Generalized Quasilinearization for the System of Fractional Differential Equations

Abstract

This paper considers the initial value problems of the system of fractional differential equations and constructs two monotone sequences of upper and lower solutions. By using quasilinearization technique, monotone sequences of approximate solutions that converge quadratically to a solution are obtained.

1. Introduction

In this paper, we consider the system of Caputo fractional differential equations: ^𝑐𝐷^𝑞𝑥=𝑓(𝑡,𝑥),𝑥𝑡₀=𝑥₀, (1) where 0<𝑞<1, 𝑓∈𝐶[Ω,𝑅^𝑛], Ω=[(𝑡,𝑥)∶𝛼(𝑡)≤𝑥≤𝛽(𝑡), 𝛼,𝛽∈𝐶^𝑞[𝐽,𝑅^𝑛],𝑡∈𝐽], and 𝐽=[𝑡₀,𝑇].

The theories and properties of fractional differential equations have received attention from some researchers because many mathematical modeling appeared in the fields of physics, chemistry, engineering and biological sciences and so on. For examples and details, we can refer to the, monographs of Miller and Ross [1], Podlubny [2], Kilbas et al. [3], and West et al [4] and the papers of Debnath [5], Rossikhin and Shitikova [6], and Ferreira et al [7]. There are many results on the basic theory of initial value and boundary value problems for fractional differential equations, which can be found in [8–10]. Meanwhile, there are some qualitative and numerical solutions for various fractional equations with delay and impulsive effects. For details, see some recent papers [11–18] and the references therein.

It is well known that the monotone iterative technique is an ingenious method providing a constructive approach to find solutions for the nonlinear problem via linear iterates. Lakshmikantham and Vatsala [19], and McRae [20] investigated the existence of minimal and maximal solutions of fractional differential equations by establishing a comparison result and using the monotone method, respectively; Benchohra and Hamani [21] used a monotone iterative technique in the presence of lower and upper solutions to discuss the existence of solutions to impulsive fractional differential inclusions.

Quasilinearization [22] provides an elegant and easier approach to obtain a sequence of approximate solutions with quadratic convergence, and the method has been extended to fractional differential equations in [23, 24]. However, to the best of our knowledge, there are few results for the system of fractional differential equations, especially results on the convergence of the system. In the present paper, we will discuss the approximate solutions of the system of fractional differential equations through the application of quasilinearization. The significance of this work lies in the fact that the system of fractional differential equations can also obtain a monotone sequence of approximate solutions converging uniformly to the solution of the problem and possessing quadratic convergence.

The nonhomogeneous linear system of Caputo fractional differential equations is given by ^𝑐𝐷^𝑞𝑥=𝐵𝑥+𝑔𝑡,𝑦,𝑥𝑡₀=𝑥₀, (2) where 𝐵 is an nth order matrix over complex field, and 𝑔 is an n-dimensional locally integrable column vector function on 𝐽.

Using the method of successive approximations, we get the solution of (2) as 𝑥(𝑡)=𝑥₀𝐸_𝑞𝐵𝑡−𝑡₀^𝑞+𝑡−𝑡₀^𝑞−1𝐸_𝑞,𝑞×𝐵𝑡−𝑡₀^𝑞𝑘𝑔𝑡,𝑦,𝑡∈𝐽, (3) where 𝐸_𝑞(𝐵)=^∞_𝑘=0𝐵^𝑘Γ𝑞𝑘+1,𝐸_𝑞,𝑞(𝐵)=^∞_𝑘=0𝐵^𝑘Γ𝑞𝑘+𝑞 (4) are Mittag-Leffler functions of one parameter and two parameters, respectively.

2. Preliminaries

Now, we present the following definition and lemma which help to prove our main result.

Definition 1. Let 𝛼,𝛽∈𝐶^𝑞[𝐽,𝑅^𝑛] be lower and upper solutions of (1) if they satisfy the inequalities ^𝑐𝐷^𝑞𝛼≤𝑓(𝑡,𝛼),𝛼₀≤𝑥₀,_𝑐𝐷^𝑞𝛽≥𝑓𝑡,𝛽,𝛽₀≥𝑥₀, (5) respectively, for 𝑡∈𝐽.

