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

Authors: Changbo He; Xueqin Lv; Jing Niu
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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]:
(1)u′(x)=g(x,u(x),u(px)),u(0)=η,
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:(2)u′(x)=F(x,u(x),v(px)),v′(x)=G(x,v(x),u(qx)),u(0)=0,v(0)=0,
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 Section2, 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)).
LetM 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.
W
2
[
0
,
+
∞
)
=
{
u
(
x
)
∣
u
′ is absolutely continuous real value functions, u′′∈L2[0,+∞),u(0)=0}.
W
2
[
0
,
+
∞
) is a Hilbert space, for u(x),v(x)∈W2[0,+∞); the inner product and norm in W2[0,+∞) are given by
(3)〈u(x),v(x)〉W2=∫0+∞‍(4uv+5u′v′+u′′v′′)dx,∥u∥W2=〈u,u〉W2,
respectively.Theorem 3.
The spaceW2[0,+∞) is a reproducing kernel space that is, for any u(y)∈W2[0,+∞) and each fixed x∈[0,+∞), there exists Rx(y)∈W2[0,+∞), y∈[0,+∞), such that u(x)=〈Rx(y),u(y)〉. And the corresponding reproducing kernel can be represented as follows [26]:
(4)Rx(y)={-112e-2(x+y)(-1+e2y)(1+e2y-2ex+y),y≤x,-112e-2(x+y)(-1+e2x)(1+e2x-2ex+y),y>x.Definition 4.
W
1
[
0
,
+
∞
)
=
{
u
(
x
)
∣
u is absolutely continuous real-valued function, u′∈L2[0,+∞)}.
The inner product and norm inW1[0,+∞) can be defined by
(5)〈u(x),v(x)〉W1=∫0+∞‍(uv+u′v′)dx,∥u∥W1=〈u,u〉W1,
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]:
(6)Qx(y)={12e-x-y(1+e2x),x<y,12e-x-y(1+e2y),x≥y.


## 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 operatorℒ=d/dx; then we can convert (2) into the following form:
(7)ℒu(x)=F(x,u(x),v(px)),ℒv(x)=G(x,v(x),u(qx)),u(0)=v(0)=0,
where x∈[0,+∞), and
(8)F(x,u(x),v(px))≝N(u(x),v(px))+f(x),(9)G(x,v(x),u(qx))≝D(v(x),u(qx))+g(x).
It is clear that ℒ:W2[0,+∞)→W1[0,+∞) is a bounded linear operator. Let φi(x)=Qxi(x), where {xi}i=1∞ is dense in the interval [0,+∞), and ψi(x)=ℒ*φi(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 {ψi(x)}i=1∞,
(10)ψi-(x)=∑k=1i‍βikψk(x),βii>0,i=1,2,…Lemma 5.
Assume that{xi}i=1∞ are dense in [0,+∞); then {ψi(x)}i=1∞ is a complete system in W2[0,+∞) and {ψi(x)}=(d/dy)Rx(y)|y=xi.Proof.
Ones have(11)ψi(x)=〈ψi(y),Rx(y)〉W2=〈ℒ*φi(y),Rx(y)〉W2=〈φi(y),ℒRx(y)〉W1=ℒRx(xi)=ddyRx(y)|y=xi.
Clearly,ψi(x)∈W2[0,+∞).
For anyu(x)∈W2[0,+∞), let 〈u(x),ψi(x)〉=0,  i=1,2,…, which means that
(12)〈u(x),(ℒ*φi(x)〉=〈ℒu(·),φi(·)〉=(ℒu)(xi)=0.
Note that {xi}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 Sequenceu~n(x) and v~n(x)

Next we construct the iterative sequenceu~n(x) and v~n(x), putting
(13)ℒMn(x)=F(x,u~n-1(x),v~n-1(px)),u~n(x)=PnMn(x),ℒNn(x)=G(x,v~n-1(x),u~n-1(qx)),v~n(x)=PnNn(x),
where Mn(x),Nn(x)∈W2[0,+∞) and Pn:W2→Span{ψ-1,ψ-2,…,ψ-n} is a orthogonal projection operator. Then by (13) it followed that:
(14)Mn(x)=∫0x‍F(μ,u~n-1(μ),v~n-1(pμ))dμ,Mn′(x)=F(x,u~n-1(x),v~n-1(px)),Mn′′(x)=ddxF(x,u~n-1(x),v~n-1(px)),(15)Nn(x)=∫0x‍G(μ,v~n-1(μ),u~n-1(qμ))dμ,Nn′(x)=G(x,v~n-1(x),u~n-1(qx)),Nn′′(x)=ddxG(x,v~n-1(x),u~n-1(qx)).Lemma 6.
Let{xi}i=1∞ be dense on [0,+∞); if the solution of (2) is unique, then the solution satisfies the form
(16)Mn(x)=∑i=1∞‍∑k=1i‍βikF(xk,u~n-1(xk),v~n-1(pxk))ψ-i(x),Nn(x)=∑i=1∞‍∑k=1i‍βikG(xk,v~n-1(xk),u~n-1(qxk))ψ-i(x).Proof.
Note that〈u(x),φi(x)〉=u(xi) and {ψ-i(x)}i=1∞ is an orthonormal basis of W2[0,+∞); hence according to Lemma 5 we have
(17)Mn(x)=∑i=1∞‍〈Mn(x),ψ-i(x)〉ψ-i(x)=∑i=1∞‍〈Mn(x),∑k=1i‍βikψk(x)〉ψ-i(x)=∑i=1∞‍∑k=1i‍βik〈Mn(x),ψk(x)〉ψ-i(x)=∑i=1∞‍∑k=1i‍βik〈ℒMn(x),φk(x)〉ψ-i(x)=∑i=1∞‍∑k=1i‍βikF(xk,un-1(xk),vn-1(pxk))ψ-i(x).
In the same manner
(18)Nn(x)=∑i=1∞‍∑k=1i‍βikG(xk,vn-1(xk),un-1(qxk))ψ-i(x).Takeu0(x)=v0(x)=0; define the iterative sequence
(19)un(x)=PnMn(x)=∑i=1n‍∑k=1i‍βikF(xk,un-1(xk),vn-1(pxk))ψ-i(x),vn(x)=PnNn(x)=∑i=1n‍∑k=1i‍βikG(xk,vn-1(xk),un-1(qxk))ψ-i(x).


