research article implicit numerical solutions for solving stochastic differential … · 2019. 7....

Research ArticleImplicit Numerical Solutions for Solving Stochastic DifferentialEquations with Jumps

Ying Du and Changlin Mei

School of Mathematics and Statistics Xirsquoan Jiaotong University Xirsquoan Shaanxi 710049 China

Correspondence should be addressed to Ying Du duyingxjtualiyuncom

Received 25 January 2014 Accepted 26 June 2014 Published 24 July 2014

Academic Editor Maoan Han

Copyright copy 2014 Y Du and C Mei This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To realize the applications of stochastic differential equations with jumps much attention has recently been paid to the constructionof efficient numerical solutions of the equations Considering the fact that the use of the explicit methods often results in instabilityand inaccurate approximations in solving stochastic differential equations we propose two implicit methods the 120579-Taylor methodand the balanced 120579-Taylor method for numerically solving the stochastic differential equation with jumps and prove that thenumerical solutions are convergent with strong order 10 For a linear scalar test equation the mean-square stability regions ofthe methods are derived Finally numerical examples are given to evaluate the performance of the methods

1 Introduction

Stochastic differential equations (SDEs) have been one of themost important mathematical tools for dealing with manyproblems in a variety of practical areas However SDEs arein general so complex that the analytical solutions can rarelybe obtained Thus it is a common way to numerically solveSDEs Since the explicit numerical methods often result ininstability and inaccurate approximations to the solutionsunless the step-size is very small it is often necessary to usesome implicit methods in numerically solving SDEs

Generally speaking there are two kinds of implicitnumerical methods One is the semi-implicit methods inwhich the drift components are computed implicitly whilethe diffusion components are computed explicitly Higham[1 2] studied the stochastic 120579-method for SDEs and SDEswith jumps (SDEJs) When 120579 = 1 the stochastic 120579-methodis the backward Euler method The backward Euler methodis discussed in [3ndash5] and the references therein Hu andGan [6] proposed a class of drift-implicit one-step methodsfor neutral stochastic delay differential equations with jumpdiffusion Higham and Kloeden [3 7] constructed the split-step backward Euler method and the compensated split-step backward Euler method for SDEJs Ding et al [8]

introduced the split-step 120579-method which is more generalthan the split-step backward Euler method Wang and Gan[9] studied split-step one-leg 120579 methods for SDEs Buckwarand Sickenberger [10] compared the mean-square stabilityproperties of the 120579-Maruyama and 120579-Milstein methods forSDEs

The other is the fully implicit methods in which boththe drift components and the diffusion components arecomputed implicitly Since implicit stochastic terms in theimplicit methods lead to infinite absolute moments of thenumerical solution extensive research has been done toaddress this issue [11ndash26] For example Milstein et al [11]proposed the balanced implicit method for the numericalsolutions of SDEs Burrage and Tian [12] suggested threeimplicit Taylor methods the implicit Euler-Taylor methodwith strong order 05 the implicit Milstein-Taylor methodwith strong order 10 and the implicit Taylor method withstrong order 15 Kahl and Schurz [16] introduced the bal-anced Milstein method for ordinary SDEs Wang and Liu[20 21] proposed the semi-implicit Milstein method andthe split-step backward balanced Milstein method for stiffstochastic systems Furthermore Haghighi andHosseini [23]developed a class of general split-step balanced numericalmethods for SDEs

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 159107 11 pageshttpdxdoiorg1011552014159107

2 Abstract and Applied Analysis

Let (ΩF F119905119905isin[1199050 119879]

P) be a complete probability spacewith the filtration F

119905119905isin[1199050 119879]

satisfying the usual conditionsthat F

119905is right-continuous and F

0contains all P-null sets

In this paper we consider the stochastic differential equationswith jumps of the form

d119909 (119905) = 119891 (119909 (119905)) d119905 + 119892 (119909 (119905)) d119882(119905) + ℎ (119909 (119905)) d119873(119905)

119905 isin [1199050 119879]

119909 (1199050) = 1199090

(1)

where119882(119905) isF119905-adaptedWiener process and119873(119905) is a scalar

poisson process with intensity 120582 and is independent of119882(119905)Hu andGan [22 25] proposed the balancedmethod for SDEJs(1) and stochastic pantograph equations with jumps respec-tively and proved that the numerical solution converges tothe analytical solution with rate 12 The asymptotic stabilityof the balanced method for SDEJs (1) was obtained in [26]To obtain higher order numerical schemes and improve theaccuracy of the numerical solutions we propose two kindsof implicit Taylor methods and prove that the numericalsolutions converge to the true solutions of SDEJs (1) with rate10

The rest of the paper is arranged as follows In Section 2we introduce the 120579-Taylor methods and the fully implicitbalanced 120579-Taylor methods for SDEJs (1) The strong con-vergence properties of these implicit methods are proved inSection 3 The mean-square stability of the numerical solu-tions is discussed in Section 4 Some numerical experimentsare performed in Section 5 to evaluate the performance of theproposed numerical methods

2 The Numerical Methods

Define a mesh 0 le 1199050lt 1199051lt sdot sdot sdot lt 119905

119899lt 119905119899+1

lt sdot sdot sdot lt 119905119873= 119879

on the time interval [1199050 119879] with 119905

119899= 119899Δ and the step-size

Δ = 119879119873 119909119899is the numerical approximation to 119909(119905

119899) Based

on appropriate stochastic Taylor expansions Maghsoodi [27]generalized the Milstein scheme to SDEJs and obtained theorder 10 strong Taylor scheme (Taylor for short) as

119909119899+1

= 119909119899+ 119891 (119909

119899) Δ + 119892 (119909

119899) Δ119882119899

+1

2119892 (119909119899) 1198921015840(119909119899) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909

119899) minus ℎℎ(119909119899)) Δ119873

119899

+ (119892ℎ(119909119899) minus 119892 (119909

119899)) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909119899) minus ℎ (119909

119899)) (Δ119873

119899)2

+ (119892 (119909119899) ℎ1015840(119909119899) minus 119892ℎ(119909119899) + 119892 (119909

119899)) Δ119885

119899

(2)

where 119892ℎ(119909) = 119892(119909 + ℎ(119909)) ℎ

ℎ(119909) = ℎ(119909 + ℎ(119909)) and Δ119885

119899=

int119905119899+1

119905119899

int119904

119905119899

d119882119905d119873119904= int119905119899+1

119905119899

(119882119904minus119882119905119899)d119873119904

Note thatΔ119885119899= sum119895(119882(120591119895)minus119882(119905

119896)) = sum

119873119899+1

119895=119873119899+1(119873119899+1minus119895+

1)(119882(120591119895) minus 119882(120591

119895minus1)) [28] Given a jump time 120591

119895in [119905119899 119905119899+1)

Δ1198851119899(120591119895) = 119882(120591

119895) minus 119882(120591

119895minus1) sim 119873(0 120591

119895minus 120591119895minus1) (119873119899+ 1 le 119895 le

119873119899+1

) In addition the random variable Δ119882119899= 119882(119905

119899+1) minus

119882(119905119899) is dependent on Δ119885

1119899(120591119895) and its sample values can be

calculated by Δ119882119899= sum119873119899+1

119895=119873119899+1Δ1198851119899(120591119895) + Δ119885

1119899(119905119899+1) where

Δ1198851119899(119905119899+1) = 119882(119905

119899+1) minus 119882(120591

119873119899+1) sim 119873(0 119905

119899+1minus 120591119873119899+1

)By changing the explicit deterministic term into implicit

term we have the following 120579-Taylor method

119909119899+1

= 119909119899+ Δ [(1 minus 120579) 119891 (119909

119899) + 120579119891 (119909

119899+1)]

+ 119892 (119909119899) Δ119882119899+1

2119892 (119909119899) 1198921015840(119909119899) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909

119899) minus ℎℎ(119909119899)) Δ119873

119899

+ (119892ℎ(119909119899) minus 119892 (119909

119899)) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909119899) minus ℎ (119909

119899)) (Δ119873

119899)2

+ (119892 (119909119899) ℎ1015840(119909119899) minus 119892ℎ(119909119899) + 119892 (119909

119899)) Δ119885

119899

(3)

Note that the 120579-Taylor method (3) becomes the Taylormethod (2) when 120579 = 0

Using the idea of the balanced implicit method andcombining it with the 120579-Taylormethod we have the followingbalanced 120579-Taylor method

119909119899+1

= 119909119899+ Δ [(1 minus 120579) 119891 (119909

119899) + 120579119891 (119909

119899+1)]

+ 119892 (119909119899) Δ119882119899+1

2119892 (119909119899) 1198921015840(119909119899) [(Δ119882

119899)2minus Δ]

+1

2(3ℎ (119909

119899) minus ℎℎ(119909119899)) Δ119873

119899

+ (119892ℎ(119909119899) minus 119892 (119909

119899)) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909119899) minus ℎ (119909

119899)) (Δ119873

119899)2

+ (119892 (119909119899) ℎ1015840(119909119899) minus 119892ℎ(119909119899) + 119892 (119909

119899)) Δ119885

119899

+ 119862119899(119909119899minus 119909119899+1)

(4)

where 119862119899= 119862(119909

119899) = 119862

0(119909119899)Δ + 119862

1(119909119899)[(Δ119882

119899)2minus Δ] with

1198620(sdot) and 119862

1(sdot) called control functions

3 Convergence of the Implicit Taylor Methods

Let | sdot | be the Euclidean norm in R119889 If 119860 is a matrix |119860| =radictrace(119860119879119860) Denote 119911

1198712= (E|119911|

2)12 for 119911 isin R119889 To

prove the convergence of the numerical solutions we makethe following assumptions

Abstract and Applied Analysis 3

Assumption 1 The coefficient functions119891 119892 and ℎ satisfy theglobal Lipschitz condition

1003816100381610038161003816119891 (1199091) minus 119891 (1199092)1003816100381610038161003816 +1003816100381610038161003816119892 (1199091) minus 119892 (1199092)

1003816100381610038161003816

+1003816100381610038161003816ℎ (1199091) minus ℎ (1199092)

1003816100381610038161003816

+10038161003816100381610038161003816119892 (1199091) 1198921015840(1199091) minus 119892 (119909

2) 1198921015840(1199092)10038161003816100381610038161003816

+10038161003816100381610038161003816119892 (1199091) ℎ1015840(1199091) minus 119892 (119909

2) ℎ1015840(1199092)10038161003816100381610038161003816le 119871

10038161003816100381610038161199091 minus 11990921003816100381610038161003816

(5)

for a positive constant 119871 and any 1199091 1199092isin R119889 and the linear

growth condition

1003816100381610038161003816119891(119909)10038161003816100381610038162

+1003816100381610038161003816119892(119909)

10038161003816100381610038162

+ |ℎ(119909)|2

+10038161003816100381610038161003816119892(119909)119892

1015840(119909)10038161003816100381610038161003816

2

+10038161003816100381610038161003816119892(119909)ℎ

1015840(119909)10038161003816100381610038161003816

2

le 1198711015840(1 + |119909|

2)

(6)

for a positive constant 1198711015840 and any 119909 isin R119889

Assumption 2 The1198620(sdot) and119862

1(sdot) are bounded 119889times119889-matrix-

valued functions For any real numbers 1205720isin [0 120572

0] and 120572

1isin

[minus1205721 1205721] with 120572

0ge Δ and 120572

1ge |(Δ119882

119899)2minus Δ| for all step-size

Δ and 119909 isin R119889 the matrix119872(119909) = 119868 + 12057201198620(119909) + 120572

11198621(119909) is

reversible and satisfies |(119872(119909))minus1| le 119861 lt infin where 119868 is a unitmatrix and 119861 is a positive constant

In what follows we will derive the strong convergenceorders of the implicit Taylor methods for SDEJs (1)

31 Convergence of the 120579-Taylor Method Define

119909120579(119905119899+1) = 119909 (119905

119899)

+ Δ [(1 minus 120579) 119891 (119909 (119905119899))

+ 120579119891 (119909 (119905119899+1))] + 119892 (119909 (119905

119899)) Δ119882

119899

+1

2119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899))

minus119892ℎ(119909 (119905119899)) + 119892 (119909 (119905

119899))) Δ119885

119899

(7)

by replacing the numerical approximations with the exactsolution values on the right-hand side of equation (3) Then

the local error ofmethod (3) is defined by 120575120579(119905119899+1) = 119909(119905

119899+1)minus

119909120579(119905119899+1) and the global error of method (3) is defined by

120598119899= 119909(119905119899) minus 119909119899

Theorem 3 Under Assumption 1 the 120579-Taylor method (3) isconsistent with order 2 in the mean and with order 15 in themean square That is the local mean error and mean-squareerror of the 120579-Taylor method (3) satisfy

max0le119899le119873minus1

10038171003817100381710038171003817E(120575120579(119905119899+1) | F119905119899)100381710038171003817100381710038171198712

le 1198671Δ2

119886119904 Δ 997888rarr 0 (8)


10038171003817100381710038171003817120575120579(119905119899+1)100381710038171003817100381710038171198712

le 1198672Δ32

119886119904 Δ 997888rarr 0 (9)

where the constants1198671and119867

2are independent of Δ

Proof To obtain the convergence rate of the 120579-Taylormethodwe firstly introduce the local Taylor numerical approximation119909119860

119899+1which is defined by

119909119860

119899+1= 119909 (119905

119899) + 119891 (119909 (119905

119899)) Δ + 119892 (119909 (119905

119899)) Δ119882

119899

+1

2119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+ 119892 (119909 (119905119899))) Δ119885

119899

(10)

Then there exists some constant1198701gt 0 such that

E [10038161003816100381610038161003816E [(119909120579

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816E [(119909119860

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

]

+ 2E [10038161003816100381610038161003816E [(119909120579

119899+1minus 119909119860

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

le 1198701Δ4+ 2E [

10038161003816100381610038161003816E [(119909120579

119899+1minus 119909119860

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

(11)

Since

E [10038161003816100381610038161003816E [(119909120579

119899+1minus 119909119860

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

= E [10038161003816100381610038161003816E [120579Δ (119891 (119909 (119905

119899+1)) minus 119891 (119909 (119905

119899))) | F

119905119899]10038161003816100381610038161003816

2

]

le 119871120579ΔE[E [1003816100381610038161003816119909(119905119899+1) minus 119909(119905119899)

1003816100381610038161003816 | F119905119899]]2

le 119874 (Δ4)

(12)

we obtain (E[|E[(119909120579119899+1

minus 119909(119905119899+1)) | F

119905119899]|2

])12

le 119874(Δ2)


On the other hand since

E [10038161003816100381610038161003816119909120579

119899+1minus 119909119860

119899+1

10038161003816100381610038161003816

2

| 119909119899= 119909 (119905

119899)]

= 11987121205792Δ2E [1003816100381610038161003816119909 (119905119899+1) minus 119909 (119905119899)

10038161003816100381610038162

]

le 119874 (Δ3)

(13)

we have

E [10038161003816100381610038161003816119909120579

119899+1minus 119909 (119905

119899+1)10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816119909119860

119899+1minus 119909 (119905

119899+1)10038161003816100381610038161003816

2

]

+ 2E [10038161003816100381610038161003816119909120579

119899+1minus 119909119860

119899+1

10038161003816100381610038161003816

2

]

le 119874 (Δ3)

(14)

Therefore the result (9) is obtained

Theorem 4 Under Assumption 1 the 120579-Taylor method (3) isconvergent with order 1 in the mean square That is the globalerror satisfies


100381710038171003817100381710038171205982

119899+1

100381710038171003817100381710038171198712le 1198673Δ 119886119904 Δ 997888rarr 0 (15)

where1198673is independent of Δ

Proof From the definitions of 120575119899and 120598119899 we have

120598119899+1

= 120598119899+ 119906119899+ 120575119899+1 (16)

where119906119899= Δ (1 minus 120579) (119891 (119909 (119905

119899)) minus 119891 (119909

119899))

+ Δ120579 (119891 (119909 (119905119899+1)) minus 119891 (119909

119899+1))

+ (119892 (119909 (119905119899)) minus 119892 (119909

119899)) Δ119882

119899

+1

2(119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899))

minus 119892 (119909119899) 1198921015840(119909119899)) [(Δ119882)

2minus Δ]

+1

2[3 (ℎ (119909 (119905

119899)) minus ℎ (119909

119899))

minus (ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899))] Δ119873

119899

+ [(119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899))

minus (119892 (119909 (119905119899)) minus 119892 (119909

119899))] Δ119882

119899Δ119873119899

+1

2[(ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899))

minus (ℎ (119909 (119905119899)) minus ℎ (119909

119899))] (Δ119873

119899)2

+ [(119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892 (119909

119899) ℎ1015840(119909119899))

minus (119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899))

+ (119892 (119909 (119905119899)) minus 119892 (119909

119899))] Δ119885

119899

(17)

Since 120598119899isF119905119899-measurable we have fromTheorem 3 that

1003816100381610038161003816E ⟨120575119899+1 120598119899⟩1003816100381610038161003816

=10038161003816100381610038161003816E [E (⟨120575

119899+1 120598119899⟩ | F119905119899)]10038161003816100381610038161003816

le E10038161003816100381610038161003816⟨E (120575

119899+1| F119905119899) 120598119899⟩10038161003816100381610038161003816

le [Δminus1(E10038161003816100381610038161003816E (120575119899+1

| F119905119899)10038161003816100381610038161003816

2

)]12

times (ΔE100381610038161003816100381612059811989910038161003816100381610038162

)12

le Δminus1(E10038161003816100381610038161003816E (120575119899+1

| F119905119899)10038161003816100381610038161003816

2

)

+ ΔE100381610038161003816100381612059811989910038161003816100381610038162

le 1198671Δ3+ ΔE

100381610038161003816100381612059811989910038161003816100381610038162

(18)

where ⟨sdot sdot⟩ indicates the scalar productNoting that E|Δ119882

119899|2= Δ E|Δ119882

119899|4= 3Δ2 E(Δ119873

119899)2=

120582Δ(1 + 120582Δ) E(Δ119873119899)4= 120582Δ(1 + 7120582Δ + 6(120582Δ)

2+ (120582Δ)

3) and

Δ119882119899is independent of Δ119873

119899 we have from Assumption 1 that

E1003816100381610038161003816(119892 (119909 (119905119899)) minus 119892 (119909119899)) Δ119882119899

10038161003816100381610038162

le ΔE1003816100381610038161003816119892 (119909 (119905119899)) minus 119892 (119909119899)

10038161003816100381610038162

le 1198712ΔE|120598119899|2

E10038161003816100381610038161003816(119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) minus 119892 (119909

119899) 1198921015840(119909119899)) [(Δ119882)

2minus Δ]

10038161003816100381610038161003816

2

le 2Δ2E10038161003816100381610038161003816119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) minus 119892 (119909

119899) 1198921015840(119909119899)10038161003816100381610038161003816

le 21198712Δ2E|120598119899|2

E1003816100381610038161003816[3 (ℎ (119909 (119905119899))minus ℎ (119909119899))minus (ℎℎ (119909 (119905119899))minus ℎℎ (119909119899))] Δ119873119899

10038161003816100381610038162

le 2120582Δ (1 + 120582Δ) [3E1003816100381610038161003816ℎ (119909 (119905119899)) minus ℎ (119909119899)

10038161003816100381610038162

+ E1003816100381610038161003816ℎ (119909 (119905119899) + ℎ (119909 (119905119899))) minus ℎ (119909119899 + ℎ (119909119899))

10038161003816100381610038162

]

le 21198712120582Δ (1 + 120582Δ) (5 + 2119871

2)E100381610038161003816100381612059811989910038161003816100381610038162

E1003816100381610038161003816[(119892ℎ (119909 (119905119899))minus 119892ℎ (119909119899))minus (119892 (119909 (119905119899))minus 119892 (119909119899))] Δ119882119899Δ119873119899

10038161003816100381610038162

le E1003816100381610038161003816(119892ℎ (119909 (119905119899)) minus 119892ℎ (119909119899)) minus (119892 (119909 (119905119899)) minus 119892 (119909119899))

10038161003816100381610038162

sdot E|Δ119882119899|2sdot E(Δ119873

119899)2

le 21198712120582Δ2(1 + 120582Δ) (3 + 2119871

2)E|120598119899|2

E10038161003816100381610038161003816[(ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899)) minus (ℎ (119909 (119905

119899)) minus ℎ (119909

119899))] (Δ119873

119899)210038161003816100381610038161003816

2

le 2120582Δ (1 + 7120582Δ + 6(120582Δ)2+ (120582Δ)

3) 1198712(3 + 2119871

2)E100381610038161003816100381612059811989910038161003816100381610038162

E10038161003816100381610038161003816[ (119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899)) minus 119892 (119909

119899) ℎ1015840(119909119899))

minus (119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899)) + (119892 (119909 (119905

119899)) minus 119892 (119909

119899))] Δ119885

119899

10038161003816100381610038162


le 61198712(2 + 119871

2)E100381610038161003816100381612059811989910038161003816100381610038162

sdot E1003816100381610038161003816Δ119885119899

10038161003816100381610038162

le 61198712(2 + 119871

2)E100381610038161003816100381612059811989910038161003816100381610038162

sdot E1003816100381610038161003816Δ119882119899 minus Δ1198851119899(119905119899+1)

10038161003816100381610038162

le 241198712(2 + 119871

2) ΔE

100381610038161003816100381612059811989910038161003816100381610038162

(19)

Hence

E100381610038161003816100381611990611989910038161003816100381610038162

le 1198691ΔE100381610038161003816100381612059811989910038161003816100381610038162

+ 81198712Δ21205792E1003816100381610038161003816120598119899+1

10038161003816100381610038162

(20)

where 1198691= 8[4119871

2+ 21198712120582(1 + 120582)(5 + 2119871

2) + 2119871

2120582(1 + 120582)(3 +

21198712) + 2120582(1 + 7120582 + 6120582

2+ 1205823)1198712(3 + 2119871

2) + 24119871

2(2 + 119871

2)

Noting that 119909(119905119899) and 119909

119899are F

119905119899-measurable and Δ119882

119899

and Δ119873119899are independent ofF

119905119899 we have

E [(119892 (119909 (119905119899)) minus 119892 (119909

119899)) Δ119882

119899F119905119899] = 0

E [(119892 (119909 (119905119899)) 1198921015840(119909 (119905119899)) minus 119892 (119909

119899) 1198921015840(119909119899))

times ((Δ119882)2minus Δ) | F

119905119899] = 0

E [[(119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899)) minus (119892 (119909 (119905

119899)) minus 119892 (119909

119899))]

timesΔ119882119899Δ119873119899| F119905119899] = 0

E [[(119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892 (119909

119899) ℎ1015840(119909119899))

minus (119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899))

+ (119892 (119909 (119905119899)) minus 119892 (119909

119899)) ] Δ119885

119899| F119905119899] = 0

(21)

Therefore10038161003816100381610038161003816E (119906119899| F119905119899)10038161003816100381610038161003816

2

le

1003816100381610038161003816100381610038161003816Δ (1 minus 120579)E [119891 (119909 (119905

119899)) minus 119891 (119909

119899) | F119905119899]

+ 120579ΔE [119891 (119909 (119905119899+1)) minus 119891 (119909

119899+1) | F119905119899]

+3

2E [(ℎ (119909 (119905

119899)) minus ℎ (119909

119899)) Δ119873

119899| F119905119899]

minus1

2E [(ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899)) Δ119873

119899| F119905119899]

+1

2E [(ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899)) (Δ119873

119899)2

| F119905119899]

minus1

2E [(ℎ (119909 (119905

119899)) minus ℎ (119909

119899)) (Δ119873

119899)2

| F119905119899]

1003816100381610038161003816100381610038161003816

2

le 1198692Δ2E100381610038161003816100381612059811989910038161003816100381610038162

+ 611987121205792Δ2E [1003816100381610038161003816120598119899+1

10038161003816100381610038162

| F119905119899]

(22)

where 1198692= 31198712[2 + 3120582

2+ (1 + 119871)

21205822+ (1 + 119871)

2(1 + 120582)

2+

1205822(1 + 120582)

2] Thus

1003816100381610038161003816E ⟨120598119899 119906119899⟩1003816100381610038161003816 le Δminus1(E10038161003816100381610038161003816E (119906119899| F119905119899)10038161003816100381610038161003816

2

) + ΔE100381610038161003816100381612059811989910038161003816100381610038162

le (1 + 1198692) ΔE

100381610038161003816100381612059811989910038161003816100381610038162

+ 611987121205792ΔE1003816100381610038161003816120598119899+1

10038161003816100381610038162

(23)

From the above arguments we obtain

E1003816100381610038161003816120598119899+1

10038161003816100381610038162

le E100381610038161003816100381612059811989910038161003816100381610038162

+ E1003816100381610038161003816120575119899+1

10038161003816100381610038162

+ E100381610038161003816100381611990611989910038161003816100381610038162

+ 21003816100381610038161003816E ⟨120575119899+1 120598119899⟩

1003816100381610038161003816 + 21003816100381610038161003816E ⟨120598119899 119906119899⟩

1003816100381610038161003816

le [1 + 2 (2 + 1198691+ 1198692) Δ]E

100381610038161003816100381612059811989910038161003816100381610038162

+ 211987121205792Δ (Δ + 6)E

1003816100381610038161003816120598119899+110038161003816100381610038162

+ (1198672+ 21198671) Δ3

(24)

