The Existence and Uniqueness of a Class of Fractional Differential Equations

Abstract

By using inequalities, fixed point theorems, and lower and upper solution method, the existence and uniqueness of a class of fractional initial value problems, 𝐷^𝑞₀₊𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡),𝐷^𝑞−1₀₊𝑥(𝑡)),𝑡∈(0,𝑇), 𝑥(0)=0,𝐷^𝑞−1₀₊𝑥(0)=𝑥₀, are discussed, where 𝑓∈𝐶([0,𝑇]×ℝ²,ℝ),𝐷^𝑞₀₊𝑥(𝑡) is the standard Riemann-Liouville fractional derivative, 1<𝑞<2. Some mistakes in the literature are pointed out and some new inequalities and existence and uniqueness results are obtained.

1. Introduction

Once the models of fractional differential equation for the actual problem have been established, people immediately faced the problem of how to solve these models. In many cases, it is very difficult to obtain the exact solution of the fractional differential equation. So it requires researchers to find as many characteristics of the solution of the problem as possible. For example, does the equation have a solution? If there is one solution, is the solution unique? How can we compare the size of the solution? We noted that although there were many works with respect to fractional differential equations, which were shown in [1–10] and the references therein, the basic theory of the problem is still not perfect.

Al-Bassam [11] (1965) first considered the following Cauchy-type initial value problem (IVP): 𝐷^𝛼_𝑎+𝑦(𝑡)(𝑡,(),𝐼=𝑓𝑡,𝑦)0<𝛼≤1_{1−𝛼𝑎+}𝑦(𝑡)_𝑡=𝑎=𝑏₁,𝑏₁∈ℝ, (1) in the space of continuous functions 𝐶[𝑎,𝑏] provided that 𝑓(𝑡,𝑦) is real-valued, continuous, and Lipschitzian in a domain 𝐺⊂ℝ×ℝ such that sup_{(𝑡,𝑦)∈𝐺}|𝑓(𝑡,𝑦)|=𝑏₀<∞. Applying the operator 𝐼^𝛼_𝑎+ he reduced problem (1) to the Volterra nonlinear integral equation: 𝑦(𝑡)=𝑏₁(𝑡−𝑎)^𝛼−1Γ(𝛼)+1Γ(𝛼)^𝑡_𝑎𝑓𝑠,𝑦(𝑠)𝑑𝑠(𝑡−𝑠)^1−𝛼,(𝑡>𝑎). (2) By the use of the method of successive approximation he established the existence of the continuous solution of (2). He probably first indicated that the method of contracting mapping can be applied to prove the uniqueness of the solution of (2) and gave such a formal proof. However, from (2) one has lim_𝑡→𝑎+𝑦(𝑡)=∞, so in space 𝐶[𝑎,𝑏], the Cauchy-type problem (1) cannot be reduced to the integral equation (2) except that 𝑏₁=0.

Delbosco and Rodino [6] (1996) considered the nonlinear fractional differential equation 𝐷^𝛼₀₊𝑦(𝑡)=𝑓𝑡,𝑦(𝑡),0≤𝑡≤1. (3) Using Schauder’s fixed point theorem to the integral operator in (2) with 𝑎=𝑏₁=0, they proved that the equation considered has at least one continuous solution 𝑦∈𝐶[0,𝛿] for a suitable 0≤𝛿≤1 provided that 𝑡^𝜎𝑓(𝑡,𝑦) is continuous on [0,1]×ℝ for some 𝜎(0≤𝜎<𝛼<1). Applying the contractive mapping method, they showed that if additionally 𝑓(≤𝑀𝑡,𝑦−𝑓𝑡,𝑧)𝑡^𝜎𝑦−𝑧, (4) then (3) has a unique solution 𝑦(𝑡)∈𝐶[0,1]. Clearly, the solution satisfies 𝑦(0)=0. They also proved that if 𝑓(𝑡,𝑦)=𝑓(𝑦) is such that 𝑓(0)=0 and the Lipschitz condition holds, then the weighted Cauchy-type IVP 𝐷^𝛼₀₊𝑦(𝑡)=𝑓𝑦(𝑡),lim_𝑡→0𝑡^1−𝛼𝑦(𝑡)()=𝑐,0<𝛼<1,𝑐∈ℝ (5) has a unique solution 𝑦(𝑡) such that 𝑡^1−𝛼𝑦(𝑡)∈𝐶[0,ℎ] for any ℎ>0.