### 3.2. The Boundedness of Sequenceun(x) and vn(x)

From(20)∂y3Rx(x+0)-∂y3Rx(x-0)=1,
we have
(21)∂y4Rx(y)=gx(y)+(∂y3Rx(x+0)-∂y3Rx(x-0))×δ(x-y)=gx(y)+δ(x-y),
where
(22)gx(y)={gx1,y<x,gx2,y>x,0,y=x,gx1(y)=∂y4(a1ey+a2e-y+a3e2y+a4e-2y); gx2(y)=∂y4(b1ey+b2e-y+b3e2y+b4e-2y); and gx1(y),gx2(y),∂ygx1(y),∂ygx2(y) are bounded functions with respect to x, y, respectively.Lemma 7.
Forx∈[0,+∞),  y,z∈(-∞,+∞),  and  F(x,y,z), G(x,y,z) are continuous bounded functions on [0,+∞), we have that Mn(i)(x) and Nn(i)(x)(i=0,1,2) are bounded.Proof.
By the expression ofMn(x), Mn′(x) and the assumptions, we know that Mn(x) and Mn′(x) are bounded. In the following, we will discuss the boundedness of Mn′′(x).
Since the function which is inC[0,+∞)2 is dense in W2[0,+∞), without loss of generality, we assume that Mn′′(x) is continuous.
Note that(23)4Rx(y)-5∂y2Rx(y)+∂y4Rx(y)=δ(x-y).
One gets
(24)Mn′′(x)=∫0∞‍Mn′′(y)δ(x-y)dy=∫0∞‍Mn′′(y)[4Rx(y)-5∂y2Rx(y)+∂y4Rx(y)]dy=∫0∞‍Mn′′(y)[4Rx(y)-5∂y2Rx(y)]dy+∫0∞‍Mn′′(y)∂y4Rx(y)dy=Mn′(y)[4Rx(y)-5∂y2Rx(y)]∫0∞‍|0∞-∫0+∞‍Mn′(y)[4∂yRx(y)-5∂y3Rx(y)]dy+∫0∞‍Mn′′(y)[gx(y)+δ(x-y)]dy=m(x)+∫0∞‍Mn′′(y)[gx(y)+δ(x-y)]dy=m(x)+∫0∞‍Mn′′(y)gx(y)dy+Mn′′(x),
where m(x)=Mn′(y)[4Rx(y)-5∂y2Rx(y)]|0∞-∫0∞‍Mn′(y)[4∂yRx(y)-5∂y3Rx(y)]dy. Thus, we have
(25)0=m(x)-∫0∞‍Mn′′(y)gx(y)dy=m(x)-∫0x‍Mn′′(y)gx1(y)dy-∫x∞‍Mn′′(y)gx2(y)dy=m(x)-Mn′(y)gx1(y)∫0x‍|0x+∫0x‍Mn′(y)∂ygx1(y)dy-Mn′(y)gx2(y)∫0x‍|xM+∫x∞‍Mn′(y)∂ygx2(y)dy=m(x)-Mn′(x)(gx1(x)-gx2(x))+(Mn′(0)gx1(0)-Mn′(M)gx2(M))+∫0x‍Mn′(y)∂ygx1(y)dy+∫x∞‍Mn′(y)∂ygx2(y)dy.
So
(26)Mn′(x)(gx1(x)-gx2(x))=m(x)+(Mn′(0)gx1(0)-Mn′(M)gx2(M))+∫0x‍Mn′(y)∂ygx1(y)+∫x∞‍Mn′(y)∂ygx2(y)dy.
Furthermore, we see that
(27)Mn′′(x)=1gx1(x)-gx2(x)×{∂x[∫x∞‍m(x)+(Mn′(0)gx1(0)-Mn′(M)gx2(M))+∫0x‍Mn′(y)∂ygx1(y)+∫x∞‍Mn′(y)∂ygx2(y)dy]-Mn′(x)∂x(gx1(x)-gx2(x))∫x∞‍}.
In view of the expression of Rx(y), we know that ∂ymRx(y) are bounded for m=1,2,3. It follows that Mn′′(x) is bounded from the boundedness of gx1(y), gx2(y), ∂ygx1(y), ∂ygx2(y), and Mn′(x). In the same way, Nn(x), Nn′(x), and Nn′′(x) are bounded.Lemma 8.
Assume thatF(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
(28)∥un∥W2≤C,∥vn∥W2≤D,(C,Dareconstants).Proof.
(29)
∥
M
n
∥
W
2
2
=
〈
M
n
,
M
n
〉
=
∫
0
∞
‍
(
4
M
n
2
+
5
M
n
′
2
+
M
n
′′
2
)
d
x
.
Note that (14) and the assumptions, by Lemma 7, |Mn(i)(x)|≤Ci,  i=0,1,2, thus ∥Mn∥≤C, where C=max{C0,C1,C2}, from (19), we have ∥un∥W2≤∥Mn∥W2≤C. In the same way, we obtain ∥vn∥W2≤D.