Lemma 2. Suppose that 𝛼,𝛽∈𝐶^𝑞[𝐽,𝑅^𝑛] are lower and upper solutions of (1), and (H₁)𝑓∈𝐶[Ω,𝑅^𝑛] are quasimonotone nondecreasing in 𝑥 for each 𝑡∈𝐽 and for each 𝑖∈𝐼={1,2,…𝑛}, 𝑓_𝑖(𝑡,𝑥)−𝑓_𝑖𝑡,𝑦≤𝐿^𝑛_𝑗=1𝑥_𝑗−𝑦_𝑗, (6)where 𝑥≥𝑦,??𝐿≥0 is a constant.
Then, 𝛼(𝑡₀)≤𝛽(𝑡₀) implies that 𝛼(𝑡)≤𝛽(𝑡).

Proof. Firstly, suppose that ^𝑐𝐷^𝑞𝛽>𝑓𝑡,𝛽,^𝑐𝐷^𝑞𝛼≤𝑓(𝑡,𝛼),𝑡∈𝐽 (7) and 𝛼(𝑡₀)<𝛽(𝑡₀). We will prove that 𝛼(𝑡)<𝛽(𝑡),𝑡∈𝐽. Suppose the conclusion is not true, then the set ℤ=^𝑛_𝑖=1𝑡∈𝐽,𝛽_𝑖(𝑡)≥𝛼_𝑖(𝑡) (8) is nonempty.
Let 𝑡₁=infℤ. Certainly, 𝑡₁>0. Since the set ℤ is closed, 𝑡₁∈ℤ and consequently there exists a 𝑗 such that 𝛽_𝑗(𝑡₁)=𝛼_𝑗(𝑡₁). Moreover, 𝛽_𝑖(𝑡₁)≥𝛼_𝑖(𝑡₁), for 𝑖≠𝑗, and 𝛼_𝑗(𝑡)<𝛽_𝑗(𝑡),𝑡∈0,𝑡₁. (9) Hence, it easily follows that ^𝑐𝐷^𝑞𝛼_𝑗𝑡₁≥^𝑐𝐷^𝑞𝛽_𝑗𝑡₁. (10) This together with the quasimonotonicity of 𝑓 yields 𝑓_𝑗𝑡₁,𝛼𝑡₁≥^𝑐𝐷^𝑞𝛼_𝑗𝑡₁≥^𝑐𝐷^𝑞𝑥𝛽_𝑗𝑡₁>𝑓_𝑗𝑡₁,𝛽_𝑗𝑡₁≥𝑓_𝑗𝑡₁,𝛼𝑡₁, (11) which leads to a contradiction.
In order to prove the case of nonstrict inequalities, consider the functions 𝛽_𝑖(𝑡)=𝛽_𝑖(𝑡)+𝜀𝑒^{(𝑛+1)𝐿𝑖𝑡}, (12) where 𝜀>0 is sufficiently small constant. Then using (6), we have ^𝑐𝐷^𝑞𝛽_𝑖(𝑡)=^𝑐𝐷^𝑞𝛽_𝑖(𝑡)+(𝑛+1)𝜀𝐿_𝑖𝑒^{(𝑛+1)𝐿𝑖𝑡}≥𝑓_𝑖𝑡,𝛽(𝑡)+(𝑛+1)𝜀𝐿_𝑖𝑒^{(𝑛+1)𝐿𝑖𝑡}≥𝑓_𝑖𝑡,𝛽(𝑡)−𝑛𝜀𝐿_𝑖𝑒^{(𝑛+1)𝐿𝑖𝑡}+(𝑛+1)𝜀𝐿_𝑖𝑒^{(𝑛+1)𝐿𝑖𝑡}>𝑓_𝑖𝑡,𝛽(𝑡). (13) Also 𝛽_𝑖(𝑡₀)>𝛼_𝑖(𝑡₀). Now using the result corresponding to strict inequalities, we get 𝛼_𝑖(𝑡)<𝛽_𝑖(𝑡)=𝛽_𝑖(𝑡)+𝜀𝑒^{(𝑛+1)𝐿𝑖𝑡},𝑡∈𝐽. (14) Letting 𝜀→0, we obtain the required result and the proof is complete.