In [12] (2008), Lakshmikantham and Vatsala considered the IVP for fractional differential equations given by 𝐷^𝛼₀₊𝑦(𝑡)=𝑓𝑡,𝑦(𝑡),𝑦(0)=𝑦₀. (6) The basic theory for the IVP of fractional differential equations was discussed by employing the classical approach. The theory of inequalities, local existence, extremal solutions, comparison result, and global existence of solutions was considered. The idea of this paper is very interesting.

In [13] (2009), Zhang considered the existence and uniqueness of the solution of the following IVP for fractional differential equation: 𝐷^𝛼₀₊𝑦(𝑡)(𝑡𝑡=𝑓𝑡,𝑦),0≤𝑡≤𝑇,_1−𝛼𝑦(𝑡)_𝑡=0=𝑦₀, (7) using the method of upper and lower solutions and its associated monotone iterative technique. However, the paper did not explain why the pointwise convergence can be used instead of the convergence with norm in the space 𝐶_1−𝛼[0,𝑇].

We refer the readers to monographs [8, 10] for other arguments about the fractional IVP. We noted that on one hand there are some confusions about the initial value of the solution in some of the above works. On the other hand there is no contribution about the basic theory for the following fractional differential equation IVP: 𝐷^𝑞₀₊𝑥(𝑡)=𝑓𝑡,𝑥(𝑡),𝐷^𝑞−1₀₊𝑥(𝑡)(),𝑥(0),𝑡∈0,𝑇=0,𝐷^𝑞−1₀₊𝑥(0)=𝑥₀, (8) where 𝑓∈𝐶([0,𝑇]×ℝ²,ℝ), 𝐷^𝑞₀₊𝑥(𝑡) is the standard Riemann-Liouville fractional derivative, 1<𝑞<2. This problem is very important in many models of physics phenomena [7, 9, 10, 14–16], so it is worth studying the parallel theory to the known theory of ordinary differential equations.

The rest of the paper is organized as follows. In Section 2, some related basic lemmas and definitions are given. Section 3 contains the uniqueness result by means of contracting mapping. The existence of the minimal and maximal solutions is given in Section 4 using lower and upper solution method.

2. Preliminaries

Lemma 1 (see [3, 8]). For 𝑦∈𝐶^𝑛−1(0,𝑇)∩𝐿¹[0,𝑇], the fractional differential equation 𝐷^𝛼₀₊𝑦(𝑡)=0 (9) has a unique solution 𝑦(𝑡)=^𝑛−1_𝑘=0𝐷^{𝛼−𝑘−1}₀₊𝑦(𝑡)_𝑡=0Γ()𝑡𝛼−𝑘^{𝛼−𝑘−1}. (10)

Lemma 2 (see [10]). The following relation holds 𝐼^𝛼₀₊𝐼^𝛽₀₊𝜑=𝐼^𝛼+𝛽₀₊𝜑, (11) in the case of(1)𝛽≥0,𝛼+𝛽≥0,𝜑(𝑡)∈𝐿¹(0,𝑇);(2)𝛽≤0,𝛼≥0,𝜑(𝑡)∈𝐼^−𝛽₀₊(𝐿¹(0,𝑇));(3)𝛼≤0,𝛼+𝛽≤0,𝜑(𝑡)∈𝐼^{−𝛼−𝛽}₀₊(𝐿¹(0,𝑇)).

Lemma 3 (see [9]). Supposing that 𝑛∈𝑁,0<𝛼∈ℝ, then(1)𝑑^𝑛𝑑𝑡^𝑛𝐷^𝛼_𝑎+𝑓(𝑡)=𝐷^𝑛+𝛼_𝑎+𝑓(𝑡); (12)(2)𝐷^𝛼_𝑎+𝑑^𝑛𝑓(𝑡)𝑑𝑡^𝑛=𝐷^𝑛+𝛼_𝑎+𝑓(𝑡)−^𝑛−1_𝑗=0𝑓^(𝑗)(𝑎)(𝑡−𝑎)^{𝑗−𝛼−𝑛}Γ.1+𝑗−𝛼−𝑛 (13)