### 3.3. Construction the of Another Iterative Sequenceun(x) and vn(x)

Theorem 9.
Let{xi}i=1∞ be dense on [0,+∞); if the solution of (2) exists and unique, then the solution satisfies the form
(30)u(x)=∑i=1∞‍∑k=1i‍βikF(xk,u(xk),v(pxk))ψ-i(x),v(x)=∑i=1∞‍∑k=1i‍βikG(xk,v(xk),u(qxk))ψ-i(x).Proof.
Note that〈u(x),φi(x)〉=u(xi) and {ψ-i(x)}i=1∞ is an orthonormal basis of W2[0,+∞); hence we have
(31)u(x)=∑i=1∞‍〈u(x),ψ-i(x)〉ψ-i(x)=∑i=1∞‍〈u(x),∑k=1i‍βikψk(x)〉ψ-i(x)=∑i=1∞‍∑k=1i‍βik〈u(x),ψk(x)〉ψ-i(x)=∑i=1∞‍∑k=1i‍βik〈ℒu(x),φk(x)〉ψ-i(x)=∑i=1∞‍∑k=1i‍βikF(xk,u(xk),v(pxk))ψ-i(x).
In the same manner
(32)v(x)=∑i=1∞‍∑k=1i‍βikG(xk,v(xk),u(qxk))ψ-i(x).In the following, a new method of solving (2) is presented. Equations (31) and (32) can be denoted by
(33)u(x)=∑i=1∞‍Aiψi-(x),(34)v(x)=∑i=1∞‍Biψi-(x),
where Ai=∑k=1iβikF(xk,u(xk),v(pxk)) and Bi=∑k=1iβikG(xk,v(xk),u(qxk)). In fact, Ai and Bi are unknown; we will approximate Ai and Bi by using the known A-i and B-i.We takeu1(x)=0, v1(x)=0 and define the following iterative sequence
(35)un(x)=∑i=1n-1‍A-iψi-(x),vn(x)=∑i=1n-1‍B-iψi-(x),
where
(36)A-1=β11F(x1,u1(x1),v1(px1))A-2=∑k=12‍β2kF(xk,uk(xk),vk(pxk))⋮A-n-1=∑k=1n-1‍βnkF(xk,uk(xk),vk(pxk)),(37)B-1=β11G(x1,v1(x1),u1(qx1))B-2=∑k=12‍β2kG(xk,vk(xk),uk(qxk))⋮B-n-1=∑k=1n-1‍βnkG(xk,vk(xk),uk(qxk)).
Next, lemmas are given.Lemma 10.
The following iterative sequences(38)un(x)=∑i=1n-1‍βikF(xk,uk(xk),vk(pxk))ψi-(x),vn(x)=∑i=1n-1‍βikG(xk,vk(xk),uk(qxk))ψi-(x)
satisfy
(39)ℒun(xj)=F(xj,uj(xj),vj(pxj)),ℒvn(xj)=G(xj,vj(xj),uj(qxj)),j≤n-1,
respectively.Proof.
Ifj=1,
(40)ℒun(x1)=∑i=1n-1‍βikF(xk,uk(xk),vk(pxk))ℒψ-i(x1)=∑i=1n-1‍βikF(xk,uk(xk),vk(pxk))×〈ℒψ-i(x),φ1(x)〉=∑i=1n-1‍βikF(xk,uk(xk),vk(pxk))×〈ψ-i(x),ψ1(x)〉,β11ℒun(x1)=∑i=1n-1‍βikF(xk,uk(xk),vk(pxk))×〈ψ-i(x),ψ-1(x)〉=β11F(x1,u1(x1),v1(px1)),
so,
(41)ℒun(x1)=F(x1,u1(x1),v1(px1)).
If  j=2,
(42)ℒun(x2)=∑i=1n-1‍βikF(xk,uk(xk),vk(pxk))ℒψ-i(x2)=∑i=1n-1‍βikF(xk,uk(xk),vk(pxk))〈ψ-i(x),ψ2(x)〉,β21× (40) +β22× (42),  we have
(43)β21ℒun(x1)+β22ℒun(x2)=∑i=1n-1‍βikF(xk,uk(xk),vk(pxk))〈ψ-i(x),ψ-2(x)〉=β21F(x1,u1(x1),v1(px1))+β22F(x2,u2(x2),v2(px2)),
by (41), ℒun(x2)=F(x2,u2(x2),v2(px2)). In the same way, we have
(44)ℒun(xj)=F(xj,uj(xj),vj(pxj)),j≤n-1.
Similarly,
(45)ℒvn(xj)=F(xj,vj(xj),uj(qxj)),j≤n-1.Theorem 11.
The iterative form(46)u~n(x)=∑i=1n‍∑k=1i‍βikF(xk,u~n-1(xk),v~n-1(pxk))ψ-i(x),v~n(x)=∑i=1n‍∑k=1i‍βikG(xk,v~n-1(xk),u~n-1(qxk))ψ-i(x),u~1(0)=v~1(0)=0
and the iterative form
(47)un(x)=∑i=1n-1‍∑k=1i‍βikF(xk,uk(xk),vk(pxk))ψ-i(x),vn(x)=∑i=1n-1‍∑k=1i‍βikG(xk,vk(xk),uk(qxk))ψ-i(x),u1(0)=v1(0)=0
are the same.Proof.
In Lemma10, let n-1=k; then
(48)ℒuk+1(xk)=F(xk,uk(xk),vk(pxk)),
but
(49)ℒun(xk)=F(xk,uk(xk),vk(pxk)),
thus
(50)un(x)=∑i=1n-1‍∑k=1i‍βikLuk+1(xk)ψ-i(x)=∑i=1n-1‍∑k=1i‍βikℒun(xk)ψ-i(x)=∑i=1n-1‍∑k=1i‍βik〈ℒun(x),φk(x)〉ψ-i(x)=∑i=1n-1‍∑k=1i‍βik〈un(x),ψk(x)〉ψ-i(x)=∑i=1n-1‍〈un(x),ψ-i(x)〉ψ-i(x).
Equation (46) can be written as
(51)ℒMn(x)=F(x,u~n-1(x),v~n-1(px)),u~n(x)=PnMn(x).
In fact,  Mn(x)=∑i=1∞‍∑k=1i‍βikF(xk,u~n-1(xk),v~n-1(pxk))ψ-i(x); then
(52)ℒu~n(xj)=ℒPnMn(xj)=〈ℒPnMn(x),φj(x)〉=〈PnMn(x),ψj(x)〉=〈Mn(x),ψj(x)〉=〈ℒMn(x),φj(x)〉=ℒMn(xj),
by (51) and (52), we have
(53)u~n(x)=∑i=1n-1‍∑k=1i‍βikF(xk,u~n-1(xk),v~n-1(pxk))ψ-i(x)=∑i=1n-1‍∑k=1i‍βikℒMn(xk)ψ-i(x)=∑i=1n-1‍∑k=1i‍βikℒu~n(xk)ψ-i(x)=∑i=1n-1‍∑k=1i‍βik〈ℒu~n(x),φk(x)〉ψ-i(x)=∑i=1n-1‍∑k=1i‍βik〈u~n(x),ψk(x)〉ψ-i(x)=∑i=1n-1‍〈u~n(x),ψ-i(x)〉ψ-i(x).
Equation (53) is the same as (50). We may prove for v~n(x) similarly.So, by Theorem11 and Lemma 8, we have the following Theorem.Theorem 12.
Under the conditions of Lemma8,
(54)un(x)=∑i=1n-1‍βikF(xk,uk(xk),vk(pxk))ψi-(x),(55)vn(x)=∑i=1n-1‍βikG(xk,vk(xk),uk(qxk))ψi-(x)
satisfy ∥un(x)∥≤C,  ∥vn(x)∥≤D.Lemma 13.
Ifu(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 Lemma13 and Theorem 12, it is easy to obtain the following Lemma 14.Lemma 14.