Corollary 3. The function 𝑓(𝑡,𝛼)=𝜎(𝑡)𝛼 where 𝜎(𝑡)≤𝐿 is admissible in Lemma 2 to yield 𝛼(𝑡)≤0,𝑡∈𝐽.

3. Main Result

Theorem 4. Assume that 𝛼₀,𝛽₀∈𝐶^𝑞[𝐽,𝑅^𝑛] are lower and upper solutions of (1) such that 𝛼₀(𝑡)≤𝛽₀(𝑡),𝑡∈𝐽, and (H₁)𝑓,𝜙∈𝐶[Ω,𝑅^𝑛] are quasimonotone nondecreasing in 𝑥 for 𝑡∈𝐽, 𝑓_𝑥, 𝜙_𝑥, 𝑓_𝑥𝑥, 𝜙_𝑥𝑥 exist and are continuous on Ω, satisfying 𝑓_𝑥𝑥+𝜙_𝑥𝑥≥0, 𝜙_𝑥𝑥≥0; (H₂)𝑎_𝑖𝑗(𝑡,𝛼₀,𝛽₀)≥0,𝑖≠𝑗, where 𝐴(𝑡,𝛼₀,𝛽₀)=[𝑎_𝑖𝑗(𝑡,𝛼₀,𝛽₀)] is an 𝑛×𝑛 matrix given by 𝐴𝑡,𝛼₀,𝛽₀≡𝑓_𝑥𝑡,𝛼₀+𝜙_𝑥𝑡,𝛼₀−𝜙_𝑥𝑡,𝛽₀=𝐹_𝑥𝑡,𝛼₀−𝜙_𝑥𝑡,𝛽₀, (15)where 𝐹_𝑥=𝑓_𝑥+𝜙_𝑥.
Then, there exist monotone sequences {𝛼_𝑛(𝑡)}, {𝛽_𝑛(𝑡)} which converge uniformly to the solution of (1) and the convergence is quadratic.