BecauseΔ rarr 0 we can assume 1minus211987121205792Δ(Δ+6) gt 0withoutloss of generality Let 119869

3= 2(2 + 119869

1+ 1198692) Then

E1003816100381610038161003816120598119899+1

10038161003816100381610038162

le (1 + 1198693Δ)E

100381610038161003816100381612059811989910038161003816100381610038162

+ (1198672+ 21198671) Δ3

= (1198672+ 21198671) Δ2(1 + 119869

3Δ)119899+1

minus 1

1198693

le 1198694Δ2

(25)

where 1198694= (1198672+ 21198671)((1198901198693119879 minus 1)119869

3)

32 Convergence of the Balanced 120579-Taylor Method Define

119909119861(119905119899+1)

= 119909 (119905119899) + Δ [(1 minus 120579) 119891 (119909 (119905

119899)) + 120579119891 (119909 (119905

119899+1))]

+ 119892 (119909 (119905119899)) Δ119882

119899

+1

2119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+ 119892 (119909 (119905119899))) Δ119885

119899+ 119862119899(119909 (119905119899) minus 119909119861(119905119899+1))

(26)

by replacing the numerical approximations with the exactsolution values on the right-hand side of (4) Then the localerror of method (4) is 120575119861(119905

119899+1) = 119909(119905

119899+1) minus 119909119861(119905119899+1) and the

global error of method (4) is 120598119899= 119909(119905119899) minus 119909119899

Theorem5 UnderAssumptions 1 and 2 the balanced 120579-Taylormethod (4) is consistent with order 2 in the mean and with


order 15 in the mean square That is the local mean error andmean-square error of the balanced 120579-Taylor method (4) satisfy


10038171003817100381710038171003817E (120575119861(119905119899+1) | F119905119899)100381710038171003817100381710038171198712

le 1198674Δ2

119886119904 Δ 997888rarr 0


10038171003817100381710038171003817120575119861(119905119899+1)100381710038171003817100381710038171198712

le 1198675Δ32

119886119904 Δ 997888rarr 0

(27)



Proof FromTheorem 3 we have

E [10038161003816100381610038161003816E [(119909119861

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816E [(119909120579

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

]

+ 2E [10038161003816100381610038161003816E [(119909119861

119899+1minus 119909120579

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

le 1198672

1Δ4+ 2E [

10038161003816100381610038161003816E [(119909119861

119899+1minus 119909120579

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

(28)

From the definitions of 119909120579119899+1

and 119909119861119899+1

in (7) and (26) we canwrite

119909119861

119899+1minus 119909120579

119899+1

= minus119862119899(119909 (119905119899) minus 119909119861(119905119899+1))

= 119862119899[Δ [(1 minus 120579) 119891 (119909 (119905

119899)) + 120579119891 (119909 (119905

119899+1))]

+ 119892 (119909 (119905119899)) Δ119882

119899

+1

2119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+119892 (119909 (119905119899))) Δ119885

119899+ 119862119899(119909119899minus 119909119861

119899+1)]

= (119868 + 119862119899)minus1

119862119899

times [Δ [(1 minus 120579) 119891 (119909 (119905119899)) + 120579119891 (119909 (119905

119899+1))]

+ 119892 (119909 (119905119899)) Δ119882

119899+1

2119892 (119909 (119905

119899))

times 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+119892 (119909 (119905119899))) Δ119885

119899]

(29)

Since the components of the matrices 1198620(sdot) and 119862

1(sdot) in 119862

119899(sdot)

are bounded there exists a positive constant 119872 such that|119862119894| le 119872 (119894 = 0 1) Under Assumptions 1 and 2 we have10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899Δ119891 (119909 (119905

119899)) | F

119905119899]10038161003816100381610038161003816

le 119872Δ1003816100381610038161003816119891 (119909119899)

1003816100381610038161003816E [100381610038161003816100381610038161198620Δ + 119862

1((Δ119882

119899)2

minus Δ)10038161003816100381610038161003816| F119905119899]

le 1198711015840119872119861(1 +

1003816100381610038161003816119909 (119905119899)10038161003816100381610038162

)12

Δ2

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899119892 (119909 (119905

119899)) Δ119882

119899| F119905119899]10038161003816100381610038161003816= 0

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899))

times [(Δ119882119899)2

minus Δ] | F119905119899]10038161003816100381610038161003816

le 11986111987210038161003816100381610038161003816119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899))10038161003816100381610038161003816

E [1003816100381610038161003816100381610038161003816((Δ119882

119899)2

minus Δ)21003816100381610038161003816100381610038161003816| F119905119899]

le 21198711015840119872119861(1 +

100381610038161003816100381611990911989910038161003816100381610038162

)12

Δ2

10038161003816100381610038161003816[E [(119868 + 119862

119899)minus1

119862119899(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899] | F119905119899]10038161003816100381610038161003816

le 12058211986111987210038161003816100381610038163ℎ (119909 (119905119899)) minus ℎℎ (119909 (119905119899))

1003816100381610038161003816 Δ2

le (2 + 119871) 1198711015840119861119872(1 +

1003816100381610038161003816119909 (119905119899)10038161003816100381610038162

)12

Δ2

10038161003816100381610038161003816E [(119868+ 119862

119899)minus1

119862119899(119892ℎ(119909 (119905119899))minus 119892 (119909 (119905

119899)))

times Δ119882119899Δ119873119899| F119905119899]10038161003816100381610038161003816= 0

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899)))

times (Δ119873119899)2

| F119905119899]10038161003816100381610038161003816

le 1198611198721003816100381610038161003816ℎℎ (119909 (119905119899)) minus ℎ (119909 (119905119899))

1003816100381610038161003816 120582Δ2(1 + 120582Δ)

le 1198711015840119861119872120582(1 +

1003816100381610038161003816119909 (119905119899)10038161003816100381610038162

)12

Δ2(1 + 120582Δ)

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899(119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899))

minus 119892ℎ(119909 (119905119899)) + 119892 (119909 (119905

119899))) Δ119885

119899| F119905119899]10038161003816100381610038161003816

=10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899(119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899))

minus 119892ℎ(119909 (119905119899)) + 119892 (119909 (119905

119899)))

times (Δ119882119899minus Δ1198851119896(119905119899+1)) | F

119905119899]10038161003816100381610038161003816

= 0

(30)


Therefore

E [10038161003816100381610038161003816E [(119909119861

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

] le 119874 (Δ4) (31)


E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899Δ [(1 minus 120579) 119891 (119909 (119905

119899)) + 120579119891 (119909 (119905

119899+1))]10038161003816100381610038161003816

2

]

le 119874 (Δ4)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899119892 (119909119899) Δ119882119899

10038161003816100381610038161003816

2

] le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]10038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

10038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

10038161003816100381610038161003816

2

]

le 119874 (Δ4)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)210038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+ 119892 (119909 (119905119899))) (Δ119885

119899)10038161003816100381610038161003816

2

]

le 119874 (Δ3)

(32)

we have

E [10038161003816100381610038161003816119909119861

119899+1minus 119909 (119905

119899+1)10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816119909 (119905119899+1) minus 119909120579

119899+1

10038161003816100381610038161003816

2

] + 2E [10038161003816100381610038161003816119909120579

119899+1minus 119909119861

119899+1

10038161003816100381610038161003816

2

]

le 119874 (Δ3)

(33)

Theorem 6 Under Assumptions 1 and 2 the balanced 120579-Taylor method (4) is convergent with order 1 in the meansquare That is the global error satisfies


100381710038171003817100381710038171205982

119899+1

100381710038171003817100381710038171198712le 1198676Δ 119886119904 Δ 997888rarr 0 (34)



120598119899+1

= 120598119899+ 119875119899+ 120575119899+1 (35)

where

119875119899= 119906119899+ 119862 (119909 (119905

119899)) (119909 (119905

119899) minus 119909119861(119905119899+1))

minus 119862 (119909119899) (119909119899minus 119909119899+1)

= 119906119899+ 119862 (119909 (119905

119899)) (119909 (119905

119899) minus 119909119899)

minus 119862 (119909 (119905119899)) (119909119861(119905119899+1) minus 119909119899+1)

+ (119862 (119909 (119905119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)

= 119906119899minus 119862 (119909 (119905

119899)) 119875119899

+ (119862 (119909 (119905119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)

= (119868 + 119862 (119909 (119905119899)))minus1

times [119906119899+ (119862 (119909 (119905

119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)]

(36)

Thus there exists a constant 1198695such that

E100381610038161003816100381611987511989910038161003816100381610038162

le 21198612E100381610038161003816100381611990611989910038161003816100381610038162

+ 21198612E1003816100381610038161003816[(1198620 (119909 (119905119899)) minus 1198620 (119909119899)) Δ

+ (1198621(119909 (119905119899)) minus 119862

1(119909119899)) ((Δ119882

119899)2

minus Δ)] (119909119899minus 119909119899+1)10038161003816100381610038161003816

2

le 21198612E100381610038161003816100381611990611989910038161003816100381610038162

+ 411986121198722Δ2E1003816100381610038161003816(119909119899 minus 119909119899+1)

10038161003816100381610038162

le 211986121198691ΔE100381610038161003816100381612059811989910038161003816100381610038162

+ 211986121198712Δ21205792E1003816100381610038161003816120598119899+1

10038161003816100381610038162

+ 1198695Δ3

(37)

and there exists a constant 1198696such that

10038161003816100381610038161003816E (119875119899| F119905119899)10038161003816100381610038161003816

2

=10038161003816100381610038161003816E ((119868 + 119862 (119909 (119905

119899)))minus1

times [119906119899+ (119862 (119909 (119905

119899)) minus 119862 (119909

119899))

times (119909119899minus 119909119899+1)] | F

119905119899)10038161003816100381610038161003816

2

le 2119861210038161003816100381610038161003816E (119906119899| F119905119899)10038161003816100381610038161003816

2

+ 21198612

times10038161003816100381610038161003816E ((119862 (119909 (119905

119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1) | F119905119899)10038161003816100381610038161003816

2

le 21198612(1198692Δ2E100381610038161003816100381612059811989910038161003816100381610038162

+ 1198712Δ21205792E (1003816100381610038161003816120598119899+1

10038161003816100381610038162

| F119905119899))

+ 21198612Δ10038161003816100381610038161003816E ((119862

0(119909 (119905119899))minus 119862

0(119909119899)) (119909119899minus 119909119899+1) | F119905119899)10038161003816100381610038161003816

2

le 211986121198692Δ2E100381610038161003816100381612059811989910038161003816100381610038162

+ 211986121198712Δ21205792E (1003816100381610038161003816120598119899+1

10038161003816100381610038162

| F119905119899) + 1198696Δ4

(38)


Thus1003816100381610038161003816E ⟨120598119899 119875119899⟩

1003816100381610038161003816

le Δminus1(E10038161003816100381610038161003816E (119875119899| F119905119899)10038161003816100381610038161003816

2

) + ΔE100381610038161003816100381612059811989910038161003816100381610038162

le (1 + 211986121198692) ΔE

100381610038161003816100381612059811989910038161003816100381610038162

+ 12119861211987121205792ΔE1003816100381610038161003816120598119899+1

10038161003816100381610038162

+ 1198696Δ3

(39)

FromTheorem 5 we have1003816100381610038161003816E ⟨120575119899+1 120598119899⟩

1003816100381610038161003816 le 1198674Δ3+ ΔE

100381610038161003816100381612059811989910038161003816100381610038162

(40)

Therefore

E1003816100381610038161003816120598119899+1

10038161003816100381610038162

le (1 + 1198697Δ)E

100381610038161003816100381612059811989910038161003816100381610038162

+ 2119861211987121205792Δ (Δ + 12)E

1003816100381610038161003816120598119899+110038161003816100381610038162

+ (1198695+ 1198675+ 21198674+ 21198696) Δ3

(41)

where 1198697= 2(119861

21198691+ 2 + 2119861

21198692) Because Δ rarr 0 we can

assume 1 minus 2119861211987121205792Δ(Δ + 12) gt 0 without loss of generalityLet 1198698= 1198695+ 1198675+ 21198674+ 21198696 Then

E10038161003816100381610038161003816120598119899+1|2le (1 + 119869

7Δ)E

10038161003816100381610038161003816120598119899|2+ 1198698Δ3

= 1198698Δ2(1 + 119869

7Δ)119899+1

minus 1

1198697

le 1198699Δ2

(42)

where 1198699= 1198698((1198901198697119879 minus 1)119869

7)

4 Stability of the Implicit Taylor Methods

In this section we will discuss the stability properties of thenumericalmethods introduced in Section 2 Consider a scalarlinear test equation

d119909 (119905) = 119886119909 (119905) d119905 + 119887119909 (119905) d119882(119905) + 119888119909 (119905) d119873(119905)

119909 (1199050) = 1199090

(43)

where 119886 119887 and 119888 are real constants The solution of (43) is119909(119905) = 119909

0119890(119886minus(12)119887

2)119905+119887119882(119905)

(1 + 119888)119873(119905) and is mean-square (MS)

stable if 2119886 + 1198872 + 120582119888(2 + 119888) lt 0 [2]The one-step scheme of the test equation (43) is

119909119899+1

= 119877 (119886 119887 119888 Δ Δ119882119899 Δ119873119899) 119909119899 (44)

The numerical method is MS-stable if

119877 (119886 119887 119888 Δ 120582) = E (1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)) lt 1 (45)

where 119877(119886 119887 119888 Δ 120582) is called theMS-stability function of thenumerical method

If the Taylor method (2) is applied to the test equation(43) we obtain

119909119899+1

= 1198771(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (46)

where

1198771(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)

= 1 + (119886 minus1

21198872)Δ + 119887Δ119882

119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899+ 119887119888Δ119882

119899Δ119873119899

+1

21198882(Δ119873119899)2

(47)

Let 119901 = 119886Δ 119902 = 119887radicΔ and 119911 = 119888120582Δ Then the MS-stabilityfunction of the Taylor method is

1198771(119901 119902 119911 119888)

= E (11987721(119886 119887 119888 Δ Δ119882

119899 Δ119873119899))

= 1 + 2119901 + 1199022+ 1199012+1

21199011199022+1

21199024

+ (2 + 119888 + 2119901 + 1198881199022+ 21199022) 119911

+ (2 + 2119888 +1

21198882+ 1199022+ 119901) 119911

2

+ (119888 + 1) 1199113+1

41199114

(48)

Thus the strong Taylormethod (2) for the linear test equation(43) is MS-stable if 119877

1(119901 119902 119911 119888) lt 1

Applying the 120579-Taylor method (3) to the test equation(43) we obtain

119909119899+1

= 1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899 120579) 119909119899 (49)

where

1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899 120579)

=1

1 minus 119886120579Δ[1 + ((1 minus 120579) 119886 minus

1

21198872) Δ

+ 119887Δ119882119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

]

(50)

Then the MS-stability function of the 120579-Taylor method is

1198772(119901 119902 119911 119888 120579)

=1

(1 minus 119901120579)2[1198771(119901 119902 119911 119888)

minus 2119901120579 + 11990121205792minus 21199012120579

minus2119901119911120579 minus 1199011199112120579]

(51)


Thus the 120579-Taylormethod (3) for the linear test equation (43)is MS-stable if 119877

2(119901 119902 119911 119888) lt 1

Applying the balanced 120579-Taylor method (4) to the testequation (43) we obtain

119909119899+1

= 1198773(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (52)

where1198773(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)

= ((1 minus 119886120579Δ) 119868 + 119862119899)minus1

times [1 + ((1 minus 120579) 119886 minus1

21198872)Δ

+ 119887Δ119882119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

+ 119862119899]

(53)

Since E(11987723) is rather complex in the general case we try

to investigate the stability of balanced 120579-method (4) for thefollowing two typical cases

Case 1 Let 1198620= minus119886 and 119862

1= 0 Then applying the balanced

120579-Taylor method (4) with 119862119899= minus119886Δ to the test equation (43)

we obtain

119909119899+1

= 1198771015840

3(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (54)

where1198771015840

3(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)

= (1 minus 119886 (1 + 120579) Δ)minus1[1 minus 119886120579Δ minus

1

21198872Δ + 119887Δ119882

119899

+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

]

(55)

Then the MS-stability function of the balanced 120579-Taylormethod (4) with 119862

119899= minus119886Δ is

1198771015840

3(119901 119902 119911 119888)

=1

(1 minus 119901 (1 + 120579))2

times [1 + 1199022+1

21199011199022+1

21199024minus 2119901120579 + 119901

21205792

+ (2 + 119888 + 1198881199022+ 21199022minus 2119901120579) 119911

+ (2 + 2119888 +1

21198882+ 1199022minus 119901120579) 119911

2

+ (119888 + 1) 1199113+1

41199114]

(56)

Thus the balanced 120579-Taylor method (4) with 119862119899= minus119886Δ for

the linear test equation (43) is MS-stable if 11987710158403(119901 119902 119911 119888) lt 1


1= 1198872 Then applying the

balanced 120579-Taylormethod (4)with119862119899= minus119886Δ+119887

2((Δ119882119899)2minusΔ)

to the test equation (43) we have

119909119899+1

= 11987710158401015840

3(119901 119902 119888 119869 Δ119873

119899) 119909119899 (57)

where11987710158401015840

3(119901 119902 119888 119869 Δ119873

119899)

= (1 minus 119901120579 + 1199022(1198692minus 1))minus1

times [1 minus 119901120579 minus3

21199022+ 119902119869 +

3

211990221198692

+1

2(2119888 minus 119888

2) Δ119873119899+ 119902119888119869Δ119873

119899

+1

21198882(Δ119873119899)2

]

(58)

and 119869 is the standard Gaussian random variable 119869 =

Δ119882119899radicΔ sim 119873(0 1) Then the MS-stability function of the


2((Δ119882119899)2minusΔ)

is

11987710158401015840

3(119901 119902 119911 119888 119869 Δ119873

119899)

=1

radic2120587

+infin

sum

minusinfin

(120582Δ)119898119890minus120582Δ

119898

times int

+infin

minusinfin

(11987710158401015840

3(119901 119902 119888 119909119898))

2

119890minus(11990922)d119909

(59)

Thus the balanced 120579-Taylor method (4) with 119862119899= minus119886Δ +

1198872((Δ119882119899)2minus Δ) for the linear test equation (43) is MS-stable

if 119877101584010158403(119901 119902 119911 119888) lt 1For the case of 119888 = minus1 and 119911 = minus1 the MS-stable regions

of the numerical methods for the test equation are plottedin Figures 1 and 2 Figure 1 shows the MS-stable regions ofTaylor method the 120579-Taylor method and the balanced 120579-Taylor method with 119862

0= minus119886 and 119862

1= 0 when 120579 = 12 and

120579 = 1 Figure 2 shows theMS-stable regions of Taylormethodthe 120579-Taylor method and the balanced 120579-Taylor method with1198620= minus119886 and 119862

1= 1198872 when 120579 = 14 and 120579 = 12 It should

be noted that the MS-stable regions are the areas below theplotted curves and symmetric about the 119901-axis From Figures1 and 2 it is observed that the MS-stable regions of the 120579-Taylor method and the balanced 120579-Taylor method increaseas the parameter 120579 increases The MS-stable properties ofthe 120579-Taylor method and the balanced 120579-Taylor method arebetter than the Taylor method Furthermore the MS-stableproperties of the balanced 120579-Taylor method with 119862

0= minus119886

and 1198621= 0 are better than those of the 120579-Taylor method for

all 120579 isin [0 1] In addition the MS-stable properties of thebalanced 120579-Taylor method with 119862

0= minus119886 and 119862

1= 1198872 are

better than those of the 120579-Taylor method when 120579 le 14 andthe MS-stable properties of the 120579-Taylor method are betterthan those of the balanced 120579-Taylor method with 119862

0= minus119886

and 1198621= 1198872 when 120579 ge 12


Table 1 Mean of the absolute errors for different values of Δ and different methods

Methods Δ 2minus8

2minus7

2minus6

2minus5

2minus4

2minus3

2minus2

2minus1

120579-Taylor120579 = 0 00024 00048 00099 00186 00389 00911 06853 30473120579 = 12 00022 00043 00090 00177 00368 00805 02548 07407120579 = 1 00042 00083 00170 00342 00699 01454 03025 05568

Balanced120579-Taylor

120579 = 0 00021 00041 00083 00152 00317 00665 03852 15570120579 = 12 00076 00134 00223 00361 00619 01504 07024 31984120579 = 1 00075 00132 00217 00347 00582 01360 06285 26648

q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 1

Taylor

Balanced 120579-Taylor with 120579 = 12


2

18

16

14

12

1

08

06

04

02

0minus2 minus15 minus1 minus05 0

Figure 1 MS-stable regions of the 120579-Taylor methods and thebalanced 120579-Taylor method with 119862

0= minus119886 and 119862

1= 0

5 Numerical Examples

In this section we conduct some simulation to demonstratethe convergence of the proposed implicit Taylor numericalsolutions (3) and (4) for the equation system (43) with thecoefficients 119886 = minus4 119887 = 1 119888 = minus05 and the jump intensity120582 = 2 We compare the explicit solutions with the numericalapproximations for the step-sizes Δ = 2

minus1 2minus2 2

minus8To measure the accuracy and convergence property of theproposed methods we compute mean of the absolute errorsas

119890 =1

2000

2000

sum

119894=1

10038161003816100381610038161003816119909(119894)

119873minus 119909(119894)(119905119873)10038161003816100381610038161003816 (60)

In Table 1 we report the simulated errors of the 120579-Taylormethod and the balanced 120579-Taylor method with 119862

0= 1

and 1198621= 1 for different values of 120579 and Δ Note that the

Taylor method is a special case of the 120579-Taylor method with120579 = 0 FromTable 1 we know that the accuracy of the 120579-Taylormethod with 120579 = 12 and the balanced 120579-Taylor method with120579 = 0 is higher than that of the Taylor method The accuracyof the balanced 120579-Taylor method with 120579 = 0 is the highestfor Δ le 2

minus2 When 120579 ge 12 the accuracy of the 120579-Taylor

q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 14

Taylor



2

18

16

14

12

1

08

06

04

02

0minus6 minus5 minus4 minus3 minus2 minus1 0


0= minus119886 and 119862

1= 1198872

method is higher than that of the balanced 120579-Taylor methodwith 119862

0= 1 and 119862

1= 1

6 Conclusions

In this paper we introduce two kinds of the implicit methodsthe 120579-Taylor method and the balanced 120579-Taylor method forsolving stochastic differential equations with Poisson jumpsIt is proved that the proposed numerical methods have astrong convergence order of 10 Moreover the MS-stableregions of the proposed numerical methods are derived fora linear scalar test equation and it is demonstrated that the120579-Taylor method and the balanced 120579-Taylor method havebetter stable properties than the Taylor method As hasbeen confirmed by the theoretical and the numerical resultsthe proposed numerical methods perform satisfactorily insolving SDEJs

Conflict of Interests

The authors declare that they have no conflict of interestsregarding to the publication of this paper


References

[1] D J Higham ldquoMean-square and asymptotic stability of thestochastic theta methodrdquo SIAM Journal on Numerical Analysisvol 38 no 3 pp 753ndash769 2000

[2] D J Higham and P E Kloeden ldquoConvergence and stabilityof implicit methods for jump-diffusion systemsrdquo InternationalJournal of Numerical Analysis and Modeling vol 3 no 2 pp125ndash140 2006

[3] D J Higham and P E Kloeden ldquoStrong convergence ratesfor backward Euler on a class of nonlinear jump-diffusionproblemsrdquo Journal of Computational and Applied Mathematicsvol 205 no 2 pp 949ndash956 2007

[4] X Mao Y Shen and A Gray ldquoAlmost sure exponential sta-bility of backward Euler-Maruyama discretizations for hybridstochastic differential equationsrdquo Journal of Computational andApplied Mathematics vol 235 no 5 pp 1213ndash1226 2011

[5] L Chen and FWu ldquoAlmost sure decay stability of the backwardEuler-Maruyama scheme for stochastic differential equationswith unbounded delayrdquo Applied Mechanics and Materials vol235 pp 39ndash44 2012

[6] L Hu and S Gan ldquoMean-square convergence of drift-implicitone-step methods for neutral stochastic delay differential equa-tions with jump diffusionrdquo Discrete Dynamics in Nature andSociety vol 2011 Article ID 917892 22 pages 2011

[7] D J Higham and P E Kloeden ldquoNumerical methods for non-linear stochastic differential equations with jumpsrdquoNumerischeMathematik vol 101 no 1 pp 101ndash119 2005