Lemma 4 (see [17]). Suppose that 𝐸 is an ordered Banach space, 𝑥₀,𝑦₀∈𝐸,𝑥₀≤𝑦₀,𝐷=[𝑥₀,𝑦₀],𝐴∶𝐷→𝐸 is an increasing completely continuous operator, and 𝑥₀≤𝐴𝑥₀,𝑦₀≥𝐴𝑦₀. Then the operator 𝐴 has a minimal fixed point 𝑥^∗ and a maximal fixed point 𝑦^∗. If one lets 𝑥_𝑛=𝐴𝑥_𝑛−1,𝑦_𝑛=𝐴𝑦_𝑛−1,𝑛=1,2,3,…, (14) then 𝑥₀≤𝑥₁≤𝑥₂≤⋯≤𝑥_𝑛≤⋯≤𝑦_𝑛≤⋯≤𝑦₂≤𝑦₁≤𝑦₀,𝑥_𝑛⟶𝑥^∗,𝑦_𝑛⟶𝑦^∗. (15)

By the use of the continuity of 𝑓 and Lemma 1, the IVP (8) is equivalent to the following Volterra integral equation [8]: 𝑥(𝑡)=𝑥₀Γ𝑞𝑡^𝑞−1+1Γ𝑞×^𝑡₀(𝑡−𝑠)^𝑞−1𝑓𝑠,𝑥(𝑠),𝐷^𝑞−1₀₊𝑥(𝑠)𝑑𝑠,0≤𝑡≤𝑇. (16) Define the space 𝐶^𝑞[0,𝑇]∶=𝑥∣𝑥∈𝐶[0,𝑇],𝐷^𝑞−1₀₊𝑥∈𝐶[0,𝑇]. (17) For 𝑥∈𝐶^𝑞[0,𝑇], define an operator 𝐴∶𝐶^𝑞[0,𝑇]→𝐶[0,𝑇] by (𝑡)=𝑥𝐴𝑥)(⁰Γ𝑞𝑡^𝑞−1+1Γ𝑞×^𝑡₀(𝑡−𝑠)^𝑞−1𝑓𝑠,𝑥(𝑠),𝐷^𝑞−1₀₊𝑥(𝑠)𝑑𝑠. (18) Then the fixed point of 𝐴 solves IVP (8) and vice versa.

Definition 5. A function 𝑣(𝑡)∈𝐶^𝑞[0,𝑇] is called a lower solution of problem (8), if it satisfies 𝐷^𝑞₀₊𝑣(𝑡)≤𝑓𝑡,𝑣(𝑡),𝐷^𝑞−1₀₊𝑣(𝑡)(),𝑣(0),𝑡∈0,𝑇=0,𝐷^𝑞−1₀₊𝑣(0)≤𝑥₀. (19)

Definition 6. A function 𝑤(𝑡)∈𝐶^𝑞[0,𝑇] is called an upper solution of problem (8), if it satisfies 𝐷^𝑞₀₊𝑤(𝑡)≥𝑓𝑡,𝑤(𝑡),𝐷^𝑞−1₀₊𝑤(𝑡)(),𝑤(0),𝑡∈0,𝑇=0,𝐷^𝑞−1₀₊𝑤(0)≥𝑥₀. (20)

If one of the above inequalities is strict, then we call it as a strict lower (upper) solution.

Remark 7. Clearly, if functions 𝑣,𝑤 are lower and upper solutions (or strict) of IVP (8), then there are 𝑣≤𝐴𝑣,𝑤≥𝐴𝑤 (or the inequality is strict).

3. The Uniqueness of the Solution

Many methods can be applied to study the existence of solution. However, generally speaking, it is nothing more than two ways. One is based on the method of the approximate solution of exact solution to prove the existence of the solution, namely, classical successive approximation method. A. Cauchy, R. Lipschitz, G. Peano, and so forth used this method to solve the existence of some special types of differential equations. In 1893, C. Picard applied this method to study the general nonlinear differential equation and established the existence and uniqueness of solutions, named the Cauchy-Picard Theorem. This method itself also contains a structural method to obtain the exact solution and thus provides a way for the approximate solution. Another method is transforming the solution into the fixed point of some maps. Although the method cannot give the approximate solution, it is the abstraction and generalization of the former method and is simple to use. In this section, we will establish the uniqueness of the solution for fractional IVP (8) by the use of the second method.

Theorem 8. Assume that 𝑓∶[0,𝑇]×ℝ²→ℝ is continuous and Lipschitzian with respect to the second and the third variables; that is to say, there exist constants 𝐿₁,𝐿₂>0 such that for all 𝑥₁,𝑦₁,𝑥₂,𝑦₂∈ℝ,𝑡∈[0,𝑇]𝑓𝑡,𝑥₁,𝑦₁−𝑓𝑡,𝑥₂,𝑦₂≤𝐿₁𝑥₁−𝑥₂+𝐿₂𝑦₁−𝑦₂. (21) Then the fractional IVP (8) has a unique solution 𝑢∈𝐶[0,𝑇].

