A New Method Based on the RKHSM for Solving Systems of Nonlinear IDDEs with Proportional Delays

Abstract

An efficient computational method is given in order to solve the systems of nonlinear infinite-delay-differential equations (IDDEs) with proportional delays. Representation of the solution and an iterative method are established in the reproducing kernel space. Some examples are displayed to demonstrate the computation efficiency of the method.

1. Introduction

In functional-differential equations (FDEs), there is a class of infinite-delay-differential equations (IDDEs) with proportional delays such systems are often encountered in many scientific fields such as electric mechanics, quantum mechanics, and optics. In view of this, developing the research for this class of IDDEs possesses great significance on theory and practice, for this attracts constant interest of researchers. Ones have found that there exist very different mathematical challenges between FDEs with proportional delays and those with constant delays. Some researches on the numerical solutions and the corresponding analysis for the linear FDEs with proportional delays have been presented by several authors. In the last few years, there has been a growing interest in studying the existence of solutions of functional differential equations with state dependent delay [1–7]. Initial-value problem for neutral functional-differential equations with proportional time delays had been studied in [8–11]; in the literature [11] authors had discussed the existence and uniqueness of analytic solution of linear proportional delays equations.

Ishiwata et al. used the rational approximation method and the collocation method to compute numerical solutions of delay-differential equations with proportional delays in [12, 13]. At [14–17], Hu et al. gave the numerical method to compute numerical solutions of neutral delay differential equations. For neutral delay differential equations with proportional delays, Chen and Wang proposed the variational iteration method [18] and the homotopy perturbation method [19]. Recently, time-delay systems become interested in applications like population growth models, transportation, communications, and agricultural models so those systems were widely studied both in a theoretical aspect and in that of related applications [20–22].

We consider the following nonlinear infinite delay-differential equation (IDDE) with proportional delay [23]: 𝑢^′(𝑥)(𝑥),𝑢(0)=𝑔𝑥,𝑢,𝑢𝑝𝑥=𝜂, (1) where 𝑝∈(0,1), 𝜂 is a given initial value, 𝑢(𝑥)∈𝑊₂[0,+∞), and for 𝑥∈[0,+∞), 𝑦,𝑧∈(−∞,+∞), 𝑔(𝑥,𝑦,𝑧) is a continuous function; 𝑔(𝑥,𝑦,𝑧)∈𝑊₁[0,+∞) as 𝑦=𝑦(𝑥), 𝑧=𝑧(𝑥)∈𝑊₁[0,+∞).

Next, the following system of nonlinear infinite-delay-differential equations (IDDEs) with proportional delays will be studied: 𝑢^′(𝑥)(𝑥),𝑣=𝐹𝑥,𝑢,𝑣𝑝𝑥_′(𝑥)(𝑥),𝑢(0)(0)=𝐺𝑥,𝑣,𝑢𝑞𝑥=0,𝑣=0, (2) where 𝑝,𝑞∈(0,1), for 𝑥∈[0,+∞), 𝑦,𝑧∈(−∞,+∞), 𝑓(𝑥,𝑦,𝑧), 𝑔(𝑥,𝑦,𝑧) are continuous bounded function, and 𝑓(𝑥,𝑦,𝑧)∈𝑊₁[0,+∞), 𝑔(𝑥,𝑦,𝑧)∈𝑊₁[0,+∞) as 𝑦=𝑦(𝑥), 𝑧=𝑧(𝑥). 𝑊₂[0,+∞) and 𝑊₁[0,∞) are reproducing kernel spaces. Equations (2) are obtained through homogenization of initial condition for model in [24]. In this study, existence and a new iterative algorithm are established for the nonlinear infinite-delay-differential equations (IDDEs) with proportional delays in the reproducing kernel spaces.

The paper is organized as follows. In Section 2, some definitions of the reproducing kernel spaces are introduced. In Section 3, main results and the structure of the solution for operator equation are discussed. Existence of the solution to (2) and an iterative method are developed for the kind of problems in the reproducing kernel space. We verify that the approximate solution converges to the exact solution uniformly. In Section 4, some experiments are given to demonstrate the computation efficiency of the algorithm. The conclusion is given in Section 5.

2. Preliminaries

Definition 1 (see [25] (reproducing kernel)). Let 𝑀 be a nonempty abstract set. A function 𝐾∶𝑀×𝑀→𝐶 is a reproducing kernel of the Hilbert space 𝐻 if and only if(a)∀𝑦∈𝑀, 𝐾(⋅,𝑥)∈𝐻,(b)∀𝑦∈𝑀, ∀𝑢∈𝐻, ⟨𝑢(⋅),𝐾(⋅,𝑦)⟩=𝑢(𝑦).
The condition (b) is called “the reproducing property”; a Hilbert space which possesses a reproducing kernel is called a Reproducing Kernel Hilbert Space (RKHS).

Next, two reproducing kernel spaces are given.

Definition 2. 𝑊₂[0,+∞)={𝑢(𝑥)∣𝑢^′ is absolutely continuous real value functions, 𝑢^′′∈𝐿²[0,+∞),𝑢(0)=0}.
𝑊₂[0,+∞) is a Hilbert space, for 𝑢(𝑥),𝑣(𝑥)∈𝑊₂[0,+∞); the inner product and norm in 𝑊₂[0,+∞) are given by ⟨𝑢(𝑥),𝑣(𝑥)⟩_𝑊2=₀^+∞4𝑢𝑣+5𝑢^′𝑣^′+𝑢^′′𝑣^′′‖𝑢‖𝑑𝑥,_𝑊2=⟨𝑢,𝑢⟩_𝑊2, (3) respectively.

Theorem 3. The space 𝑊₂[0,+∞) is a reproducing kernel space that is, for any 𝑢(𝑦)∈𝑊₂[0,+∞) and each fixed 𝑥∈[0,+∞), there exists 𝑅_𝑥(𝑦)∈𝑊₂[0,+∞), 𝑦∈[0,+∞), such that 𝑢(𝑥)=⟨𝑅_𝑥(𝑦),𝑢(𝑦)⟩. And the corresponding reproducing kernel can be represented as follows [26]: 𝑅_𝑥𝑦=−1𝑒12^{−2(𝑥+𝑦)}−1+𝑒^2𝑦1+𝑒^2𝑦−2𝑒^𝑥+𝑦−1,𝑦≤𝑥,𝑒12^{−2(𝑥+𝑦)}−1+𝑒^2𝑥1+𝑒^2𝑥−2𝑒^𝑥+𝑦,𝑦>𝑥. (4)

Definition 4. 𝑊₁[0,+∞)={𝑢(𝑥)∣𝑢 is absolutely continuous real-valued function, 𝑢^′∈𝐿²[0,+∞)}.
The inner product and norm in 𝑊₁[0,+∞) can be defined by ⟨𝑢(𝑥),𝑣(𝑥)⟩_𝑊1=₀^+∞𝑢𝑣+𝑢^′𝑣^′‖𝑢‖𝑑𝑥,_𝑊1=⟨𝑢,𝑢⟩_𝑊1, (5) respectively, where 𝑢(𝑥),𝑣(𝑥)∈𝑊₁[0,+∞). It has been proved that 𝑊₁[0,+∞) is a complete reproducing kernel space and its reproducing kernel is as follows [27]: 𝑄_𝑥𝑦=12𝑒^{−𝑥−𝑦}1+𝑒^2𝑥1,𝑥<𝑦,2𝑒^{−𝑥−𝑦}1+𝑒^2𝑦,𝑥≥𝑦. (6)

3. Statements of the Main Results

In this section, the implementation method of obtaining the solution of (2) is proposed in the reproducing kernel space 𝑊₂[0,+∞).