Ifun(x)⟶∥·∥W2u-(x)(n→∞), vn(x)⟶∥·∥W2v-(x)(n→∞), xn→y(n→∞), and F(x,y,z) and G(x,y,z) satisfy the conditions of Lemma 8, then
(56)F(xn,un-1(xn),vn-1(pxn))⟶F(y,u-(y),v-(py))asn⟶∞,G(xn,vn-1(xn),un-1(qxn))⟶G(y,v-(y),u-(qy))asn⟶∞.Theorem 15.
Let{xi}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 un(x) and vn(x) in (35) converge to the exact solution u(x) and v(x) of (2), respectively, and u(x)=∑i=1∞A-iψ-i(x), v(x)=∑i=1∞B-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 un(x),vn(x).
By (35), we infer that
(57)un+1(x)=un(x)+A-nψ-n(x),(58)vn+1(x)=vn(x)+B-nψ-n(x).
From the orthogonality of {ψ-i(x)}i=1∞, it follows that
(59)∥un+1∥W22=∥un∥W22+(A-n)2=∥un-1∥W22+(A-n-1)2+(A-n)2⋮=∥u1∥W22+∑i=1n‍(A-i)2,(60)∥vn+1∥W22=∥vn∥W22+(B-n)2=∥vn-1∥W22+(B-n-1)2+(B-n)2⋮=∥v1∥W22+∑i=1n‍(B-i)2.
From (59) and (60), we know sequence ∥un∥W2 and ∥vn∥W2 is increasing. By Theorem 12, ∥un∥W2 and ∥vn∥W2 are bounded; hence ∥un∥W2 and ∥vn∥W2 are convergent such that
(61)∑i=1∞‍(A-i)2<∞,∑i=1∞‍(B-i)2<∞.
This implies that
(62)A-i=∑k=1i‍βikF(xk,uk(xk),vk(pxk))∈l2(i=1,2,…),(63)B-i=∑k=1i‍βikG(xk,vk(xk),uk(qxk))∈l2(i=1,2,…).
Without loss of generality, assume m>n; we have
(64)∥um(x)-un(x)∥W22=∥um(x)-um-1(x)+um-1(x)-um-2(x)+···+un+1(x)-un(x)∥W22≤∥um(x)-um-1(x)∥W22+⋯+∥un+1(x)-un(x)∥W22=∑i=n+1m‍(A-i)2⟶0,(n⟶∞),∥vm(x)-vn(x)∥W22=∥vm(x)-vm-1(x)+vm-1(x)-vm-2(x)+⋯+vn+1(x)-vn(x)∥W22≤∥vm(x)-vm-1(x)∥W22+…+∥vn+1(x)-vn(x)∥W22=∑i=n+1m‍(B-i)2⟶0,(n⟶∞).
Considering the completeness of W2[0,+∞), there exist u-(x) and v-(x) in W2[0,+∞) such that
(65)un(x)⟶∥·∥u-(x)asn⟶∞,vn(x)⟶∥·∥v-(x)asn⟶∞.
(
2
) Secondly, we will prove that u-(x) and v-(x) are the solutions of (2).
By Lemma14 and the proof of (1), we may know that un(x), and vn(x) respectively, converge uniformly to u-(x) and v-(x)(n→∞). Taking limits in (35), we have
(66)u-(x)=∑i=1∞‍A-iψ-i(x),v-(x)=∑i=1∞‍B-iψ-i(x).
Since
(67)(ℒu-)(xj)=∑i=1∞‍A-i〈ℒψ-i,φj〉=∑i=1∞‍A-i〈ψ-i,ℒ*φj〉=∑i=1∞‍A-i〈ψ-i,ψj〉,
in the same way, we have
(68)(ℒv-)(xj)=∑i=1∞‍B-i〈ψ-i,ψj〉,
it follows that
(69)∑j=1n‍βnj(ℒu-)(xj)=∑i=1∞‍A-i〈ψ-i,∑j=1n‍βnjψj〉=∑i=1∞‍A-i〈ψ-i,ψ-n〉=A-n,∑j=1nβnj(ℒv-)(xj)‍=∑i=1∞‍B-i〈ψ-i,∑j=1n‍βnjψj〉=∑i=1∞‍B-i〈ψ-i,ψ-n〉=B-n.
If  n=1, then
(70)(ℒu-)(x1)=F(x1,u1(x1),v1(px1)),(71)(ℒv-)(x1)=G(x1,v1(x1),u1(qx1)).
Ifn=2, then
(72)β21(ℒu-)(x1)+β22(ℒu-)(x2)=β21F(x1,u1(x1),v1(px1))+β22F(x2,u2(x2),v2(px2)),β21(ℒv-)(x1)+β22(ℒv-)(x2)=β21G(x1,v1(x1),u1(qx1))+β22G(x2,v2(x2),u2(qx2)).
From (70) and (71), it is clear that
(73)(ℒu-)(x2)=F(x2,u2(x2),v2(px2)),(ℒv-)(x2)=G(x2,v2(x2),u2(qx2)).
Furthermore, it is easy to see by induction that
(74)(ℒu-)(xj)=F(xj,uj(xj),vj(pxj)),j=1,2,…,(75)(ℒv-)(xj)=G(xj,vj(xj),uj(qxj)),j=1,2,….
Since {xi}i=1∞ is dense in [0,+∞), for any y∈[0,+∞), there exists subsequence {xnj} such that
(76)xnj⟶yasj⟶∞.
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
(77)(ℒu-)(y)=F(y,u-(y),v-(py)),(78)(ℒv-)(y)=G(y,v-(y),u-(qy)).
That is, u-(x) and v-(x) are the solutions of (2) and
(79)u-(x)=∑i=1∞‍A-iψ-i(x),(80)v-(x)=∑i=1∞‍B-iψ-i(x),
where A-i=∑k=1iβikF(xk,u-k(xk),v-k(pxk))  and  B-i=∑k=1iβikG(xk,v-k(xk),u-k(qxk)).In the proof of the convergence in Theorem15 we only use ∥un∥≤C, ∥vn∥≤D; thus we obtain the following corollary.Corollary 16.
Suppose∥un∥ and ∥vn∥  are bounded; then the iterative sequence (35) is convergent to the exact solution of (2).Theorem 17.
Assumeu(x) and v(x) are the solutions of (2), rn1=∥u(x)-un(x)∥W2 and rn2=∥v(x)-vn(x)∥W2 are the approximate errors of un(x),vn(x), where un(x) and vn(x) are given by (35). Then the errors  rn1, rn2 are monotone decreasing in the sense of ∥·∥W2.Proof.
From (35), and (79), it follows that
(81)∥rn1(x)∥W22=∥∑i=n+1∞‍A-iψ-i(x)∥W22=∑i=n+1∞‍(A-i)2.
In the same way, we obtain from (35), (80)
(82)∥rn2(x)∥W22=∥∑i=n+1∞‍B-iψ-i(x)∥W22=∑i=n+1∞‍(B-i)2.
Equations (81) and (82) show that the errors rn1 and rn2 are monotone decreasing in the sense of ∥·∥W2.