Proof. For 𝜑∈𝐶^𝑞[0,𝑇], the norm is defined as 𝜑=max_{𝑡∈[0,𝑇]}𝑒^−𝛽𝑡𝜑(𝑡)+max_{𝑡∈[0,𝑇]}𝑒^−𝛽𝑡𝐷^𝑞−1₀₊𝜑(𝑡), (22) where 𝛽 is a positive constant such that 𝐿₁+𝐿₂∫^𝑇₀(𝑇−𝑠)^2𝑞−2𝑑𝑠^1/2Γ𝑞2𝛽^1/2<1. (23) Then (𝐶^𝑞[0,𝑇],‖⋅‖) is a Banach space.
Clearly, the operator 𝐴 defined by (18) maps 𝐶^𝑞[0,𝑇] to 𝐶^𝑞[0,𝑇].
Now we prove that operator 𝐴 is a compressed map on (𝐶^𝑞[0,𝑇],‖⋅‖). Let 𝑦,𝑧∈𝐶^𝑞[0,𝑇]; then, for 𝑡∈[0,𝑇], (𝑡)−(𝑡)=1𝐴𝑦𝐴𝑧)(Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑓𝑠,𝑦(𝑠),𝐷^𝑞−1₀₊𝑦(𝑠)−𝑓𝑠,𝑧(𝑠),𝐷^𝑞−1₀₊𝑧(𝑠)𝑑𝑠. (24)
Taking into account that the function 𝑓 is Lipschitzian, by the use of the Cauchy-Schwartz inequality, we have 𝑒^−𝛽𝑡𝐴𝑦−𝐴𝑧(𝑡)≤𝑒^−𝛽𝑡1Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑒^𝛽𝑠𝑒^−𝛽𝑠×𝐿₁𝑦(𝑠)−𝑧(𝑠)+𝐿₂𝐷^𝑞−1₀₊𝑦(𝑠)−𝐷^𝑞−1₀₊𝑧(𝑠≤𝐿)𝑑𝑠₁+𝐿₂𝑒^−𝛽𝑡Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑒^𝛽𝑠=𝐿𝑦−𝑧𝑑𝑠₁+𝐿₂𝑒^−𝛽𝑡𝑦−𝑧Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑒^𝛽𝑠≤𝐿𝑑𝑠₁+𝐿₂𝑒^−𝛽𝑡𝑦−𝑧Γ𝑞^𝑡₀(𝑡−𝑠)^2𝑞−2𝑑𝑠^1/2×^𝑡₀𝑒^2𝛽𝑠𝑑𝑠^1/2≤𝐿₁+𝐿₂𝑒^−𝛽𝑡𝑦−𝑧Γ𝑞^𝑡₀(𝑡−𝑠)^2𝑞−2𝑑𝑠^1/2×𝑒^𝛽𝑡2𝛽^1/2≤𝐿₁+𝐿₂∫^𝑇₀(𝑇−𝑠)^2𝑞−2𝑑𝑠^1/2Γ𝑞2𝛽^1/2𝑦−𝑧. (25) According to the definition of 𝛽, we know that 𝐴∶𝐶^𝑞[0,𝑇]→𝐶^𝑞[0,𝑇] is a compressed map. Banach fixed point theorem shows that there exists a unique 𝑦∈𝐶^𝑞[0,𝑇] such that 𝐴𝑦=𝑦; equivalently, IVP (8) has a unique solution 𝑦∈𝐶[0,𝑇].

Remark 9. Similar to paper [6], we can permit function 𝑓(𝑡,𝑥,𝑦) to have some singularity on 𝑡.

Remark 10. The study about the following problem is meaningful: 𝐷^𝑞₀₊𝑥(𝑡)=𝑓𝑡,𝑥(𝑡),𝐷^𝑝₀₊𝑥(𝑡(),𝑥(0)),𝑡∈0,𝑇=0,𝐷^𝑞−1₀₊𝑥(0)=𝑥₀, (26) where 𝑓∈𝐶([0,𝑇]×ℝ²,ℝ),𝐷^𝑞₀₊𝑥(𝑡) is the standard Riemann-Liouville fractional derivative, 1<𝑝<𝑞<2.

4. Some Inequalities and the Existence of the Solution

Firstly, let us discuss the result about the strict inequalities for fractional IVP.