[8] X-H Ding Q Ma and L Zhang ldquoConvergence and stabilityof the split-step 120579-method for stochastic differential equationsrdquoComputers amp Mathematics with Applications vol 60 no 5 pp1310ndash1321 2010

[9] X Wang and S Gan ldquoB-convergence of split-step one-leg thetamethods for stochastic differential equationsrdquo Journal of AppliedMathematics and Computing vol 38 no 1-2 pp 489ndash503 2012

[10] E Buckwar and T Sickenberger ldquoA comparative linear mean-square stability analysis of Maruyama- and MILstein-typemethodsrdquo Mathematics and Computers in Simulation vol 81no 6 pp 1110ndash1127 2011

[11] G N Milstein E Platen and H Schurz ldquoBalanced implicitmethods for stiff stochastic systemsrdquo SIAM Journal on Numeri-cal Analysis vol 35 no 3 pp 1010ndash1019 1998

[12] K Burrage and T Tian ldquoThe composite Euler method for stiffstochastic differential equationsrdquo Journal of Computational andApplied Mathematics vol 131 no 1-2 pp 407ndash426 2001

[13] T H Tian and K Burrage ldquoImplicit Taylor methods for stiffstochastic differential equationsrdquoAppliedNumericalMathemat-ics vol 38 no 1-2 pp 167ndash185 2001

[14] G N Milstein and M V Tretyakov Stochastic Numerics forMathematical Physics Scientific Computation Springer BerlinGermany 2004

[15] H Schurz ldquoConvergence and stability of balanced implicitmethods for systems of SDEsrdquo International Journal of Numeri-cal Analysis and Modeling vol 2 no 2 pp 197ndash220 2005

[16] C Kahl and H Schurz ldquoBalanced Milstein methods for ordi-nary SDEsrdquoMonte Carlo Methods and Applications vol 12 no2 pp 143ndash170 2006

[17] J Alcock and K Burrage ldquoA note on the balancedmethodrdquo BITNumerical Mathematics vol 46 no 4 pp 689ndash710 2006

[18] S S AhmadN Chandra Parida and S Raha ldquoThe fully implicitstochastic-120572 method for stiff stochastic differential equationsrdquo

Journal of Computational Physics vol 228 no 22 pp 8263ndash8282 2009

[19] M A Omar A Aboul-Hassan and S I Rabia ldquoThe compositeMilstein methods for the numerical solution of Stratonovichstochastic differential equationsrdquo Applied Mathematics andComputation vol 215 no 2 pp 727ndash745 2009

[20] P Wang and Z-X Liu ldquoStabilized Milstein type methods forstiff stochastic systemsrdquo Journal of Numerical Mathematics andStochastics vol 1 no 1 pp 33ndash44 2009

[21] P Wang and Z Liu ldquoSplit-step backward balanced Milsteinmethods for stiff stochastic systemsrdquo Applied Numerical Math-ematics vol 59 no 6 pp 1198ndash1213 2009

[22] L Hu and S Gan ldquoConvergence and stability of the balancedmethods for stochastic differential equations with jumpsrdquoInternational Journal of Computer Mathematics vol 88 no 10pp 2089ndash2108 2011

[23] A Haghighi and S M Hosseini ldquoA class of split-step balancedmethods for stiff stochastic differential equationsrdquo NumericalAlgorithms vol 61 no 1 pp 141ndash162 2012

[24] X Wang S Gan and D Wang ldquoA family of fully implicitMilstein methods for stiff stochastic differential equations withmultiplicative noiserdquo BIT Numerical Mathematics vol 52 no3 pp 741ndash772 2012

[25] L Hu and S Gan ldquoNumerical analysis of the balanced implicitmethods for stochastic pantograph equations with jumpsrdquoApplied Mathematics and Computation vol 223 pp 281ndash2972013

[26] L Hu S Gan and X Wang ldquoAsymptotic stability of balancedmethods for stochastic jump-diffusion differential equationsrdquoJournal of Computational andAppliedMathematics vol 238 pp126ndash143 2013

[27] Y Maghsoodi ldquoMean square efficient numerical solution ofjump-diffusion stochastic differential equationsrdquo Sankhya TheIndian Journal of Statistics vol 58 no 1 pp 25ndash47 1996

[28] Y Maghsoodi and C J Harris ldquoIn-probability approximationand simulation of nonlinear jump-diffusion stochastic differ-ential equationsrdquo IMA Journal of Mathematical Control andInformation vol 4 no 1 pp 65ndash92 1987

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Let (ΩF F119905119905isin[1199050 119879]

P) be a complete probability spacewith the filtration F

119905119905isin[1199050 119879]

satisfying the usual conditionsthat F

119905is right-continuous and F

0contains all P-null sets

In this paper we consider the stochastic differential equationswith jumps of the form

d119909 (119905) = 119891 (119909 (119905)) d119905 + 119892 (119909 (119905)) d119882(119905) + ℎ (119909 (119905)) d119873(119905)

119905 isin [1199050 119879]

119909 (1199050) = 1199090

(1)

where119882(119905) isF119905-adaptedWiener process and119873(119905) is a scalar

poisson process with intensity 120582 and is independent of119882(119905)Hu andGan [22 25] proposed the balancedmethod for SDEJs(1) and stochastic pantograph equations with jumps respec-tively and proved that the numerical solution converges tothe analytical solution with rate 12 The asymptotic stabilityof the balanced method for SDEJs (1) was obtained in [26]To obtain higher order numerical schemes and improve theaccuracy of the numerical solutions we propose two kindsof implicit Taylor methods and prove that the numericalsolutions converge to the true solutions of SDEJs (1) with rate10

The rest of the paper is arranged as follows In Section 2we introduce the 120579-Taylor methods and the fully implicitbalanced 120579-Taylor methods for SDEJs (1) The strong con-vergence properties of these implicit methods are proved inSection 3 The mean-square stability of the numerical solu-tions is discussed in Section 4 Some numerical experimentsare performed in Section 5 to evaluate the performance of theproposed numerical methods

2 The Numerical Methods

Define a mesh 0 le 1199050lt 1199051lt sdot sdot sdot lt 119905

119899lt 119905119899+1

lt sdot sdot sdot lt 119905119873= 119879

on the time interval [1199050 119879] with 119905

119899= 119899Δ and the step-size

Δ = 119879119873 119909119899is the numerical approximation to 119909(119905

119899) Based

on appropriate stochastic Taylor expansions Maghsoodi [27]generalized the Milstein scheme to SDEJs and obtained theorder 10 strong Taylor scheme (Taylor for short) as

119909119899+1

= 119909119899+ 119891 (119909

119899) Δ + 119892 (119909

119899) Δ119882119899

+1

2119892 (119909119899) 1198921015840(119909119899) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909

119899) minus ℎℎ(119909119899)) Δ119873

119899

+ (119892ℎ(119909119899) minus 119892 (119909

119899)) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909119899) minus ℎ (119909

119899)) (Δ119873

119899)2

+ (119892 (119909119899) ℎ1015840(119909119899) minus 119892ℎ(119909119899) + 119892 (119909

119899)) Δ119885

119899

(2)

where 119892ℎ(119909) = 119892(119909 + ℎ(119909)) ℎ

ℎ(119909) = ℎ(119909 + ℎ(119909)) and Δ119885

119899=

int119905119899+1

119905119899

int119904

119905119899

d119882119905d119873119904= int119905119899+1

119905119899

(119882119904minus119882119905119899)d119873119904

Note thatΔ119885119899= sum119895(119882(120591119895)minus119882(119905

119896)) = sum

119873119899+1

119895=119873119899+1(119873119899+1minus119895+

1)(119882(120591119895) minus 119882(120591

119895minus1)) [28] Given a jump time 120591

119895in [119905119899 119905119899+1)

Δ1198851119899(120591119895) = 119882(120591

119895) minus 119882(120591

119895minus1) sim 119873(0 120591

119895minus 120591119895minus1) (119873119899+ 1 le 119895 le

119873119899+1

) In addition the random variable Δ119882119899= 119882(119905

119899+1) minus

119882(119905119899) is dependent on Δ119885

1119899(120591119895) and its sample values can be

calculated by Δ119882119899= sum119873119899+1

119895=119873119899+1Δ1198851119899(120591119895) + Δ119885

1119899(119905119899+1) where

Δ1198851119899(119905119899+1) = 119882(119905

119899+1) minus 119882(120591

119873119899+1) sim 119873(0 119905

119899+1minus 120591119873119899+1

)By changing the explicit deterministic term into implicit

term we have the following 120579-Taylor method

119909119899+1

= 119909119899+ Δ [(1 minus 120579) 119891 (119909

119899) + 120579119891 (119909

119899+1)]

+ 119892 (119909119899) Δ119882119899+1

2119892 (119909119899) 1198921015840(119909119899) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909

119899) minus ℎℎ(119909119899)) Δ119873

119899

+ (119892ℎ(119909119899) minus 119892 (119909

119899)) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909119899) minus ℎ (119909

119899)) (Δ119873

119899)2

+ (119892 (119909119899) ℎ1015840(119909119899) minus 119892ℎ(119909119899) + 119892 (119909

119899)) Δ119885

119899

(3)

Note that the 120579-Taylor method (3) becomes the Taylormethod (2) when 120579 = 0

Using the idea of the balanced implicit method andcombining it with the 120579-Taylormethod we have the followingbalanced 120579-Taylor method

119909119899+1

= 119909119899+ Δ [(1 minus 120579) 119891 (119909

119899) + 120579119891 (119909

119899+1)]

+ 119892 (119909119899) Δ119882119899+1

2119892 (119909119899) 1198921015840(119909119899) [(Δ119882

119899)2minus Δ]

+1

2(3ℎ (119909

119899) minus ℎℎ(119909119899)) Δ119873

119899

+ (119892ℎ(119909119899) minus 119892 (119909

119899)) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909119899) minus ℎ (119909

119899)) (Δ119873

119899)2

+ (119892 (119909119899) ℎ1015840(119909119899) minus 119892ℎ(119909119899) + 119892 (119909

119899)) Δ119885

119899

+ 119862119899(119909119899minus 119909119899+1)

(4)

where 119862119899= 119862(119909

119899) = 119862

0(119909119899)Δ + 119862

1(119909119899)[(Δ119882

119899)2minus Δ] with

1198620(sdot) and 119862

1(sdot) called control functions

3 Convergence of the Implicit Taylor Methods

Let | sdot | be the Euclidean norm in R119889 If 119860 is a matrix |119860| =radictrace(119860119879119860) Denote 119911

1198712= (E|119911|

2)12 for 119911 isin R119889 To

prove the convergence of the numerical solutions we makethe following assumptions



1003816100381610038161003816119891 (1199091) minus 119891 (1199092)1003816100381610038161003816 +1003816100381610038161003816119892 (1199091) minus 119892 (1199092)

1003816100381610038161003816

+1003816100381610038161003816ℎ (1199091) minus ℎ (1199092)

1003816100381610038161003816

+10038161003816100381610038161003816119892 (1199091) 1198921015840(1199091) minus 119892 (119909

2) 1198921015840(1199092)10038161003816100381610038161003816

+10038161003816100381610038161003816119892 (1199091) ℎ1015840(1199091) minus 119892 (119909

2) ℎ1015840(1199092)10038161003816100381610038161003816le 119871

10038161003816100381610038161199091 minus 11990921003816100381610038161003816

(5)


growth condition

1003816100381610038161003816119891(119909)10038161003816100381610038162

+1003816100381610038161003816119892(119909)

10038161003816100381610038162

+ |ℎ(119909)|2

+10038161003816100381610038161003816119892(119909)119892

1015840(119909)10038161003816100381610038161003816

2

+10038161003816100381610038161003816119892(119909)ℎ

1015840(119909)10038161003816100381610038161003816

2

le 1198711015840(1 + |119909|

2)

(6)





0] and 120572

1isin

[minus1205721 1205721] with 120572

0ge Δ and 120572

1ge |(Δ119882



11198621(119909) is




119909120579(119905119899+1) = 119909 (119905

119899)

+ Δ [(1 minus 120579) 119891 (119909 (119905119899))

+ 120579119891 (119909 (119905119899+1))] + 119892 (119909 (119905

119899)) Δ119882

119899

+1

2119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899))

minus119892ℎ(119909 (119905119899)) + 119892 (119909 (119905

119899))) Δ119885

119899

(7)



119899+1)minus


120598119899= 119909(119905119899) minus 119909119899



10038171003817100381710038171003817E(120575120579(119905119899+1) | F119905119899)100381710038171003817100381710038171198712

le 1198671Δ2

119886119904 Δ 997888rarr 0 (8)


10038171003817100381710038171003817120575120579(119905119899+1)100381710038171003817100381710038171198712

le 1198672Δ32

119886119904 Δ 997888rarr 0 (9)





119909119860

119899+1= 119909 (119905

119899) + 119891 (119909 (119905

119899)) Δ + 119892 (119909 (119905

119899)) Δ119882

119899

+1

2119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+ 119892 (119909 (119905119899))) Δ119885

119899

(10)


E [10038161003816100381610038161003816E [(119909120579

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816E [(119909119860

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

]

+ 2E [10038161003816100381610038161003816E [(119909120579

119899+1minus 119909119860

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

le 1198701Δ4+ 2E [

10038161003816100381610038161003816E [(119909120579

119899+1minus 119909119860

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

(11)

Since

E [10038161003816100381610038161003816E [(119909120579

119899+1minus 119909119860

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

= E [10038161003816100381610038161003816E [120579Δ (119891 (119909 (119905

119899+1)) minus 119891 (119909 (119905

119899))) | F

119905119899]10038161003816100381610038161003816

2

]

le 119871120579ΔE[E [1003816100381610038161003816119909(119905119899+1) minus 119909(119905119899)

1003816100381610038161003816 | F119905119899]]2

le 119874 (Δ4)

(12)

we obtain (E[|E[(119909120579119899+1

minus 119909(119905119899+1)) | F

119905119899]|2

])12

le 119874(Δ2)



E [10038161003816100381610038161003816119909120579

119899+1minus 119909119860

119899+1

10038161003816100381610038161003816

2

| 119909119899= 119909 (119905

119899)]

= 11987121205792Δ2E [1003816100381610038161003816119909 (119905119899+1) minus 119909 (119905119899)

10038161003816100381610038162

]

le 119874 (Δ3)

(13)

we have

E [10038161003816100381610038161003816119909120579

119899+1minus 119909 (119905

119899+1)10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816119909119860

119899+1minus 119909 (119905

119899+1)10038161003816100381610038161003816

2

]

+ 2E [10038161003816100381610038161003816119909120579

119899+1minus 119909119860

119899+1

10038161003816100381610038161003816

2

]

le 119874 (Δ3)

(14)




100381710038171003817100381710038171205982

119899+1

100381710038171003817100381710038171198712le 1198673Δ 119886119904 Δ 997888rarr 0 (15)



120598119899+1

= 120598119899+ 119906119899+ 120575119899+1 (16)

where119906119899= Δ (1 minus 120579) (119891 (119909 (119905

119899)) minus 119891 (119909

119899))

+ Δ120579 (119891 (119909 (119905119899+1)) minus 119891 (119909

119899+1))

+ (119892 (119909 (119905119899)) minus 119892 (119909

119899)) Δ119882

119899

+1

2(119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899))

minus 119892 (119909119899) 1198921015840(119909119899)) [(Δ119882)

2minus Δ]

+1

2[3 (ℎ (119909 (119905

119899)) minus ℎ (119909

119899))

minus (ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899))] Δ119873

119899

+ [(119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899))

minus (119892 (119909 (119905119899)) minus 119892 (119909

119899))] Δ119882

119899Δ119873119899

+1

2[(ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899))

minus (ℎ (119909 (119905119899)) minus ℎ (119909

119899))] (Δ119873

119899)2

+ [(119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892 (119909

119899) ℎ1015840(119909119899))

minus (119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899))

+ (119892 (119909 (119905119899)) minus 119892 (119909

119899))] Δ119885

119899

(17)


1003816100381610038161003816E ⟨120575119899+1 120598119899⟩1003816100381610038161003816

=10038161003816100381610038161003816E [E (⟨120575

119899+1 120598119899⟩ | F119905119899)]10038161003816100381610038161003816

le E10038161003816100381610038161003816⟨E (120575

119899+1| F119905119899) 120598119899⟩10038161003816100381610038161003816

le [Δminus1(E10038161003816100381610038161003816E (120575119899+1

| F119905119899)10038161003816100381610038161003816

2

)]12

times (ΔE100381610038161003816100381612059811989910038161003816100381610038162

)12

le Δminus1(E10038161003816100381610038161003816E (120575119899+1

| F119905119899)10038161003816100381610038161003816

2

)

+ ΔE100381610038161003816100381612059811989910038161003816100381610038162

le 1198671Δ3+ ΔE

100381610038161003816100381612059811989910038161003816100381610038162

(18)


119899|2= Δ E|Δ119882

119899|4= 3Δ2 E(Δ119873

119899)2=

120582Δ(1 + 120582Δ) E(Δ119873119899)4= 120582Δ(1 + 7120582Δ + 6(120582Δ)

2+ (120582Δ)

3) and



E1003816100381610038161003816(119892 (119909 (119905119899)) minus 119892 (119909119899)) Δ119882119899

10038161003816100381610038162

le ΔE1003816100381610038161003816119892 (119909 (119905119899)) minus 119892 (119909119899)

10038161003816100381610038162

le 1198712ΔE|120598119899|2

E10038161003816100381610038161003816(119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) minus 119892 (119909

119899) 1198921015840(119909119899)) [(Δ119882)

2minus Δ]

10038161003816100381610038161003816

2

le 2Δ2E10038161003816100381610038161003816119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) minus 119892 (119909

119899) 1198921015840(119909119899)10038161003816100381610038161003816

le 21198712Δ2E|120598119899|2


10038161003816100381610038162

le 2120582Δ (1 + 120582Δ) [3E1003816100381610038161003816ℎ (119909 (119905119899)) minus ℎ (119909119899)

10038161003816100381610038162

+ E1003816100381610038161003816ℎ (119909 (119905119899) + ℎ (119909 (119905119899))) minus ℎ (119909119899 + ℎ (119909119899))

10038161003816100381610038162

]

le 21198712120582Δ (1 + 120582Δ) (5 + 2119871

2)E100381610038161003816100381612059811989910038161003816100381610038162


10038161003816100381610038162


10038161003816100381610038162

sdot E|Δ119882119899|2sdot E(Δ119873

119899)2

le 21198712120582Δ2(1 + 120582Δ) (3 + 2119871

2)E|120598119899|2

E10038161003816100381610038161003816[(ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899)) minus (ℎ (119909 (119905

119899)) minus ℎ (119909

119899))] (Δ119873

119899)210038161003816100381610038161003816

2

le 2120582Δ (1 + 7120582Δ + 6(120582Δ)2+ (120582Δ)

3) 1198712(3 + 2119871

2)E100381610038161003816100381612059811989910038161003816100381610038162

E10038161003816100381610038161003816[ (119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899)) minus 119892 (119909

119899) ℎ1015840(119909119899))

minus (119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899)) + (119892 (119909 (119905

119899)) minus 119892 (119909

119899))] Δ119885

119899

10038161003816100381610038162


le 61198712(2 + 119871

2)E100381610038161003816100381612059811989910038161003816100381610038162

sdot E1003816100381610038161003816Δ119885119899

10038161003816100381610038162

le 61198712(2 + 119871

2)E100381610038161003816100381612059811989910038161003816100381610038162

sdot E1003816100381610038161003816Δ119882119899 minus Δ1198851119899(119905119899+1)

10038161003816100381610038162

le 241198712(2 + 119871

2) ΔE

100381610038161003816100381612059811989910038161003816100381610038162

(19)

Hence

E100381610038161003816100381611990611989910038161003816100381610038162

le 1198691ΔE100381610038161003816100381612059811989910038161003816100381610038162

+ 81198712Δ21205792E1003816100381610038161003816120598119899+1

10038161003816100381610038162

(20)

where 1198691= 8[4119871

2+ 21198712120582(1 + 120582)(5 + 2119871

2) + 2119871

2120582(1 + 120582)(3 +

21198712) + 2120582(1 + 7120582 + 6120582

2+ 1205823)1198712(3 + 2119871

2) + 24119871

2(2 + 119871

2)

Noting that 119909(119905119899) and 119909

119899are F


119899


119905119899 we have

E [(119892 (119909 (119905119899)) minus 119892 (119909

119899)) Δ119882

119899F119905119899] = 0

E [(119892 (119909 (119905119899)) 1198921015840(119909 (119905119899)) minus 119892 (119909

119899) 1198921015840(119909119899))


119905119899] = 0

E [[(119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899)) minus (119892 (119909 (119905

119899)) minus 119892 (119909

119899))]

timesΔ119882119899Δ119873119899| F119905119899] = 0

E [[(119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892 (119909

119899) ℎ1015840(119909119899))

minus (119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899))

+ (119892 (119909 (119905119899)) minus 119892 (119909

119899)) ] Δ119885

119899| F119905119899] = 0

(21)

Therefore10038161003816100381610038161003816E (119906119899| F119905119899)10038161003816100381610038161003816

2

le

1003816100381610038161003816100381610038161003816Δ (1 minus 120579)E [119891 (119909 (119905

119899)) minus 119891 (119909

119899) | F119905119899]

+ 120579ΔE [119891 (119909 (119905119899+1)) minus 119891 (119909

119899+1) | F119905119899]

+3

2E [(ℎ (119909 (119905

119899)) minus ℎ (119909

119899)) Δ119873

119899| F119905119899]

minus1

2E [(ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899)) Δ119873

119899| F119905119899]

+1

2E [(ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899)) (Δ119873

119899)2

| F119905119899]

minus1

2E [(ℎ (119909 (119905

119899)) minus ℎ (119909

119899)) (Δ119873

119899)2

| F119905119899]

1003816100381610038161003816100381610038161003816

2

le 1198692Δ2E100381610038161003816100381612059811989910038161003816100381610038162

+ 611987121205792Δ2E [1003816100381610038161003816120598119899+1

10038161003816100381610038162

| F119905119899]

(22)

where 1198692= 31198712[2 + 3120582

2+ (1 + 119871)

21205822+ (1 + 119871)

2(1 + 120582)

2+

1205822(1 + 120582)

2] Thus

1003816100381610038161003816E ⟨120598119899 119906119899⟩1003816100381610038161003816 le Δminus1(E10038161003816100381610038161003816E (119906119899| F119905119899)10038161003816100381610038161003816

2

) + ΔE100381610038161003816100381612059811989910038161003816100381610038162

le (1 + 1198692) ΔE

100381610038161003816100381612059811989910038161003816100381610038162

+ 611987121205792ΔE1003816100381610038161003816120598119899+1

10038161003816100381610038162

(23)


E1003816100381610038161003816120598119899+1

10038161003816100381610038162

le E100381610038161003816100381612059811989910038161003816100381610038162

+ E1003816100381610038161003816120575119899+1

10038161003816100381610038162

+ E100381610038161003816100381611990611989910038161003816100381610038162

+ 21003816100381610038161003816E ⟨120575119899+1 120598119899⟩

1003816100381610038161003816 + 21003816100381610038161003816E ⟨120598119899 119906119899⟩

1003816100381610038161003816

le [1 + 2 (2 + 1198691+ 1198692) Δ]E

100381610038161003816100381612059811989910038161003816100381610038162

+ 211987121205792Δ (Δ + 6)E

1003816100381610038161003816120598119899+110038161003816100381610038162

+ (1198672+ 21198671) Δ3

(24)


3= 2(2 + 119869

1+ 1198692) Then

E1003816100381610038161003816120598119899+1

10038161003816100381610038162

le (1 + 1198693Δ)E

100381610038161003816100381612059811989910038161003816100381610038162

+ (1198672+ 21198671) Δ3

= (1198672+ 21198671) Δ2(1 + 119869

3Δ)119899+1

minus 1

1198693

le 1198694Δ2

(25)

where 1198694= (1198672+ 21198671)((1198901198693119879 minus 1)119869

3)


119909119861(119905119899+1)

= 119909 (119905119899) + Δ [(1 minus 120579) 119891 (119909 (119905

119899)) + 120579119891 (119909 (119905

119899+1))]

+ 119892 (119909 (119905119899)) Δ119882

119899

+1

2119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+ 119892 (119909 (119905119899))) Δ119885

119899+ 119862119899(119909 (119905119899) minus 119909119861(119905119899+1))

(26)


119899+1) = 119909(119905

119899+1) minus 119909119861(119905119899+1) and the






10038171003817100381710038171003817E (120575119861(119905119899+1) | F119905119899)100381710038171003817100381710038171198712

le 1198674Δ2

119886119904 Δ 997888rarr 0


10038171003817100381710038171003817120575119861(119905119899+1)100381710038171003817100381710038171198712

le 1198675Δ32

119886119904 Δ 997888rarr 0

(27)




E [10038161003816100381610038161003816E [(119909119861

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816E [(119909120579

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

]