## 3.1.  Construction the of Iterative Sequenceu~n(x) and v~n(x)

Next we construct the iterative sequenceu~n(x) and v~n(x), putting
(13)ℒMn(x)=F(x,u~n-1(x),v~n-1(px)),u~n(x)=PnMn(x),ℒNn(x)=G(x,v~n-1(x),u~n-1(qx)),v~n(x)=PnNn(x),
where Mn(x),Nn(x)∈W2[0,+∞) and Pn:W2→Span{ψ-1,ψ-2,…,ψ-n} is a orthogonal projection operator. Then by (13) it followed that:
(14)Mn(x)=∫0x‍F(μ,u~n-1(μ),v~n-1(pμ))dμ,Mn′(x)=F(x,u~n-1(x),v~n-1(px)),Mn′′(x)=ddxF(x,u~n-1(x),v~n-1(px)),(15)Nn(x)=∫0x‍G(μ,v~n-1(μ),u~n-1(qμ))dμ,Nn′(x)=G(x,v~n-1(x),u~n-1(qx)),Nn′′(x)=ddxG(x,v~n-1(x),u~n-1(qx)).Lemma 6.
Let{xi}i=1∞ be dense on [0,+∞); if the solution of (2) is unique, then the solution satisfies the form
(16)Mn(x)=∑i=1∞‍∑k=1i‍βikF(xk,u~n-1(xk),v~n-1(pxk))ψ-i(x),Nn(x)=∑i=1∞‍∑k=1i‍βikG(xk,v~n-1(xk),u~n-1(qxk))ψ-i(x).Proof.
Note that〈u(x),φi(x)〉=u(xi) and {ψ-i(x)}i=1∞ is an orthonormal basis of W2[0,+∞); hence according to Lemma 5 we have
(17)Mn(x)=∑i=1∞‍〈Mn(x),ψ-i(x)〉ψ-i(x)=∑i=1∞‍〈Mn(x),∑k=1i‍βikψk(x)〉ψ-i(x)=∑i=1∞‍∑k=1i‍βik〈Mn(x),ψk(x)〉ψ-i(x)=∑i=1∞‍∑k=1i‍βik〈ℒMn(x),φk(x)〉ψ-i(x)=∑i=1∞‍∑k=1i‍βikF(xk,un-1(xk),vn-1(pxk))ψ-i(x).
In the same manner
(18)Nn(x)=∑i=1∞‍∑k=1i‍βikG(xk,vn-1(xk),un-1(qxk))ψ-i(x).Takeu0(x)=v0(x)=0; define the iterative sequence
(19)un(x)=PnMn(x)=∑i=1n‍∑k=1i‍βikF(xk,un-1(xk),vn-1(pxk))ψ-i(x),vn(x)=PnNn(x)=∑i=1n‍∑k=1i‍βikG(xk,vn-1(xk),un-1(qxk))ψ-i(x).


## 3.2. The Boundedness of Sequenceun(x) and vn(x)

From(20)∂y3Rx(x+0)-∂y3Rx(x-0)=1,
we have
(21)∂y4Rx(y)=gx(y)+(∂y3Rx(x+0)-∂y3Rx(x-0))×δ(x-y)=gx(y)+δ(x-y),
where
(22)gx(y)={gx1,y<x,gx2,y>x,0,y=x,gx1(y)=∂y4(a1ey+a2e-y+a3e2y+a4e-2y); gx2(y)=∂y4(b1ey+b2e-y+b3e2y+b4e-2y); and gx1(y),gx2(y),∂ygx1(y),∂ygx2(y) are bounded functions with respect to x, y, respectively.Lemma 7.
Forx∈[0,+∞),  y,z∈(-∞,+∞),  and  F(x,y,z), G(x,y,z) are continuous bounded functions on [0,+∞), we have that Mn(i)(x) and Nn(i)(x)(i=0,1,2) are bounded.Proof.
By the expression ofMn(x), Mn′(x) and the assumptions, we know that Mn(x) and Mn′(x) are bounded. In the following, we will discuss the boundedness of Mn′′(x).
Since the function which is inC[0,+∞)2 is dense in W2[0,+∞), without loss of generality, we assume that Mn′′(x) is continuous.
Note that(23)4Rx(y)-5∂y2Rx(y)+∂y4Rx(y)=δ(x-y).
One gets
(24)Mn′′(x)=∫0∞‍Mn′′(y)δ(x-y)dy=∫0∞‍Mn′′(y)[4Rx(y)-5∂y2Rx(y)+∂y4Rx(y)]dy=∫0∞‍Mn′′(y)[4Rx(y)-5∂y2Rx(y)]dy+∫0∞‍Mn′′(y)∂y4Rx(y)dy=Mn′(y)[4Rx(y)-5∂y2Rx(y)]∫0∞‍|0∞-∫0+∞‍Mn′(y)[4∂yRx(y)-5∂y3Rx(y)]dy+∫0∞‍Mn′′(y)[gx(y)+δ(x-y)]dy=m(x)+∫0∞‍Mn′′(y)[gx(y)+δ(x-y)]dy=m(x)+∫0∞‍Mn′′(y)gx(y)dy+Mn′′(x),
where m(x)=Mn′(y)[4Rx(y)-5∂y2Rx(y)]|0∞-∫0∞‍Mn′(y)[4∂yRx(y)-5∂y3Rx(y)]dy. Thus, we have
(25)0=m(x)-∫0∞‍Mn′′(y)gx(y)dy=m(x)-∫0x‍Mn′′(y)gx1(y)dy-∫x∞‍Mn′′(y)gx2(y)dy=m(x)-Mn′(y)gx1(y)∫0x‍|0x+∫0x‍Mn′(y)∂ygx1(y)dy-Mn′(y)gx2(y)∫0x‍|xM+∫x∞‍Mn′(y)∂ygx2(y)dy=m(x)-Mn′(x)(gx1(x)-gx2(x))+(Mn′(0)gx1(0)-Mn′(M)gx2(M))+∫0x‍Mn′(y)∂ygx1(y)dy+∫x∞‍Mn′(y)∂ygx2(y)dy.
So
(26)Mn′(x)(gx1(x)-gx2(x))=m(x)+(Mn′(0)gx1(0)-Mn′(M)gx2(M))+∫0x‍Mn′(y)∂ygx1(y)+∫x∞‍Mn′(y)∂ygx2(y)dy.
Furthermore, we see that
(27)Mn′′(x)=1gx1(x)-gx2(x)×{∂x[∫x∞‍m(x)+(Mn′(0)gx1(0)-Mn′(M)gx2(M))+∫0x‍Mn′(y)∂ygx1(y)+∫x∞‍Mn′(y)∂ygx2(y)dy]-Mn′(x)∂x(gx1(x)-gx2(x))∫x∞‍}.
In view of the expression of Rx(y), we know that ∂ymRx(y) are bounded for m=1,2,3. It follows that Mn′′(x) is bounded from the boundedness of gx1(y), gx2(y), ∂ygx1(y), ∂ygx2(y), and Mn′(x). In the same way, Nn(x), Nn′(x), and Nn′′(x) are bounded.Lemma 8.
Assume thatF(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
(28)∥un∥W2≤C,∥vn∥W2≤D,(C,Dareconstants).Proof.
(29)
∥
M
n
∥
W
2
2
=
〈
M
n
,
M
n
〉
=
∫
0
∞
‍
(
4
M
n
2
+
5
M
n
′
2
+
M
n
′′
2
)
d
x
.
Note that (14) and the assumptions, by Lemma 7, |Mn(i)(x)|≤Ci,  i=0,1,2, thus ∥Mn∥≤C, where C=max{C0,C1,C2}, from (19), we have ∥un∥W2≤∥Mn∥W2≤C. In the same way, we obtain ∥vn∥W2≤D.