Theorem 11. Assume that the functions 𝑣,𝑤∈𝐶^𝑞[0,𝑇] are lower and upper solutions of problem (8) and at least one of them is strict. For every 𝑡∈[0,𝑇],𝑓(𝑡,𝑥,𝑦) is nondecreasing about 𝑥,𝑦. Then 𝐷^𝑞−1₀₊𝑣(𝑡)<𝐷^𝑞−1₀₊𝑤(𝑡),0<𝑡≤𝑇;𝑣(𝑡)<𝑤(𝑡),0<𝑡≤𝑇. (27)(28) Furthermore, the fractional IVP (8) has a minimal solution 𝑥^∗ and a maximal solution 𝑦^∗ such that 𝑥^∗=lim_𝑛→∞𝐴^𝑛𝑣,𝑦^∗=lim_𝑛→∞𝐴^𝑛𝑤. (29)

Proof. Without loss of generality, suppose that 𝐷^𝑞−1₀₊𝑤(0)>𝑥₀. Let 𝑦(𝑡)=𝐷^𝑞−1₀₊𝑣(𝑡). By the use of Lemma 3 and the definition of lower solution 𝑣, one has 𝑦^′(𝑡)(𝑡)(𝑡)(),𝑦(0)≤𝑓𝑡,𝑣,𝑦,𝑡∈0,𝑇≤𝑥₀. (30) Integration from 0 to 𝑡 yields 𝑦(𝑡)≤𝑥₀+^𝑡₀𝑓𝑠,𝑣(𝑠),𝑦(𝑠)𝑑𝑠. (31)
Similarly, let 𝑧(𝑡)=𝐷^𝑞−1₀₊𝑤(𝑡); we get 𝑧(𝑡)≥𝑥₀+^𝑡₀𝑓(𝑠,𝑤(𝑠),𝑧(𝑠))𝑑𝑠. (32)
Suppose for contradiction that conclusion (27) is not true. Combining the fact that 𝐷^𝑞−1₀₊𝑣, 𝐷^𝑞−1₀₊𝑤 is continuous on [0,𝑇] and 𝑦(0)=𝐷^𝑞−1₀₊𝑣(0)≤𝑥₀<𝐷^𝑞−1₀₊𝑤(0)=𝑧(0), there exists 𝑡₁∈(0,𝑇] such that 𝑦𝑡₁=𝑧𝑡₁,𝑦(𝑡)<𝑧(𝑡),0≤𝑡<𝑡₁. (33)
Taking into account that 𝐷^𝑞−1₀₊(𝑤(𝑡)−𝑣(𝑡))∶=ℎ(𝑡)>0 for 0<𝑡<𝑡₁ and 𝑤(0)−𝑣(0)=0, by the use of the monotonicity of integral operator 𝐼^𝑞−1₀₊, one has 𝑣(𝑡)≤𝑤(𝑡),0≤𝑡≤𝑡₁. (34) With inequalities (33) and (34), 𝑓 is nondecreasing and above arguments give 𝑧𝑡₁≥𝑥₀+^𝑡1₀𝑓(𝑠,𝑤(𝑠),𝑧(𝑠))𝑑𝑠>𝑥₀+^𝑡1₀𝑓𝑠,𝑣(𝑠),𝑦(𝑠)𝑑𝑠≥𝑦𝑡₁, (35) which is a contradiction to (33). Thus, conclusion (27) holds. Furthermore, combining 𝑣(0)=𝑤(0)=0 and the monotonicity of integral 𝐼^𝑞−1₀₊ yields that (28) also holds.
A standard proof can show that 𝐴∶𝐶^𝑞[0,𝑇]→𝐶^𝑞[0,𝑇] is an increasing completely continuous operator. Setting 𝐷∶=[𝑣,𝑤], by the use of Lemma 4, the existence of 𝑥^∗,𝑦^∗ is obtained. The proof is complete.

The following conclusion is about the nonstrict inequalities.

