Γ(21/20)Γ(4/5)=7Γ(1−𝜌+𝛼)𝑡Γ(1−𝜌)_1/4,𝑔(𝑡,𝑥_𝑡,𝑦_𝑡,𝑧_𝑡)=Γ(5/4)Γ(3/4)^−𝑡₋₁{1+2𝑥²(𝑡+s)+11+𝑥²=(𝑡+𝑠)𝑑𝑠}Γ(5/4)Γ(3/4)⁰_𝑡−1{1+2𝑥²₁(𝑠)+11+[𝑥₁(𝑠)]²𝑑𝑠}≤Γ(5/4)Γ(3/4){1+2⁰_𝑡−1𝑑𝑠}≤3Γ(5/4)Γ(3/4)=3Γ(1−𝜎+𝛽)𝑡Γ(1−𝜎)_1/2,ℎ(𝑡,𝑥_𝑡,𝑦_𝑡,𝑧_𝑡)=Γ(5/4)Γ(1/4){2+^−𝑡₋₁3𝑦²(𝑡+𝑠)+11+𝑦²=(𝑡+𝑠)𝑑𝑠}Γ(5/4)Γ(1/4){2+⁰_𝑡−13𝑥²₂(𝑠)+11+[𝑥₂(𝑠)]²𝑑𝑠}≤Γ(5/4)Γ(1/4){2+3⁰_𝑡−1𝑑𝑠}≤5Γ(5/4)Γ(1/4)=5Γ(1−𝜐+𝛾)Γ(1−𝜐). (3.35)(1)(2)(3)(4)(5)(6)(7)(8) Hence, 𝜌 = max{3,5,7} = 7 and the above calculations satisfy condition (H3). On the other hand, 𝑡^1/5𝑓(𝑡,𝑥_𝑡,𝑦_𝑡,𝑧_𝑡)=Γ(21/20)Γ(4/5){3+^−𝑡₋₁4𝑧²(𝑡+𝑠)+11+𝑧²=(𝑡+𝑠)𝑑𝑠}Γ(21/20)Γ(4/5){3+⁰_𝑡−14𝑧²(𝑠)+11+𝑧²(𝑠)𝑑𝑠}≥Γ(21/20)Γ(4/5){3+3⁰_𝑡−1𝑑𝑠}≥3Γ(21/20)Γ(4/5)=3Γ(1−𝜌+𝛼),𝑡Γ(1−𝜌)_1/4𝑔(𝑡,𝑥_𝑡,𝑦_𝑡,𝑧_𝑡)=Γ(5/4)Γ(3/4){1+^−𝑡₋₁2𝑥²(𝑡+𝑠)+11+𝑥²=(𝑡+𝑠)𝑑𝑠}Γ(5/4)Γ(3/4){1+⁰_𝑡−12𝑥²(𝑠)+11+𝑥²(𝑠)𝑑𝑠}≥Γ(5/4)Γ(3/4){1+⁰_𝑡−1≥𝑑𝑠}Γ(5/4)=Γ(3/4)Γ(1−𝜎+𝛽),𝑡Γ(1−𝜎)_1/2ℎ(𝑡,𝑥_𝑡,𝑦_𝑡,𝑧_𝑡)=Γ(5/4)Γ(1/4){2+^−𝑡₋₁3𝑦²(𝑡+𝑠)+11+𝑦²=(𝑡+𝑠)𝑑𝑠}Γ(5/4)Γ(1/4){2+⁰_𝑡−13𝑦²(𝑠)+11+𝑦²(𝑠)𝑑𝑠}≥Γ(5/4)Γ(1/4){2+2⁰_𝑡−1𝑑𝑠}≥2Γ(5/4)Γ(1/4)=2Γ(1−𝜐+𝛾).Γ(1−𝜐) (3.36) Hence 𝜇=min{1,2,3}=1 and the above calculations satisfy condition (H4). Then, (3.33) has at least one positive solution.