+ 2E [10038161003816100381610038161003816E [(119909119861

119899+1minus 119909120579

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

le 1198672

1Δ4+ 2E [

10038161003816100381610038161003816E [(119909119861

119899+1minus 119909120579

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

(28)


and 119909119861119899+1


119909119861

119899+1minus 119909120579

119899+1

= minus119862119899(119909 (119905119899) minus 119909119861(119905119899+1))

= 119862119899[Δ [(1 minus 120579) 119891 (119909 (119905

119899)) + 120579119891 (119909 (119905

119899+1))]

+ 119892 (119909 (119905119899)) Δ119882

119899

+1

2119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+119892 (119909 (119905119899))) Δ119885

119899+ 119862119899(119909119899minus 119909119861

119899+1)]

= (119868 + 119862119899)minus1

119862119899

times [Δ [(1 minus 120579) 119891 (119909 (119905119899)) + 120579119891 (119909 (119905

119899+1))]

+ 119892 (119909 (119905119899)) Δ119882

119899+1

2119892 (119909 (119905

119899))

times 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+119892 (119909 (119905119899))) Δ119885

119899]

(29)


1(sdot) in 119862

119899(sdot)


119899)minus1

119862119899Δ119891 (119909 (119905

119899)) | F

119905119899]10038161003816100381610038161003816

le 119872Δ1003816100381610038161003816119891 (119909119899)

1003816100381610038161003816E [100381610038161003816100381610038161198620Δ + 119862

1((Δ119882

119899)2

minus Δ)10038161003816100381610038161003816| F119905119899]

le 1198711015840119872119861(1 +

1003816100381610038161003816119909 (119905119899)10038161003816100381610038162

)12

Δ2

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899119892 (119909 (119905

119899)) Δ119882

119899| F119905119899]10038161003816100381610038161003816= 0

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899))

times [(Δ119882119899)2

minus Δ] | F119905119899]10038161003816100381610038161003816

le 11986111987210038161003816100381610038161003816119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899))10038161003816100381610038161003816

E [1003816100381610038161003816100381610038161003816((Δ119882

119899)2

minus Δ)21003816100381610038161003816100381610038161003816| F119905119899]

le 21198711015840119872119861(1 +

100381610038161003816100381611990911989910038161003816100381610038162

)12

Δ2

10038161003816100381610038161003816[E [(119868 + 119862

119899)minus1

119862119899(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899] | F119905119899]10038161003816100381610038161003816

le 12058211986111987210038161003816100381610038163ℎ (119909 (119905119899)) minus ℎℎ (119909 (119905119899))

1003816100381610038161003816 Δ2

le (2 + 119871) 1198711015840119861119872(1 +

1003816100381610038161003816119909 (119905119899)10038161003816100381610038162

)12

Δ2

10038161003816100381610038161003816E [(119868+ 119862

119899)minus1

119862119899(119892ℎ(119909 (119905119899))minus 119892 (119909 (119905

119899)))

times Δ119882119899Δ119873119899| F119905119899]10038161003816100381610038161003816= 0

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899)))

times (Δ119873119899)2

| F119905119899]10038161003816100381610038161003816

le 1198611198721003816100381610038161003816ℎℎ (119909 (119905119899)) minus ℎ (119909 (119905119899))

1003816100381610038161003816 120582Δ2(1 + 120582Δ)

le 1198711015840119861119872120582(1 +

1003816100381610038161003816119909 (119905119899)10038161003816100381610038162

)12

Δ2(1 + 120582Δ)

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899(119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899))

minus 119892ℎ(119909 (119905119899)) + 119892 (119909 (119905

119899))) Δ119885

119899| F119905119899]10038161003816100381610038161003816

=10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899(119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899))

minus 119892ℎ(119909 (119905119899)) + 119892 (119909 (119905

119899)))

times (Δ119882119899minus Δ1198851119896(119905119899+1)) | F

119905119899]10038161003816100381610038161003816

= 0

(30)


Therefore

E [10038161003816100381610038161003816E [(119909119861

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

] le 119874 (Δ4) (31)


E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899Δ [(1 minus 120579) 119891 (119909 (119905

119899)) + 120579119891 (119909 (119905

119899+1))]10038161003816100381610038161003816

2

]

le 119874 (Δ4)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899119892 (119909119899) Δ119882119899

10038161003816100381610038161003816

2

] le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]10038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

10038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

10038161003816100381610038161003816

2

]

le 119874 (Δ4)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)210038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+ 119892 (119909 (119905119899))) (Δ119885

119899)10038161003816100381610038161003816

2

]

le 119874 (Δ3)

(32)

we have

E [10038161003816100381610038161003816119909119861

119899+1minus 119909 (119905

119899+1)10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816119909 (119905119899+1) minus 119909120579

119899+1

10038161003816100381610038161003816

2

] + 2E [10038161003816100381610038161003816119909120579

119899+1minus 119909119861

119899+1

10038161003816100381610038161003816

2

]

le 119874 (Δ3)

(33)



100381710038171003817100381710038171205982

119899+1

100381710038171003817100381710038171198712le 1198676Δ 119886119904 Δ 997888rarr 0 (34)



120598119899+1

= 120598119899+ 119875119899+ 120575119899+1 (35)

where

119875119899= 119906119899+ 119862 (119909 (119905

119899)) (119909 (119905

119899) minus 119909119861(119905119899+1))

minus 119862 (119909119899) (119909119899minus 119909119899+1)

= 119906119899+ 119862 (119909 (119905

119899)) (119909 (119905

119899) minus 119909119899)

minus 119862 (119909 (119905119899)) (119909119861(119905119899+1) minus 119909119899+1)

+ (119862 (119909 (119905119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)

= 119906119899minus 119862 (119909 (119905

119899)) 119875119899

+ (119862 (119909 (119905119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)

= (119868 + 119862 (119909 (119905119899)))minus1

times [119906119899+ (119862 (119909 (119905

119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)]

(36)


E100381610038161003816100381611987511989910038161003816100381610038162

le 21198612E100381610038161003816100381611990611989910038161003816100381610038162

+ 21198612E1003816100381610038161003816[(1198620 (119909 (119905119899)) minus 1198620 (119909119899)) Δ

+ (1198621(119909 (119905119899)) minus 119862

1(119909119899)) ((Δ119882

119899)2

minus Δ)] (119909119899minus 119909119899+1)10038161003816100381610038161003816

2

le 21198612E100381610038161003816100381611990611989910038161003816100381610038162

+ 411986121198722Δ2E1003816100381610038161003816(119909119899 minus 119909119899+1)

10038161003816100381610038162

le 211986121198691ΔE100381610038161003816100381612059811989910038161003816100381610038162

+ 211986121198712Δ21205792E1003816100381610038161003816120598119899+1

10038161003816100381610038162

+ 1198695Δ3

(37)


10038161003816100381610038161003816E (119875119899| F119905119899)10038161003816100381610038161003816

2

=10038161003816100381610038161003816E ((119868 + 119862 (119909 (119905

119899)))minus1

times [119906119899+ (119862 (119909 (119905

119899)) minus 119862 (119909

119899))

times (119909119899minus 119909119899+1)] | F

119905119899)10038161003816100381610038161003816

2

le 2119861210038161003816100381610038161003816E (119906119899| F119905119899)10038161003816100381610038161003816

2

+ 21198612

times10038161003816100381610038161003816E ((119862 (119909 (119905

119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1) | F119905119899)10038161003816100381610038161003816

2

le 21198612(1198692Δ2E100381610038161003816100381612059811989910038161003816100381610038162

+ 1198712Δ21205792E (1003816100381610038161003816120598119899+1

10038161003816100381610038162

| F119905119899))

+ 21198612Δ10038161003816100381610038161003816E ((119862

0(119909 (119905119899))minus 119862

0(119909119899)) (119909119899minus 119909119899+1) | F119905119899)10038161003816100381610038161003816

2

le 211986121198692Δ2E100381610038161003816100381612059811989910038161003816100381610038162

+ 211986121198712Δ21205792E (1003816100381610038161003816120598119899+1

10038161003816100381610038162

| F119905119899) + 1198696Δ4

(38)


Thus1003816100381610038161003816E ⟨120598119899 119875119899⟩

1003816100381610038161003816

le Δminus1(E10038161003816100381610038161003816E (119875119899| F119905119899)10038161003816100381610038161003816

2

) + ΔE100381610038161003816100381612059811989910038161003816100381610038162

le (1 + 211986121198692) ΔE

100381610038161003816100381612059811989910038161003816100381610038162

+ 12119861211987121205792ΔE1003816100381610038161003816120598119899+1

10038161003816100381610038162

+ 1198696Δ3

(39)


1003816100381610038161003816 le 1198674Δ3+ ΔE

100381610038161003816100381612059811989910038161003816100381610038162

(40)

Therefore

E1003816100381610038161003816120598119899+1

10038161003816100381610038162

le (1 + 1198697Δ)E

100381610038161003816100381612059811989910038161003816100381610038162

+ 2119861211987121205792Δ (Δ + 12)E

1003816100381610038161003816120598119899+110038161003816100381610038162

+ (1198695+ 1198675+ 21198674+ 21198696) Δ3

(41)

where 1198697= 2(119861

21198691+ 2 + 2119861



E10038161003816100381610038161003816120598119899+1|2le (1 + 119869

7Δ)E

10038161003816100381610038161003816120598119899|2+ 1198698Δ3

= 1198698Δ2(1 + 119869

7Δ)119899+1

minus 1

1198697

le 1198699Δ2

(42)

where 1198699= 1198698((1198901198697119879 minus 1)119869

7)



d119909 (119905) = 119886119909 (119905) d119905 + 119887119909 (119905) d119882(119905) + 119888119909 (119905) d119873(119905)

119909 (1199050) = 1199090

(43)


0119890(119886minus(12)119887

2)119905+119887119882(119905)



119909119899+1

= 119877 (119886 119887 119888 Δ Δ119882119899 Δ119873119899) 119909119899 (44)


119877 (119886 119887 119888 Δ 120582) = E (1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)) lt 1 (45)



119909119899+1

= 1198771(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (46)

where

1198771(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)

= 1 + (119886 minus1

21198872)Δ + 119887Δ119882

119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899+ 119887119888Δ119882

119899Δ119873119899

+1

21198882(Δ119873119899)2

(47)


1198771(119901 119902 119911 119888)

= E (11987721(119886 119887 119888 Δ Δ119882

119899 Δ119873119899))

= 1 + 2119901 + 1199022+ 1199012+1

21199011199022+1

21199024

+ (2 + 119888 + 2119901 + 1198881199022+ 21199022) 119911

+ (2 + 2119888 +1

21198882+ 1199022+ 119901) 119911

2

+ (119888 + 1) 1199113+1

41199114

(48)


1(119901 119902 119911 119888) lt 1


119909119899+1

= 1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899 120579) 119909119899 (49)

where

1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899 120579)

=1


1

21198872) Δ

+ 119887Δ119882119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

]

(50)


1198772(119901 119902 119911 119888 120579)

=1

(1 minus 119901120579)2[1198771(119901 119902 119911 119888)

minus 2119901120579 + 11990121205792minus 21199012120579

minus2119901119911120579 minus 1199011199112120579]

(51)



2(119901 119902 119911 119888) lt 1


119909119899+1

= 1198773(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (52)

where1198773(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)

= ((1 minus 119886120579Δ) 119868 + 119862119899)minus1


21198872)Δ

+ 119887Δ119882119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

+ 119862119899]

(53)






we obtain

119909119899+1

= 1198771015840

3(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (54)

where1198771015840

3(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)


1

21198872Δ + 119887Δ119882

119899

+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

]

(55)


119899= minus119886Δ is

1198771015840

3(119901 119902 119911 119888)

=1

(1 minus 119901 (1 + 120579))2

times [1 + 1199022+1

21199011199022+1

21199024minus 2119901120579 + 119901

21205792

+ (2 + 119888 + 1198881199022+ 21199022minus 2119901120579) 119911

+ (2 + 2119888 +1

21198882+ 1199022minus 119901120579) 119911

2

+ (119888 + 1) 1199113+1

41199114]

(56)






2((Δ119882119899)2minusΔ)


119909119899+1

= 11987710158401015840

3(119901 119902 119888 119869 Δ119873

119899) 119909119899 (57)

where11987710158401015840

3(119901 119902 119888 119869 Δ119873

119899)



21199022+ 119902119869 +

3

211990221198692

+1

2(2119888 minus 119888

2) Δ119873119899+ 119902119888119869Δ119873

119899

+1

21198882(Δ119873119899)2

]

(58)




2((Δ119882119899)2minusΔ)

is

11987710158401015840

3(119901 119902 119911 119888 119869 Δ119873

119899)

=1

radic2120587

+infin

sum

minusinfin

(120582Δ)119898119890minus120582Δ

119898

times int

+infin

minusinfin

(11987710158401015840

3(119901 119902 119888 119909119898))

2

119890minus(11990922)d119909

(59)





0= minus119886 and 119862

1= 0 when 120579 = 12 and




0= minus119886



0= minus119886 and 119862

1= 1198872 are


0= minus119886

and 1198621= 1198872 when 120579 ge 12



Methods Δ 2minus8

2minus7

2minus6

2minus5

2minus4

2minus3

2minus2

2minus1

120579-Taylor120579 = 0 00024 00048 00099 00186 00389 00911 06853 30473120579 = 12 00022 00043 00090 00177 00368 00805 02548 07407120579 = 1 00042 00083 00170 00342 00699 01454 03025 05568


120579 = 0 00021 00041 00083 00152 00317 00665 03852 15570120579 = 12 00076 00134 00223 00361 00619 01504 07024 31984120579 = 1 00075 00132 00217 00347 00582 01360 06285 26648

q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 1

Taylor



2

18

16

14

12

1

08

06

04

02



0= minus119886 and 119862

1= 0



minus1 2minus2 2


119890 =1

2000

2000

sum

119894=1

10038161003816100381610038161003816119909(119894)

119873minus 119909(119894)(119905119873)10038161003816100381610038161003816 (60)


0= 1




q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 14

Taylor



2

18

16

14

12

1

08

06

04

02



0= minus119886 and 119862

1= 1198872


0= 1 and 119862

1= 1

6 Conclusions





References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1003816100381610038161003816119891 (1199091) minus 119891 (1199092)1003816100381610038161003816 +1003816100381610038161003816119892 (1199091) minus 119892 (1199092)

1003816100381610038161003816

+1003816100381610038161003816ℎ (1199091) minus ℎ (1199092)

1003816100381610038161003816

+10038161003816100381610038161003816119892 (1199091) 1198921015840(1199091) minus 119892 (119909

2) 1198921015840(1199092)10038161003816100381610038161003816

+10038161003816100381610038161003816119892 (1199091) ℎ1015840(1199091) minus 119892 (119909

2) ℎ1015840(1199092)10038161003816100381610038161003816le 119871

10038161003816100381610038161199091 minus 11990921003816100381610038161003816

(5)


growth condition

1003816100381610038161003816119891(119909)10038161003816100381610038162

+1003816100381610038161003816119892(119909)

10038161003816100381610038162

+ |ℎ(119909)|2

+10038161003816100381610038161003816119892(119909)119892

1015840(119909)10038161003816100381610038161003816

2

+10038161003816100381610038161003816119892(119909)ℎ

1015840(119909)10038161003816100381610038161003816

2

le 1198711015840(1 + |119909|

2)

(6)





0] and 120572

1isin

[minus1205721 1205721] with 120572

0ge Δ and 120572

1ge |(Δ119882



11198621(119909) is




119909120579(119905119899+1) = 119909 (119905

119899)

+ Δ [(1 minus 120579) 119891 (119909 (119905119899))

+ 120579119891 (119909 (119905119899+1))] + 119892 (119909 (119905

119899)) Δ119882

119899

+1

2119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899))

minus119892ℎ(119909 (119905119899)) + 119892 (119909 (119905

119899))) Δ119885

119899

(7)



119899+1)minus


120598119899= 119909(119905119899) minus 119909119899



10038171003817100381710038171003817E(120575120579(119905119899+1) | F119905119899)100381710038171003817100381710038171198712

le 1198671Δ2

119886119904 Δ 997888rarr 0 (8)


10038171003817100381710038171003817120575120579(119905119899+1)100381710038171003817100381710038171198712

le 1198672Δ32

119886119904 Δ 997888rarr 0 (9)





119909119860

119899+1= 119909 (119905

119899) + 119891 (119909 (119905

119899)) Δ + 119892 (119909 (119905

119899)) Δ119882

119899

+1

2119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+ 119892 (119909 (119905119899))) Δ119885

119899

(10)


E [10038161003816100381610038161003816E [(119909120579

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816E [(119909119860

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

]

+ 2E [10038161003816100381610038161003816E [(119909120579

119899+1minus 119909119860

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

le 1198701Δ4+ 2E [

10038161003816100381610038161003816E [(119909120579

119899+1minus 119909119860

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

(11)

Since

E [10038161003816100381610038161003816E [(119909120579

119899+1minus 119909119860

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

= E [10038161003816100381610038161003816E [120579Δ (119891 (119909 (119905

119899+1)) minus 119891 (119909 (119905

119899))) | F

119905119899]10038161003816100381610038161003816

2

]

le 119871120579ΔE[E [1003816100381610038161003816119909(119905119899+1) minus 119909(119905119899)

1003816100381610038161003816 | F119905119899]]2

le 119874 (Δ4)

(12)

we obtain (E[|E[(119909120579119899+1

minus 119909(119905119899+1)) | F

119905119899]|2

])12

le 119874(Δ2)



E [10038161003816100381610038161003816119909120579

119899+1minus 119909119860

119899+1

10038161003816100381610038161003816

2

| 119909119899= 119909 (119905

119899)]

= 11987121205792Δ2E [1003816100381610038161003816119909 (119905119899+1) minus 119909 (119905119899)

10038161003816100381610038162

]

le 119874 (Δ3)

(13)

we have

E [10038161003816100381610038161003816119909120579

119899+1minus 119909 (119905

119899+1)10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816119909119860

119899+1minus 119909 (119905

119899+1)10038161003816100381610038161003816

2

]

+ 2E [10038161003816100381610038161003816119909120579

119899+1minus 119909119860

119899+1

10038161003816100381610038161003816

2

]

le 119874 (Δ3)

(14)




100381710038171003817100381710038171205982

119899+1

100381710038171003817100381710038171198712le 1198673Δ 119886119904 Δ 997888rarr 0 (15)



120598119899+1

= 120598119899+ 119906119899+ 120575119899+1 (16)

where119906119899= Δ (1 minus 120579) (119891 (119909 (119905

119899)) minus 119891 (119909

119899))

+ Δ120579 (119891 (119909 (119905119899+1)) minus 119891 (119909

119899+1))

+ (119892 (119909 (119905119899)) minus 119892 (119909

119899)) Δ119882

119899

+1

2(119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899))

minus 119892 (119909119899) 1198921015840(119909119899)) [(Δ119882)

2minus Δ]

+1

2[3 (ℎ (119909 (119905

119899)) minus ℎ (119909

119899))

minus (ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899))] Δ119873

119899

+ [(119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899))

minus (119892 (119909 (119905119899)) minus 119892 (119909

119899))] Δ119882

119899Δ119873119899

+1

2[(ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899))

minus (ℎ (119909 (119905119899)) minus ℎ (119909

119899))] (Δ119873

119899)2

+ [(119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892 (119909

119899) ℎ1015840(119909119899))

minus (119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899))

+ (119892 (119909 (119905119899)) minus 119892 (119909

119899))] Δ119885

119899

(17)


1003816100381610038161003816E ⟨120575119899+1 120598119899⟩1003816100381610038161003816

=10038161003816100381610038161003816E [E (⟨120575

119899+1 120598119899⟩ | F119905119899)]10038161003816100381610038161003816

le E10038161003816100381610038161003816⟨E (120575

119899+1| F119905119899) 120598119899⟩10038161003816100381610038161003816

le [Δminus1(E10038161003816100381610038161003816E (120575119899+1

| F119905119899)10038161003816100381610038161003816

2

)]12

times (ΔE100381610038161003816100381612059811989910038161003816100381610038162

)12

le Δminus1(E10038161003816100381610038161003816E (120575119899+1

| F119905119899)10038161003816100381610038161003816

2

)

+ ΔE100381610038161003816100381612059811989910038161003816100381610038162

le 1198671Δ3+ ΔE

100381610038161003816100381612059811989910038161003816100381610038162

(18)


119899|2= Δ E|Δ119882

119899|4= 3Δ2 E(Δ119873

119899)2=

120582Δ(1 + 120582Δ) E(Δ119873119899)4= 120582Δ(1 + 7120582Δ + 6(120582Δ)

2+ (120582Δ)

3) and



E1003816100381610038161003816(119892 (119909 (119905119899)) minus 119892 (119909119899)) Δ119882119899

10038161003816100381610038162

le ΔE1003816100381610038161003816119892 (119909 (119905119899)) minus 119892 (119909119899)

10038161003816100381610038162

le 1198712ΔE|120598119899|2

E10038161003816100381610038161003816(119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) minus 119892 (119909

119899) 1198921015840(119909119899)) [(Δ119882)

2minus Δ]

10038161003816100381610038161003816

2

le 2Δ2E10038161003816100381610038161003816119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) minus 119892 (119909

119899) 1198921015840(119909119899)10038161003816100381610038161003816

le 21198712Δ2E|120598119899|2


10038161003816100381610038162

le 2120582Δ (1 + 120582Δ) [3E1003816100381610038161003816ℎ (119909 (119905119899)) minus ℎ (119909119899)

10038161003816100381610038162

+ E1003816100381610038161003816ℎ (119909 (119905119899) + ℎ (119909 (119905119899))) minus ℎ (119909119899 + ℎ (119909119899))

10038161003816100381610038162

]

le 21198712120582Δ (1 + 120582Δ) (5 + 2119871

2)E100381610038161003816100381612059811989910038161003816100381610038162


10038161003816100381610038162


10038161003816100381610038162

sdot E|Δ119882119899|2sdot E(Δ119873

119899)2

le 21198712120582Δ2(1 + 120582Δ) (3 + 2119871

2)E|120598119899|2

E10038161003816100381610038161003816[(ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899)) minus (ℎ (119909 (119905

119899)) minus ℎ (119909

119899))] (Δ119873

119899)210038161003816100381610038161003816

2

le 2120582Δ (1 + 7120582Δ + 6(120582Δ)2+ (120582Δ)

3) 1198712(3 + 2119871

2)E100381610038161003816100381612059811989910038161003816100381610038162

E10038161003816100381610038161003816[ (119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899)) minus 119892 (119909

119899) ℎ1015840(119909119899))

minus (119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899)) + (119892 (119909 (119905

119899)) minus 119892 (119909

119899))] Δ119885

119899

10038161003816100381610038162


le 61198712(2 + 119871

2)E100381610038161003816100381612059811989910038161003816100381610038162

sdot E1003816100381610038161003816Δ119885119899

10038161003816100381610038162

le 61198712(2 + 119871

2)E100381610038161003816100381612059811989910038161003816100381610038162

sdot E1003816100381610038161003816Δ119882119899 minus Δ1198851119899(119905119899+1)

10038161003816100381610038162

le 241198712(2 + 119871

2) ΔE

100381610038161003816100381612059811989910038161003816100381610038162

(19)

Hence

E100381610038161003816100381611990611989910038161003816100381610038162

le 1198691ΔE100381610038161003816100381612059811989910038161003816100381610038162

+ 81198712Δ21205792E1003816100381610038161003816120598119899+1

10038161003816100381610038162

(20)

where 1198691= 8[4119871

2+ 21198712120582(1 + 120582)(5 + 2119871

2) + 2119871

2120582(1 + 120582)(3 +

21198712) + 2120582(1 + 7120582 + 6120582

2+ 1205823)1198712(3 + 2119871

2) + 24119871

2(2 + 119871

2)

Noting that 119909(119905119899) and 119909

119899are F


119899


119905119899 we have

E [(119892 (119909 (119905119899)) minus 119892 (119909

119899)) Δ119882

119899F119905119899] = 0

E [(119892 (119909 (119905119899)) 1198921015840(119909 (119905119899)) minus 119892 (119909

119899) 1198921015840(119909119899))


119905119899] = 0

E [[(119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899)) minus (119892 (119909 (119905

119899)) minus 119892 (119909

119899))]

timesΔ119882119899Δ119873119899| F119905119899] = 0

E [[(119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892 (119909

119899) ℎ1015840(119909119899))

minus (119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899))

+ (119892 (119909 (119905119899)) minus 119892 (119909

119899)) ] Δ119885

119899| F119905119899] = 0

(21)

Therefore10038161003816100381610038161003816E (119906119899| F119905119899)10038161003816100381610038161003816

2

le

1003816100381610038161003816100381610038161003816Δ (1 minus 120579)E [119891 (119909 (119905

119899)) minus 119891 (119909

119899) | F119905119899]

+ 120579ΔE [119891 (119909 (119905119899+1)) minus 119891 (119909

119899+1) | F119905119899]