Put the differential operator ℒ=𝑑/𝑑𝑥; then we can convert (2) into the following form: (𝑥)(𝑥),(𝑥)(𝑥),𝑢(0)(0)ℒ𝑢=𝐹𝑥,𝑢,𝑣𝑝𝑥ℒ𝑣=𝐺𝑥,𝑣,𝑢𝑞𝑥=𝑣=0, (7) where 𝑥∈[0,+∞), and 𝐹𝑥,𝑢(𝑥),𝑣𝑝𝑥^def𝑢(𝑥)(𝑥),𝐺(𝑥)=𝑁,𝑣𝑝𝑥+𝑓𝑥,𝑣,𝑢𝑞𝑥=^def𝐷𝑣(𝑥),𝑢𝑞𝑥+𝑔(𝑥). (8)(9) It is clear that ℒ∶𝑊₂[0,+∞)→𝑊₁[0,+∞) is a bounded linear operator. Let 𝜑_𝑖(𝑥)=𝑄_𝑥𝑖(𝑥), where {𝑥_𝑖}^∞_𝑖=1 is dense in the interval [0,+∞), and 𝜓_𝑖(𝑥)=ℒ^∗𝜑_𝑖(𝑥), where ℒ^∗ is the conjugate operator of ℒ. Define the normal orthogonal system 𝜓_𝑖(𝑥) in 𝑊₂[0,+∞), which derives from Gram-Schmidt orthogonalization process of {𝜓_𝑖(𝑥)}^∞_𝑖=1, 𝜓_𝑖(𝑥)=^𝑖_𝑘=1𝛽_𝑖𝑘𝜓_𝑘(𝑥),𝛽_𝑖𝑖>0,𝑖=1,2,… (10)

Lemma 5. Assume that {𝑥_𝑖}^∞_𝑖=1 are dense in [0,+∞); then {𝜓_𝑖(𝑥)}^∞_𝑖=1 is a complete system in 𝑊₂[0,+∞) and {𝜓_𝑖(𝑥)}=(𝑑/𝑑𝑦)𝑅_𝑥(𝑦)|_{𝑦=𝑥𝑖}.

Proof. Ones have 𝜓_𝑖(𝑥)=𝜓_𝑖𝑦,𝑅_𝑥𝑦_𝑊2=ℒ^∗𝜑_𝑖𝑦,𝑅_𝑥𝑦_𝑊2=𝜑_𝑖𝑦,ℒ𝑅_𝑥𝑦_𝑊1=ℒ𝑅_𝑥𝑥_𝑖=𝑑𝑅𝑑𝑦^𝑥𝑦_{𝑦=𝑥𝑖}. (11)
Clearly, 𝜓_𝑖(𝑥)∈𝑊₂[0,+∞).
For any 𝑢(𝑥)∈𝑊₂[0,+∞), let ⟨𝑢(𝑥),𝜓_𝑖(𝑥)⟩=0, 𝑖=1,2,…, which means that 𝑢(𝑥),ℒ^∗𝜑_𝑖(𝑥)=ℒ𝑢(⋅),𝜑_𝑖(⋅)=(ℒ𝑢)𝑥_𝑖=0. (12) Note that {𝑥_𝑖}^∞_𝑖=1 is dense in [0,+∞); therefore, ℒ𝑢(𝑥)=0. It follows that 𝑢(𝑥)≡0 from the existence of ℒ⁻¹.

3.1. Construction the of Iterative Sequence 𝑢^∼_𝑛(𝑥) and 𝑣^∼_𝑛(𝑥)

Next we construct the iterative sequence 𝑢^∼_𝑛(𝑥) and 𝑣^∼_𝑛(𝑥), putting ℒ𝑀_𝑛(𝑥)=𝐹𝑥,𝑢^∼_𝑛−1(𝑥),𝑣^∼_𝑛−1𝑝𝑥_∼,𝑢_𝑛(𝑥)=𝑃_𝑛𝑀_𝑛(𝑥),ℒ𝑁_𝑛(𝑥)=𝐺𝑥,𝑣^∼_𝑛−1(𝑥),𝑢^∼_𝑛−1𝑞𝑥_∼,𝑣_𝑛(𝑥)=𝑃_𝑛𝑁_𝑛(𝑥), (13) where 𝑀_𝑛(𝑥),𝑁_𝑛(𝑥)∈𝑊₂[0,+∞) and 𝑃_𝑛∶𝑊₂→Span{𝜓₁,𝜓₂,…,𝜓_𝑛} is a orthogonal projection operator. Then by (13) it followed that: 𝑀_𝑛(𝑥)=^𝑥₀𝐹𝜇,𝑢^∼_𝑛−1𝜇,𝑣^∼_𝑛−1𝑀𝑝𝜇𝑑𝜇,_′𝑛(𝑥)=𝐹𝑥,𝑢^∼_𝑛−1(𝑥),𝑣^∼_𝑛−1,𝑀𝑝𝑥_𝑛′′(𝑥)=𝑑𝐹𝑑𝑥𝑥,𝑢^∼_𝑛−1(𝑥),𝑣^∼_𝑛−1,𝑁𝑝𝑥_𝑛(𝑥)=^𝑥₀𝐺𝜇,𝑣^∼_𝑛−1𝜇,𝑢^∼_𝑛−1𝑁𝑞𝜇𝑑𝜇,_′𝑛(𝑥)=𝐺𝑥,𝑣^∼_𝑛−1(𝑥),𝑢^∼_𝑛−1,𝑁𝑞𝑥_𝑛′′(𝑥)=𝑑𝐺𝑑𝑥𝑥,𝑣^∼_𝑛−1(𝑥),𝑢^∼_𝑛−1𝑞𝑥. (14)(15)

Lemma 6. Let {𝑥_𝑖}^∞_𝑖=1 be dense on [0,+∞); if the solution of (2) is unique, then the solution satisfies the form 𝑀_𝑛(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢^∼_𝑛−1𝑥_𝑘,𝑣^∼_𝑛−1𝑝𝑥_𝑘𝜓_𝑖(𝑥),𝑁_𝑛(𝑥)=∑^∞𝑖_𝑖=1∑_𝑘=1𝛽_𝑖𝑘𝐺𝑥_𝑘,𝑣^∼_𝑛−1𝑥_𝑘,𝑢^∼_𝑛−1𝑞𝑥_𝑘𝜓_𝑖(𝑥). (16)

Proof. Note that ⟨𝑢(𝑥),𝜑_𝑖(𝑥)⟩=𝑢(𝑥_𝑖) and {𝜓_𝑖(𝑥)}^∞_𝑖=1 is an orthonormal basis of 𝑊₂[0,+∞); hence according to Lemma 5 we have 𝑀_𝑛(𝑥)=^∞_𝑖=1𝑀_𝑛(𝑥),𝜓_𝑖(𝑥)𝜓_𝑖(𝑥)=^∞_𝑖=1𝑀_𝑛(𝑥),^𝑖_𝑘=1𝛽_𝑖𝑘𝜓_𝑘(𝑥)𝜓_𝑖(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑀_𝑛(𝑥),𝜓_𝑘(𝑥)𝜓_𝑖(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘ℒ𝑀_𝑛(𝑥),𝜑_𝑘(𝑥)𝜓_𝑖(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢_𝑛−1𝑥_𝑘,𝑣_𝑛−1𝑝𝑥_𝑘𝜓_𝑖(𝑥). (17) In the same manner 𝑁_𝑛(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐺𝑥_𝑘,𝑣_𝑛−1𝑥_𝑘,𝑢_𝑛−1𝑞𝑥_𝑘𝜓_𝑖(𝑥). (18)

Take 𝑢₀(𝑥)=𝑣₀(𝑥)=0; define the iterative sequence 𝑢_𝑛(𝑥)=𝑃_𝑛𝑀_𝑛(𝑥)=^𝑛𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢_𝑛−1𝑥_𝑘,𝑣_𝑛−1𝑝𝑥_𝑘𝜓_𝑖(𝑥),𝑣_𝑛(𝑥)=𝑃_𝑛𝑁_𝑛(𝑥)=^𝑛𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐺𝑥_𝑘,𝑣_𝑛−1𝑥_𝑘,𝑢_𝑛−1𝑞𝑥_𝑘𝜓_𝑖(𝑥). (19)

3.2. The Boundedness of Sequence 𝑢_𝑛(𝑥) and 𝑣_𝑛(𝑥)

From 𝜕³_𝑦𝑅_𝑥(𝑥+0)−𝜕³_𝑦𝑅_𝑥(𝑥−0)=1, (20) we have 𝜕⁴_𝑦𝑅_𝑥𝑦=𝑔_𝑥𝑦+𝜕³_𝑦𝑅_𝑥(𝑥+0)−𝜕³_𝑦𝑅_𝑥(𝑥−0)×𝛿𝑥−𝑦=𝑔_𝑥𝑦+𝛿𝑥−𝑦, (21) where 𝑔_𝑥𝑦=𝑔¹_𝑥𝑔,𝑦<𝑥,_2𝑥,𝑦>𝑥,0,𝑦=𝑥, (22)𝑔¹_𝑥(𝑦)=𝜕⁴_𝑦(𝑎₁𝑒^𝑦+𝑎₂𝑒^−𝑦+𝑎₃𝑒^2𝑦+𝑎₄𝑒^−2𝑦); 𝑔²_𝑥(𝑦)=𝜕⁴_𝑦(𝑏₁𝑒^𝑦+𝑏₂𝑒^−𝑦+𝑏₃𝑒^2𝑦+𝑏₄𝑒^−2𝑦); and 𝑔¹_𝑥(𝑦),𝑔²_𝑥(𝑦),𝜕_𝑦𝑔¹_𝑥(𝑦),𝜕_𝑦𝑔²_𝑥(𝑦) are bounded functions with respect to 𝑥, 𝑦, respectively.