4. Unique Existence of Solution

In this section we give conditions on 𝑓 and 𝑔 which render unique positive solution to (1.2).

Theorem 4.1. Let 𝑓,𝑔 : (0,𝑇]×𝐶²→[0,∞) be continuous and lim_𝑡→0⁺𝑓(𝑡,𝑥_𝑡,𝑦_𝑡)=lim_𝑡→0⁺𝑔(𝑡,𝑥_𝑡,𝑦_𝑡)=+∞. If there exist 𝜌,𝜎∈(0,1) such that 0<𝜌<𝛼<1,0<𝜎<𝛽<1 and 𝑡^𝜌𝑓(𝑡,𝑥_𝑡,𝑦_𝑡),𝑡^𝜎𝑔(𝑡,𝑥_𝑡,𝑦_𝑡) are continuous functions on [0,𝑇]×𝐶², then (1.2) has unique positive solution with the following conditions:
(H5)|𝑡^𝜌𝑓(𝑡,𝑢)−𝑡^𝜌𝑓(𝑡,𝑣)|<𝐿₁‖𝑢−𝑣‖,|𝑡^𝜎𝑔(𝑡,𝑢)−𝑡^𝜌𝑔(𝑡,𝑣)|<𝐿₂‖𝑢−𝑣‖;(H6)𝐿=max{(𝐿₁/Γ(𝛼))𝑡^𝛼−𝜌𝐵(1−𝜌,𝛼),(𝐿₂/Γ(𝛽))𝑡^𝛽−𝜎𝐵(1−𝜎,𝛽)}<1, where 𝑢,𝑣∈𝐶², and 𝐿₁,𝐿₂ are positive constants.

Proof. As pointed out in the proceeding section, (1.2) is equivalent to the integral equation (3.1) and the solution of (3.1) is equivalent to the fixed point of operator 𝑊. Thus for 𝑢=(𝑢₁,𝑣₁),𝑣=(𝑢₂,𝑣₂)∈𝐾|𝐹𝑢₁(𝑡)−𝐹𝑢₂(𝑡)|≤𝐼^𝛼{𝑡^−𝜌|𝑡^𝜌𝑓_𝑖(𝑡,𝑢_𝑡+𝜔_𝑡)−𝑡^𝜌𝑓(𝑡,𝑣_𝑡+𝜔_𝑡)|}≤𝐿₁‖𝑢−𝑣‖𝐼^𝛼{𝑡^−𝜌𝐿}=‖𝑢−𝑣‖¹Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑠^−𝜌=𝐿₁Γ(𝛼)‖𝑢−𝑣‖𝐵(1−𝜌,𝛼)𝑡^𝛼−𝜌. (4.1) Similarly, |𝐺𝑣₁(𝑡)−𝐹𝑣₂𝐿(𝑡)|≤²Γ(𝛽)‖𝑢−𝑣‖𝐵(1−𝜎,𝛽)𝑡^𝛽−𝜎. (4.2) Hence, for each 𝑢,𝑣∈𝐾, we have ‖𝑊𝑢(𝑡)−𝑊𝑣(𝑡)‖=max{max_{0≤𝑡≤𝑇}|𝐹𝑢₁(𝑡)−𝐹𝑣₁(𝑡)|,|𝐹𝑢₁(𝑡)−𝐹𝑣₁(𝑡)|}=𝐿‖𝑢−𝑣‖. (4.3) Hence, by application of the Banach fixed point theorem, 𝑊 has unique fixed point in 𝐾, which is the unique positive solution of (1.2).