Proof. (H₁) in Theorem 4 implies for any 𝑥≥𝑦,??𝑥,𝑦∈Ω, 𝑓(𝑡,𝑥)≥𝑓𝑡,𝑦+𝑓_𝑥𝑡,𝑦+𝜙_𝑥×−𝜙().𝑡,𝑦𝑥−𝑦𝑡,𝑥−𝜙𝑡,𝑦 (16) And for any 𝛼₀(𝑡)≤𝑦≤𝑥≤𝛽₀(𝑡), (17) we have 𝑎_𝑖𝑗𝑡,𝑦,𝑥≥0,𝑖≠𝑗, (18) since 𝑎_𝑖𝑗(𝑡,𝑦,𝑥)≥𝑎_𝑖𝑗(𝑡,𝛼₀,𝛽₀) by assumption (H₁).
It is also clear that for 𝛼₀(𝑡)≤𝑦≤𝑥≤𝛽₀(𝑡), 𝑓_𝑖𝑡,𝑥₁,…,𝑥_𝑛−𝑓_𝑖𝑡,𝑦₁,…,𝑦_𝑛≤𝐿_𝑖^𝑛_𝑖=1𝑥_𝑖−𝑦_𝑖. (19)
Let 𝛼₁,𝛽₁ be the solutions of IVPs: ^𝑐𝐷^𝑞𝛼₁=𝑓𝑡,𝛼₀+𝐴𝑡,𝛼₀,𝛽₀×𝛼₁−𝛼₀,𝛼₁(0)=𝑥₀,_𝑐𝐷^𝑞𝛽₁=𝑓𝑡,𝛽₀+𝐴𝑡,𝛼₀,𝛽₀×𝛽₁−𝛽₀,𝛽₁(0)=𝑥₀, (20) where 𝛼₀(0)≤𝑥₀≤𝛽₀(0). We will prove that 𝛼₀≤𝛼₁,𝑡∈𝐽. To do this, let 𝑝=𝛼₀−𝛼₁, so that 𝑝(0)≤0. Then using (20), we obtain ^𝑐𝐷^𝑞𝑝=^𝑐𝐷^𝑞𝛼₀−^𝑐𝐷^𝑞𝛼₁≤𝑓𝑡,𝛼₀−𝑓𝑡,𝛼₀+𝐴𝑡,𝛼₀,𝛽₀𝛼₁−𝛼₀=𝐴𝑡,𝛼₀,𝛽₀𝑝. (21)
Since 𝐴(𝑡,𝛼₀,𝛽₀) is quasimonotone nondecreasing by assumption (H₁) to (H₂) and it follows from Corollary 3 that 𝑝(𝑡)≤0,𝑡∈𝐽, proving that 𝛼₀≤𝛼₁,𝑡∈𝐽.
Now we let 𝑝=𝛼₁−𝛽₀ and note that 𝑝(0)≤0. Also, ^𝑐𝐷^𝑞𝑝=^𝑐𝐷^𝑞𝛼₁−^𝑐𝐷^𝑞𝛽₀≤𝑓𝑡,𝛼₀+𝐴𝑡,𝛼₀,𝛽₀𝛼₁−𝛼₀−𝑓𝑡,𝛽₀; (22) since 𝛽₀≥𝛼₀, using (16), we get 𝑓𝑡,𝛽₀≥𝑓𝑡,𝛼₀+𝐹_𝑥𝑡,𝛼₀𝛽₀−𝛼₀−𝜙𝑡,𝛽₀−𝜙𝑡,𝛼₀. (23)
In view of (H₁), we have 𝜙𝑡,𝛽₀−𝜙𝑡,𝛼₀≤𝜙_𝑥𝑡,𝛽₀𝛽₀−𝛼₀, (24) which yields 𝑓𝑡,𝛽₀≥𝑓𝑡,𝛼₀+𝐹_𝑥𝑡,𝛼₀−𝜙_𝑥𝑡,𝛽₀𝛽₀−𝛼₀. (25) Hence, we obtain ^𝑐𝐷^𝑞𝑝≤𝐴𝑡,𝛼₀,𝛽₀𝑝; (26) this implies that 𝛼₁(𝑡)≤𝛽₀(𝑡),𝑡∈𝐽, using Corollary 3. As a result, we have 𝛼₀(𝑡)≤𝛼₁(𝑡)≤𝛽₀(𝑡),𝑡∈𝐽. (27) In a