## 3.3. Construction the of Another Iterative Sequenceun(x) and vn(x)

Theorem 9.
Let{xi}i=1∞ be dense on [0,+∞); if the solution of (2) exists and unique, then the solution satisfies the form
(30)u(x)=∑i=1∞‍∑k=1i‍βikF(xk,u(xk),v(pxk))ψ-i(x),v(x)=∑i=1∞‍∑k=1i‍βikG(xk,v(xk),u(qxk))ψ-i(x).Proof.
Note that〈u(x),φi(x)〉=u(xi) and {ψ-i(x)}i=1∞ is an orthonormal basis of W2[0,+∞); hence we have
(31)u(x)=∑i=1∞‍〈u(x),ψ-i(x)〉ψ-i(x)=∑i=1∞‍〈u(x),∑k=1i‍βikψk(x)〉ψ-i(x)=∑i=1∞‍∑k=1i‍βik〈u(x),ψk(x)〉ψ-i(x)=∑i=1∞‍∑k=1i‍βik〈ℒu(x),φk(x)〉ψ-i(x)=∑i=1∞‍∑k=1i‍βikF(xk,u(xk),v(pxk))ψ-i(x).
In the same manner
(32)v(x)=∑i=1∞‍∑k=1i‍βikG(xk,v(xk),u(qxk))ψ-i(x).In the following, a new method of solving (2) is presented. Equations (31) and (32) can be denoted by
(33)u(x)=∑i=1∞‍Aiψi-(x),(34)v(x)=∑i=1∞‍Biψi-(x),
where Ai=∑k=1iβikF(xk,u(xk),v(pxk)) and Bi=∑k=1iβikG(xk,v(xk),u(qxk)). In fact, Ai and Bi are unknown; we will approximate Ai and Bi by using the known A-i and B-i.We takeu1(x)=0, v1(x)=0 and define the following iterative sequence
(35)un(x)=∑i=1n-1‍A-iψi-(x),vn(x)=∑i=1n-1‍B-iψi-(x),
where
(36)A-1=β11F(x1,u1(x1),v1(px1))A-2=∑k=12‍β2kF(xk,uk(xk),vk(pxk))⋮A-n-1=∑k=1n-1‍βnkF(xk,uk(xk),vk(pxk)),(37)B-1=β11G(x1,v1(x1),u1(qx1))B-2=∑k=12‍β2kG(xk,vk(xk),uk(qxk))⋮B-n-1=∑k=1n-1‍βnkG(xk,vk(xk),uk(qxk)).
Next, lemmas are given.Lemma 10.
The following iterative sequences(38)un(x)=∑i=1n-1‍βikF(xk,uk(xk),vk(pxk))ψi-(x),vn(x)=∑i=1n-1‍βikG(xk,vk(xk),uk(qxk))ψi-(x)
satisfy
(39)ℒun(xj)=F(xj,uj(xj),vj(pxj)),ℒvn(xj)=G(xj,vj(xj),uj(qxj)),j≤n-1,
respectively.Proof.
Ifj=1,
(40)ℒun(x1)=∑i=1n-1‍βikF(xk,uk(xk),vk(pxk))ℒψ-i(x1)=∑i=1n-1‍βikF(xk,uk(xk),vk(pxk))×〈ℒψ-i(x),φ1(x)〉=∑i=1n-1‍βikF(xk,uk(xk),vk(pxk))×〈ψ-i(x),ψ1(x)〉,β11ℒun(x1)=∑i=1n-1‍βikF(xk,uk(xk),vk(pxk))×〈ψ-i(x),ψ-1(x)〉=β11F(x1,u1(x1),v1(px1)),
so,
(41)ℒun(x1)=F(x1,u1(x1),v1(px1)).
If  j=2,
(42)ℒun(x2)=∑i=1n-1‍βikF(xk,uk(xk),vk(pxk))ℒψ-i(x2)=∑i=1n-1‍βikF(xk,uk(xk),vk(pxk))〈ψ-i(x),ψ2(x)〉,β21× (40) +β22× (42),  we have
(43)β21ℒun(x1)+β22ℒun(x2)=∑i=1n-1‍βikF(xk,uk(xk),vk(pxk))〈ψ-i(x),ψ-2(x)〉=β21F(x1,u1(x1),v1(px1))+β22F(x2,u2(x2),v2(px2)),
by (41), ℒun(x2)=F(x2,u2(x2),v2(px2)). In the same way, we have
(44)ℒun(xj)=F(xj,uj(xj),vj(pxj)),j≤n-1.
Similarly,
(45)ℒvn(xj)=F(xj,vj(xj),uj(qxj)),j≤n-1.Theorem 11.
The iterative form(46)u~n(x)=∑i=1n‍∑k=1i‍βikF(xk,u~n-1(xk),v~n-1(pxk))ψ-i(x),v~n(x)=∑i=1n‍∑k=1i‍βikG(xk,v~n-1(xk),u~n-1(qxk))ψ-i(x),u~1(0)=v~1(0)=0
and the iterative form
(47)un(x)=∑i=1n-1‍∑k=1i‍βikF(xk,uk(xk),vk(pxk))ψ-i(x),vn(x)=∑i=1n-1‍∑k=1i‍βikG(xk,vk(xk),uk(qxk))ψ-i(x),u1(0)=v1(0)=0
are the same.Proof.
In Lemma10, let n-1=k; then
(48)ℒuk+1(xk)=F(xk,uk(xk),vk(pxk)),
but
(49)ℒun(xk)=F(xk,uk(xk),vk(pxk)),
thus
(50)un(x)=∑i=1n-1‍∑k=1i‍βikLuk+1(xk)ψ-i(x)=∑i=1n-1‍∑k=1i‍βikℒun(xk)ψ-i(x)=∑i=1n-1‍∑k=1i‍βik〈ℒun(x),φk(x)〉ψ-i(x)=∑i=1n-1‍∑k=1i‍βik〈un(x),ψk(x)〉ψ-i(x)=∑i=1n-1‍〈un(x),ψ-i(x)〉ψ-i(x).
Equation (46) can be written as
(51)ℒMn(x)=F(x,u~n-1(x),v~n-1(px)),u~n(x)=PnMn(x).
In fact,  Mn(x)=∑i=1∞‍∑k=1i‍βikF(xk,u~n-1(xk),v~n-1(pxk))ψ-i(x); then
(52)ℒu~n(xj)=ℒPnMn(xj)=〈ℒPnMn(x),φj(x)〉=〈PnMn(x),ψj(x)〉=〈Mn(x),ψj(x)〉=〈ℒMn(x),φj(x)〉=ℒMn(xj),
by (51) and (52), we have
(53)u~n(x)=∑i=1n-1‍∑k=1i‍βikF(xk,u~n-1(xk),v~n-1(pxk))ψ-i(x)=∑i=1n-1‍∑k=1i‍βikℒMn(xk)ψ-i(x)=∑i=1n-1‍∑k=1i‍βikℒu~n(xk)ψ-i(x)=∑i=1n-1‍∑k=1i‍βik〈ℒu~n(x),φk(x)〉ψ-i(x)=∑i=1n-1‍∑k=1i‍βik〈u~n(x),ψk(x)〉ψ-i(x)=∑i=1n-1‍〈u~n(x),ψ-i(x)〉ψ-i(x).
Equation (53) is the same as (50). We may prove for v~n(x) similarly.So, by Theorem11 and Lemma 8, we have the following Theorem.Theorem 12.
Under the conditions of Lemma8,
