International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 254720 | 18 pages | https://doi.org/10.1155/2009/254720

Implicit Difference Inequalities Corresponding to First-Order Partial Differential Functional Equations

Academic Editor: Donal O'Regan
Received19 Aug 2008
Accepted05 Jan 2009
Published16 Mar 2009

Abstract

We give a theorem on implicit difference functional inequalities generated by mixed problems for nonlinear systems of first-order partial differential functional equations. We apply this result in the investigations of the stability of difference methods. Classical solutions of mixed problems are approximated in the paper by solutions of suitable implicit difference schemes. The proof of the convergence of difference method is based on comparison technique, and the result on difference functional inequalities is used. Numerical examples are presented.

1. Introduction

The papers [1, 2] initiated the theory of difference inequalities generated by first-order partial differential equations. The results and the methods presented in [1, 2] were extended in [3, 4] on functional differential problems, and they were generalized in [5–8] on parabolic differential and differential functional equations. Explicit difference schemes were considered in the above papers.

Our purpose is to give a result on implicit difference inequalities corresponding to initial boundary value problems for first-order functional differential equations.

We prove also that that there are implicit difference methods which are convergent. The proof of the convergence is based on a theorem on difference functional inequalities.

We formulate our functional differential problems. For any metric spaces 𝑋 and π‘Œ we denote by 𝐢(𝑋,π‘Œ) the class of all continuous functions from 𝑋 into π‘Œ. We will use vectorial inequalities with the understanding that the same inequalities hold between their corresponding components. Write𝐸=[0,π‘Ž]Γ—(βˆ’π‘,𝑏),𝐷=βˆ’π‘‘0ξ€»,0Γ—[βˆ’π‘‘,𝑑],(1.1)where π‘Ž>0, 𝑏=(𝑏1,…,𝑏𝑛)βˆˆβ„π‘›, 𝑏𝑖>0 for 1≀𝑖≀𝑛 and 𝑑=(𝑑1,…,𝑑𝑛)βˆˆβ„π‘›+, 𝑑0βˆˆβ„+, ℝ+=[0,+∞). Let 𝑐=𝑏+𝑑 and𝐸0=ξ€Ίβˆ’π‘‘0ξ€»πœ•,0Γ—[βˆ’π‘,𝑐],0𝐸=[0,π‘Ž]Γ—([βˆ’π‘,𝑐]⧡(βˆ’π‘,𝑏)),Ξ©=𝐸βˆͺ𝐸0βˆͺπœ•0𝐸.(1.2)For a function π‘§βˆΆΞ©β†’β„π‘˜, 𝑧=(𝑧1,…,π‘§π‘˜), and for a point (𝑑,π‘₯)∈𝐸 where 𝐸 is the closure of 𝐸, we define a function 𝑧(𝑑,π‘₯)βˆΆπ·β†’β„π‘˜ by 𝑧(𝑑,π‘₯)(𝜏,𝑦)=𝑧(𝑑+𝜏,π‘₯+𝑦), (𝜏,𝑦)∈𝐷. Then 𝑧(𝑑,π‘₯) is the restriction of 𝑧 to the set [π‘‘βˆ’π‘‘0,𝑑]Γ—[π‘₯βˆ’π‘‘,π‘₯+𝑑] and this restriction is shifted to the set 𝐷. Write Ξ£=𝐸×𝐢(𝐷,β„π‘˜)×ℝ𝑛 and suppose that 𝑓=(𝑓1,…,π‘“π‘˜)βˆΆΞ£β†’β„π‘˜ and πœ‘βˆΆπΈ0βˆͺπœ•0πΈβ†’β„π‘˜, πœ‘=(πœ‘1,…,πœ‘π‘˜), are given functions. Let us denote by 𝑧=(𝑧1,…,π‘§π‘˜) an unknown function of the variables (𝑑,π‘₯), π‘₯=(π‘₯1,…,π‘₯𝑛). Write𝑓𝔽[𝑧](𝑑,π‘₯)=1𝑑,π‘₯,𝑧(𝑑,π‘₯),πœ•π‘₯𝑧1ξ€Έ(𝑑,π‘₯),…,π‘“π‘˜ξ€·π‘‘,π‘₯,𝑧(𝑑,π‘₯),πœ•π‘₯π‘§π‘˜(𝑑,π‘₯)ξ€Έξ€Έ(1.3)and πœ•π‘₯𝑧𝑖=(πœ•π‘₯1𝑧𝑖,…,πœ•π‘₯𝑛𝑧𝑖), 1β‰€π‘–β‰€π‘˜. We consider the system of functional differential equationsπœ•π‘‘π‘§(𝑑,π‘₯)=𝔽[𝑧](𝑑,π‘₯)(1.4)with the initial boundary condition𝑧(𝑑,π‘₯)=πœ‘(𝑑,π‘₯)on𝐸0βˆͺπœ•0𝐸.(1.5)In the paper we consider classical solutions of (1.4), (1.5).

We give examples of equations which can be obtained from (1.4) by specializing the operator 𝑓.