Lemma 7. For 𝑥∈[0,+∞), 𝑦,𝑧∈(−∞,+∞), and 𝐹(𝑥,𝑦,𝑧), 𝐺(𝑥,𝑦,𝑧) are continuous bounded functions on [0,+∞), we have that 𝑀_𝑛^(𝑖)(𝑥) and 𝑁_𝑛^(𝑖)(𝑥)(𝑖=0,1,2) are bounded.

Proof. By the expression of 𝑀_𝑛(𝑥), 𝑀^′_𝑛(𝑥) and the assumptions, we know that 𝑀_𝑛(𝑥) and 𝑀^′_𝑛(𝑥) are bounded. In the following, we will discuss the boundedness of 𝑀_𝑛^′′(𝑥).
Since the function which is in 𝐶²_[0,+∞) is dense in 𝑊₂[0,+∞), without loss of generality, we assume that 𝑀_𝑛^′′(𝑥) is continuous.
Note that 4𝑅_𝑥𝑦−5𝜕²_𝑦𝑅_𝑥𝑦+𝜕⁴_𝑦𝑅_𝑥𝑦=𝛿𝑥−𝑦. (23) One gets 𝑀_𝑛^′′(𝑥)=^∞₀𝑀_𝑛^′′𝑦𝛿=𝑥−𝑦𝑑𝑦_∞0𝑀_𝑛^′′𝑦4𝑅_𝑥𝑦−5𝜕²_𝑦𝑅_𝑥𝑦+𝜕⁴_𝑦𝑅_𝑥𝑦=𝑑𝑦_∞0𝑀_𝑛^′′𝑦4𝑅_𝑥𝑦−5𝜕²_𝑦𝑅_𝑥𝑦+𝑑𝑦_∞0𝑀_𝑛^′′𝑦𝜕⁴_𝑦𝑅_𝑥𝑦𝑑𝑦=𝑀^′_𝑛𝑦4𝑅_𝑥𝑦−5𝜕²_𝑦𝑅_𝑥𝑦^∞₀−₀^+∞𝑀^′_𝑛𝑦4𝜕_𝑦𝑅_𝑥𝑦−5𝜕³_𝑦𝑅_𝑥𝑦+𝑑𝑦_∞0𝑀_𝑛^′′𝑦𝑔_𝑥𝑦(𝑥)++𝛿𝑥−𝑦𝑑𝑦=𝑚^∞₀𝑀_𝑛^′′𝑦𝑔_𝑥𝑦(𝑥)++𝛿𝑥−𝑦𝑑𝑦=𝑚^∞₀𝑀_𝑛^′′𝑦𝑔_𝑥𝑦𝑑𝑦+𝑀_𝑛^′′(𝑥), (24) where 𝑚(𝑥)=𝑀^′_𝑛(𝑦)[4𝑅_𝑥(𝑦)−5𝜕²_𝑦𝑅_𝑥(𝑦)]|^∞₀−∫^∞₀𝑀^′_𝑛(𝑦)[4𝜕_𝑦𝑅_𝑥(𝑦)−5𝜕³_𝑦𝑅_𝑥(𝑦)]𝑑𝑦. Thus, we have 0=𝑚(𝑥)−^∞₀𝑀_𝑛^′′𝑦𝑔_𝑥𝑦(𝑥)−𝑑𝑦=𝑚^𝑥₀𝑀_𝑛^′′𝑦𝑔¹_𝑥𝑦𝑑𝑦−^∞_𝑥𝑀_𝑛^′′𝑦𝑔²_𝑥𝑦(𝑥)𝑑𝑦=𝑚−𝑀^′_𝑛𝑦𝑔¹_𝑥𝑦^𝑥₀+^𝑥₀𝑀^′_𝑛𝑦𝜕_𝑦𝑔¹_𝑥𝑦𝑑𝑦−𝑀^′_𝑛𝑦𝑔²_𝑥𝑦^𝑀_𝑥+^∞_𝑥𝑀^′_𝑛𝑦𝜕_𝑦𝑔²_𝑥𝑦(𝑥)𝑑𝑦=𝑚−𝑀^′_𝑛(𝑥)𝑔¹_𝑥(𝑥)−𝑔²_𝑥(𝑥)+𝑀^′_𝑛(0)𝑔¹_𝑥(0)−𝑀^′_𝑛(𝑀)𝑔²_𝑥(𝑀)+^𝑥₀𝑀^′_𝑛𝑦𝜕_𝑦𝑔¹_𝑥𝑦𝑑𝑦+^∞_𝑥𝑀^′_𝑛𝑦𝜕_𝑦𝑔²_𝑥𝑦𝑑𝑦. (25) So 𝑀^′_𝑛(𝑥)𝑔¹_𝑥(𝑥)−𝑔²_𝑥(𝑥)=𝑚(𝑥)+𝑀^′_𝑛(0)𝑔¹_𝑥(0)−𝑀^′_𝑛(𝑀)𝑔²_𝑥(𝑀)+^𝑥₀𝑀^′_𝑛𝑦𝜕_𝑦𝑔¹_𝑥𝑦+^∞_𝑥𝑀^′_𝑛𝑦𝜕_𝑦𝑔²_𝑥𝑦𝑑𝑦. (26) Furthermore, we see that 𝑀_𝑛^′′(𝑥)=1𝑔¹_𝑥(𝑥)−𝑔²_𝑥(𝑥)×𝜕_𝑥𝑚(𝑥)+𝑀^′_𝑛(0)𝑔¹_𝑥(0)−𝑀^′_𝑛(𝑀)𝑔²_𝑥(𝑀)+^𝑥₀𝑀^′_𝑛𝑦𝜕_𝑦𝑔¹_𝑥𝑦+^∞_𝑥𝑀^′_𝑛𝑦𝜕_𝑦𝑔²_𝑥𝑦𝑑𝑦−𝑀^′_𝑛(𝑥)𝜕_𝑥𝑔¹_𝑥(𝑥)−𝑔²_𝑥(𝑥). (27) In view of the expression of 𝑅_𝑥(𝑦), we know that 𝜕^𝑚_𝑦𝑅_𝑥(𝑦) are bounded for 𝑚=1,2,3. It follows that 𝑀_𝑛^′′(𝑥) is bounded from the boundedness of 𝑔¹_𝑥(𝑦), 𝑔²_𝑥(𝑦), 𝜕_𝑦𝑔¹_𝑥(𝑦), 𝜕_𝑦𝑔²_𝑥(𝑦), and 𝑀^′_𝑛(𝑥). In the same way, 𝑁_𝑛(𝑥), 𝑁^′_𝑛(𝑥), and 𝑁_𝑛^′′(𝑥) are bounded.

Lemma 8. Assume that 𝐹(𝑥,𝑦,𝑧), 𝐺(𝑥,𝑦,𝑧) are continuous bounded functions for 𝑥∈[0,+∞), 𝑦,𝑧∈(−∞,+∞), and 𝐹(𝑥,𝑦,𝑧),𝐺(𝑥,𝑦,𝑧)∈𝑊₁[0,+∞) as 𝑦=𝑦(𝑥)∈𝑊₂, 𝑧=𝑧(𝑥)∈𝑊₂, then 𝑢_𝑛𝑊2≤𝐶,𝑣_𝑛𝑊2≤𝐷,(𝐶,𝐷areconstants). (28)

Proof. 𝑀_𝑛²_𝑊2=𝑀_𝑛,𝑀_𝑛=^∞₀4𝑀²_𝑛+5𝑀_𝑛^′2+𝑀_𝑛^′′2𝑑𝑥. (29) Note that (14) and the assumptions, by Lemma 7, |𝑀_𝑛^(𝑖)(𝑥)|≤𝐶^𝑖, 𝑖=0,1,2, thus ‖𝑀_𝑛‖≤𝐶, where 𝐶=max{𝐶⁰,𝐶¹,𝐶²}, from (19), we have ‖𝑢_𝑛‖_𝑊2≤‖𝑀_𝑛‖_𝑊2≤𝐶. In the same way, we obtain ‖𝑣_𝑛‖_𝑊2≤𝐷.