+3

2E [(ℎ (119909 (119905

119899)) minus ℎ (119909

119899)) Δ119873

119899| F119905119899]

minus1

2E [(ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899)) Δ119873

119899| F119905119899]

+1

2E [(ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899)) (Δ119873

119899)2

| F119905119899]

minus1

2E [(ℎ (119909 (119905

119899)) minus ℎ (119909

119899)) (Δ119873

119899)2

| F119905119899]

1003816100381610038161003816100381610038161003816

2

le 1198692Δ2E100381610038161003816100381612059811989910038161003816100381610038162

+ 611987121205792Δ2E [1003816100381610038161003816120598119899+1

10038161003816100381610038162

| F119905119899]

(22)

where 1198692= 31198712[2 + 3120582

2+ (1 + 119871)

21205822+ (1 + 119871)

2(1 + 120582)

2+

1205822(1 + 120582)

2] Thus

1003816100381610038161003816E ⟨120598119899 119906119899⟩1003816100381610038161003816 le Δminus1(E10038161003816100381610038161003816E (119906119899| F119905119899)10038161003816100381610038161003816

2

) + ΔE100381610038161003816100381612059811989910038161003816100381610038162

le (1 + 1198692) ΔE

100381610038161003816100381612059811989910038161003816100381610038162

+ 611987121205792ΔE1003816100381610038161003816120598119899+1

10038161003816100381610038162

(23)


E1003816100381610038161003816120598119899+1

10038161003816100381610038162

le E100381610038161003816100381612059811989910038161003816100381610038162

+ E1003816100381610038161003816120575119899+1

10038161003816100381610038162

+ E100381610038161003816100381611990611989910038161003816100381610038162

+ 21003816100381610038161003816E ⟨120575119899+1 120598119899⟩

1003816100381610038161003816 + 21003816100381610038161003816E ⟨120598119899 119906119899⟩

1003816100381610038161003816

le [1 + 2 (2 + 1198691+ 1198692) Δ]E

100381610038161003816100381612059811989910038161003816100381610038162

+ 211987121205792Δ (Δ + 6)E

1003816100381610038161003816120598119899+110038161003816100381610038162

+ (1198672+ 21198671) Δ3

(24)


3= 2(2 + 119869

1+ 1198692) Then

E1003816100381610038161003816120598119899+1

10038161003816100381610038162

le (1 + 1198693Δ)E

100381610038161003816100381612059811989910038161003816100381610038162

+ (1198672+ 21198671) Δ3

= (1198672+ 21198671) Δ2(1 + 119869

3Δ)119899+1

minus 1

1198693

le 1198694Δ2

(25)

where 1198694= (1198672+ 21198671)((1198901198693119879 minus 1)119869

3)


119909119861(119905119899+1)

= 119909 (119905119899) + Δ [(1 minus 120579) 119891 (119909 (119905

119899)) + 120579119891 (119909 (119905

119899+1))]

+ 119892 (119909 (119905119899)) Δ119882

119899

+1

2119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+ 119892 (119909 (119905119899))) Δ119885

119899+ 119862119899(119909 (119905119899) minus 119909119861(119905119899+1))

(26)


119899+1) = 119909(119905

119899+1) minus 119909119861(119905119899+1) and the






10038171003817100381710038171003817E (120575119861(119905119899+1) | F119905119899)100381710038171003817100381710038171198712

le 1198674Δ2

119886119904 Δ 997888rarr 0


10038171003817100381710038171003817120575119861(119905119899+1)100381710038171003817100381710038171198712

le 1198675Δ32

119886119904 Δ 997888rarr 0

(27)




E [10038161003816100381610038161003816E [(119909119861

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816E [(119909120579

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

]

+ 2E [10038161003816100381610038161003816E [(119909119861

119899+1minus 119909120579

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

le 1198672

1Δ4+ 2E [

10038161003816100381610038161003816E [(119909119861

119899+1minus 119909120579

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

(28)


and 119909119861119899+1


119909119861

119899+1minus 119909120579

119899+1

= minus119862119899(119909 (119905119899) minus 119909119861(119905119899+1))

= 119862119899[Δ [(1 minus 120579) 119891 (119909 (119905

119899)) + 120579119891 (119909 (119905

119899+1))]

+ 119892 (119909 (119905119899)) Δ119882

119899

+1

2119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+119892 (119909 (119905119899))) Δ119885

119899+ 119862119899(119909119899minus 119909119861

119899+1)]

= (119868 + 119862119899)minus1

119862119899

times [Δ [(1 minus 120579) 119891 (119909 (119905119899)) + 120579119891 (119909 (119905

119899+1))]

+ 119892 (119909 (119905119899)) Δ119882

119899+1

2119892 (119909 (119905

119899))

times 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+119892 (119909 (119905119899))) Δ119885

119899]

(29)


1(sdot) in 119862

119899(sdot)


119899)minus1

119862119899Δ119891 (119909 (119905

119899)) | F

119905119899]10038161003816100381610038161003816

le 119872Δ1003816100381610038161003816119891 (119909119899)

1003816100381610038161003816E [100381610038161003816100381610038161198620Δ + 119862

1((Δ119882

119899)2

minus Δ)10038161003816100381610038161003816| F119905119899]

le 1198711015840119872119861(1 +

1003816100381610038161003816119909 (119905119899)10038161003816100381610038162

)12

Δ2

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899119892 (119909 (119905

119899)) Δ119882

119899| F119905119899]10038161003816100381610038161003816= 0

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899))

times [(Δ119882119899)2

minus Δ] | F119905119899]10038161003816100381610038161003816

le 11986111987210038161003816100381610038161003816119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899))10038161003816100381610038161003816

E [1003816100381610038161003816100381610038161003816((Δ119882

119899)2

minus Δ)21003816100381610038161003816100381610038161003816| F119905119899]

le 21198711015840119872119861(1 +

100381610038161003816100381611990911989910038161003816100381610038162

)12

Δ2

10038161003816100381610038161003816[E [(119868 + 119862

119899)minus1

119862119899(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899] | F119905119899]10038161003816100381610038161003816

le 12058211986111987210038161003816100381610038163ℎ (119909 (119905119899)) minus ℎℎ (119909 (119905119899))

1003816100381610038161003816 Δ2

le (2 + 119871) 1198711015840119861119872(1 +

1003816100381610038161003816119909 (119905119899)10038161003816100381610038162

)12

Δ2

10038161003816100381610038161003816E [(119868+ 119862

119899)minus1

119862119899(119892ℎ(119909 (119905119899))minus 119892 (119909 (119905

119899)))

times Δ119882119899Δ119873119899| F119905119899]10038161003816100381610038161003816= 0

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899)))

times (Δ119873119899)2

| F119905119899]10038161003816100381610038161003816

le 1198611198721003816100381610038161003816ℎℎ (119909 (119905119899)) minus ℎ (119909 (119905119899))

1003816100381610038161003816 120582Δ2(1 + 120582Δ)

le 1198711015840119861119872120582(1 +

1003816100381610038161003816119909 (119905119899)10038161003816100381610038162

)12

Δ2(1 + 120582Δ)

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899(119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899))

minus 119892ℎ(119909 (119905119899)) + 119892 (119909 (119905

119899))) Δ119885

119899| F119905119899]10038161003816100381610038161003816

=10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899(119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899))

minus 119892ℎ(119909 (119905119899)) + 119892 (119909 (119905

119899)))

times (Δ119882119899minus Δ1198851119896(119905119899+1)) | F

119905119899]10038161003816100381610038161003816

= 0

(30)


Therefore

E [10038161003816100381610038161003816E [(119909119861

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

] le 119874 (Δ4) (31)


E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899Δ [(1 minus 120579) 119891 (119909 (119905

119899)) + 120579119891 (119909 (119905

119899+1))]10038161003816100381610038161003816

2

]

le 119874 (Δ4)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899119892 (119909119899) Δ119882119899

10038161003816100381610038161003816

2

] le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]10038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

10038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

10038161003816100381610038161003816

2

]

le 119874 (Δ4)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)210038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+ 119892 (119909 (119905119899))) (Δ119885

119899)10038161003816100381610038161003816

2

]

le 119874 (Δ3)

(32)

we have

E [10038161003816100381610038161003816119909119861

119899+1minus 119909 (119905

119899+1)10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816119909 (119905119899+1) minus 119909120579

119899+1

10038161003816100381610038161003816

2

] + 2E [10038161003816100381610038161003816119909120579

119899+1minus 119909119861

119899+1

10038161003816100381610038161003816

2

]

le 119874 (Δ3)

(33)



100381710038171003817100381710038171205982

119899+1

100381710038171003817100381710038171198712le 1198676Δ 119886119904 Δ 997888rarr 0 (34)



120598119899+1

= 120598119899+ 119875119899+ 120575119899+1 (35)

where

119875119899= 119906119899+ 119862 (119909 (119905

119899)) (119909 (119905

119899) minus 119909119861(119905119899+1))

minus 119862 (119909119899) (119909119899minus 119909119899+1)

= 119906119899+ 119862 (119909 (119905

119899)) (119909 (119905

119899) minus 119909119899)

minus 119862 (119909 (119905119899)) (119909119861(119905119899+1) minus 119909119899+1)

+ (119862 (119909 (119905119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)

= 119906119899minus 119862 (119909 (119905

119899)) 119875119899

+ (119862 (119909 (119905119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)

= (119868 + 119862 (119909 (119905119899)))minus1

times [119906119899+ (119862 (119909 (119905

119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)]

(36)


E100381610038161003816100381611987511989910038161003816100381610038162

le 21198612E100381610038161003816100381611990611989910038161003816100381610038162

+ 21198612E1003816100381610038161003816[(1198620 (119909 (119905119899)) minus 1198620 (119909119899)) Δ

+ (1198621(119909 (119905119899)) minus 119862

1(119909119899)) ((Δ119882

119899)2

minus Δ)] (119909119899minus 119909119899+1)10038161003816100381610038161003816

2

le 21198612E100381610038161003816100381611990611989910038161003816100381610038162

+ 411986121198722Δ2E1003816100381610038161003816(119909119899 minus 119909119899+1)

10038161003816100381610038162

le 211986121198691ΔE100381610038161003816100381612059811989910038161003816100381610038162

+ 211986121198712Δ21205792E1003816100381610038161003816120598119899+1

10038161003816100381610038162

+ 1198695Δ3

(37)


10038161003816100381610038161003816E (119875119899| F119905119899)10038161003816100381610038161003816

2

=10038161003816100381610038161003816E ((119868 + 119862 (119909 (119905

119899)))minus1

times [119906119899+ (119862 (119909 (119905

119899)) minus 119862 (119909

119899))

times (119909119899minus 119909119899+1)] | F

119905119899)10038161003816100381610038161003816

2

le 2119861210038161003816100381610038161003816E (119906119899| F119905119899)10038161003816100381610038161003816

2

+ 21198612

times10038161003816100381610038161003816E ((119862 (119909 (119905

119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1) | F119905119899)10038161003816100381610038161003816

2

le 21198612(1198692Δ2E100381610038161003816100381612059811989910038161003816100381610038162

+ 1198712Δ21205792E (1003816100381610038161003816120598119899+1

10038161003816100381610038162

| F119905119899))

+ 21198612Δ10038161003816100381610038161003816E ((119862

0(119909 (119905119899))minus 119862

0(119909119899)) (119909119899minus 119909119899+1) | F119905119899)10038161003816100381610038161003816

2

le 211986121198692Δ2E100381610038161003816100381612059811989910038161003816100381610038162

+ 211986121198712Δ21205792E (1003816100381610038161003816120598119899+1

10038161003816100381610038162

| F119905119899) + 1198696Δ4

(38)


Thus1003816100381610038161003816E ⟨120598119899 119875119899⟩

1003816100381610038161003816

le Δminus1(E10038161003816100381610038161003816E (119875119899| F119905119899)10038161003816100381610038161003816

2

) + ΔE100381610038161003816100381612059811989910038161003816100381610038162

le (1 + 211986121198692) ΔE

100381610038161003816100381612059811989910038161003816100381610038162

+ 12119861211987121205792ΔE1003816100381610038161003816120598119899+1

10038161003816100381610038162

+ 1198696Δ3

(39)


1003816100381610038161003816 le 1198674Δ3+ ΔE

100381610038161003816100381612059811989910038161003816100381610038162

(40)

Therefore

E1003816100381610038161003816120598119899+1

10038161003816100381610038162

le (1 + 1198697Δ)E

100381610038161003816100381612059811989910038161003816100381610038162

+ 2119861211987121205792Δ (Δ + 12)E

1003816100381610038161003816120598119899+110038161003816100381610038162

+ (1198695+ 1198675+ 21198674+ 21198696) Δ3

(41)

where 1198697= 2(119861

21198691+ 2 + 2119861



E10038161003816100381610038161003816120598119899+1|2le (1 + 119869

7Δ)E

10038161003816100381610038161003816120598119899|2+ 1198698Δ3

= 1198698Δ2(1 + 119869

7Δ)119899+1

minus 1

1198697

le 1198699Δ2

(42)

where 1198699= 1198698((1198901198697119879 minus 1)119869

7)



d119909 (119905) = 119886119909 (119905) d119905 + 119887119909 (119905) d119882(119905) + 119888119909 (119905) d119873(119905)

119909 (1199050) = 1199090

(43)


0119890(119886minus(12)119887

2)119905+119887119882(119905)



119909119899+1

= 119877 (119886 119887 119888 Δ Δ119882119899 Δ119873119899) 119909119899 (44)


119877 (119886 119887 119888 Δ 120582) = E (1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)) lt 1 (45)



119909119899+1

= 1198771(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (46)

where

1198771(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)

= 1 + (119886 minus1

21198872)Δ + 119887Δ119882

119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899+ 119887119888Δ119882

119899Δ119873119899

+1

21198882(Δ119873119899)2

(47)


1198771(119901 119902 119911 119888)

= E (11987721(119886 119887 119888 Δ Δ119882

119899 Δ119873119899))

= 1 + 2119901 + 1199022+ 1199012+1

21199011199022+1

21199024

+ (2 + 119888 + 2119901 + 1198881199022+ 21199022) 119911

+ (2 + 2119888 +1

21198882+ 1199022+ 119901) 119911

2

+ (119888 + 1) 1199113+1

41199114

(48)


1(119901 119902 119911 119888) lt 1


119909119899+1

= 1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899 120579) 119909119899 (49)

where

1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899 120579)

=1


1

21198872) Δ

+ 119887Δ119882119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

]

(50)


1198772(119901 119902 119911 119888 120579)

=1

(1 minus 119901120579)2[1198771(119901 119902 119911 119888)

minus 2119901120579 + 11990121205792minus 21199012120579

minus2119901119911120579 minus 1199011199112120579]

(51)



2(119901 119902 119911 119888) lt 1


119909119899+1

= 1198773(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (52)

where1198773(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)

= ((1 minus 119886120579Δ) 119868 + 119862119899)minus1


21198872)Δ

+ 119887Δ119882119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

+ 119862119899]

(53)






we obtain

119909119899+1

= 1198771015840

3(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (54)

where1198771015840

3(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)


1

21198872Δ + 119887Δ119882

119899

+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

]

(55)


119899= minus119886Δ is

1198771015840

3(119901 119902 119911 119888)

=1

(1 minus 119901 (1 + 120579))2

times [1 + 1199022+1

21199011199022+1

21199024minus 2119901120579 + 119901

21205792

+ (2 + 119888 + 1198881199022+ 21199022minus 2119901120579) 119911

+ (2 + 2119888 +1

21198882+ 1199022minus 119901120579) 119911

2

+ (119888 + 1) 1199113+1

41199114]

(56)






2((Δ119882119899)2minusΔ)


119909119899+1

= 11987710158401015840

3(119901 119902 119888 119869 Δ119873

119899) 119909119899 (57)

where11987710158401015840

3(119901 119902 119888 119869 Δ119873

119899)



21199022+ 119902119869 +

3

211990221198692

+1

2(2119888 minus 119888

2) Δ119873119899+ 119902119888119869Δ119873

119899

+1

21198882(Δ119873119899)2

]

(58)




2((Δ119882119899)2minusΔ)

is

11987710158401015840

3(119901 119902 119911 119888 119869 Δ119873

119899)

=1

radic2120587

+infin

sum

minusinfin

(120582Δ)119898119890minus120582Δ

119898

times int

+infin

minusinfin

(11987710158401015840

3(119901 119902 119888 119909119898))

2

119890minus(11990922)d119909

(59)





0= minus119886 and 119862

1= 0 when 120579 = 12 and




0= minus119886



0= minus119886 and 119862

1= 1198872 are


0= minus119886

and 1198621= 1198872 when 120579 ge 12



Methods Δ 2minus8

2minus7

2minus6

2minus5

2minus4

2minus3

2minus2

2minus1

120579-Taylor120579 = 0 00024 00048 00099 00186 00389 00911 06853 30473120579 = 12 00022 00043 00090 00177 00368 00805 02548 07407120579 = 1 00042 00083 00170 00342 00699 01454 03025 05568


120579 = 0 00021 00041 00083 00152 00317 00665 03852 15570120579 = 12 00076 00134 00223 00361 00619 01504 07024 31984120579 = 1 00075 00132 00217 00347 00582 01360 06285 26648

q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 1

Taylor



2

18

16

14

12

1

08

06

04

02



0= minus119886 and 119862

1= 0



minus1 2minus2 2


119890 =1

2000

2000

sum

119894=1

10038161003816100381610038161003816119909(119894)

119873minus 119909(119894)(119905119873)10038161003816100381610038161003816 (60)


0= 1




q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 14

Taylor



2

18

16

14

12

1

08

06

04

02



0= minus119886 and 119862

1= 1198872


0= 1 and 119862

1= 1

6 Conclusions





References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











E [10038161003816100381610038161003816119909120579

119899+1minus 119909119860

119899+1

10038161003816100381610038161003816

2

| 119909119899= 119909 (119905

119899)]

= 11987121205792Δ2E [1003816100381610038161003816119909 (119905119899+1) minus 119909 (119905119899)

10038161003816100381610038162

]

le 119874 (Δ3)

(13)

we have

E [10038161003816100381610038161003816119909120579

119899+1minus 119909 (119905

119899+1)10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816119909119860

119899+1minus 119909 (119905

119899+1)10038161003816100381610038161003816

2

]

+ 2E [10038161003816100381610038161003816119909120579

119899+1minus 119909119860

119899+1

10038161003816100381610038161003816

2

]

le 119874 (Δ3)

(14)




100381710038171003817100381710038171205982

119899+1

100381710038171003817100381710038171198712le 1198673Δ 119886119904 Δ 997888rarr 0 (15)



120598119899+1

= 120598119899+ 119906119899+ 120575119899+1 (16)

where119906119899= Δ (1 minus 120579) (119891 (119909 (119905

119899)) minus 119891 (119909

119899))

+ Δ120579 (119891 (119909 (119905119899+1)) minus 119891 (119909

119899+1))

+ (119892 (119909 (119905119899)) minus 119892 (119909

119899)) Δ119882

119899

+1

2(119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899))

minus 119892 (119909119899) 1198921015840(119909119899)) [(Δ119882)

2minus Δ]

+1

2[3 (ℎ (119909 (119905

119899)) minus ℎ (119909

119899))

minus (ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899))] Δ119873

119899

+ [(119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899))

minus (119892 (119909 (119905119899)) minus 119892 (119909

119899))] Δ119882

119899Δ119873119899

+1

2[(ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899))

minus (ℎ (119909 (119905119899)) minus ℎ (119909

119899))] (Δ119873

119899)2

+ [(119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892 (119909

119899) ℎ1015840(119909119899))

minus (119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899))

+ (119892 (119909 (119905119899)) minus 119892 (119909

119899))] Δ119885

119899

(17)


1003816100381610038161003816E ⟨120575119899+1 120598119899⟩1003816100381610038161003816

=10038161003816100381610038161003816E [E (⟨120575

119899+1 120598119899⟩ | F119905119899)]10038161003816100381610038161003816

le E10038161003816100381610038161003816⟨E (120575

119899+1| F119905119899) 120598119899⟩10038161003816100381610038161003816

le [Δminus1(E10038161003816100381610038161003816E (120575119899+1

| F119905119899)10038161003816100381610038161003816

2

)]12

times (ΔE100381610038161003816100381612059811989910038161003816100381610038162

)12

le Δminus1(E10038161003816100381610038161003816E (120575119899+1

| F119905119899)10038161003816100381610038161003816

2

)

+ ΔE100381610038161003816100381612059811989910038161003816100381610038162

le 1198671Δ3+ ΔE

100381610038161003816100381612059811989910038161003816100381610038162

(18)


119899|2= Δ E|Δ119882

119899|4= 3Δ2 E(Δ119873

119899)2=

120582Δ(1 + 120582Δ) E(Δ119873119899)4= 120582Δ(1 + 7120582Δ + 6(120582Δ)

2+ (120582Δ)

3) and



E1003816100381610038161003816(119892 (119909 (119905119899)) minus 119892 (119909119899)) Δ119882119899

10038161003816100381610038162

le ΔE1003816100381610038161003816119892 (119909 (119905119899)) minus 119892 (119909119899)

10038161003816100381610038162

le 1198712ΔE|120598119899|2

E10038161003816100381610038161003816(119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) minus 119892 (119909

119899) 1198921015840(119909119899)) [(Δ119882)

2minus Δ]

10038161003816100381610038161003816

2

le 2Δ2E10038161003816100381610038161003816119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) minus 119892 (119909

119899) 1198921015840(119909119899)10038161003816100381610038161003816

le 21198712Δ2E|120598119899|2


10038161003816100381610038162

le 2120582Δ (1 + 120582Δ) [3E1003816100381610038161003816ℎ (119909 (119905119899)) minus ℎ (119909119899)

10038161003816100381610038162

+ E1003816100381610038161003816ℎ (119909 (119905119899) + ℎ (119909 (119905119899))) minus ℎ (119909119899 + ℎ (119909119899))

10038161003816100381610038162

]

le 21198712120582Δ (1 + 120582Δ) (5 + 2119871

2)E100381610038161003816100381612059811989910038161003816100381610038162


10038161003816100381610038162


10038161003816100381610038162

sdot E|Δ119882119899|2sdot E(Δ119873

119899)2

le 21198712120582Δ2(1 + 120582Δ) (3 + 2119871

2)E|120598119899|2

E10038161003816100381610038161003816[(ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899)) minus (ℎ (119909 (119905

119899)) minus ℎ (119909

119899))] (Δ119873

119899)210038161003816100381610038161003816

2

le 2120582Δ (1 + 7120582Δ + 6(120582Δ)2+ (120582Δ)

3) 1198712(3 + 2119871

2)E100381610038161003816100381612059811989910038161003816100381610038162

E10038161003816100381610038161003816[ (119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899)) minus 119892 (119909

119899) ℎ1015840(119909119899))

minus (119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899)) + (119892 (119909 (119905

119899)) minus 119892 (119909

119899))] Δ119885

119899

10038161003816100381610038162


le 61198712(2 + 119871

2)E100381610038161003816100381612059811989910038161003816100381610038162

sdot E1003816100381610038161003816Δ119885119899

10038161003816100381610038162

le 61198712(2 + 119871

2)E100381610038161003816100381612059811989910038161003816100381610038162

sdot E1003816100381610038161003816Δ119882119899 minus Δ1198851119899(119905119899+1)

10038161003816100381610038162

le 241198712(2 + 119871

2) ΔE

100381610038161003816100381612059811989910038161003816100381610038162

(19)

Hence

E100381610038161003816100381611990611989910038161003816100381610038162

le 1198691ΔE100381610038161003816100381612059811989910038161003816100381610038162

+ 81198712Δ21205792E1003816100381610038161003816120598119899+1

10038161003816100381610038162

(20)

where 1198691= 8[4119871

2+ 21198712120582(1 + 120582)(5 + 2119871

2) + 2119871

2120582(1 + 120582)(3 +

21198712) + 2120582(1 + 7120582 + 6120582

2+ 1205823)1198712(3 + 2119871

2) + 24119871

2(2 + 119871

2)

Noting that 119909(119905119899) and 119909

119899are F


119899


119905119899 we have

E [(119892 (119909 (119905119899)) minus 119892 (119909

119899)) Δ119882

119899F119905119899] = 0

E [(119892 (119909 (119905119899)) 1198921015840(119909 (119905119899)) minus 119892 (119909

119899) 1198921015840(119909119899))


119905119899] = 0

E [[(119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899)) minus (119892 (119909 (119905

119899)) minus 119892 (119909

119899))]

timesΔ119882119899Δ119873119899| F119905119899] = 0

E [[(119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892 (119909

119899) ℎ1015840(119909119899))

minus (119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899))

+ (119892 (119909 (119905119899)) minus 119892 (119909

119899)) ] Δ119885

119899| F119905119899] = 0

(21)

Therefore10038161003816100381610038161003816E (119906119899| F119905119899)10038161003816100381610038161003816

2

le

1003816100381610038161003816100381610038161003816Δ (1 minus 120579)E [119891 (119909 (119905