similar way, we can prove that 𝛼₀(𝑡)≤𝛽₁(𝑡)≤𝛽₀(𝑡),𝑡∈𝐽. (28) To show 𝛼₁(𝑡)≤𝛽₁(𝑡), we use (16), (19), and (H₁): ^𝑐𝐷^𝑞𝛼₁=𝑓𝑡,𝛼₀+𝐹_𝑥𝑡,𝛼₀−𝜙_𝑥𝑡,𝛽₀𝛼₁−𝛼₀≤𝑓𝑡,𝛼₁+𝜙𝑡,𝛼₁−𝜙𝑡,𝛼₀−𝜙_𝑥𝑡,𝛽₀𝛼₁−𝛼₀≤𝑓𝑡,𝛼₁+¹₀𝜙_𝑥𝑡,𝑠𝛼₁+(1−𝑠)𝛼₀×𝛼𝑑𝑠₁−𝛼₀−𝜙_𝑥𝑡,𝛽₀𝛼₁−𝛼₀≤𝑓𝑡,𝛼₁+𝜙_𝑥𝑡,𝛼₁𝛼₁−𝛼₀−𝜙_𝑥𝑡,𝛽₀𝛼₁−𝛼₀≤𝑓𝑡,𝛼₁. (29) Using a similar argument, it is easy to show that ^𝑐𝐷^𝑞𝛽₁≥𝑓𝑡,𝛽₁, (30) And, therefore, by Lemma 2 and (19), it follows that 𝛼₁(𝑡)≤𝛽₁(𝑡),𝑡∈𝐽; (31) by (19), we have 𝑓 is Lipschitzian in 𝑥 on Ω. This proves that 𝛼₀(𝑡)≤𝛼₁(𝑡)≤𝛽₁(𝑡)≤𝛽₀(𝑡),𝑡∈𝐽. (32) Now assume that for some 𝑘>1, 𝛼_𝑘−1(𝑡)≤𝛼_𝑘(𝑡)≤𝛽_𝑘(𝑡)≤𝛽_𝑘−1(𝑡),𝑡∈𝐽. (33) We now aim to show that 𝛼_𝑘(𝑡)≤𝛼_𝑘+1(𝑡)≤𝛽_𝑘+1(𝑡)≤𝛽_𝑘(𝑡),𝑡∈𝐽, (34) where 𝛼_𝑘+1(𝑡) and 𝛽_𝑘+1(𝑡) are the solutions of linear IVPs: ^𝑐𝐷^𝑞𝛼_𝑘+1=𝑓𝑡,𝛼_𝑘+𝐹_𝑥𝑡,𝛼_𝑘−𝜙_𝑥𝑡,𝛽_𝑘×𝛼_𝑘+1−𝛼_𝑘,𝛼_𝑘+1(0)=𝑥₀,_𝑐𝐷^𝑞𝛽_𝑘+1=𝑓𝑡,𝛽_𝑘+𝐹_𝑥𝑡,𝛼_𝑘−𝜙_𝑥𝑡,𝛽_𝑘×𝛽_𝑘+1−𝛽_𝑘,𝛽_𝑘+1(0)=𝑥₀. (35) Now, set 𝑝=𝛼_𝑘−𝛼_𝑘+1 so that ^𝑐𝐷^𝑞𝑝=^𝑐𝐷^𝑞𝛼_𝑘−^𝑐𝐷^𝑞𝛼_𝑘+1≤𝑓𝑡,𝛼_𝑘−𝑓𝑡,𝛼_𝑘+𝐹_𝑥𝑡,𝛼_𝑘−𝜙_𝑥𝑡,𝛽_𝑘×𝛼_𝑘+1−𝛼_𝑘=𝐹_𝑥𝑡,𝛼_𝑘−𝜙_𝑥𝑡,𝛽_𝑘𝑝=𝐴𝑡,𝛼_𝑘,𝛽_𝑘𝑝 (36) and 𝑝(0)=0. It follows from Corollary 3 and using (16) that 𝛼_𝑘(𝑡)≤𝛼_𝑘+1(𝑡),𝑡∈𝐽. (37)