3.3. Construction the of Another Iterative Sequence 𝑢_𝑛(𝑥) and 𝑣_𝑛(𝑥)

Theorem 9. Let {𝑥_𝑖}^∞_𝑖=1 be dense on [0,+∞); if the solution of (2) exists and unique, then the solution satisfies the form 𝑢(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢𝑥_𝑘,𝑣𝑝𝑥_𝑘𝜓_𝑖(𝑥),𝑣(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐺𝑥_𝑘,𝑣𝑥_𝑘,𝑢𝑞𝑥_𝑘𝜓_𝑖(𝑥). (30)

Proof. Note that ⟨𝑢(𝑥),𝜑_𝑖(𝑥)⟩=𝑢(𝑥_𝑖) and {𝜓_𝑖(𝑥)}^∞_𝑖=1 is an orthonormal basis of 𝑊₂[0,+∞); hence we have 𝑢(𝑥)=^∞_𝑖=1𝑢(𝑥),𝜓_𝑖(𝑥)𝜓_𝑖(𝑥)=^∞_𝑖=1𝑢(𝑥),^𝑖_𝑘=1𝛽_𝑖𝑘𝜓_𝑘(𝑥)𝜓_𝑖(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_i𝑘𝑢(𝑥),𝜓_𝑘(𝑥)𝜓_𝑖(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘ℒ𝑢(𝑥),𝜑_𝑘(𝑥)𝜓_𝑖(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢𝑥_𝑘,𝑣𝑝𝑥_𝑘𝜓_𝑖(𝑥). (31) In the same manner 𝑣(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐺𝑥_𝑘,𝑣𝑥_𝑘,𝑢𝑞𝑥_𝑘𝜓_𝑖(𝑥). (32)

In the following, a new method of solving (2) is presented. Equations (31) and (32) can be denoted by 𝑢(𝑥)=^∞_𝑖=1𝐴_𝑖𝜓_𝑖(𝑥),𝑣(𝑥)=^∞_𝑖=1𝐵_𝑖𝜓_𝑖(𝑥), (33)(34) where 𝐴_𝑖=∑^𝑖_𝑘=1𝛽_𝑖𝑘𝐹(𝑥_𝑘,𝑢(𝑥_𝑘),𝑣(𝑝𝑥_𝑘)) and 𝐵_𝑖=∑^𝑖_𝑘=1𝛽_𝑖𝑘𝐺(𝑥_𝑘,𝑣(𝑥_𝑘),𝑢(𝑞𝑥_𝑘)). In fact, 𝐴_𝑖 and 𝐵_𝑖 are unknown; we will approximate 𝐴_𝑖 and 𝐵_𝑖 by using the known 𝐴_𝑖 and 𝐵_𝑖.

We take 𝑢₁(𝑥)=0, 𝑣₁(𝑥)=0 and define the following iterative sequence 𝑢_𝑛(𝑥)=^𝑛−1_𝑖=1𝐴_𝑖𝜓_𝑖(𝑥),𝑣_𝑛(𝑥)=∑^𝑛−1_𝑖=1𝐵_𝑖𝜓_𝑖(𝑥), (35) where 𝐴₁=𝛽₁₁𝐹𝑥₁,𝑢₁𝑥₁,𝑣₁𝑝𝑥₁𝐴₂=²_𝑘=1𝛽_2𝑘𝐹𝑥_𝑘,𝑢_𝑘𝑥_𝑘,𝑣_𝑘𝑝𝑥_𝑘⋮𝐴_𝑛−1=^𝑛−1_𝑘=1𝛽_𝑛𝑘𝐹𝑥_𝑘,𝑢_𝑘𝑥_𝑘,𝑣_𝑘𝑝𝑥_𝑘,𝐵₁=𝛽₁₁𝐺𝑥₁,𝑣₁𝑥₁,𝑢₁𝑞𝑥₁𝐵₂=²_𝑘=1𝛽_2𝑘𝐺𝑥_𝑘,𝑣_𝑘𝑥_𝑘,𝑢_𝑘𝑞𝑥_𝑘⋮𝐵_𝑛−1=^𝑛−1_𝑘=1𝛽_𝑛𝑘𝐺𝑥_𝑘,𝑣_𝑘𝑥_𝑘,𝑢_𝑘𝑞𝑥_𝑘. (36)(37) Next, lemmas are given.

Lemma 10. The following iterative sequences 𝑢_𝑛(𝑥)=^𝑛−1_𝑖=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢_𝑘𝑥_𝑘,𝑣_𝑘𝑝𝑥_𝑘𝜓_𝑖(𝑥),𝑣_𝑛(𝑥)=^𝑛−1_𝑖=1𝛽_𝑖𝑘𝐺𝑥_𝑘,𝑣_𝑘𝑥_𝑘,𝑢_𝑘𝑞𝑥_𝑘𝜓_𝑖(𝑥) (38) satisfy ℒ𝑢_𝑛𝑥_𝑗=𝐹𝑥_𝑗,𝑢_𝑗𝑥_𝑗,𝑣_𝑗𝑝𝑥_𝑗,ℒ𝑣_𝑛𝑥_𝑗=𝐺𝑥_𝑗,𝑣_𝑗𝑥_𝑗,𝑢_𝑗𝑞𝑥_𝑗,𝑗≤𝑛−1, (39) respectively.

Proof. If 𝑗=1, ℒ𝑢_𝑛𝑥₁=^𝑛−1_𝑖=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢_𝑘𝑥_𝑘,𝑣_𝑘𝑝𝑥_𝑘ℒ𝜓_𝑖𝑥₁=^𝑛−1_𝑖=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢_𝑘𝑥_𝑘,𝑣_𝑘𝑝𝑥_𝑘×ℒ𝜓_𝑖(𝑥),𝜑₁(𝑥=_{𝑛−1𝑖=1})𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢_𝑘𝑥_𝑘,𝑣_𝑘𝑝𝑥_𝑘×𝜓_𝑖(𝑥),𝜓₁(𝑥,𝛽₁₁)ℒ𝑢_𝑛𝑥₁=^𝑛−1_𝑖=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢_𝑘𝑥_𝑘,𝑣_𝑘𝑝𝑥_𝑘×𝜓_𝑖(𝑥),𝜓₁(𝑥)=𝛽₁₁𝐹𝑥₁,𝑢₁𝑥₁,𝑣₁𝑝𝑥₁, (40) so, ℒ𝑢_𝑛𝑥₁=𝐹𝑥₁,𝑢₁𝑥₁,𝑣₁𝑝𝑥₁. (41) If 𝑗=2, ℒ𝑢_𝑛𝑥₂=^𝑛−1_𝑖=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢_𝑘𝑥_𝑘,𝑣_𝑘𝑝𝑥_𝑘ℒ𝜓_𝑖𝑥₂=^𝑛−1_𝑖=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢_𝑘𝑥_𝑘,𝑣_𝑘𝑝𝑥_𝑘𝜓_𝑖(𝑥),𝜓₂(𝑥), (42)𝛽₂₁× (40) +𝛽₂₂× (42), we have 𝛽₂₁ℒ𝑢_𝑛𝑥₁+𝛽₂₂ℒ𝑢_𝑛𝑥₂=^𝑛−1_𝑖=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢_𝑘𝑥_𝑘,𝑣_𝑘𝑝𝑥_𝑘𝜓_𝑖(𝑥),𝜓₂(𝑥)=𝛽₂₁𝐹𝑥₁,𝑢₁𝑥₁,𝑣₁𝑝𝑥₁+𝛽₂₂𝐹𝑥₂,𝑢₂𝑥₂,𝑣₂𝑝𝑥₂, (43) by (41), ℒ𝑢_𝑛(𝑥₂)=𝐹(𝑥₂,𝑢₂(𝑥₂),𝑣₂(𝑝𝑥₂)). In the same way, we have ℒ𝑢_𝑛𝑥_𝑗=𝐹𝑥_𝑗,𝑢_𝑗𝑥_𝑗,𝑣_𝑗𝑝𝑥_𝑗,𝑗≤𝑛−1. (44) Similarly, ℒ𝑣_𝑛𝑥_𝑗=𝐹𝑥_𝑗,𝑣_𝑗𝑥_𝑗,𝑢_𝑗𝑞𝑥_𝑗,𝑗≤𝑛−1. (45)