Example 4.2. Consider the system of nonlinear fractional differential equation 𝐷^4/51[𝑥(𝑡)−𝑥(0)]=(1+8𝑡²)√𝑡(𝑥_𝑡+𝑦_𝑡𝐷),𝑡∈(0,1],_3/41[𝑦(𝑡)−𝑦(0)]=(1+4𝑡²)√⁴𝑡(𝑥_𝑡+𝑦_𝑡),𝑡∈(0,1],(𝑥(𝑡),𝑦(𝑡))=(−sin𝜋𝑡,−sin(𝜋𝑡2)),𝑡∈[−1,0], (4.4) where 𝑥_𝑡=^−𝑡₋₁𝑑𝑠1+𝑥²(𝑡+𝑠),𝑦_𝑡=^−𝑡₋₁2𝑦(𝑡+𝑠)1+𝑦²(𝑡+𝑠)𝑑𝑠. (4.5) Here, 𝛼 = 4/5,𝛽 = 3/4,𝑓(𝑡,𝑥_𝑡,𝑦_𝑡) = 𝑒^−𝑡𝑡^−1/2(𝑥_𝑡+𝑦_𝑡), and 𝑔(𝑡,𝑥_𝑡,𝑦_𝑡)  = 𝑒^−𝑡𝑡^−1/4(𝑥_𝑡+𝑦_𝑡) such that lim_𝑡→0⁺𝑓(𝑡,⋅,⋅) = lim_𝑡→0⁺𝑔(𝑡,⋅,⋅) = +∞. We select 𝜌 = 3/4 and 𝜎 = 1/2. Hence 𝑡^𝜌𝑓(𝑡,⋅,⋅) and 𝑡^𝜎𝑔(𝑡,⋅,⋅) are continuous on [0,1]: |𝑡^𝜌𝑓(𝑡,𝑥_𝑡,𝑦_𝑡)−𝑡^𝜎𝑔(𝑡,𝑥_𝑡,𝑦_𝑡1)|≤max{18Γ(1/2)𝐵(4,451),15Γ(3/4)𝐵(2,34)}‖𝑢−𝑣‖<0.4‖𝑢−𝑣‖, (4.6) where 𝑢=(𝑥_𝑡,𝑦_𝑡),𝑣=(𝑥^′_𝑡,𝑦^′_𝑡). Equation (4.4) satisfies the conditions required in Theorem 4.1 and this theorem implies that the nonlinear equation (4.4) has unique solution in 𝐾. Using Theorem 3.2, (4.4) is equivalent to the system of integral equations 𝑥(𝑡)=𝑥(0)+𝐼^4/5{1(1+8𝑡²)√𝑡(𝑥_𝑡+𝑦_𝑡)},𝑡∈(0,1],𝑦(𝑡)=𝑦(0)+𝐼^3/4{1(1+4𝑡²)√⁴𝑡(𝑥_𝑡+𝑦_𝑡𝜋)},𝑡∈(0,1],(𝑥(𝑡),𝑦(𝑡))=(−sin𝜋𝑡,−sin2𝑡),𝑡∈[−1,0], (4.7) and the solution of (4.4) is (𝑥(𝑡),𝑦(𝑡))=lim_𝑛→+∞(𝑥^𝑛(𝑡),𝑦^𝑛(𝑡)) where (𝑥^𝑛+1(𝑡),𝑦^𝑛+1(𝑡))_{𝑛=0,1,2,…}=(𝐹𝑥^𝑛(𝑡),𝐺𝑦^𝑛(𝑡)),𝑥⁰(𝑡)=𝑥(0),𝑦⁰(0)=𝑦(0). (4.8)

5. Conclusions

In this paper the existence of positive solutions for system of nonlinear fractional differential equations in two dimensions with the delay comprising of standard Riemann-Liouville fractional derivatives has been discussed in Banach space. The unique solution under Lipschitz condition is also derived. For getting our results in this paper Leray-Schauder`s theorem, Krasnoselskii`s fixed point theorem, and Banach contraction principle had played important roles. Our research work in this paper can be generalized to system of nonlinear fractional differential equations in multiple dimensions with the finite delay or infinite delay. The present work provides insights in the equations encountered in fractional order dynamic systems and the control systems which further may be explored.

Acknowledgments

The authors thank the referees’ efforts for their remarks and Professor Dumitru B𝑎̆leanu for his regular contact.