119899)) minus 119891 (119909

119899) | F119905119899]

+ 120579ΔE [119891 (119909 (119905119899+1)) minus 119891 (119909

119899+1) | F119905119899]

+3

2E [(ℎ (119909 (119905

119899)) minus ℎ (119909

119899)) Δ119873

119899| F119905119899]

minus1

2E [(ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899)) Δ119873

119899| F119905119899]

+1

2E [(ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899)) (Δ119873

119899)2

| F119905119899]

minus1

2E [(ℎ (119909 (119905

119899)) minus ℎ (119909

119899)) (Δ119873

119899)2

| F119905119899]

1003816100381610038161003816100381610038161003816

2

le 1198692Δ2E100381610038161003816100381612059811989910038161003816100381610038162

+ 611987121205792Δ2E [1003816100381610038161003816120598119899+1

10038161003816100381610038162

| F119905119899]

(22)

where 1198692= 31198712[2 + 3120582

2+ (1 + 119871)

21205822+ (1 + 119871)

2(1 + 120582)

2+

1205822(1 + 120582)

2] Thus

1003816100381610038161003816E ⟨120598119899 119906119899⟩1003816100381610038161003816 le Δminus1(E10038161003816100381610038161003816E (119906119899| F119905119899)10038161003816100381610038161003816

2

) + ΔE100381610038161003816100381612059811989910038161003816100381610038162

le (1 + 1198692) ΔE

100381610038161003816100381612059811989910038161003816100381610038162

+ 611987121205792ΔE1003816100381610038161003816120598119899+1

10038161003816100381610038162

(23)


E1003816100381610038161003816120598119899+1

10038161003816100381610038162

le E100381610038161003816100381612059811989910038161003816100381610038162

+ E1003816100381610038161003816120575119899+1

10038161003816100381610038162

+ E100381610038161003816100381611990611989910038161003816100381610038162

+ 21003816100381610038161003816E ⟨120575119899+1 120598119899⟩

1003816100381610038161003816 + 21003816100381610038161003816E ⟨120598119899 119906119899⟩

1003816100381610038161003816

le [1 + 2 (2 + 1198691+ 1198692) Δ]E

100381610038161003816100381612059811989910038161003816100381610038162

+ 211987121205792Δ (Δ + 6)E

1003816100381610038161003816120598119899+110038161003816100381610038162

+ (1198672+ 21198671) Δ3

(24)


3= 2(2 + 119869

1+ 1198692) Then

E1003816100381610038161003816120598119899+1

10038161003816100381610038162

le (1 + 1198693Δ)E

100381610038161003816100381612059811989910038161003816100381610038162

+ (1198672+ 21198671) Δ3

= (1198672+ 21198671) Δ2(1 + 119869

3Δ)119899+1

minus 1

1198693

le 1198694Δ2

(25)

where 1198694= (1198672+ 21198671)((1198901198693119879 minus 1)119869

3)


119909119861(119905119899+1)

= 119909 (119905119899) + Δ [(1 minus 120579) 119891 (119909 (119905

119899)) + 120579119891 (119909 (119905

119899+1))]

+ 119892 (119909 (119905119899)) Δ119882

119899

+1

2119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+ 119892 (119909 (119905119899))) Δ119885

119899+ 119862119899(119909 (119905119899) minus 119909119861(119905119899+1))

(26)


119899+1) = 119909(119905

119899+1) minus 119909119861(119905119899+1) and the






10038171003817100381710038171003817E (120575119861(119905119899+1) | F119905119899)100381710038171003817100381710038171198712

le 1198674Δ2

119886119904 Δ 997888rarr 0


10038171003817100381710038171003817120575119861(119905119899+1)100381710038171003817100381710038171198712

le 1198675Δ32

119886119904 Δ 997888rarr 0

(27)




E [10038161003816100381610038161003816E [(119909119861

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816E [(119909120579

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

]

+ 2E [10038161003816100381610038161003816E [(119909119861

119899+1minus 119909120579

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

le 1198672

1Δ4+ 2E [

10038161003816100381610038161003816E [(119909119861

119899+1minus 119909120579

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

(28)


and 119909119861119899+1


119909119861

119899+1minus 119909120579

119899+1

= minus119862119899(119909 (119905119899) minus 119909119861(119905119899+1))

= 119862119899[Δ [(1 minus 120579) 119891 (119909 (119905

119899)) + 120579119891 (119909 (119905

119899+1))]

+ 119892 (119909 (119905119899)) Δ119882

119899

+1

2119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+119892 (119909 (119905119899))) Δ119885

119899+ 119862119899(119909119899minus 119909119861

119899+1)]

= (119868 + 119862119899)minus1

119862119899

times [Δ [(1 minus 120579) 119891 (119909 (119905119899)) + 120579119891 (119909 (119905

119899+1))]

+ 119892 (119909 (119905119899)) Δ119882

119899+1

2119892 (119909 (119905

119899))

times 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+119892 (119909 (119905119899))) Δ119885

119899]

(29)


1(sdot) in 119862

119899(sdot)


119899)minus1

119862119899Δ119891 (119909 (119905

119899)) | F

119905119899]10038161003816100381610038161003816

le 119872Δ1003816100381610038161003816119891 (119909119899)

1003816100381610038161003816E [100381610038161003816100381610038161198620Δ + 119862

1((Δ119882

119899)2

minus Δ)10038161003816100381610038161003816| F119905119899]

le 1198711015840119872119861(1 +

1003816100381610038161003816119909 (119905119899)10038161003816100381610038162

)12

Δ2

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899119892 (119909 (119905

119899)) Δ119882

119899| F119905119899]10038161003816100381610038161003816= 0

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899))

times [(Δ119882119899)2

minus Δ] | F119905119899]10038161003816100381610038161003816

le 11986111987210038161003816100381610038161003816119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899))10038161003816100381610038161003816

E [1003816100381610038161003816100381610038161003816((Δ119882

119899)2

minus Δ)21003816100381610038161003816100381610038161003816| F119905119899]

le 21198711015840119872119861(1 +

100381610038161003816100381611990911989910038161003816100381610038162

)12

Δ2

10038161003816100381610038161003816[E [(119868 + 119862

119899)minus1

119862119899(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899] | F119905119899]10038161003816100381610038161003816

le 12058211986111987210038161003816100381610038163ℎ (119909 (119905119899)) minus ℎℎ (119909 (119905119899))

1003816100381610038161003816 Δ2

le (2 + 119871) 1198711015840119861119872(1 +

1003816100381610038161003816119909 (119905119899)10038161003816100381610038162

)12

Δ2

10038161003816100381610038161003816E [(119868+ 119862

119899)minus1

119862119899(119892ℎ(119909 (119905119899))minus 119892 (119909 (119905

119899)))

times Δ119882119899Δ119873119899| F119905119899]10038161003816100381610038161003816= 0

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899)))

times (Δ119873119899)2

| F119905119899]10038161003816100381610038161003816

le 1198611198721003816100381610038161003816ℎℎ (119909 (119905119899)) minus ℎ (119909 (119905119899))

1003816100381610038161003816 120582Δ2(1 + 120582Δ)

le 1198711015840119861119872120582(1 +

1003816100381610038161003816119909 (119905119899)10038161003816100381610038162

)12

Δ2(1 + 120582Δ)

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899(119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899))

minus 119892ℎ(119909 (119905119899)) + 119892 (119909 (119905

119899))) Δ119885

119899| F119905119899]10038161003816100381610038161003816

=10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899(119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899))

minus 119892ℎ(119909 (119905119899)) + 119892 (119909 (119905

119899)))

times (Δ119882119899minus Δ1198851119896(119905119899+1)) | F

119905119899]10038161003816100381610038161003816

= 0

(30)


Therefore

E [10038161003816100381610038161003816E [(119909119861

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

] le 119874 (Δ4) (31)


E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899Δ [(1 minus 120579) 119891 (119909 (119905

119899)) + 120579119891 (119909 (119905

119899+1))]10038161003816100381610038161003816

2

]

le 119874 (Δ4)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899119892 (119909119899) Δ119882119899

10038161003816100381610038161003816

2

] le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]10038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

10038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

10038161003816100381610038161003816

2

]

le 119874 (Δ4)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)210038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+ 119892 (119909 (119905119899))) (Δ119885

119899)10038161003816100381610038161003816

2

]

le 119874 (Δ3)

(32)

we have

E [10038161003816100381610038161003816119909119861

119899+1minus 119909 (119905

119899+1)10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816119909 (119905119899+1) minus 119909120579

119899+1

10038161003816100381610038161003816

2

] + 2E [10038161003816100381610038161003816119909120579

119899+1minus 119909119861

119899+1

10038161003816100381610038161003816

2

]

le 119874 (Δ3)

(33)



100381710038171003817100381710038171205982

119899+1

100381710038171003817100381710038171198712le 1198676Δ 119886119904 Δ 997888rarr 0 (34)



120598119899+1

= 120598119899+ 119875119899+ 120575119899+1 (35)

where

119875119899= 119906119899+ 119862 (119909 (119905

119899)) (119909 (119905

119899) minus 119909119861(119905119899+1))

minus 119862 (119909119899) (119909119899minus 119909119899+1)

= 119906119899+ 119862 (119909 (119905

119899)) (119909 (119905

119899) minus 119909119899)

minus 119862 (119909 (119905119899)) (119909119861(119905119899+1) minus 119909119899+1)

+ (119862 (119909 (119905119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)

= 119906119899minus 119862 (119909 (119905

119899)) 119875119899

+ (119862 (119909 (119905119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)

= (119868 + 119862 (119909 (119905119899)))minus1

times [119906119899+ (119862 (119909 (119905

119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)]

(36)


E100381610038161003816100381611987511989910038161003816100381610038162

le 21198612E100381610038161003816100381611990611989910038161003816100381610038162

+ 21198612E1003816100381610038161003816[(1198620 (119909 (119905119899)) minus 1198620 (119909119899)) Δ

+ (1198621(119909 (119905119899)) minus 119862

1(119909119899)) ((Δ119882

119899)2

minus Δ)] (119909119899minus 119909119899+1)10038161003816100381610038161003816

2

le 21198612E100381610038161003816100381611990611989910038161003816100381610038162

+ 411986121198722Δ2E1003816100381610038161003816(119909119899 minus 119909119899+1)

10038161003816100381610038162

le 211986121198691ΔE100381610038161003816100381612059811989910038161003816100381610038162

+ 211986121198712Δ21205792E1003816100381610038161003816120598119899+1

10038161003816100381610038162

+ 1198695Δ3

(37)


10038161003816100381610038161003816E (119875119899| F119905119899)10038161003816100381610038161003816

2

=10038161003816100381610038161003816E ((119868 + 119862 (119909 (119905

119899)))minus1

times [119906119899+ (119862 (119909 (119905

119899)) minus 119862 (119909

119899))

times (119909119899minus 119909119899+1)] | F

119905119899)10038161003816100381610038161003816

2

le 2119861210038161003816100381610038161003816E (119906119899| F119905119899)10038161003816100381610038161003816

2

+ 21198612

times10038161003816100381610038161003816E ((119862 (119909 (119905

119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1) | F119905119899)10038161003816100381610038161003816

2

le 21198612(1198692Δ2E100381610038161003816100381612059811989910038161003816100381610038162

+ 1198712Δ21205792E (1003816100381610038161003816120598119899+1

10038161003816100381610038162

| F119905119899))

+ 21198612Δ10038161003816100381610038161003816E ((119862

0(119909 (119905119899))minus 119862

0(119909119899)) (119909119899minus 119909119899+1) | F119905119899)10038161003816100381610038161003816

2

le 211986121198692Δ2E100381610038161003816100381612059811989910038161003816100381610038162

+ 211986121198712Δ21205792E (1003816100381610038161003816120598119899+1

10038161003816100381610038162

| F119905119899) + 1198696Δ4

(38)


Thus1003816100381610038161003816E ⟨120598119899 119875119899⟩

1003816100381610038161003816

le Δminus1(E10038161003816100381610038161003816E (119875119899| F119905119899)10038161003816100381610038161003816

2

) + ΔE100381610038161003816100381612059811989910038161003816100381610038162

le (1 + 211986121198692) ΔE

100381610038161003816100381612059811989910038161003816100381610038162

+ 12119861211987121205792ΔE1003816100381610038161003816120598119899+1

10038161003816100381610038162

+ 1198696Δ3

(39)


1003816100381610038161003816 le 1198674Δ3+ ΔE

100381610038161003816100381612059811989910038161003816100381610038162

(40)

Therefore

E1003816100381610038161003816120598119899+1

10038161003816100381610038162

le (1 + 1198697Δ)E

100381610038161003816100381612059811989910038161003816100381610038162

+ 2119861211987121205792Δ (Δ + 12)E

1003816100381610038161003816120598119899+110038161003816100381610038162

+ (1198695+ 1198675+ 21198674+ 21198696) Δ3

(41)

where 1198697= 2(119861

21198691+ 2 + 2119861



E10038161003816100381610038161003816120598119899+1|2le (1 + 119869

7Δ)E

10038161003816100381610038161003816120598119899|2+ 1198698Δ3

= 1198698Δ2(1 + 119869

7Δ)119899+1

minus 1

1198697

le 1198699Δ2

(42)

where 1198699= 1198698((1198901198697119879 minus 1)119869

7)



d119909 (119905) = 119886119909 (119905) d119905 + 119887119909 (119905) d119882(119905) + 119888119909 (119905) d119873(119905)

119909 (1199050) = 1199090

(43)


0119890(119886minus(12)119887

2)119905+119887119882(119905)



119909119899+1

= 119877 (119886 119887 119888 Δ Δ119882119899 Δ119873119899) 119909119899 (44)


119877 (119886 119887 119888 Δ 120582) = E (1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)) lt 1 (45)



119909119899+1

= 1198771(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (46)

where

1198771(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)

= 1 + (119886 minus1

21198872)Δ + 119887Δ119882

119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899+ 119887119888Δ119882

119899Δ119873119899

+1

21198882(Δ119873119899)2

(47)


1198771(119901 119902 119911 119888)

= E (11987721(119886 119887 119888 Δ Δ119882

119899 Δ119873119899))

= 1 + 2119901 + 1199022+ 1199012+1

21199011199022+1

21199024

+ (2 + 119888 + 2119901 + 1198881199022+ 21199022) 119911

+ (2 + 2119888 +1

21198882+ 1199022+ 119901) 119911

2

+ (119888 + 1) 1199113+1

41199114

(48)


1(119901 119902 119911 119888) lt 1


119909119899+1

= 1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899 120579) 119909119899 (49)

where

1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899 120579)

=1


1

21198872) Δ

+ 119887Δ119882119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

]

(50)


1198772(119901 119902 119911 119888 120579)

=1

(1 minus 119901120579)2[1198771(119901 119902 119911 119888)

minus 2119901120579 + 11990121205792minus 21199012120579

minus2119901119911120579 minus 1199011199112120579]

(51)



2(119901 119902 119911 119888) lt 1


119909119899+1

= 1198773(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (52)

where1198773(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)

= ((1 minus 119886120579Δ) 119868 + 119862119899)minus1


21198872)Δ

+ 119887Δ119882119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

+ 119862119899]

(53)






we obtain

119909119899+1

= 1198771015840

3(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (54)

where1198771015840

3(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)


1

21198872Δ + 119887Δ119882

119899

+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

]

(55)


119899= minus119886Δ is

1198771015840

3(119901 119902 119911 119888)

=1

(1 minus 119901 (1 + 120579))2

times [1 + 1199022+1

21199011199022+1

21199024minus 2119901120579 + 119901

21205792

+ (2 + 119888 + 1198881199022+ 21199022minus 2119901120579) 119911

+ (2 + 2119888 +1

21198882+ 1199022minus 119901120579) 119911

2

+ (119888 + 1) 1199113+1

41199114]

(56)






2((Δ119882119899)2minusΔ)


119909119899+1

= 11987710158401015840

3(119901 119902 119888 119869 Δ119873

119899) 119909119899 (57)

where11987710158401015840

3(119901 119902 119888 119869 Δ119873

119899)



21199022+ 119902119869 +

3

211990221198692

+1

2(2119888 minus 119888

2) Δ119873119899+ 119902119888119869Δ119873

119899

+1

21198882(Δ119873119899)2

]

(58)




2((Δ119882119899)2minusΔ)

is

11987710158401015840

3(119901 119902 119911 119888 119869 Δ119873

119899)

=1

radic2120587

+infin

sum

minusinfin

(120582Δ)119898119890minus120582Δ

119898

times int

+infin

minusinfin

(11987710158401015840

3(119901 119902 119888 119909119898))

2

119890minus(11990922)d119909

(59)





0= minus119886 and 119862

1= 0 when 120579 = 12 and




0= minus119886



0= minus119886 and 119862

1= 1198872 are


0= minus119886

and 1198621= 1198872 when 120579 ge 12



Methods Δ 2minus8

2minus7

2minus6

2minus5

2minus4

2minus3

2minus2

2minus1

120579-Taylor120579 = 0 00024 00048 00099 00186 00389 00911 06853 30473120579 = 12 00022 00043 00090 00177 00368 00805 02548 07407120579 = 1 00042 00083 00170 00342 00699 01454 03025 05568


120579 = 0 00021 00041 00083 00152 00317 00665 03852 15570120579 = 12 00076 00134 00223 00361 00619 01504 07024 31984120579 = 1 00075 00132 00217 00347 00582 01360 06285 26648

q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 1

Taylor



2

18

16

14

12

1

08

06

04

02



0= minus119886 and 119862

1= 0



minus1 2minus2 2


119890 =1

2000

2000

sum

119894=1

10038161003816100381610038161003816119909(119894)

119873minus 119909(119894)(119905119873)10038161003816100381610038161003816 (60)


0= 1




q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 14

Taylor



2

18

16

14

12

1

08

06

04

02



0= minus119886 and 119862

1= 1198872


0= 1 and 119862

1= 1

6 Conclusions





References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le 61198712(2 + 119871

2)E100381610038161003816100381612059811989910038161003816100381610038162

sdot E1003816100381610038161003816Δ119885119899

10038161003816100381610038162

le 61198712(2 + 119871

2)E100381610038161003816100381612059811989910038161003816100381610038162

sdot E1003816100381610038161003816Δ119882119899 minus Δ1198851119899(119905119899+1)

10038161003816100381610038162

le 241198712(2 + 119871

2) ΔE

100381610038161003816100381612059811989910038161003816100381610038162

(19)

Hence

E100381610038161003816100381611990611989910038161003816100381610038162

le 1198691ΔE100381610038161003816100381612059811989910038161003816100381610038162

+ 81198712Δ21205792E1003816100381610038161003816120598119899+1

10038161003816100381610038162

(20)

where 1198691= 8[4119871

2+ 21198712120582(1 + 120582)(5 + 2119871

2) + 2119871

2120582(1 + 120582)(3 +

21198712) + 2120582(1 + 7120582 + 6120582

2+ 1205823)1198712(3 + 2119871

2) + 24119871

2(2 + 119871

2)

Noting that 119909(119905119899) and 119909

119899are F


119899


119905119899 we have

E [(119892 (119909 (119905119899)) minus 119892 (119909

119899)) Δ119882

119899F119905119899] = 0

E [(119892 (119909 (119905119899)) 1198921015840(119909 (119905119899)) minus 119892 (119909

119899) 1198921015840(119909119899))


119905119899] = 0

E [[(119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899)) minus (119892 (119909 (119905

119899)) minus 119892 (119909

119899))]

timesΔ119882119899Δ119873119899| F119905119899] = 0

E [[(119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892 (119909

119899) ℎ1015840(119909119899))

minus (119892ℎ(119909 (119905119899)) minus 119892

ℎ(119909119899))

+ (119892 (119909 (119905119899)) minus 119892 (119909

119899)) ] Δ119885

119899| F119905119899] = 0

(21)

Therefore10038161003816100381610038161003816E (119906119899| F119905119899)10038161003816100381610038161003816

2

le

1003816100381610038161003816100381610038161003816Δ (1 minus 120579)E [119891 (119909 (119905

119899)) minus 119891 (119909

119899) | F119905119899]

+ 120579ΔE [119891 (119909 (119905119899+1)) minus 119891 (119909

119899+1) | F119905119899]

+3

2E [(ℎ (119909 (119905

119899)) minus ℎ (119909

119899)) Δ119873

119899| F119905119899]

minus1

2E [(ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899)) Δ119873

119899| F119905119899]

+1

2E [(ℎℎ(119909 (119905119899)) minus ℎ

ℎ(119909119899)) (Δ119873

119899)2

| F119905119899]

minus1

2E [(ℎ (119909 (119905

119899)) minus ℎ (119909

119899)) (Δ119873

119899)2

| F119905119899]

1003816100381610038161003816100381610038161003816

2

le 1198692Δ2E100381610038161003816100381612059811989910038161003816100381610038162

+ 611987121205792Δ2E [1003816100381610038161003816120598119899+1

10038161003816100381610038162

| F119905119899]

(22)

where 1198692= 31198712[2 + 3120582

2+ (1 + 119871)

21205822+ (1 + 119871)

2(1 + 120582)

2+

1205822(1 + 120582)

2] Thus

1003816100381610038161003816E ⟨120598119899 119906119899⟩1003816100381610038161003816 le Δminus1(E10038161003816100381610038161003816E (119906119899| F119905119899)10038161003816100381610038161003816

2

) + ΔE100381610038161003816100381612059811989910038161003816100381610038162

le (1 + 1198692) ΔE

100381610038161003816100381612059811989910038161003816100381610038162

+ 611987121205792ΔE1003816100381610038161003816120598119899+1

10038161003816100381610038162

(23)


E1003816100381610038161003816120598119899+1

10038161003816100381610038162

le E100381610038161003816100381612059811989910038161003816100381610038162

+ E1003816100381610038161003816120575119899+1

10038161003816100381610038162

+ E100381610038161003816100381611990611989910038161003816100381610038162

+ 21003816100381610038161003816E ⟨120575119899+1 120598119899⟩

1003816100381610038161003816 + 21003816100381610038161003816E ⟨120598119899 119906119899⟩

1003816100381610038161003816

le [1 + 2 (2 + 1198691+ 1198692) Δ]E

100381610038161003816100381612059811989910038161003816100381610038162

+ 211987121205792Δ (Δ + 6)E

1003816100381610038161003816120598119899+110038161003816100381610038162

+ (1198672+ 21198671) Δ3

(24)


3= 2(2 + 119869

1+ 1198692) Then

E1003816100381610038161003816120598119899+1

10038161003816100381610038162

le (1 + 1198693Δ)E

100381610038161003816100381612059811989910038161003816100381610038162

+ (1198672+ 21198671) Δ3

= (1198672+ 21198671) Δ2(1 + 119869

3Δ)119899+1

minus 1

1198693

le 1198694Δ2

(25)

where 1198694= (1198672+ 21198671)((1198901198693119879 minus 1)119869

3)


119909119861(119905119899+1)

= 119909 (119905119899) + Δ [(1 minus 120579) 119891 (119909 (119905

119899)) + 120579119891 (119909 (119905

119899+1))]

+ 119892 (119909 (119905119899)) Δ119882

119899

+1

2119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+ 119892 (119909 (119905119899))) Δ119885

119899+ 119862119899(119909 (119905119899) minus 119909119861(119905119899+1))

(26)


119899+1) = 119909(119905

119899+1) minus 119909119861(119905119899+1) and the






10038171003817100381710038171003817E (120575119861(119905119899+1) | F119905119899)100381710038171003817100381710038171198712

le 1198674Δ2

119886119904 Δ 997888rarr 0


10038171003817100381710038171003817120575119861(119905119899+1)100381710038171003817100381710038171198712

le 1198675Δ32

119886119904 Δ 997888rarr 0

(27)




E [10038161003816100381610038161003816E [(119909119861

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816E [(119909120579

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

]

+ 2E [10038161003816100381610038161003816E [(119909119861

119899+1minus 119909120579

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

le 1198672

1Δ4+ 2E [

10038161003816100381610038161003816E [(119909119861

119899+1minus 119909120579

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

(28)


and 119909119861119899+1


119909119861

119899+1minus 119909120579

119899+1

= minus119862119899(119909 (119905119899) minus 119909119861(119905119899+1))

= 119862119899[Δ [(1 minus 120579) 119891 (119909 (119905

119899)) + 120579119891 (119909 (119905

119899+1))]

+ 119892 (119909 (119905119899)) Δ119882

119899

+1

2119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+119892 (119909 (119905119899))) Δ119885

119899+ 119862119899(119909119899minus 119909119861

119899+1)]

= (119868 + 119862119899)minus1

119862119899

times [Δ [(1 minus 120579) 119891 (119909 (119905119899)) + 120579119891 (119909 (119905

119899+1))]

+ 119892 (119909 (119905119899)) Δ119882

119899+1

2119892 (119909 (119905

119899))

times 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+119892 (119909 (119905119899))) Δ119885

119899]

(29)