Theorem 11. The iterative form 𝑢^∼_𝑛(𝑥)=^𝑛𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢^∼_𝑛−1𝑥_𝑘,𝑣^∼_𝑛−1𝑝𝑥_𝑘𝜓_𝑖(𝑥),_∼𝑣_𝑛(𝑥)=^𝑛𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐺𝑥_𝑘,𝑣^∼_𝑛−1𝑥_𝑘,𝑢^∼_𝑛−1𝑞𝑥_𝑘𝜓_𝑖(𝑥),_∼𝑢₁(0)=𝑣^∼₁(0)=0 (46) and the iterative form 𝑢_𝑛(𝑥)=^𝑛−1𝑖_𝑖=1𝑘=1𝛽_i𝑘𝐹𝑥_𝑘,𝑢_𝑘𝑥_𝑘,𝑣_𝑘𝑝𝑥_𝑘𝜓_𝑖(𝑥),𝑣_𝑛(𝑥)=^𝑛−1𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐺𝑥_𝑘,𝑣_𝑘𝑥_𝑘,𝑢_𝑘𝑞𝑥_𝑘𝜓_𝑖(𝑥),𝑢₁(0)=𝑣₁(0)=0 (47) are the same.

Proof. In Lemma 10, let 𝑛−1=𝑘; then ℒ𝑢_𝑘+1𝑥_𝑘=𝐹𝑥_𝑘,𝑢_𝑘𝑥_𝑘,𝑣_𝑘𝑝𝑥_𝑘, (48) but ℒ𝑢_𝑛𝑥_𝑘=𝐹𝑥_𝑘,𝑢_𝑘𝑥_𝑘,𝑣_𝑘𝑝𝑥_𝑘, (49) thus 𝑢_𝑛(𝑥)=^𝑛−1𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐿𝑢_𝑘+1𝑥_𝑘𝜓_𝑖(𝑥)=^𝑛−1𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘ℒ𝑢_𝑛𝑥_𝑘𝜓_𝑖(𝑥)=^𝑛−1𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘ℒ𝑢_𝑛(𝑥),𝜑_𝑘(𝑥)𝜓_𝑖(𝑥)=^𝑛−1𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑢_𝑛(𝑥),𝜓_𝑘(𝑥)𝜓_𝑖(𝑥)=^𝑛−1_𝑖=1𝑢_𝑛(𝑥),𝜓_𝑖(𝑥)𝜓_𝑖(𝑥). (50) Equation (46) can be written as ℒ𝑀_𝑛(𝑥)=𝐹𝑥,𝑢^∼_𝑛−1(𝑥),𝑣^∼_𝑛−1𝑝𝑥_∼,𝑢_𝑛(𝑥)=𝑃_𝑛𝑀_𝑛(𝑥). (51) In fact, 𝑀_𝑛(𝑥)=∑^∞_𝑖=1∑^𝑖_𝑘=1𝛽_𝑖𝑘𝐹(𝑥_𝑘,𝑢^∼_𝑛−1(𝑥_𝑘),𝑣^∼_𝑛−1(𝑝𝑥_𝑘))𝜓_𝑖(𝑥); then ℒ𝑢^∼_𝑛𝑥_𝑗=ℒ𝑃_𝑛𝑀_𝑛𝑥_𝑗=ℒ𝑃_𝑛𝑀_𝑛(𝑥),𝜑_𝑗(𝑥=𝑃_𝑛)𝑀_𝑛(𝑥),𝜓_𝑗(𝑥=𝑀_𝑛)(𝑥),𝜓_𝑗(𝑥=)ℒ𝑀_𝑛(𝑥),𝜑_𝑗(𝑥)=ℒ𝑀_𝑛𝑥_𝑗, (52) by (51) and (52), we have 𝑢^∼_𝑛(𝑥)=^𝑛−1𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢^∼_𝑛−1𝑥_𝑘,𝑣^∼_𝑛−1𝑝𝑥_𝑘𝜓_𝑖(𝑥)=^𝑛−1𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘ℒ𝑀_𝑛𝑥_𝑘𝜓_𝑖(𝑥)=^𝑛−1𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘ℒ𝑢^∼_𝑛𝑥_𝑘𝜓_𝑖(𝑥)=^𝑛−1𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘ℒ𝑢^∼_𝑛(𝑥),𝜑_𝑘(𝑥)𝜓_𝑖(𝑥)=^𝑛−1𝑖_𝑖=1𝑘=1𝛽𝑢_𝑖𝑘^∼_𝑛(𝑥),𝜓_𝑘(𝑥)𝜓_𝑖(𝑥)=𝑢^𝑛−1_𝑖=1^∼_𝑛(𝑥),𝜓_𝑖(𝑥)𝜓_𝑖(𝑥). (53) Equation (53) is the same as (50). We may prove for 𝑣^∼_𝑛(𝑥) similarly.

So, by Theorem 11 and Lemma 8, we have the following Theorem.

Theorem 12. Under the conditions of Lemma 8, 𝑢_𝑛(𝑥)=^𝑛−1_𝑖=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢_𝑘𝑥_𝑘,𝑣_𝑘𝑝𝑥_𝑘𝜓_𝑖(𝑥),𝑣_𝑛(𝑥)=^𝑛−1_𝑖=1𝛽_𝑖𝑘𝐺𝑥_𝑘,𝑣_𝑘𝑥_𝑘,𝑢_𝑘𝑞𝑥_𝑘𝜓_𝑖(𝑥) (54)(55) satisfy ‖𝑢_𝑛(𝑥)‖≤𝐶, ‖𝑣_𝑛(𝑥)‖≤𝐷.

Lemma 13. If 𝑢(𝑥) and 𝑣(𝑥)∈𝑊₂[0,+∞), then there exists 𝑀₁,𝑀₂>0, such that |𝑢(𝑥)|≤𝑀₁‖𝑢(𝑥)‖_𝑊2 and |𝑣(𝑥)|≤𝑀₂‖𝑣(𝑥)‖_𝑊2.

Proof. It is easy to obtain from the properties in the reproducing kernel space.

By Lemma 13 and Theorem 12, it is easy to obtain the following Lemma 14.

Lemma 14. If 𝑢_𝑛(𝑥)⟶^{‖⋅‖𝑊2}𝑢(𝑥)(𝑛→∞), 𝑣_𝑛(𝑥)⟶^{‖⋅‖𝑊2}𝑣(𝑥)(𝑛→∞), 𝑥_𝑛→𝑦(𝑛→∞), and 𝐹(𝑥,𝑦,𝑧) and 𝐺(𝑥,𝑦,𝑧) satisfy the conditions of Lemma 8, then 𝐹𝑥_𝑛,𝑢_𝑛−1𝑥_𝑛,𝑣_𝑛−1𝑝𝑥_𝑛⟶𝐹𝑦,𝑢𝑦,𝑣𝐺𝑥𝑝𝑦as𝑛⟶∞,_𝑛,𝑣_𝑛−1𝑥_𝑛,𝑢_𝑛−1𝑞𝑥_𝑛⟶𝐺𝑦,𝑣𝑦,𝑢𝑞𝑦as𝑛⟶∞. (56)

Theorem 15. Let {𝑥_𝑖}^∞_𝑖=1 be dense in [0,+∞), and 𝐹(𝑥,𝑦,𝑧) and 𝐺(𝑥,𝑦,𝑧) satisfy the conditions of Lemma 8, then the n-term approximate solutions 𝑢_𝑛(𝑥) and 𝑣_𝑛(𝑥) in (35) converge to the exact solution u(𝑥) and 𝑣(𝑥) of (2), respectively, and 𝑢(𝑥)=∑^∞_𝑖=1𝐴_𝑖𝜓_𝑖(𝑥), 𝑣(𝑥)=∑^∞_𝑖=1𝐵_𝑖𝜓_𝑖(𝑥), where 𝐴_𝑖 and 𝐵_𝑖 are given, respectively, by (36), and (37).