Theorem 12. Assume that the functions 𝑣,𝑤∈𝐶^𝑞[0,𝑇] are lower and upper solutions of problem (8). If there exist two real numbers 𝐿₁,𝐿₂>0 such that, for 𝑡∈[0,𝑇], 𝑥₁≥𝑥₂,𝑦₁≥𝑦₂, there holds 0≤𝑓𝑡,𝑥₁,𝑦₁−𝑓𝑡,𝑥₂,𝑦₂≤𝐿₁𝑡^𝑞+𝑡^𝑞−1𝑥₁−𝑥₂+𝐿₂Γ𝑞𝑡𝑦+Γ𝑞+1¹−𝑦₂, (36) then 𝐿₁+𝐿₂≤Γ(𝑞+1) implies 𝑣(𝑡)≤𝑤(𝑡),0≤𝑡≤𝑇. (37) Furthermore, the fractional IVP (8) has a minimal solution 𝑥^∗ and a maximal solution 𝑦^∗ such that 𝑥^∗=lim_𝑛→∞𝐴^𝑛𝑣,𝑦^∗=lim_𝑛→∞𝐴^𝑛𝑤. (38)

Proof. Given 𝜖>0, let 𝑤_𝜖(𝑡)=𝑤(𝑡)+𝜖(𝑡^𝑞+𝑡^𝑞−1). Then, for 0≤𝑡≤𝑇, 𝐷^𝑞−1₀₊𝑤_𝜖(𝑡)=𝐷^𝑞−1₀₊𝑤(𝑡)+Γ𝑞+1𝜖𝑡+Γ𝑞𝜖, (39) and specially, 𝐷^𝑞−1₀₊𝑤_𝜖(0)=𝐷^𝑞−1₀₊𝑤(0)𝑞𝑤+Γ𝜖,_𝜖(0)=𝑤(0),𝑤_𝜖(𝑡)>𝑤(𝑡),for0<𝑡≤𝑇. (40)
From (39) and condition (36), combining the fact that ^𝑡₀(𝑡−𝑠)^𝑞−1𝑑𝑠=𝑡^𝑞1₀(1−𝜎)^𝑞−1Γ𝑞𝑑𝜎=Γ𝑡𝑞+1^𝑞 (41) with the condition 𝐿₁+𝐿₂<Γ(𝑞+1), one has 𝐷^𝑞₀₊𝑤_𝜖(𝑡)=𝑤(𝑡)+𝜖𝑡^𝑞+𝑡^𝑞−1≥𝑓𝑡,𝑤(𝑡),𝐷^𝑞−1₀₊𝑤(𝑡)+Γ𝑞+1𝜖≥𝑓𝑡,𝑤_𝜖(𝑡),𝐷^𝑞−1₀₊𝑤_𝜖(𝑡)𝜖−𝐿+Γ𝑞+1₁𝑤_𝜖(𝑡)−𝑤(𝑡)𝑡^𝑞+𝑡^𝑞−1−𝐿₂𝐷^𝑞−1₀₊𝑤_𝜖(𝑡)−𝐷^𝑞−1₀₊𝑤(𝑡)Γ𝑞+Γ𝑞+1𝑡≥𝑓𝑡,𝑤_𝜖(𝑡),𝐷^𝑞−1₀₊𝑤_𝜖(𝑡). (42) This inequality combined with 𝑤_𝜖(0)=0,𝐷^𝑞−1₀₊𝑤_𝜖(0)=𝐷^𝑞−1₀₊𝑤(0)+Γ(𝑞)𝜖>𝑥₀ shows that 𝑤_𝜖(𝑡) is a strict upper solution of problem (8).
For 𝑤_𝜖(𝑡),𝑣(𝑡), by the use of Theorem 11, we get 𝑣(𝑡)<𝑤_𝜖(𝑡),0<𝑡≤𝑇. As 𝜖>0 is arbitrary, (37) holds.
The rest of the proof is just similar to Theorem 11.

Remark 13. If we instead use condition (36) with the condition that there exists real number 𝐿₁,𝐿₂>0 such that, for 𝑡∈[0,𝑇],𝑥₁≥𝑥₂,𝑦₁≥𝑦₂, 0≤𝑓𝑡,𝑥₁,𝑦₁−𝑓𝑡,𝑥₂,𝑦₂≤𝐿₁𝑥₁−𝑥₂+𝐿₂𝑦₁−𝑦₂, (43) then Theorem 12 still holds as long as the condition 𝐿₁Γ𝑞Γ𝑇2𝑞2^𝑞+𝑇^𝑞−1+𝐿₂Γ𝑞Γ𝑞+1𝑞𝑇𝑞+1+1≤1 (44) is used instead of the condition Γ(𝑞+1)≥𝐿₁+𝐿₂.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by NNSF (61174078, 61201431), SDNSF (ZR2010AM035), a project of Shandong Higher Educational Science and Technology program (J11LA07), the Taishan Scholar project, Research Award Fund for Outstanding Young Scientists of Shandong (BS2012SF022), and SDUST Research Fund (2011KYTD105). The authors thank the referee for his/her valuable comments and constructive suggestions.