1(sdot) in 119862

119899(sdot)


119899)minus1

119862119899Δ119891 (119909 (119905

119899)) | F

119905119899]10038161003816100381610038161003816

le 119872Δ1003816100381610038161003816119891 (119909119899)

1003816100381610038161003816E [100381610038161003816100381610038161198620Δ + 119862

1((Δ119882

119899)2

minus Δ)10038161003816100381610038161003816| F119905119899]

le 1198711015840119872119861(1 +

1003816100381610038161003816119909 (119905119899)10038161003816100381610038162

)12

Δ2

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899119892 (119909 (119905

119899)) Δ119882

119899| F119905119899]10038161003816100381610038161003816= 0

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899))

times [(Δ119882119899)2

minus Δ] | F119905119899]10038161003816100381610038161003816

le 11986111987210038161003816100381610038161003816119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899))10038161003816100381610038161003816

E [1003816100381610038161003816100381610038161003816((Δ119882

119899)2

minus Δ)21003816100381610038161003816100381610038161003816| F119905119899]

le 21198711015840119872119861(1 +

100381610038161003816100381611990911989910038161003816100381610038162

)12

Δ2

10038161003816100381610038161003816[E [(119868 + 119862

119899)minus1

119862119899(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899] | F119905119899]10038161003816100381610038161003816

le 12058211986111987210038161003816100381610038163ℎ (119909 (119905119899)) minus ℎℎ (119909 (119905119899))

1003816100381610038161003816 Δ2

le (2 + 119871) 1198711015840119861119872(1 +

1003816100381610038161003816119909 (119905119899)10038161003816100381610038162

)12

Δ2

10038161003816100381610038161003816E [(119868+ 119862

119899)minus1

119862119899(119892ℎ(119909 (119905119899))minus 119892 (119909 (119905

119899)))

times Δ119882119899Δ119873119899| F119905119899]10038161003816100381610038161003816= 0

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899)))

times (Δ119873119899)2

| F119905119899]10038161003816100381610038161003816

le 1198611198721003816100381610038161003816ℎℎ (119909 (119905119899)) minus ℎ (119909 (119905119899))

1003816100381610038161003816 120582Δ2(1 + 120582Δ)

le 1198711015840119861119872120582(1 +

1003816100381610038161003816119909 (119905119899)10038161003816100381610038162

)12

Δ2(1 + 120582Δ)

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899(119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899))

minus 119892ℎ(119909 (119905119899)) + 119892 (119909 (119905

119899))) Δ119885

119899| F119905119899]10038161003816100381610038161003816

=10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899(119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899))

minus 119892ℎ(119909 (119905119899)) + 119892 (119909 (119905

119899)))

times (Δ119882119899minus Δ1198851119896(119905119899+1)) | F

119905119899]10038161003816100381610038161003816

= 0

(30)


Therefore

E [10038161003816100381610038161003816E [(119909119861

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

] le 119874 (Δ4) (31)


E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899Δ [(1 minus 120579) 119891 (119909 (119905

119899)) + 120579119891 (119909 (119905

119899+1))]10038161003816100381610038161003816

2

]

le 119874 (Δ4)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899119892 (119909119899) Δ119882119899

10038161003816100381610038161003816

2

] le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]10038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

10038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

10038161003816100381610038161003816

2

]

le 119874 (Δ4)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)210038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+ 119892 (119909 (119905119899))) (Δ119885

119899)10038161003816100381610038161003816

2

]

le 119874 (Δ3)

(32)

we have

E [10038161003816100381610038161003816119909119861

119899+1minus 119909 (119905

119899+1)10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816119909 (119905119899+1) minus 119909120579

119899+1

10038161003816100381610038161003816

2

] + 2E [10038161003816100381610038161003816119909120579

119899+1minus 119909119861

119899+1

10038161003816100381610038161003816

2

]

le 119874 (Δ3)

(33)



100381710038171003817100381710038171205982

119899+1

100381710038171003817100381710038171198712le 1198676Δ 119886119904 Δ 997888rarr 0 (34)



120598119899+1

= 120598119899+ 119875119899+ 120575119899+1 (35)

where

119875119899= 119906119899+ 119862 (119909 (119905

119899)) (119909 (119905

119899) minus 119909119861(119905119899+1))

minus 119862 (119909119899) (119909119899minus 119909119899+1)

= 119906119899+ 119862 (119909 (119905

119899)) (119909 (119905

119899) minus 119909119899)

minus 119862 (119909 (119905119899)) (119909119861(119905119899+1) minus 119909119899+1)

+ (119862 (119909 (119905119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)

= 119906119899minus 119862 (119909 (119905

119899)) 119875119899

+ (119862 (119909 (119905119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)

= (119868 + 119862 (119909 (119905119899)))minus1

times [119906119899+ (119862 (119909 (119905

119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)]

(36)


E100381610038161003816100381611987511989910038161003816100381610038162

le 21198612E100381610038161003816100381611990611989910038161003816100381610038162

+ 21198612E1003816100381610038161003816[(1198620 (119909 (119905119899)) minus 1198620 (119909119899)) Δ

+ (1198621(119909 (119905119899)) minus 119862

1(119909119899)) ((Δ119882

119899)2

minus Δ)] (119909119899minus 119909119899+1)10038161003816100381610038161003816

2

le 21198612E100381610038161003816100381611990611989910038161003816100381610038162

+ 411986121198722Δ2E1003816100381610038161003816(119909119899 minus 119909119899+1)

10038161003816100381610038162

le 211986121198691ΔE100381610038161003816100381612059811989910038161003816100381610038162

+ 211986121198712Δ21205792E1003816100381610038161003816120598119899+1

10038161003816100381610038162

+ 1198695Δ3

(37)


10038161003816100381610038161003816E (119875119899| F119905119899)10038161003816100381610038161003816

2

=10038161003816100381610038161003816E ((119868 + 119862 (119909 (119905

119899)))minus1

times [119906119899+ (119862 (119909 (119905

119899)) minus 119862 (119909

119899))

times (119909119899minus 119909119899+1)] | F

119905119899)10038161003816100381610038161003816

2

le 2119861210038161003816100381610038161003816E (119906119899| F119905119899)10038161003816100381610038161003816

2

+ 21198612

times10038161003816100381610038161003816E ((119862 (119909 (119905

119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1) | F119905119899)10038161003816100381610038161003816

2

le 21198612(1198692Δ2E100381610038161003816100381612059811989910038161003816100381610038162

+ 1198712Δ21205792E (1003816100381610038161003816120598119899+1

10038161003816100381610038162

| F119905119899))

+ 21198612Δ10038161003816100381610038161003816E ((119862

0(119909 (119905119899))minus 119862

0(119909119899)) (119909119899minus 119909119899+1) | F119905119899)10038161003816100381610038161003816

2

le 211986121198692Δ2E100381610038161003816100381612059811989910038161003816100381610038162

+ 211986121198712Δ21205792E (1003816100381610038161003816120598119899+1

10038161003816100381610038162

| F119905119899) + 1198696Δ4

(38)


Thus1003816100381610038161003816E ⟨120598119899 119875119899⟩

1003816100381610038161003816

le Δminus1(E10038161003816100381610038161003816E (119875119899| F119905119899)10038161003816100381610038161003816

2

) + ΔE100381610038161003816100381612059811989910038161003816100381610038162

le (1 + 211986121198692) ΔE

100381610038161003816100381612059811989910038161003816100381610038162

+ 12119861211987121205792ΔE1003816100381610038161003816120598119899+1

10038161003816100381610038162

+ 1198696Δ3

(39)


1003816100381610038161003816 le 1198674Δ3+ ΔE

100381610038161003816100381612059811989910038161003816100381610038162

(40)

Therefore

E1003816100381610038161003816120598119899+1

10038161003816100381610038162

le (1 + 1198697Δ)E

100381610038161003816100381612059811989910038161003816100381610038162

+ 2119861211987121205792Δ (Δ + 12)E

1003816100381610038161003816120598119899+110038161003816100381610038162

+ (1198695+ 1198675+ 21198674+ 21198696) Δ3

(41)

where 1198697= 2(119861

21198691+ 2 + 2119861



E10038161003816100381610038161003816120598119899+1|2le (1 + 119869

7Δ)E

10038161003816100381610038161003816120598119899|2+ 1198698Δ3

= 1198698Δ2(1 + 119869

7Δ)119899+1

minus 1

1198697

le 1198699Δ2

(42)

where 1198699= 1198698((1198901198697119879 minus 1)119869

7)



d119909 (119905) = 119886119909 (119905) d119905 + 119887119909 (119905) d119882(119905) + 119888119909 (119905) d119873(119905)

119909 (1199050) = 1199090

(43)


0119890(119886minus(12)119887

2)119905+119887119882(119905)



119909119899+1

= 119877 (119886 119887 119888 Δ Δ119882119899 Δ119873119899) 119909119899 (44)


119877 (119886 119887 119888 Δ 120582) = E (1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)) lt 1 (45)



119909119899+1

= 1198771(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (46)

where

1198771(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)

= 1 + (119886 minus1

21198872)Δ + 119887Δ119882

119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899+ 119887119888Δ119882

119899Δ119873119899

+1

21198882(Δ119873119899)2

(47)


1198771(119901 119902 119911 119888)

= E (11987721(119886 119887 119888 Δ Δ119882

119899 Δ119873119899))

= 1 + 2119901 + 1199022+ 1199012+1

21199011199022+1

21199024

+ (2 + 119888 + 2119901 + 1198881199022+ 21199022) 119911

+ (2 + 2119888 +1

21198882+ 1199022+ 119901) 119911

2

+ (119888 + 1) 1199113+1

41199114

(48)


1(119901 119902 119911 119888) lt 1


119909119899+1

= 1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899 120579) 119909119899 (49)

where

1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899 120579)

=1


1

21198872) Δ

+ 119887Δ119882119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

]

(50)


1198772(119901 119902 119911 119888 120579)

=1

(1 minus 119901120579)2[1198771(119901 119902 119911 119888)

minus 2119901120579 + 11990121205792minus 21199012120579

minus2119901119911120579 minus 1199011199112120579]

(51)



2(119901 119902 119911 119888) lt 1


119909119899+1

= 1198773(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (52)

where1198773(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)

= ((1 minus 119886120579Δ) 119868 + 119862119899)minus1


21198872)Δ

+ 119887Δ119882119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

+ 119862119899]

(53)






we obtain

119909119899+1

= 1198771015840

3(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (54)

where1198771015840

3(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)


1

21198872Δ + 119887Δ119882

119899

+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

]

(55)


119899= minus119886Δ is

1198771015840

3(119901 119902 119911 119888)

=1

(1 minus 119901 (1 + 120579))2

times [1 + 1199022+1

21199011199022+1

21199024minus 2119901120579 + 119901

21205792

+ (2 + 119888 + 1198881199022+ 21199022minus 2119901120579) 119911

+ (2 + 2119888 +1

21198882+ 1199022minus 119901120579) 119911

2

+ (119888 + 1) 1199113+1

41199114]

(56)






2((Δ119882119899)2minusΔ)


119909119899+1

= 11987710158401015840

3(119901 119902 119888 119869 Δ119873

119899) 119909119899 (57)

where11987710158401015840

3(119901 119902 119888 119869 Δ119873

119899)



21199022+ 119902119869 +

3

211990221198692

+1

2(2119888 minus 119888

2) Δ119873119899+ 119902119888119869Δ119873

119899

+1

21198882(Δ119873119899)2

]

(58)




2((Δ119882119899)2minusΔ)

is

11987710158401015840

3(119901 119902 119911 119888 119869 Δ119873

119899)

=1

radic2120587

+infin

sum

minusinfin

(120582Δ)119898119890minus120582Δ

119898

times int

+infin

minusinfin

(11987710158401015840

3(119901 119902 119888 119909119898))

2

119890minus(11990922)d119909

(59)





0= minus119886 and 119862

1= 0 when 120579 = 12 and




0= minus119886



0= minus119886 and 119862

1= 1198872 are


0= minus119886

and 1198621= 1198872 when 120579 ge 12



Methods Δ 2minus8

2minus7

2minus6

2minus5

2minus4

2minus3

2minus2

2minus1

120579-Taylor120579 = 0 00024 00048 00099 00186 00389 00911 06853 30473120579 = 12 00022 00043 00090 00177 00368 00805 02548 07407120579 = 1 00042 00083 00170 00342 00699 01454 03025 05568


120579 = 0 00021 00041 00083 00152 00317 00665 03852 15570120579 = 12 00076 00134 00223 00361 00619 01504 07024 31984120579 = 1 00075 00132 00217 00347 00582 01360 06285 26648

q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 1

Taylor



2

18

16

14

12

1

08

06

04

02



0= minus119886 and 119862

1= 0



minus1 2minus2 2


119890 =1

2000

2000

sum

119894=1

10038161003816100381610038161003816119909(119894)

119873minus 119909(119894)(119905119873)10038161003816100381610038161003816 (60)


0= 1




q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 14

Taylor



2

18

16

14

12

1

08

06

04

02



0= minus119886 and 119862

1= 1198872


0= 1 and 119862

1= 1

6 Conclusions





References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












10038171003817100381710038171003817E (120575119861(119905119899+1) | F119905119899)100381710038171003817100381710038171198712

le 1198674Δ2

119886119904 Δ 997888rarr 0


10038171003817100381710038171003817120575119861(119905119899+1)100381710038171003817100381710038171198712

le 1198675Δ32

119886119904 Δ 997888rarr 0

(27)




E [10038161003816100381610038161003816E [(119909119861

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816E [(119909120579

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

]

+ 2E [10038161003816100381610038161003816E [(119909119861

119899+1minus 119909120579

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

le 1198672

1Δ4+ 2E [

10038161003816100381610038161003816E [(119909119861

119899+1minus 119909120579

119899+1) | F119905119899]10038161003816100381610038161003816

2

]

(28)


and 119909119861119899+1


119909119861

119899+1minus 119909120579

119899+1

= minus119862119899(119909 (119905119899) minus 119909119861(119905119899+1))

= 119862119899[Δ [(1 minus 120579) 119891 (119909 (119905

119899)) + 120579119891 (119909 (119905

119899+1))]

+ 119892 (119909 (119905119899)) Δ119882

119899

+1

2119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+119892 (119909 (119905119899))) Δ119885

119899+ 119862119899(119909119899minus 119909119861

119899+1)]

= (119868 + 119862119899)minus1

119862119899

times [Δ [(1 minus 120579) 119891 (119909 (119905119899)) + 120579119891 (119909 (119905

119899+1))]

+ 119892 (119909 (119905119899)) Δ119882

119899+1

2119892 (119909 (119905

119899))

times 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]

+1

2(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

+ (119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

+1

2(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)2

+ (119892 (119909 (119905119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+119892 (119909 (119905119899))) Δ119885

119899]

(29)


1(sdot) in 119862

119899(sdot)


119899)minus1

119862119899Δ119891 (119909 (119905

119899)) | F

119905119899]10038161003816100381610038161003816

le 119872Δ1003816100381610038161003816119891 (119909119899)

1003816100381610038161003816E [100381610038161003816100381610038161198620Δ + 119862

1((Δ119882

119899)2

minus Δ)10038161003816100381610038161003816| F119905119899]

le 1198711015840119872119861(1 +

1003816100381610038161003816119909 (119905119899)10038161003816100381610038162

)12

Δ2

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899119892 (119909 (119905

119899)) Δ119882

119899| F119905119899]10038161003816100381610038161003816= 0

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899))

times [(Δ119882119899)2

minus Δ] | F119905119899]10038161003816100381610038161003816

le 11986111987210038161003816100381610038161003816119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899))10038161003816100381610038161003816

E [1003816100381610038161003816100381610038161003816((Δ119882

119899)2

minus Δ)21003816100381610038161003816100381610038161003816| F119905119899]

le 21198711015840119872119861(1 +

100381610038161003816100381611990911989910038161003816100381610038162

)12

Δ2

10038161003816100381610038161003816[E [(119868 + 119862

119899)minus1

119862119899(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899] | F119905119899]10038161003816100381610038161003816

le 12058211986111987210038161003816100381610038163ℎ (119909 (119905119899)) minus ℎℎ (119909 (119905119899))

1003816100381610038161003816 Δ2

le (2 + 119871) 1198711015840119861119872(1 +

1003816100381610038161003816119909 (119905119899)10038161003816100381610038162

)12

Δ2

10038161003816100381610038161003816E [(119868+ 119862

119899)minus1

119862119899(119892ℎ(119909 (119905119899))minus 119892 (119909 (119905

119899)))

times Δ119882119899Δ119873119899| F119905119899]10038161003816100381610038161003816= 0

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899)))

times (Δ119873119899)2

| F119905119899]10038161003816100381610038161003816

le 1198611198721003816100381610038161003816ℎℎ (119909 (119905119899)) minus ℎ (119909 (119905119899))

1003816100381610038161003816 120582Δ2(1 + 120582Δ)

le 1198711015840119861119872120582(1 +

1003816100381610038161003816119909 (119905119899)10038161003816100381610038162

)12

Δ2(1 + 120582Δ)

10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899(119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899))

minus 119892ℎ(119909 (119905119899)) + 119892 (119909 (119905

119899))) Δ119885

119899| F119905119899]10038161003816100381610038161003816

=10038161003816100381610038161003816E [(119868 + 119862

119899)minus1

119862119899(119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899))

minus 119892ℎ(119909 (119905119899)) + 119892 (119909 (119905

119899)))

times (Δ119882119899minus Δ1198851119896(119905119899+1)) | F

119905119899]10038161003816100381610038161003816

= 0

(30)


Therefore

E [10038161003816100381610038161003816E [(119909119861

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

] le 119874 (Δ4) (31)


E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899Δ [(1 minus 120579) 119891 (119909 (119905

119899)) + 120579119891 (119909 (119905

119899+1))]10038161003816100381610038161003816

2

]

le 119874 (Δ4)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899119892 (119909119899) Δ119882119899

10038161003816100381610038161003816

2

] le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]10038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

10038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

10038161003816100381610038161003816

2

]

le 119874 (Δ4)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)210038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+ 119892 (119909 (119905119899))) (Δ119885

119899)10038161003816100381610038161003816

2

]

le 119874 (Δ3)

(32)

we have

E [10038161003816100381610038161003816119909119861

119899+1minus 119909 (119905

119899+1)10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816119909 (119905119899+1) minus 119909120579

119899+1

10038161003816100381610038161003816

2

] + 2E [10038161003816100381610038161003816119909120579

119899+1minus 119909119861

119899+1

10038161003816100381610038161003816

2

]

le 119874 (Δ3)

(33)



100381710038171003817100381710038171205982

119899+1

100381710038171003817100381710038171198712le 1198676Δ 119886119904 Δ 997888rarr 0 (34)



120598119899+1

= 120598119899+ 119875119899+ 120575119899+1 (35)

where

119875119899= 119906119899+ 119862 (119909 (119905

119899)) (119909 (119905

119899) minus 119909119861(119905119899+1))

minus 119862 (119909119899) (119909119899minus 119909119899+1)

= 119906119899+ 119862 (119909 (119905

119899)) (119909 (119905

119899) minus 119909119899)

minus 119862 (119909 (119905119899)) (119909119861(119905119899+1) minus 119909119899+1)

+ (119862 (119909 (119905119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)

= 119906119899minus 119862 (119909 (119905

119899)) 119875119899

+ (119862 (119909 (119905119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)

= (119868 + 119862 (119909 (119905119899)))minus1

times [119906119899+ (119862 (119909 (119905

119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)]

(36)


E100381610038161003816100381611987511989910038161003816100381610038162

le 21198612E100381610038161003816100381611990611989910038161003816100381610038162

+ 21198612E1003816100381610038161003816[(1198620 (119909 (119905119899)) minus 1198620 (119909119899)) Δ

+ (1198621(119909 (119905119899)) minus 119862

1(119909119899)) ((Δ119882

119899)2

minus Δ)] (119909119899minus 119909119899+1)10038161003816100381610038161003816

2

le 21198612E100381610038161003816100381611990611989910038161003816100381610038162

+ 411986121198722Δ2E1003816100381610038161003816(119909119899 minus 119909119899+1)

10038161003816100381610038162

le 211986121198691ΔE100381610038161003816100381612059811989910038161003816100381610038162

+ 211986121198712Δ21205792E1003816100381610038161003816120598119899+1

10038161003816100381610038162

+ 1198695Δ3

(37)


10038161003816100381610038161003816E (119875119899| F119905119899)10038161003816100381610038161003816

2

=10038161003816100381610038161003816E ((119868 + 119862 (119909 (119905

119899)))minus1

times [119906119899+ (119862 (119909 (119905

119899)) minus 119862 (119909

119899))

times (119909119899minus 119909119899+1)] | F

119905119899)10038161003816100381610038161003816

2

le 2119861210038161003816100381610038161003816E (119906119899| F119905119899)10038161003816100381610038161003816

2

+ 21198612

times10038161003816100381610038161003816E ((119862 (119909 (119905

119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1) | F119905119899)10038161003816100381610038161003816

2

le 21198612(1198692Δ2E100381610038161003816100381612059811989910038161003816100381610038162

+ 1198712Δ21205792E (1003816100381610038161003816120598119899+1

10038161003816100381610038162

| F119905119899))

+ 21198612Δ10038161003816100381610038161003816E ((119862

0(119909 (119905119899))minus 119862

0(119909119899)) (119909119899minus 119909119899+1) | F119905119899)10038161003816100381610038161003816

2

le 211986121198692Δ2E100381610038161003816100381612059811989910038161003816100381610038162

+ 211986121198712Δ21205792E (1003816100381610038161003816120598119899+1

10038161003816100381610038162

| F119905119899) + 1198696Δ4

(38)


Thus1003816100381610038161003816E ⟨120598119899 119875119899⟩

1003816100381610038161003816

le Δminus1(E10038161003816100381610038161003816E (119875119899| F119905119899)10038161003816100381610038161003816

2

) + ΔE100381610038161003816100381612059811989910038161003816100381610038162

le (1 + 211986121198692) ΔE

100381610038161003816100381612059811989910038161003816100381610038162

+ 12119861211987121205792ΔE1003816100381610038161003816120598119899+1

10038161003816100381610038162

+ 1198696Δ3

(39)


1003816100381610038161003816 le 1198674Δ3+ ΔE

100381610038161003816100381612059811989910038161003816100381610038162

(40)

Therefore

E1003816100381610038161003816120598119899+1

10038161003816100381610038162

le (1 + 1198697Δ)E

100381610038161003816100381612059811989910038161003816100381610038162

+ 2119861211987121205792Δ (Δ + 12)E

1003816100381610038161003816120598119899+110038161003816100381610038162

+ (1198695+ 1198675+ 21198674+ 21198696) Δ3

(41)

where 1198697= 2(119861

21198691+ 2 + 2119861



E10038161003816100381610038161003816120598119899+1|2le (1 + 119869

7Δ)E

10038161003816100381610038161003816120598119899|2+ 1198698Δ3

= 1198698Δ2(1 + 119869

7Δ)119899+1

minus 1

1198697

le 1198699Δ2

(42)

where 1198699= 1198698((1198901198697119879 minus 1)119869

7)



d119909 (119905) = 119886119909 (119905) d119905 + 119887119909 (119905) d119882(119905) + 119888119909 (119905) d119873(119905)

119909 (1199050) = 1199090

(43)


0119890(119886minus(12)119887

2)119905+119887119882(119905)



119909119899+1

= 119877 (119886 119887 119888 Δ Δ119882119899 Δ119873119899) 119909119899 (44)


119877 (119886 119887 119888 Δ 120582) = E (1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)) lt 1 (45)



119909119899+1

= 1198771(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (46)

where

1198771(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)

= 1 + (119886 minus1

21198872)Δ + 119887Δ119882

119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899+ 119887119888Δ119882

119899Δ119873119899

+1

21198882(Δ119873119899)2

(47)


1198771(119901 119902 119911 119888)

= E (11987721(119886 119887 119888 Δ Δ119882

119899 Δ119873119899))

= 1 + 2119901 + 1199022+ 1199012+1

21199011199022+1

21199024

+ (2 + 119888 + 2119901 + 1198881199022+ 21199022) 119911

+ (2 + 2119888 +1

21198882+ 1199022+ 119901) 119911

2

+ (119888 + 1) 1199113+1

41199114

(48)


1(119901 119902 119911 119888) lt 1


119909119899+1

= 1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899 120579) 119909119899 (49)

where

1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899 120579)

=1


1

21198872) Δ

+ 119887Δ119882119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

]

(50)


1198772(119901 119902 119911 119888 120579)

=1

(1 minus 119901120579)2[1198771(119901 119902 119911 119888)

minus 2119901120579 + 11990121205792minus 21199012120579

minus2119901119911120579 minus 1199011199112120579]

(51)



2(119901 119902 119911 119888) lt 1


119909119899+1

= 1198773(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (52)

where1198773(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)

= ((1 minus 119886120579Δ) 119868 + 119862119899)minus1


21198872)Δ

+ 119887Δ119882119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

+ 119862119899]