(54)un(x)=∑i=1n-1‍βikF(xk,uk(xk),vk(pxk))ψi-(x),(55)vn(x)=∑i=1n-1‍βikG(xk,vk(xk),uk(qxk))ψi-(x)
satisfy ∥un(x)∥≤C,  ∥vn(x)∥≤D.Lemma 13.
Ifu(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 Lemma13 and Theorem 12, it is easy to obtain the following Lemma 14.Lemma 14.
Ifun(x)⟶∥·∥W2u-(x)(n→∞), vn(x)⟶∥·∥W2v-(x)(n→∞), xn→y(n→∞), and F(x,y,z) and G(x,y,z) satisfy the conditions of Lemma 8, then
(56)F(xn,un-1(xn),vn-1(pxn))⟶F(y,u-(y),v-(py))asn⟶∞,G(xn,vn-1(xn),un-1(qxn))⟶G(y,v-(y),u-(qy))asn⟶∞.Theorem 15.
Let{xi}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 un(x) and vn(x) in (35) converge to the exact solution u(x) and v(x) of (2), respectively, and u(x)=∑i=1∞A-iψ-i(x), v(x)=∑i=1∞B-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 un(x),vn(x).
By (35), we infer that
(57)un+1(x)=un(x)+A-nψ-n(x),(58)vn+1(x)=vn(x)+B-nψ-n(x).
From the orthogonality of {ψ-i(x)}i=1∞, it follows that
(59)∥un+1∥W22=∥un∥W22+(A-n)2=∥un-1∥W22+(A-n-1)2+(A-n)2⋮=∥u1∥W22+∑i=1n‍(A-i)2,(60)∥vn+1∥W22=∥vn∥W22+(B-n)2=∥vn-1∥W22+(B-n-1)2+(B-n)2⋮=∥v1∥W22+∑i=1n‍(B-i)2.
From (59) and (60), we know sequence ∥un∥W2 and ∥vn∥W2 is increasing. By Theorem 12, ∥un∥W2 and ∥vn∥W2 are bounded; hence ∥un∥W2 and ∥vn∥W2 are convergent such that
(61)∑i=1∞‍(A-i)2<∞,∑i=1∞‍(B-i)2<∞.
This implies that
(62)A-i=∑k=1i‍βikF(xk,uk(xk),vk(pxk))∈l2(i=1,2,…),(63)B-i=∑k=1i‍βikG(xk,vk(xk),uk(qxk))∈l2(i=1,2,…).
Without loss of generality, assume m>n; we have
(64)∥um(x)-un(x)∥W22=∥um(x)-um-1(x)+um-1(x)-um-2(x)+···+un+1(x)-un(x)∥W22≤∥um(x)-um-1(x)∥W22+⋯+∥un+1(x)-un(x)∥W22=∑i=n+1m‍(A-i)2⟶0,(n⟶∞),∥vm(x)-vn(x)∥W22=∥vm(x)-vm-1(x)+vm-1(x)-vm-2(x)+⋯+vn+1(x)-vn(x)∥W22≤∥vm(x)-vm-1(x)∥W22+…+∥vn+1(x)-vn(x)∥W22=∑i=n+1m‍(B-i)2⟶0,(n⟶∞).
Considering the completeness of W2[0,+∞), there exist u-(x) and v-(x) in W2[0,+∞) such that
(65)un(x)⟶∥·∥u-(x)asn⟶∞,vn(x)⟶∥·∥v-(x)asn⟶∞.
(
2
) Secondly, we will prove that u-(x) and v-(x) are the solutions of (2).
By Lemma14 and the proof of (1), we may know that un(x), and vn(x) respectively, converge uniformly to u-(x) and v-(x)(n→∞). Taking limits in (35), we have
(66)u-(x)=∑i=1∞‍A-iψ-i(x),v-(x)=∑i=1∞‍B-iψ-i(x).
Since
(67)(ℒu-)(xj)=∑i=1∞‍A-i〈ℒψ-i,φj〉=∑i=1∞‍A-i〈ψ-i,ℒ*φj〉=∑i=1∞‍A-i〈ψ-i,ψj〉,
in the same way, we have
(68)(ℒv-)(xj)=∑i=1∞‍B-i〈ψ-i,ψj〉,
it follows that
(69)∑j=1n‍βnj(ℒu-)(xj)=∑i=1∞‍A-i〈ψ-i,∑j=1n‍βnjψj〉=∑i=1∞‍A-i〈ψ-i,ψ-n〉=A-n,∑j=1nβnj(ℒv-)(xj)‍=∑i=1∞‍B-i〈ψ-i,∑j=1n‍βnjψj〉=∑i=1∞‍B-i〈ψ-i,ψ-n〉=B-n.
If  n=1, then
(70)(ℒu-)(x1)=F(x1,u1(x1),v1(px1)),(71)(ℒv-)(x1)=G(x1,v1(x1),u1(qx1)).
Ifn=2, then
(72)β21(ℒu-)(x1)+β22(ℒu-)(x2)=β21F(x1,u1(x1),v1(px1))+β22F(x2,u2(x2),v2(px2)),β21(ℒv-)(x1)+β22(ℒv-)(x2)=β21G(x1,v1(x1),u1(qx1))+β22G(x2,v2(x2),u2(qx2)).
From (70) and (71), it is clear that
(73)(ℒu-)(x2)=F(x2,u2(x2),v2(px2)),(ℒv-)(x2)=G(x2,v2(x2),u2(qx2)).
Furthermore, it is easy to see by induction that
(74)(ℒu-)(xj)=F(xj,uj(xj),vj(pxj)),j=1,2,…,(75)(ℒv-)(xj)=G(xj,vj(xj),uj(qxj)),j=1,2,….
Since {xi}i=1∞ is dense in [0,+∞), for any y∈[0,+∞), there exists subsequence {xnj} such that
(76)xnj⟶yasj⟶∞.
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
(77)(ℒu-)(y)=F(y,u-(y),v-(py)),(78)(ℒv-)(y)=G(y,v-(y),u-(qy)).
That is, u-(x) and v-(x) are the solutions of (2) and
(79)u-(x)=∑i=1∞‍A-iψ-i(x),(80)v-(x)=∑i=1∞‍B-iψ-i(x),
where A-i=∑k=1iβikF(xk,u-k(xk),v-k(pxk))  and  B-i=∑k=1iβikG(xk,v-k(xk),u-k(qxk)).In the proof of the convergence in Theorem15 we only use ∥un∥≤C, ∥vn∥≤D; thus we obtain the following corollary.Corollary 16.
Suppose∥un∥ and ∥vn∥  are bounded; then the iterative sequence (35) is convergent to the exact solution of (2).Theorem 17.