On the other hand, letting 𝑝=𝛼_𝑘+1−𝛽_𝑘 yields ^𝑐𝐷^𝑞𝑝=^𝑐𝐷^𝑞𝛼_𝑘+1−^𝑐𝐷^𝑞𝛽_𝑘≤𝑓𝑡,𝛼_𝑘+𝐹_𝑥𝑡,𝛼_𝑘−𝜙_𝑥𝑡,𝛽_𝑘×𝛼_𝑘+1−𝛼_𝑘−𝑓𝑡,𝛽_𝑘. (38) Since 𝛽_𝑘≥𝛼_𝑘, (16) and (H₁) give, as before, 𝑓𝑡,𝛽_𝑘≥𝑓𝑡,𝛼_𝑘+𝐹_𝑥𝑡,𝛼_𝑘−𝜙_𝑥𝑡,𝛽_𝑘𝛽_𝑘−𝛼_𝑘, (39) which shows that ^𝑐𝐷^𝑞𝑝≤𝐹_𝑥𝑡,𝛼_𝑘−𝜙_𝑥𝑡,𝛽_𝑘𝑝=𝐴𝑡,𝛼_𝑘,𝛽_𝑘𝑝. (40) This proves that 𝑝(𝑡)≤0 using Corollary 3 and (16), since 𝑝(0)=0. Hence, we get 𝛼_𝑘(𝑡)≤𝛼_𝑘+1(𝑡)≤𝛽_𝑘(𝑡),𝑡∈𝐽. (41) Similarly, we can prove that 𝛼_𝑘(𝑡)≤𝛽_𝑘+1(𝑡)≤𝛽_𝑘(𝑡),𝑡∈𝐽. (42) Also, by (16), (35), and the fact that 𝛼_𝑘+1≥𝛼_𝑘, we obtain ^𝑐𝐷^𝑞𝛼_𝑘+1≤𝑓𝑡,𝛼_𝑘+1+𝜙𝑡,𝛼_𝑘+1−𝜙𝑡,𝛼_𝑘−𝜙_𝑥𝑡,𝛽_𝑘𝛼_𝑘+1−𝛼_𝑘≤𝑓𝑡,𝛼_𝑘+1+¹₀𝜙_𝑥𝑡,𝑠𝛼_𝑘+1+(1−𝑠)𝛼_𝑘×𝛼𝑑𝑠_𝑘+1−𝛼_𝑘−𝜙_𝑥𝑡,𝛽_𝑘𝛼_𝑘+1−𝛼_𝑘≤𝑓𝑡,𝛼_𝑘+1+𝜙_𝑥𝑡,𝛼_𝑘+1𝛼_𝑘+1−𝛼_𝑘−𝜙_𝑥𝑡,𝛽_𝑘𝛼_𝑘+1−𝛼_𝑘≤𝑓𝑡,𝛼_𝑘+1. (43)
Using a similar argument, we have, as before, ^𝑐𝐷^𝑞𝛽_𝑘+1≥𝑓(𝑡,𝛽_𝑘+1), and hence Lemma 2 shows that 𝛼_𝑘+1(𝑡)≤𝛽_𝑘+1(𝑡),𝑡∈𝐽. (44) By (19), we have that 𝑓 is Lipschitzian in 𝑥 on Ω and is quasimonotone nondecreasing in 𝑥. This proves (34). Therefore, by induction, we have for all 𝑛: 𝛼₀(𝑡)≤𝛼₁(𝑡)≤⋯≤𝛼_𝑛(𝑡)≤𝛽_𝑛(𝑡)≤⋯𝛽₁(𝑡)≤𝛽₀(𝑡),𝑡∈𝐽. (45)
Employing the standard procedure, it is now easy to prove that the sequences {𝛼_𝑛(𝑡)} and {𝛽_𝑛(𝑡)} converge uniformly and monotonically to the unique solution of (1) on 𝐽.