(53)






we obtain

119909119899+1

= 1198771015840

3(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (54)

where1198771015840

3(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)


1

21198872Δ + 119887Δ119882

119899

+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

]

(55)


119899= minus119886Δ is

1198771015840

3(119901 119902 119911 119888)

=1

(1 minus 119901 (1 + 120579))2

times [1 + 1199022+1

21199011199022+1

21199024minus 2119901120579 + 119901

21205792

+ (2 + 119888 + 1198881199022+ 21199022minus 2119901120579) 119911

+ (2 + 2119888 +1

21198882+ 1199022minus 119901120579) 119911

2

+ (119888 + 1) 1199113+1

41199114]

(56)






2((Δ119882119899)2minusΔ)


119909119899+1

= 11987710158401015840

3(119901 119902 119888 119869 Δ119873

119899) 119909119899 (57)

where11987710158401015840

3(119901 119902 119888 119869 Δ119873

119899)



21199022+ 119902119869 +

3

211990221198692

+1

2(2119888 minus 119888

2) Δ119873119899+ 119902119888119869Δ119873

119899

+1

21198882(Δ119873119899)2

]

(58)




2((Δ119882119899)2minusΔ)

is

11987710158401015840

3(119901 119902 119911 119888 119869 Δ119873

119899)

=1

radic2120587

+infin

sum

minusinfin

(120582Δ)119898119890minus120582Δ

119898

times int

+infin

minusinfin

(11987710158401015840

3(119901 119902 119888 119909119898))

2

119890minus(11990922)d119909

(59)





0= minus119886 and 119862

1= 0 when 120579 = 12 and




0= minus119886



0= minus119886 and 119862

1= 1198872 are


0= minus119886

and 1198621= 1198872 when 120579 ge 12



Methods Δ 2minus8

2minus7

2minus6

2minus5

2minus4

2minus3

2minus2

2minus1

120579-Taylor120579 = 0 00024 00048 00099 00186 00389 00911 06853 30473120579 = 12 00022 00043 00090 00177 00368 00805 02548 07407120579 = 1 00042 00083 00170 00342 00699 01454 03025 05568


120579 = 0 00021 00041 00083 00152 00317 00665 03852 15570120579 = 12 00076 00134 00223 00361 00619 01504 07024 31984120579 = 1 00075 00132 00217 00347 00582 01360 06285 26648

q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 1

Taylor



2

18

16

14

12

1

08

06

04

02



0= minus119886 and 119862

1= 0



minus1 2minus2 2


119890 =1

2000

2000

sum

119894=1

10038161003816100381610038161003816119909(119894)

119873minus 119909(119894)(119905119873)10038161003816100381610038161003816 (60)


0= 1




q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 14

Taylor



2

18

16

14

12

1

08

06

04

02



0= minus119886 and 119862

1= 1198872


0= 1 and 119862

1= 1

6 Conclusions





References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Therefore

E [10038161003816100381610038161003816E [(119909119861

119899+1minus 119909 (119905

119899+1)) | F

119905119899]10038161003816100381610038161003816

2

] le 119874 (Δ4) (31)


E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899Δ [(1 minus 120579) 119891 (119909 (119905

119899)) + 120579119891 (119909 (119905

119899+1))]10038161003816100381610038161003816

2

]

le 119874 (Δ4)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899119892 (119909119899) Δ119882119899

10038161003816100381610038161003816

2

] le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899119892 (119909 (119905

119899)) 1198921015840(119909 (119905119899)) [(Δ119882

119899)2

minus Δ]10038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(3ℎ (119909 (119905

119899)) minus ℎ

ℎ(119909 (119905119899))) Δ119873

119899

10038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(119892ℎ(119909 (119905119899)) minus 119892 (119909 (119905

119899))) Δ119882

119899Δ119873119899

10038161003816100381610038161003816

2

]

le 119874 (Δ4)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(ℎℎ(119909 (119905119899)) minus ℎ (119909 (119905

119899))) (Δ119873

119899)210038161003816100381610038161003816

2

]

le 119874 (Δ3)

E [10038161003816100381610038161003816(119868 + 119862

119899)minus1

119862119899(119892 (119909 (119905

119899)) ℎ1015840(119909 (119905119899)) minus 119892

ℎ(119909 (119905119899))

+ 119892 (119909 (119905119899))) (Δ119885

119899)10038161003816100381610038161003816

2

]

le 119874 (Δ3)

(32)

we have

E [10038161003816100381610038161003816119909119861

119899+1minus 119909 (119905

119899+1)10038161003816100381610038161003816

2

]

le 2E [10038161003816100381610038161003816119909 (119905119899+1) minus 119909120579

119899+1

10038161003816100381610038161003816

2

] + 2E [10038161003816100381610038161003816119909120579

119899+1minus 119909119861

119899+1

10038161003816100381610038161003816

2

]

le 119874 (Δ3)

(33)



100381710038171003817100381710038171205982

119899+1

100381710038171003817100381710038171198712le 1198676Δ 119886119904 Δ 997888rarr 0 (34)



120598119899+1

= 120598119899+ 119875119899+ 120575119899+1 (35)

where

119875119899= 119906119899+ 119862 (119909 (119905

119899)) (119909 (119905

119899) minus 119909119861(119905119899+1))

minus 119862 (119909119899) (119909119899minus 119909119899+1)

= 119906119899+ 119862 (119909 (119905

119899)) (119909 (119905

119899) minus 119909119899)

minus 119862 (119909 (119905119899)) (119909119861(119905119899+1) minus 119909119899+1)

+ (119862 (119909 (119905119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)

= 119906119899minus 119862 (119909 (119905

119899)) 119875119899

+ (119862 (119909 (119905119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)

= (119868 + 119862 (119909 (119905119899)))minus1

times [119906119899+ (119862 (119909 (119905

119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1)]

(36)


E100381610038161003816100381611987511989910038161003816100381610038162

le 21198612E100381610038161003816100381611990611989910038161003816100381610038162

+ 21198612E1003816100381610038161003816[(1198620 (119909 (119905119899)) minus 1198620 (119909119899)) Δ

+ (1198621(119909 (119905119899)) minus 119862

1(119909119899)) ((Δ119882

119899)2

minus Δ)] (119909119899minus 119909119899+1)10038161003816100381610038161003816

2

le 21198612E100381610038161003816100381611990611989910038161003816100381610038162

+ 411986121198722Δ2E1003816100381610038161003816(119909119899 minus 119909119899+1)

10038161003816100381610038162

le 211986121198691ΔE100381610038161003816100381612059811989910038161003816100381610038162

+ 211986121198712Δ21205792E1003816100381610038161003816120598119899+1

10038161003816100381610038162

+ 1198695Δ3

(37)


10038161003816100381610038161003816E (119875119899| F119905119899)10038161003816100381610038161003816

2

=10038161003816100381610038161003816E ((119868 + 119862 (119909 (119905

119899)))minus1

times [119906119899+ (119862 (119909 (119905

119899)) minus 119862 (119909

119899))

times (119909119899minus 119909119899+1)] | F

119905119899)10038161003816100381610038161003816

2

le 2119861210038161003816100381610038161003816E (119906119899| F119905119899)10038161003816100381610038161003816

2

+ 21198612

times10038161003816100381610038161003816E ((119862 (119909 (119905

119899)) minus 119862 (119909

119899)) (119909119899minus 119909119899+1) | F119905119899)10038161003816100381610038161003816

2

le 21198612(1198692Δ2E100381610038161003816100381612059811989910038161003816100381610038162

+ 1198712Δ21205792E (1003816100381610038161003816120598119899+1

10038161003816100381610038162

| F119905119899))

+ 21198612Δ10038161003816100381610038161003816E ((119862

0(119909 (119905119899))minus 119862

0(119909119899)) (119909119899minus 119909119899+1) | F119905119899)10038161003816100381610038161003816

2

le 211986121198692Δ2E100381610038161003816100381612059811989910038161003816100381610038162

+ 211986121198712Δ21205792E (1003816100381610038161003816120598119899+1

10038161003816100381610038162

| F119905119899) + 1198696Δ4

(38)


Thus1003816100381610038161003816E ⟨120598119899 119875119899⟩

1003816100381610038161003816

le Δminus1(E10038161003816100381610038161003816E (119875119899| F119905119899)10038161003816100381610038161003816

2

) + ΔE100381610038161003816100381612059811989910038161003816100381610038162

le (1 + 211986121198692) ΔE

100381610038161003816100381612059811989910038161003816100381610038162

+ 12119861211987121205792ΔE1003816100381610038161003816120598119899+1

10038161003816100381610038162

+ 1198696Δ3

(39)


1003816100381610038161003816 le 1198674Δ3+ ΔE

100381610038161003816100381612059811989910038161003816100381610038162

(40)

Therefore

E1003816100381610038161003816120598119899+1

10038161003816100381610038162

le (1 + 1198697Δ)E

100381610038161003816100381612059811989910038161003816100381610038162

+ 2119861211987121205792Δ (Δ + 12)E

1003816100381610038161003816120598119899+110038161003816100381610038162

+ (1198695+ 1198675+ 21198674+ 21198696) Δ3

(41)

where 1198697= 2(119861

21198691+ 2 + 2119861



E10038161003816100381610038161003816120598119899+1|2le (1 + 119869

7Δ)E

10038161003816100381610038161003816120598119899|2+ 1198698Δ3

= 1198698Δ2(1 + 119869

7Δ)119899+1

minus 1

1198697

le 1198699Δ2

(42)

where 1198699= 1198698((1198901198697119879 minus 1)119869

7)



d119909 (119905) = 119886119909 (119905) d119905 + 119887119909 (119905) d119882(119905) + 119888119909 (119905) d119873(119905)

119909 (1199050) = 1199090

(43)


0119890(119886minus(12)119887

2)119905+119887119882(119905)



119909119899+1

= 119877 (119886 119887 119888 Δ Δ119882119899 Δ119873119899) 119909119899 (44)


119877 (119886 119887 119888 Δ 120582) = E (1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)) lt 1 (45)



119909119899+1

= 1198771(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (46)

where

1198771(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)

= 1 + (119886 minus1

21198872)Δ + 119887Δ119882

119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899+ 119887119888Δ119882

119899Δ119873119899

+1

21198882(Δ119873119899)2

(47)


1198771(119901 119902 119911 119888)

= E (11987721(119886 119887 119888 Δ Δ119882

119899 Δ119873119899))

= 1 + 2119901 + 1199022+ 1199012+1

21199011199022+1

21199024

+ (2 + 119888 + 2119901 + 1198881199022+ 21199022) 119911

+ (2 + 2119888 +1

21198882+ 1199022+ 119901) 119911

2

+ (119888 + 1) 1199113+1

41199114

(48)


1(119901 119902 119911 119888) lt 1


119909119899+1

= 1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899 120579) 119909119899 (49)

where

1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899 120579)

=1


1

21198872) Δ

+ 119887Δ119882119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

]

(50)


1198772(119901 119902 119911 119888 120579)

=1

(1 minus 119901120579)2[1198771(119901 119902 119911 119888)

minus 2119901120579 + 11990121205792minus 21199012120579

minus2119901119911120579 minus 1199011199112120579]

(51)



2(119901 119902 119911 119888) lt 1


119909119899+1

= 1198773(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (52)

where1198773(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)

= ((1 minus 119886120579Δ) 119868 + 119862119899)minus1


21198872)Δ

+ 119887Δ119882119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

+ 119862119899]

(53)






we obtain

119909119899+1

= 1198771015840

3(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (54)

where1198771015840

3(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)


1

21198872Δ + 119887Δ119882

119899

+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

]

(55)


119899= minus119886Δ is

1198771015840

3(119901 119902 119911 119888)

=1

(1 minus 119901 (1 + 120579))2

times [1 + 1199022+1

21199011199022+1

21199024minus 2119901120579 + 119901

21205792

+ (2 + 119888 + 1198881199022+ 21199022minus 2119901120579) 119911

+ (2 + 2119888 +1

21198882+ 1199022minus 119901120579) 119911

2

+ (119888 + 1) 1199113+1

41199114]

(56)






2((Δ119882119899)2minusΔ)


119909119899+1

= 11987710158401015840

3(119901 119902 119888 119869 Δ119873

119899) 119909119899 (57)

where11987710158401015840

3(119901 119902 119888 119869 Δ119873

119899)



21199022+ 119902119869 +

3

211990221198692

+1

2(2119888 minus 119888

2) Δ119873119899+ 119902119888119869Δ119873

119899

+1

21198882(Δ119873119899)2

]

(58)




2((Δ119882119899)2minusΔ)

is

11987710158401015840

3(119901 119902 119911 119888 119869 Δ119873

119899)

=1

radic2120587

+infin

sum

minusinfin

(120582Δ)119898119890minus120582Δ

119898

times int

+infin

minusinfin

(11987710158401015840

3(119901 119902 119888 119909119898))

2

119890minus(11990922)d119909

(59)





0= minus119886 and 119862

1= 0 when 120579 = 12 and




0= minus119886



0= minus119886 and 119862

1= 1198872 are


0= minus119886

and 1198621= 1198872 when 120579 ge 12



Methods Δ 2minus8

2minus7

2minus6

2minus5

2minus4

2minus3

2minus2

2minus1

120579-Taylor120579 = 0 00024 00048 00099 00186 00389 00911 06853 30473120579 = 12 00022 00043 00090 00177 00368 00805 02548 07407120579 = 1 00042 00083 00170 00342 00699 01454 03025 05568


120579 = 0 00021 00041 00083 00152 00317 00665 03852 15570120579 = 12 00076 00134 00223 00361 00619 01504 07024 31984120579 = 1 00075 00132 00217 00347 00582 01360 06285 26648

q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 1

Taylor



2

18

16

14

12

1

08

06

04

02



0= minus119886 and 119862

1= 0



minus1 2minus2 2


119890 =1

2000

2000

sum

119894=1

10038161003816100381610038161003816119909(119894)

119873minus 119909(119894)(119905119873)10038161003816100381610038161003816 (60)


0= 1




q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 14

Taylor



2

18

16

14

12

1

08

06

04

02



0= minus119886 and 119862

1= 1198872


0= 1 and 119862

1= 1

6 Conclusions





References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Thus1003816100381610038161003816E ⟨120598119899 119875119899⟩

1003816100381610038161003816

le Δminus1(E10038161003816100381610038161003816E (119875119899| F119905119899)10038161003816100381610038161003816

2

) + ΔE100381610038161003816100381612059811989910038161003816100381610038162

le (1 + 211986121198692) ΔE

100381610038161003816100381612059811989910038161003816100381610038162

+ 12119861211987121205792ΔE1003816100381610038161003816120598119899+1

10038161003816100381610038162

+ 1198696Δ3

(39)


1003816100381610038161003816 le 1198674Δ3+ ΔE

100381610038161003816100381612059811989910038161003816100381610038162

(40)

Therefore

E1003816100381610038161003816120598119899+1

10038161003816100381610038162

le (1 + 1198697Δ)E

100381610038161003816100381612059811989910038161003816100381610038162

+ 2119861211987121205792Δ (Δ + 12)E

1003816100381610038161003816120598119899+110038161003816100381610038162

+ (1198695+ 1198675+ 21198674+ 21198696) Δ3

(41)

where 1198697= 2(119861

21198691+ 2 + 2119861



E10038161003816100381610038161003816120598119899+1|2le (1 + 119869

7Δ)E

10038161003816100381610038161003816120598119899|2+ 1198698Δ3

= 1198698Δ2(1 + 119869

7Δ)119899+1

minus 1

1198697

le 1198699Δ2

(42)

where 1198699= 1198698((1198901198697119879 minus 1)119869

7)



d119909 (119905) = 119886119909 (119905) d119905 + 119887119909 (119905) d119882(119905) + 119888119909 (119905) d119873(119905)

119909 (1199050) = 1199090

(43)


0119890(119886minus(12)119887

2)119905+119887119882(119905)



119909119899+1

= 119877 (119886 119887 119888 Δ Δ119882119899 Δ119873119899) 119909119899 (44)


119877 (119886 119887 119888 Δ 120582) = E (1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)) lt 1 (45)



119909119899+1

= 1198771(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (46)

where

1198771(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)

= 1 + (119886 minus1

21198872)Δ + 119887Δ119882

119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899+ 119887119888Δ119882

119899Δ119873119899

+1

21198882(Δ119873119899)2

(47)


1198771(119901 119902 119911 119888)

= E (11987721(119886 119887 119888 Δ Δ119882

119899 Δ119873119899))

= 1 + 2119901 + 1199022+ 1199012+1

21199011199022+1

21199024

+ (2 + 119888 + 2119901 + 1198881199022+ 21199022) 119911

+ (2 + 2119888 +1

21198882+ 1199022+ 119901) 119911

2

+ (119888 + 1) 1199113+1

41199114

(48)


1(119901 119902 119911 119888) lt 1


119909119899+1

= 1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899 120579) 119909119899 (49)

where

1198772(119886 119887 119888 Δ Δ119882

119899 Δ119873119899 120579)

=1


1

21198872) Δ

+ 119887Δ119882119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

]

(50)


1198772(119901 119902 119911 119888 120579)

=1

(1 minus 119901120579)2[1198771(119901 119902 119911 119888)

minus 2119901120579 + 11990121205792minus 21199012120579

minus2119901119911120579 minus 1199011199112120579]

(51)



2(119901 119902 119911 119888) lt 1


119909119899+1

= 1198773(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (52)

where1198773(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)

= ((1 minus 119886120579Δ) 119868 + 119862119899)minus1


21198872)Δ

+ 119887Δ119882119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

+ 119862119899]

(53)






we obtain

119909119899+1

= 1198771015840

3(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (54)

where1198771015840

3(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)


1

21198872Δ + 119887Δ119882

119899

+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

]

(55)


119899= minus119886Δ is

1198771015840

3(119901 119902 119911 119888)

=1

(1 minus 119901 (1 + 120579))2

times [1 + 1199022+1

21199011199022+1

21199024minus 2119901120579 + 119901

21205792

+ (2 + 119888 + 1198881199022+ 21199022minus 2119901120579) 119911

+ (2 + 2119888 +1

21198882+ 1199022minus 119901120579) 119911

2

+ (119888 + 1) 1199113+1

41199114]

(56)






2((Δ119882119899)2minusΔ)


119909119899+1

= 11987710158401015840

3(119901 119902 119888 119869 Δ119873

119899) 119909119899 (57)

where11987710158401015840

3(119901 119902 119888 119869 Δ119873

119899)



21199022+ 119902119869 +

3

211990221198692

+1

2(2119888 minus 119888

2) Δ119873119899+ 119902119888119869Δ119873

119899

+1

21198882(Δ119873119899)2

]

(58)




2((Δ119882119899)2minusΔ)

is

11987710158401015840

3(119901 119902 119911 119888 119869 Δ119873

119899)

=1

radic2120587

+infin

sum

minusinfin

(120582Δ)119898119890minus120582Δ

119898

times int

+infin

minusinfin

(11987710158401015840

3(119901 119902 119888 119909119898))

2

119890minus(11990922)d119909

(59)





0= minus119886 and 119862

1= 0 when 120579 = 12 and




0= minus119886



0= minus119886 and 119862

1= 1198872 are


0= minus119886

and 1198621= 1198872 when 120579 ge 12



Methods Δ 2minus8

2minus7

2minus6

2minus5

2minus4

2minus3

2minus2

2minus1

120579-Taylor120579 = 0 00024 00048 00099 00186 00389 00911 06853 30473120579 = 12 00022 00043 00090 00177 00368 00805 02548 07407120579 = 1 00042 00083 00170 00342 00699 01454 03025 05568


120579 = 0 00021 00041 00083 00152 00317 00665 03852 15570120579 = 12 00076 00134 00223 00361 00619 01504 07024 31984120579 = 1 00075 00132 00217 00347 00582 01360 06285 26648

q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 1

Taylor



2

18

16

14

12

1

08

06

04

02



0= minus119886 and 119862

1= 0



minus1 2minus2 2


119890 =1

2000

2000

sum

119894=1

10038161003816100381610038161003816119909(119894)

119873minus 119909(119894)(119905119873)10038161003816100381610038161003816 (60)


0= 1




q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 14

Taylor



2

18

16

14

12

1

08

06

04

02



0= minus119886 and 119862

1= 1198872


0= 1 and 119862

1= 1

6 Conclusions





References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2(119901 119902 119911 119888) lt 1


119909119899+1

= 1198773(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (52)

where1198773(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)

= ((1 minus 119886120579Δ) 119868 + 119862119899)minus1


21198872)Δ

+ 119887Δ119882119899+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

+ 119862119899]

(53)






we obtain

119909119899+1

= 1198771015840

3(119886 119887 119888 Δ Δ119882

119899 Δ119873119899) 119909119899 (54)

where1198771015840

3(119886 119887 119888 Δ Δ119882

119899 Δ119873119899)


1

21198872Δ + 119887Δ119882

119899

+1

21198872(Δ119882119899)2

+1

2(2119888 minus 119888

2) Δ119873119899

+ 119887119888Δ119882119899Δ119873119899+1

21198882(Δ119873119899)2

]

(55)


119899= minus119886Δ is

1198771015840

3(119901 119902 119911 119888)

=1

(1 minus 119901 (1 + 120579))2

times [1 + 1199022+1

21199011199022+1

21199024minus 2119901120579 + 119901

21205792

+ (2 + 119888 + 1198881199022+ 21199022minus 2119901120579) 119911

+ (2 + 2119888 +1

21198882+ 1199022minus 119901120579) 119911

2

+ (119888 + 1) 1199113+1

41199114]

(56)






2((Δ119882119899)2minusΔ)


119909119899+1

= 11987710158401015840

3(119901 119902 119888 119869 Δ119873

119899) 119909119899 (57)

where11987710158401015840

3(119901 119902 119888 119869 Δ119873

119899)



21199022+ 119902119869 +

3

211990221198692

+1

2(2119888 minus 119888

2) Δ119873119899+ 119902119888119869Δ119873

119899

+1

21198882(Δ119873119899)2

]

(58)




2((Δ119882119899)2minusΔ)

is

11987710158401015840

3(119901 119902 119911 119888 119869 Δ119873

119899)

=1

radic2120587

+infin

sum

minusinfin

(120582Δ)119898119890minus120582Δ

119898

times int

+infin

minusinfin

(11987710158401015840

3(119901 119902 119888 119909119898))

2

119890minus(11990922)d119909

(59)





0= minus119886 and 119862

1= 0 when 120579 = 12 and




0= minus119886



0= minus119886 and 119862

1= 1198872 are


0= minus119886

and 1198621= 1198872 when 120579 ge 12



Methods Δ 2minus8

2minus7

2minus6

2minus5

2minus4

2minus3

2minus2

2minus1

120579-Taylor120579 = 0 00024 00048 00099 00186 00389 00911 06853 30473120579 = 12 00022 00043 00090 00177 00368 00805 02548 07407120579 = 1 00042 00083 00170 00342 00699 01454 03025 05568


120579 = 0 00021 00041 00083 00152 00317 00665 03852 15570120579 = 12 00076 00134 00223 00361 00619 01504 07024 31984120579 = 1 00075 00132 00217 00347 00582 01360 06285 26648

q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 1

Taylor



2

18

16

14

12

1

08

06

04

02



0= minus119886 and 119862

1= 0



minus1 2minus2 2


119890 =1

2000

2000

sum

119894=1

10038161003816100381610038161003816119909(119894)

119873minus 119909(119894)(119905119873)10038161003816100381610038161003816 (60)


0= 1




q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 14

Taylor



2

18

16

14

12

1

08

06

04

02



0= minus119886 and 119862

1= 1198872


0= 1 and 119862

1= 1

6 Conclusions





References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Methods Δ 2minus8

2minus7

2minus6

2minus5

2minus4

2minus3

2minus2

2minus1

120579-Taylor120579 = 0 00024 00048 00099 00186 00389 00911 06853 30473120579 = 12 00022 00043 00090 00177 00368 00805 02548 07407120579 = 1 00042 00083 00170 00342 00699 01454 03025 05568


120579 = 0 00021 00041 00083 00152 00317 00665 03852 15570120579 = 12 00076 00134 00223 00361 00619 01504 07024 31984120579 = 1 00075 00132 00217 00347 00582 01360 06285 26648

q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 1

Taylor



2

18

16

14

12

1

08

06

04

02



0= minus119886 and 119862

1= 0



minus1 2minus2 2


119890 =1

2000

2000

sum

119894=1

10038161003816100381610038161003816119909(119894)

119873minus 119909(119894)(119905119873)10038161003816100381610038161003816 (60)


0= 1




q

p

120579-Taylor with 120579 = 12

120579-Taylor with 120579 = 14

Taylor



2

18

16

14

12

1

08

06

04

02



0= minus119886 and 119862

1= 1198872


0= 1 and 119862

1= 1

6 Conclusions





References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article implicit numerical solutions for solving stochastic differential … · 2019. 7....

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