Proof. (1) Firstly, we will prove the convergence of 𝑢_𝑛(𝑥),𝑣_𝑛(𝑥).
By (35), we infer that 𝑢_𝑛+1(𝑥)=𝑢_𝑛(𝑥)+𝐴_𝑛𝜓_𝑛(𝑥),𝑣_𝑛+1(𝑥)=𝑣_𝑛(𝑥)+𝐵_𝑛𝜓_𝑛(𝑥). (57)(58) From the orthogonality of {𝜓_𝑖(𝑥)}^∞_𝑖=1, it follows that 𝑢_𝑛+1²_𝑊2=𝑢_𝑛²_𝑊2+𝐴_𝑛²=𝑢_𝑛−1²_𝑊2+𝐴_𝑛−1²+𝐴_𝑛²⋮=𝑢₁²_𝑊2+^𝑛_𝑖=1𝐴_𝑖²,𝑣_𝑛+1²_𝑊2=𝑣_𝑛²_𝑊2+𝐵_𝑛²=𝑣_𝑛−1²_𝑊2+𝐵_𝑛−1²+𝐵_𝑛²⋮=𝑣₁²_𝑊2+^𝑛_𝑖=1𝐵_𝑖². (59)(60) From (59) and (60), we know sequence ‖𝑢_𝑛‖_𝑊2 and ‖𝑣_𝑛‖_𝑊2 is increasing. By Theorem 12, ‖𝑢_𝑛‖_𝑊2 and ‖𝑣_𝑛‖_𝑊2 are bounded; hence ‖𝑢_𝑛‖_𝑊2 and ‖𝑣_𝑛‖_𝑊2 are convergent such that ^∞_𝑖=1𝐴_𝑖²<∞,_∞𝑖=1𝐵_𝑖²<∞. (61) This implies that 𝐴_𝑖=^𝑖_𝑘=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢_𝑘𝑥_𝑘,𝑣_𝑘𝑝𝑥_𝑘∈𝑙²(𝑖=1,2,…),𝐵_𝑖=^𝑖_𝑘=1𝛽_𝑖𝑘𝐺𝑥_𝑘,𝑣_𝑘𝑥_𝑘,𝑢_𝑘𝑞𝑥_𝑘∈𝑙²(𝑖=1,2,…). (62)(63) Without loss of generality, assume 𝑚>𝑛; we have 𝑢_𝑚(𝑥)−𝑢_𝑛(𝑥)²_𝑊2=𝑢_𝑚(𝑥)−𝑢_𝑚−1(𝑥)+𝑢_𝑚−1(𝑥)−𝑢_𝑚−2(𝑥)+⋅⋅⋅+𝑢_𝑛+1(𝑥)−𝑢_𝑛(𝑥)²_𝑊2≤𝑢_𝑚(𝑥)−𝑢_𝑚−1(𝑥)²_𝑊2+⋯+𝑢_𝑛+1(𝑥)−𝑢_𝑛(𝑥)²_𝑊2=^𝑚_𝑖=𝑛+1𝐴_𝑖²(),𝑣⟶0,𝑛⟶∞_𝑚(𝑥)−𝑣_𝑛(𝑥)²_𝑊2=𝑣_𝑚(𝑥)−𝑣_𝑚−1(𝑥)+𝑣_𝑚−1(𝑥)−𝑣_𝑚−2(𝑥)+⋯+𝑣_𝑛+1(𝑥)−𝑣_𝑛(𝑥)²_𝑊2≤𝑣_𝑚(𝑥)−𝑣_𝑚−1(𝑥)²_𝑊2+…+𝑣_𝑛+1(𝑥)−𝑣_𝑛(𝑥)²_𝑊2=^𝑚_𝑖=𝑛+1𝐵_i²⟶0,(𝑛⟶∞). (64) Considering the completeness of 𝑊₂[0,+∞), there exist 𝑢(𝑥) and 𝑣(𝑥) in 𝑊₂[0,+∞) such that 𝑢_𝑛(𝑥)⟶^‖⋅‖𝑢(𝑥)𝑣as𝑛⟶∞,_𝑛(𝑥)⟶^‖⋅‖𝑣(𝑥)as𝑛⟶∞. (65)
(2) Secondly, we will prove that 𝑢(𝑥) and 𝑣(𝑥) are the solutions of (2).
By Lemma 14 and the proof of (1), we may know that 𝑢_𝑛(𝑥), and 𝑣_𝑛(𝑥) respectively, converge uniformly to 𝑢(𝑥) and 𝑣(𝑥)(𝑛→∞). Taking limits in (35), we have 𝑢(𝑥)=^∞_𝑖=1𝐴_𝑖𝜓_𝑖(𝑥),𝑣(𝑥)=^∞_𝑖=1𝐵_𝑖𝜓_𝑖(𝑥). (66) Since (ℒ𝑢)𝑥_𝑗=^∞_𝑖=1𝐴_𝑖ℒ𝜓_𝑖,𝜑_𝑗=^∞_𝑖=1𝐴_𝑖𝜓_𝑖,ℒ^∗𝜑_𝑗=^∞_𝑖=1𝐴_𝑖𝜓_𝑖,𝜓_𝑗, (67) in the same way, we have (ℒ𝑣)𝑥_𝑗=^∞_𝑖=1𝐵_𝑖𝜓_𝑖,𝜓_𝑗, (68) it follows that ^𝑛_𝑗=1𝛽_𝑛𝑗(ℒ𝑢)𝑥_𝑗=^∞_𝑖=1𝐴_𝑖𝜓_𝑖,^𝑛_𝑗=1𝛽_𝑛𝑗𝜓_𝑗=^∞_𝑖=1𝐴_𝑖𝜓_𝑖,𝜓_𝑛=𝐴_𝑛,_𝑛𝑗=1𝛽_𝑛𝑗(ℒ𝑣)𝑥_𝑗=^∞_𝑖=1𝐵_𝑖𝜓_𝑖,^𝑛_𝑗=1𝛽_𝑛𝑗𝜓_𝑗=^∞_𝑖=1𝐵_𝑖𝜓_𝑖,𝜓_𝑛=𝐵_𝑛. (69) If 𝑛=1, then (ℒ𝑢)𝑥₁=𝐹𝑥₁,𝑢₁𝑥₁,𝑣₁𝑝𝑥₁,(ℒ𝑣)𝑥₁=𝐺𝑥₁,𝑣₁𝑥₁,𝑢₁𝑞𝑥₁. (70)(71)
If 𝑛=2, then 𝛽₂₁(ℒ𝑢)𝑥₁+𝛽₂₂(ℒ𝑢)𝑥₂=𝛽₂₁𝐹𝑥₁,𝑢₁𝑥₁,𝑣₁𝑝𝑥₁+𝛽₂₂𝐹𝑥₂,𝑢₂𝑥₂,𝑣₂𝑝𝑥₂,𝛽₂₁(ℒ𝑣)𝑥₁+𝛽₂₂(ℒ𝑣)𝑥₂=𝛽₂₁𝐺𝑥₁,𝑣₁𝑥₁,𝑢₁𝑞𝑥₁+𝛽₂₂𝐺𝑥₂,𝑣₂𝑥₂,𝑢₂𝑞𝑥₂. (72)
From (70) and (71), it is clear that (ℒ𝑢)𝑥₂=𝐹𝑥₂,𝑢₂𝑥₂,𝑣₂𝑝𝑥₂,(ℒ𝑣)𝑥₂=𝐺𝑥₂,𝑣₂𝑥₂,𝑢₂𝑞𝑥₂. (73) Furthermore, it is easy to see by induction that (ℒ𝑢)𝑥_𝑗=𝐹𝑥_𝑗,𝑢_𝑗𝑥_𝑗,𝑣_𝑗𝑝𝑥_𝑗(ℒ,𝑗=1,2,…,𝑣)𝑥_𝑗=𝐺𝑥_𝑗,𝑣_𝑗𝑥_𝑗,𝑢_𝑗𝑞𝑥_𝑗,𝑗=1,2,…. (74)(75) Since {𝑥_𝑖}^∞_𝑖=1 is dense in [0,+∞), for any 𝑦∈[0,+∞), there exists subsequence {𝑥_𝑛𝑗} such that 𝑥_𝑛𝑗⟶𝑦as𝑗⟶∞. (76) Hence, let 𝑗→∞ in (74), and (75); by Lemma 14 and the continuity of 𝐹(𝑥,𝑢(𝑥),𝑣(𝑝𝑥)) and 𝐺(𝑥,𝑣(𝑥),𝑢(𝑞𝑥)), we have (ℒ𝑢)𝑦=𝐹𝑦,𝑢𝑦,𝑣,(ℒ𝑝𝑦𝑣)𝑦=𝐺𝑦,𝑣𝑦,𝑢𝑞𝑦. (77)(78) That is, 𝑢(𝑥) and 𝑣(𝑥) are the solutions of (2) and 𝑢(𝑥)=^∞_𝑖=1𝐴_𝑖𝜓_𝑖(𝑥),𝑣(𝑥)=^∞_𝑖=1𝐵_𝑖𝜓_𝑖(𝑥), (79)(80) where 𝐴_𝑖=∑^𝑖_𝑘=1𝛽_𝑖𝑘𝐹(𝑥_𝑘,𝑢_𝑘(𝑥_𝑘),𝑣_𝑘(𝑝𝑥_𝑘)) and 𝐵_𝑖=∑^𝑖_𝑘=1𝛽_𝑖𝑘𝐺(𝑥_𝑘,𝑣_𝑘(𝑥_𝑘),𝑢_𝑘(𝑞𝑥_𝑘)).