Assumeu(x) and v(x) are the solutions of (2), rn1=∥u(x)-un(x)∥W2 and rn2=∥v(x)-vn(x)∥W2 are the approximate errors of un(x),vn(x), where un(x) and vn(x) are given by (35). Then the errors  rn1, rn2 are monotone decreasing in the sense of ∥·∥W2.Proof.
From (35), and (79), it follows that
(81)∥rn1(x)∥W22=∥∑i=n+1∞‍A-iψ-i(x)∥W22=∑i=n+1∞‍(A-i)2.
In the same way, we obtain from (35), (80)
(82)∥rn2(x)∥W22=∥∑i=n+1∞‍B-iψ-i(x)∥W22=∑i=n+1∞‍(B-i)2.
Equations (81) and (82) show that the errors rn1 and rn2 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 Enu(x)≝|(u(x)-un(x))/u(x)|, Env(x)≝|(v(x)-vn(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(83)u′(x)=31+u(x)2+2xcos(v(13x))+f(x),v′(x)=2e-v(x)+4x3sin(u(12x))+g(x),u(0)=0,v(0)=0,
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 βik.Step 3. According to (10), we can get the normal orthogonal systems {ψi-(x)}i=150.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 Figure1. At the same time, we have computed the approximate solutions un(x) and vn(x)(n=300,900) in [0,3] and also calculated the relative errors Enu and Env in Table 1. The root mean square errors (RMSE) about u(x) with un and v(x) with vn are shown in Table 2.Table 1
Relative errors in[0,3] for Example 18.
Node
E
300
u
E
300
v
E
900
u
E
300
v
0.3
4.97833
e
-
5
8.4527
e
-
5
5.53163
e
-
6
9.39806
e
-
6
0.6
4.43080
e
-
5
7.3330
e
-
5
4.92288
e
-
6
8.15664
e
-
6
0.9
3.86801
e
-
5
6.6731
e
-
5
4.29733
e
-
6
7.42512
e
-
6
1.2
3.41875
e
-
5
6.1845
e
-
5
3.79797
e
-
6
6.88400
e
-
6
1.5
3.09088
e
-
5
5.7095
e
-
5
3.43349
e
-
6
6.35872
e
-
6
1.8
2.87391
e
-
5
5.1303
e
-
5
3.19221
e
-
6
5.71898
e
-
6
2.1
2.75649
e
-
5
4.3153
e
-
5
3.06149
e
-
6
4.81984
e
-
6
2.4
2.7288
e
-
5
3.0422
e
-
5
3.03034
e
-
6
3.41736
e
-
6
2.7
2.7829
e
-
5
6.7566
e
-
5
3.09008
e
-
6
8.12936
e
-
7
3.0
2.90670
e
-
5
7.5017
e
-
5
3.23206
e
-
6
8.72219
e
-
6Table 2
The RMS errors in[0,3] for Example 18.
n
∑
i
=
1
n
‍
(
u
n
(
x
i
)
-
u
(
x
i
)
)
2
/
n
∑
i
=
1
n
‍
(
v
n
(
x
i
)
-
v
(
x
i
)
)
2
/
n
300
9.95194
e
-
6
4.00007
e
-
5
900
1.10551
e
-
6
4.45554
e
-
6The left is the superimposed image ofu(x) with u50(x) in [0,1]. The right is the superimposed image of v(x) with v50(x) in [0,1].
(a)
(b)Example 19.
Considering equations(84)u′(x)=3cos(u(x))+2xsin(v(15x))+f(x),v′(x)=2sin(v(x))+4x3cos(u(13x))+g(x),u(0)=0,v(0)=0.
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.
Node
E
800
u
E
800
v
E
1200
u
E
1200
u
0.4
8.06470
e
-
5
4.94886
e
-
5
3.58334
e
-
5
2.20033
e
-
5
0.8
1.07473
e
-
4
5.27211
e
-
5
4.77632
e
-
5
2.34475
e
-
5
1.2
2.56877
e
-
4
5.32904
e
-
5
1.14199
e
-
4
2.37040
e
-
5
1.6
1.19110
e
-
4
5.01608
e
-
5
5.29170
e
-
5
2.23098
e
-
5
2.0
8.56122
e
-
5
4.35915
e
-
5
3.80383
e
-
5
1.93810
e
-
5
2.4
2.27843
e
-
4
3.52666
e
-
5
1.01280
e
-
4
1.56705
e
-
5
2.8
1.38237
e
-
4
2.74018
e
-
5
6.13912
e
-
5
1.21676
e
-
5
3.2
7.26035
e
-
5
2.15572
e
-
5
3.22429
e
-
5
9.56745
e
-
6
3.6
1.86684
e
-
5
1.80587
e
-
5
8.29627
e
-
5
8.01449
e
-
6
4.0
1.50042
e
-
4
1.64230
e
-
5
6.66348
e
-
5
7.14957
e
-
6Table 4
The RMS errors in[0,4] for Example 19.
n
∑
i
=
1
n
‍
(
u
n
(
x
i
)
-
u
(
x
i
)
)
2
/
n
∑
i
=
1
n
‍
(
v
n
(
x
i
)
-
v
(
x
i
)
)
2
/
n
800
1.87807
e
-
3
7.85856
e
-
5
1200
8.34276
e
-
4
3.48826
e
-
5The left is the superimposed image ofu(x) with u50(x) in [0,1]. The right is the superimposed image of v(x) with v50(x) in [0,1].
(a)
(b)


## 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.



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*Source: 541935-2013-04-24.xml*