We will now show that the convergence of {𝛼_𝑛(𝑡)} and {𝛽_𝑛(𝑡)} to 𝑥(𝑡) is quadratic. First set 𝑝_𝑛+1=𝑥−𝛼_𝑛+1,𝑟_𝑛+1=𝛽_𝑛+1−𝑥,𝑡∈𝐽, (46) so that 𝑝_𝑛+1(0)=𝑟_𝑛+1(0)=0. Then using integral mean value theorem and the fact that 𝑥≥𝛼_𝑛+1 and (H₁), we get ^𝑐𝐷^𝑞𝑝_𝑛+1=^𝑐𝐷^𝑞𝑥−^𝑐𝐷^𝑞𝛼_𝑛+1=𝑓(𝑡,𝑥)−𝑓𝑡,𝛼_𝑛+𝐹_𝑥𝑡,𝛼_𝑛−𝜙_𝑥𝑡,𝛽_𝑛×𝛼_𝑛+1−𝛼_𝑛=𝐹(𝑡,𝑥)−𝐹𝑡,𝛼_𝑛−𝐹_𝑥𝑡,𝛼_𝑛−𝜙_𝑥𝑡,𝛽_𝑛×𝛼_𝑛+1−𝛼_𝑛−𝜙(𝑡,𝑥)−𝜙𝑡,𝛼_𝑛=¹₀𝐹_𝑥𝑡,𝑠𝑥+(1−𝑠)𝛼_𝑛𝑑𝑠𝑝_𝑛−𝐹_𝑥𝑡,𝛼_𝑛−𝜙_𝑥𝑡,𝛽_𝑛−𝑝_𝑛+1+𝑝_𝑛−¹₀𝜙_𝑥𝑡,𝑠𝑥+(1−𝑠)𝛼_𝑛𝑑𝑠𝑥−𝛼_𝑛≤𝐹_𝑥(𝑡,𝑥)−𝐹_𝑥𝑡,𝛼_𝑛𝑝_𝑛+𝜙_𝑥𝑡,𝛽_𝑛−𝜙_𝑥𝑡,𝛼_𝑛𝑝_𝑛+𝑓_𝑥𝑡,𝛼_𝑛𝑝_𝑛+1. (47) So, we get ^𝑐𝐷^𝑞𝑝_𝑛+1≤𝑝^𝑇_𝑛¹₀𝐹_𝑥𝑥𝑡,𝑠𝑥+(1−𝑠)𝛼_𝑛𝑑𝑠𝑝_𝑛+𝑟_𝑛+𝑝_𝑛^𝑇1₀𝜙_𝑥𝑥𝑡,𝑠𝛽_𝑛+(1−𝑠)𝛼_𝑛𝑑𝑠+𝑓_𝑥𝑡,𝛼_𝑛𝑝_𝑛+1≤𝑀𝑝_𝑛+1+𝑊𝑝_𝑛⋅𝑝_𝑛+𝑁𝑟_𝑛+𝑝_𝑛𝑝_𝑛≤𝑀𝑝_𝑛+1+𝑊+3𝑁2𝑝_𝑛²+𝑁2𝑟_𝑛², (48) where |𝑝_𝑛|²=(|𝑝_𝑛1|²,|𝑝_𝑛2|²,…,|𝑝_𝑛𝑛|²), |𝑟_𝑛|=(|𝑟_𝑛1|²,|𝑟_𝑛2|²,…,|𝑟_𝑛𝑛|²), 𝑓_𝑥≤𝑀, ∑^𝑛_𝑖=1𝑓^𝑖_𝑥𝑥≤𝑊 and ∑^𝑛_𝑖=1𝜙^𝑖_𝑥𝑥≤𝑁 in Ω, and 𝑀, 𝑊, and 𝑁 are 𝑛×𝑛 positive matrices. Using (2) to (4), we can get 𝑝_𝑛+1(𝑡)≤𝑊+3𝑁2𝑝_𝑛²+𝑁2𝑟_𝑛²×𝑇−𝑡₀^𝑞−1𝐸_𝑞,𝑞𝑀𝑇−𝑡₀^𝑞𝑘≤𝑉𝑊+3𝑁2𝑝_𝑛²+𝑁2𝑟_𝑛², (49) where 𝑉=(𝑇−𝑡₀)^𝑞−1𝐸_𝑞,𝑞(𝑀(𝑇−𝑡₀)^𝑞)^𝑘. Thus, we have 𝑝_𝑛+1≤𝑉𝑊+3𝑁2𝑝_𝑛²+𝑁2𝑟_𝑛². (50) Similarly, we have 𝑟_𝑛+1𝑊≤𝑉2𝑝_𝑛²+3𝑊2+5𝑁2𝑟_𝑛². (51) The proof is complete.

Acknowledgments

The authors would like to thank the reviewers for their valuable suggestions and comments. This work is supported by the National Natural Science Foundation of China (11271106) and the Natural Science Foundation of Hebei Province of China (A2013201232).