In the proof of the convergence in Theorem 15 we only use ‖𝑢_𝑛‖≤𝐶, ‖𝑣_𝑛‖≤𝐷; thus we obtain the following corollary.

Corollary 16. Suppose ‖𝑢_𝑛‖ and ‖𝑣_𝑛‖ are bounded; then the iterative sequence (35) is convergent to the exact solution of (2).

Theorem 17. Assume 𝑢(𝑥) and 𝑣(𝑥) are the solutions of (2), 𝑟¹_𝑛=‖𝑢(𝑥)−𝑢_𝑛(𝑥)‖_𝑊2 and 𝑟²_𝑛=‖𝑣(𝑥)−𝑣_𝑛(𝑥)‖_𝑊2 are the approximate errors of 𝑢_𝑛(𝑥),𝑣_𝑛(𝑥), where 𝑢_𝑛(𝑥) and 𝑣_𝑛(𝑥) are given by (35). Then the errors 𝑟¹_𝑛, 𝑟²_𝑛 are monotone decreasing in the sense of ‖⋅‖_𝑊2.

Proof. From (35), and (79), it follows that 𝑟¹_𝑛(𝑥)²_𝑊2=^∞_𝑖=𝑛+1𝐴_𝑖𝜓_𝑖(𝑥)²_𝑊2=^∞_𝑖=𝑛+1𝐴_𝑖². (81) In the same way, we obtain from (35), (80) 𝑟²_𝑛(𝑥)²_𝑊2=^∞_𝑖=𝑛+1𝐵_𝑖𝜓_𝑖(𝑥)²_𝑊2=^∞_𝑖=𝑛+1𝐵_𝑖². (82) Equations (81) and (82) show that the errors 𝑟¹_𝑛 and 𝑟²_𝑛 are monotone decreasing in the sense of ‖⋅‖_𝑊2.

4. Numerical Examples

In order to demonstrate the efficiency of our algorithm for solving (2), we will present two numerical examples in the reproducing kernel space 𝑊₂[0,+∞). Let 𝑛 be the number of discrete points in [0,+∞). Denote 𝐸^𝑢_𝑛(𝑥)=^def|(𝑢(𝑥)−𝑢_𝑛(𝑥))/𝑢(𝑥)|, 𝐸^𝑣_𝑛(𝑥)=^def|(𝑣(𝑥)−𝑣_𝑛(𝑥))/𝑣(𝑥)|. All computations are performed by the Mathematica 5.0 software package. Results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other.

Example 18. In this example we consider the problem 𝑢^′(𝑥)=31+𝑢(𝑥)²𝑣1+2𝑥cos3𝑥(𝑥),𝑣+𝑓_′(𝑥)=2𝑒^{−𝑣(𝑥)}+4𝑥³𝑢1sin2𝑥(𝑥),𝑢(0)(0)+𝑔=0,𝑣=0, (83) with exact solution 𝑢(𝑥)=𝑥𝑒^−𝑥, 𝑣(𝑥)=sin𝑥, and 𝑓(𝑥)=𝑒^−𝑥−𝑒^−𝑥𝑥−(3/(1+𝑒⁻²𝑥²))−2𝑥cos(sin(𝑥/3)), 𝑔(𝑥)=−2𝑒^−sin𝑥+cos𝑥−4𝑥³sin((1/2)𝑥𝑒^−𝑥/2). Applying the presented method in Section 3, we calculate the approximate solution 𝑢₅₀(𝑥) and 𝑣₅₀(𝑥) in [0,1] as follows.

 Step 1. By the method of the appendix, the corresponding reproducing kernel functions can be obtained.

 Step 2. Choosing a dense subset in [0,1], then we get the orthogonalization coefficients 𝛽_𝑖𝑘.

Step 3. According to (10), we can get the normal orthogonal systems {𝜓_𝑖(𝑥)}⁵⁰_𝑖=1.

Step 4. Selecting the initial value 𝑢₀(𝑥)=𝑣₀(𝑥)=0, we obtain 𝑢₁(𝑥),𝑢₂(𝑥),…,𝑢₅₀(𝑥) and 𝑣₁(𝑥),𝑣₂(𝑥),…,𝑣₅₀(𝑥) by (19) developed in the paper.

The graphs of the superimposed image emerge in Figure 1. At the same time, we have computed the approximate solutions 𝑢_𝑛(𝑥) and 𝑣_𝑛(𝑥)(𝑛=300,900) in [0,3] and also calculated the relative errors 𝐸^𝑢_𝑛 and 𝐸^𝑣_𝑛 in Table 1. The root mean square errors (RMSE) about 𝑢(𝑥) with 𝑢_𝑛 and 𝑣(𝑥) with 𝑣_𝑛 are shown in Table 2.

Table 1 

Relative errors in [0,3] for Example 18.

Table 2 

The RMS errors in [0,3] for Example 18.

(a)

(b)

Figure 1 

The left is the superimposed image of 𝑢(𝑥) with 𝑢50(𝑥) in [0,1]. The right is the superimposed image of 𝑣(𝑥) with 𝑣50(𝑥) in [0,1].

Example 19. Considering equations 𝑢^′(𝑥)(𝑢(𝑥𝑣1=3cos))+2𝑥sin5𝑥(𝑥),𝑣+𝑓_′(𝑥)=2sin(𝑣(𝑥))+4𝑥³𝑢1cos3𝑥(𝑥),𝑢(0)(0)+𝑔=0,𝑣=0. (84) The true solutions are 𝑢(𝑥)=5𝑥, 𝑣(𝑥)=4tan(𝑥), 𝑓(𝑥)=5−3cos(5𝑥)−2𝑥sin(4tan(1/5)), 𝑔(𝑥)=−4𝑥³cos(5𝑥/3)+4sec𝑥²−2sin(4tan𝑥). The numerical results are given in Figure 2 and Tables 3 and 4. The figures and tables illustrate that the method given in the paper is efficient.

Table 3 

Relative errors in [0,4] for Example 19.

Table 4 

The RMS errors in [0,4] for Example 19.

(a)

(b)

Figure 2 

The left is the superimposed image of 𝑢(𝑥) with 𝑢50(𝑥) in [0,1]. The right is the superimposed image of 𝑣(𝑥) with 𝑣50(𝑥) in [0,1].

5. Conclusion

In this paper, RKHSM has been successfully applied to find the solutions of systems of nonlinear IDDEs with proportional delays. The efficiency and accuracy of the proposed decomposition method were demonstrated by two test problems. It is concluded from above tables and figures that the RKHSM is an accurate and efficient method to solve IDDEs with proportional delays. Moreover, the method is also effective for solving some nonlinear initial-boundary value problems and nonlocal boundary value problems.

Appendix

The Reproducing Kernel Space 𝑊₂[0,𝑚]

𝑊₂[0,𝑚] is defined as 𝑊₂[0,𝑚]={𝑢(𝑥)∣𝑢^′ are absolutely continuous real value functions, 𝑢^′′∈𝐿²[0,𝑚], 𝑢(0)=0}. The inner product in 𝑊₂[0,𝑚] is given by 𝑢𝑦,𝑣𝑦_𝑊2=^𝑚₀4𝑢𝑣+5𝑢^′𝑣^′+𝑢^′′𝑣^′′𝑑𝑦, (A.1) where 𝑢,𝑣∈𝑊₂[0,𝑚] and the norm ‖𝑢‖_𝑊2 is denoted by ‖𝑢‖_𝑊2=⟨𝑢,𝑢⟩_𝑊2.

Theorem A.1. The space 𝑊₂[0,𝑚] is a reproducing kernel space; that is, for any 𝑢(𝑦)∈𝑊₂[0,𝑚] and each fixed 𝑥∈[0,𝑚], there exists 𝑅_𝑥(𝑦)∈𝑊₂[0,𝑚], 𝑦∈[0,𝑚], such that ⟨𝑢(𝑦),𝑅_𝑥(𝑦)⟩_{𝑊2[0,𝑚]}=𝑢(𝑥). The reproducing kernel 𝑅_𝑥(𝑦) can be denoted by 𝑅_𝑥𝑦=𝑐₁𝑒^𝑦+𝑐₂𝑒^−𝑦+𝑐₃𝑒^2𝑦+𝑐₄𝑒^−2𝑦𝑑,𝑦≤𝑥,₁𝑒^𝑦+𝑑₂𝑒^−𝑦+𝑑₃𝑒^2𝑦+𝑑₄𝑒^−2𝑦,𝑦>𝑥. (A.2)

Proof. Applying to the integrations by parts for (A.1), we have ⟨𝑢𝑦,𝑅_𝑥𝑦⟩_{𝑊2[0,𝑚]}=^𝑚₀𝑢𝑦4𝑅_𝑥𝑦−5𝑅_𝑥^′′𝑦+𝑅_𝑥⁽⁴⁾𝑦𝑦𝑅𝑑𝑦+5𝑢^′_𝑥𝑦^𝑚₀+𝑢^′𝑦𝑅_𝑥^′′𝑦^𝑚₀−𝑢𝑦𝑅_𝑥⁽³⁾𝑦^𝑚₀. (A.3)
Since 𝑅_𝑥(𝑦)∈𝑊₂[0,𝑚], it follows that 𝑅_𝑥(0)=0. (A.4)
For 𝑢(𝑦)∈𝑊₂[0,𝑚], thus, 𝑢(0)=0.
Suppose that 𝑅_𝑥(𝑦) satisfies the following generalized differential equations: 4𝑅_𝑥𝑦−5𝑅_𝑥^′′𝑦+𝑅_𝑥⁽⁴⁾𝑦=𝛿𝑦−𝑥,5𝑅^′_𝑥(𝑚)−𝑅_𝑥⁽³⁾(𝑚)𝑅=0,_𝑥′′(𝑚)𝑅=0,_𝑥′′(0)=0. (A.5) Then ⟨𝑢(𝑦),𝑅_𝑥(𝑦)⟩_{𝑊2[0,𝑚]}=∫^𝑚₀𝑢(𝑦)𝛿(𝑦−𝑥)𝑑𝑦=𝑢(𝑥). Hence, 𝑅_𝑥(𝑦) is the reproducing kernel of space 𝑊₂[0,𝑚].
In the following, we will get the expression of the reproducing kernel 𝑅_𝑥(𝑦).
The characteristic equation of 𝑅_𝑥(𝑦)−5𝑅_𝑥^′′(𝑦)+𝑅_𝑥⁽⁴⁾(𝑦)=𝛿(𝑦−𝑥) is given by 𝜆⁴−5𝜆²+4=0, and the characteristic roots are 𝜆_1,2=±1, 𝜆_3,4=±2.
We denote 𝑅_𝑥(𝑦) by 𝑅_𝑥𝑦=𝑐₁𝑒^𝑦+𝑐₂𝑒^−𝑦+𝑐₃𝑒^2𝑦+𝑐₄𝑒^−2𝑦𝑑,𝑦≤𝑥,₁𝑒^𝑦+d₂𝑒^−𝑦+𝑑₃𝑒^2𝑦+𝑑₄𝑒^−2𝑦,𝑦>𝑥. (A.6) By the definition of space 𝑊₂[0,𝑚], coefficients 𝑐₁,…,𝑐₄, 𝑑₁,…,𝑑₄ satisfy 𝑅_𝑥^(𝑘)(𝑥+0)=𝑅_𝑥^(𝑘)(),𝑅𝑥−0)(𝑘=0,1,2_𝑥(3)(𝑚)−5𝑅^′_𝑥(𝑚)𝑅=0,_𝑥(3)(𝑥+0)−𝑅_𝑥⁽³⁾()𝑅𝑥−0=1,_𝑥′′(𝑚)𝑅=0,_𝑥′′(0)𝑅=0,_𝑥(0)=0, (A.7) from which, the unknown coefficients of (A.6) can be obtained: 𝑐₁=𝑒^−2𝑥4𝑒^3𝑚−7𝑒^3𝑥−9𝑒^2𝑚+𝑥+7𝑒^6𝑚+𝑥+9𝑒^4𝑚+3𝑥−4𝑒^3𝑚+4𝑥6−7−9𝑒^2𝑚+9𝑒^4𝑚+7𝑒^6𝑚𝑐₂=𝑒^−2𝑥−4𝑒^3𝑚+7𝑒^3𝑥+9𝑒^2𝑚+𝑥−7𝑒^6𝑚+𝑥−9𝑒^4𝑚+3𝑥+4𝑒^3𝑚+4𝑥6−7−9𝑒^2𝑚+9𝑒^4𝑚+7𝑒^6m𝑐₃=𝑒^−2𝑥−9𝑒^4𝑚−7𝑒^6𝑚+7𝑒^4𝑥+8𝑒^3𝑚+𝑥−8𝑒^3(𝑚+𝑥)+9𝑒^2𝑚+4𝑥12−7−9𝑒^2𝑚+9𝑒^4𝑚+7𝑒^6𝑚𝑐₄=−𝑒^−2𝑥−9𝑒^4𝑚−7𝑒^6𝑚+7𝑒^4𝑥+8𝑒^3𝑚+𝑥−8𝑒^3(𝑚+𝑥)+9𝑒^2𝑚+4𝑥12−7−9𝑒^2𝑚+9𝑒^4𝑚+7𝑒^6𝑚𝑑₁=−𝑒^−2𝑥−1+𝑒^2𝑥4𝑒^3𝑚+7𝑒^𝑥−9𝑒^4𝑚+𝑥+4𝑒^3𝑚+2𝑥6−7−9𝑒^2𝑚+9𝑒^4𝑚+7𝑒^6𝑚𝑑₂=𝑒^{2𝑚−2𝑥}−1+𝑒^2𝑥4𝑒^𝑚+7𝑒^4𝑚+𝑥−9𝑒^𝑥+4𝑒^𝑚+2𝑥6−7−9𝑒^2𝑚+9𝑒^4𝑚+7𝑒^6𝑚𝑑₃=𝑒^−2𝑥−1+𝑒^2𝑥7+9𝑒^2𝑚+7𝑒^2𝑥+9𝑒^2(𝑚+𝑥)−8𝑒^3𝑚+𝑥12−7−9𝑒^2𝑚+9𝑒^4𝑚+7𝑒^6𝑚𝑑₄=−𝑒^{3𝑚−2𝑥}−1+𝑒^2𝑥9𝑒^𝑚+7𝑒^3𝑚+9𝑒^𝑚+2𝑥−8𝑒^𝑥+7𝑒^3𝑚+2𝑥12−7−9𝑒^2𝑚+9𝑒^4𝑚+7𝑒^6𝑚. (A.8)

Acknowledgments

This research is supported by the National Natural Science Foundation of China (61071181), the Educational Department Scientific Technology Program of Heilongjiang Province (12531180, 12512133, and 12521148), and the Academic Foundation for Youth of Harbin Normal University (KGB201226).