Example 1.1. Suppose that the function π›ΌβˆΆπΈβ†’β„1+𝑛 satisfies the condition: 𝛼(𝑑,π‘₯)βˆ’(𝑑,π‘₯)∈𝐷 for (𝑑,π‘₯)∈𝐸. For a given 𝑓𝑓=(1𝑓,…,π‘˜)βˆΆπΈΓ—β„π‘˜Γ—β„π‘˜Γ—β„π‘›β†’β„π‘˜ we put 𝑓𝑓(𝑑,π‘₯,𝑀,π‘ž)=𝑑,π‘₯,𝑀(0,πœƒ),𝑀𝛼(𝑑,π‘₯)βˆ’(𝑑,π‘₯),π‘žonΞ£,(1.6)where πœƒ=(0,…,0)βˆˆβ„π‘›. Then (1.4) is reduced to the system of differential equations with deviated variablesπœ•π‘‘π‘§π‘–ξ‚π‘“(𝑑,π‘₯)=𝑖𝑑,π‘₯,𝑧(𝑑,π‘₯),𝑧𝛼(𝑑,π‘₯),πœ•π‘₯𝑧𝑖(𝑑,π‘₯),𝑖=1,…,π‘˜.(1.7)

Example 1.2. For the above 𝑓 we define ξ‚π‘“ξ‚€ξ€œπ‘“(𝑑,π‘₯,𝑀,π‘ž)=𝑑,π‘₯,𝑀(0,πœƒ),𝐷𝑀(𝜏,𝑦)π‘‘π‘¦π‘‘πœ,π‘žonΞ£.(1.8)Then (1.4) is equivalent to the system of differential integral equationsπœ•π‘‘π‘§π‘–ξ‚π‘“(𝑑,π‘₯)=π‘–ξ‚€ξ€œπ‘‘,π‘₯,𝑧(𝑑,π‘₯),𝐷𝑧(𝑑+𝜏,π‘₯+𝑦)π‘‘π‘¦π‘‘πœ,πœ•π‘₯𝑧𝑖(𝑑,π‘₯),𝑖=1,…,π‘˜.(1.9)

It is clear that more complicated differential systems with deviated variables and differential integral problems can be obtained from (1.4) by a suitable definition of 𝑓. Sufficient conditions for the existence and uniqueness of classical or generalized solutions of (1.4), (1.5) can be found in [9, 10].

Our motivations for investigations of implicit difference functional inequalities and for the construction of implicit difference schemes are the following. Two types of assumptions are needed in theorems on the stability of difference functional equations generated by (1.4), (1.5). The first type conditions concern regularity of 𝑓. It is assumed that

(i)the function 𝑓 of the variables (𝑑,π‘₯,𝑀,π‘ž), π‘ž=(π‘ž1,…,π‘žπ‘›), is of class 𝐢1 with respect to π‘ž and the functions πœ•π‘žπ‘“π‘–=(πœ•π‘ž1𝑓𝑖,…,πœ•π‘žπ‘›π‘“π‘–), 1β‰€π‘–β‰€π‘˜, are bounded,(ii)𝑓 satisfies the Perron type estimates with respect to the functional variable 𝑀. The second type conditions concern the mesh. It is required that difference schemes generated by (1.4), (1.5) satisfy the condition1βˆ’β„Ž0𝑛𝑗=11β„Žπ‘—||πœ•π‘žπ‘—π‘“π‘–||(𝑑,π‘₯,𝑀,π‘ž)β‰₯0onΞ£for𝑖=1,…,π‘˜,(1.10)where β„Ž0 and β„Žβ€²=(β„Ž1,…,β„Žπ‘›) are steps of the mesh with respect to 𝑑 and (π‘₯1,…,π‘₯𝑛) respectively. The above assumption is known as a generalized Courant-Friedrichs-Levy (CFL) condition for (1.4), (1.5) (see [11, Chapter 3] and [10, Chapter 5]). It is clear that strong assumptions on relations between β„Ž0 and β„Žξ…ž are required in (1.10). It is important in our considerations that assumption (1.10) is omitted in a theorem on difference inequalities and in a theorem on the convergence of difference schemes.

We show that there are implicit difference methods for (1.4), (1.5) which are convergent while the corresponding explicit difference schemes are not convergent. We give suitable numerical examples.

The paper is organized as follows. A theorem on implicit difference functional inequalities with unknown function of several variables is proved in Section 2. We propose in Section 3 implicit difference schemes for the numerical solving of functional differential equations. Convergence results and error estimates are presented. A theorem on difference inequalities is used in the investigation of the stability of implicit difference methods. Numerical examples are given in the last part of the paper.

We use in the paper general ideas for finite difference equations which were introduced in [12–14]. For further bibliographic informations concerning differential and functional differential inequalities and applications see the survey paper [15] and the monographs [16, 17].

2. Functional Difference Inequalities

For any two sets π‘ˆ and π‘Š we denote by 𝐹(π‘ˆ,π‘Š) the class of all functions defined on π‘ˆ and taking values in π‘Š. Let β„• and β„€ be the sets of natural numbers and integers, respectively. For π‘₯=(π‘₯1,…,π‘₯𝑛)βˆˆβ„π‘›, 𝑝=(𝑝1,…,π‘π‘˜)βˆˆβ„π‘˜ we put||π‘₯β€–π‘₯β€–=1||||π‘₯+β‹―+𝑛||,β€–π‘β€–βˆžξ€½||𝑝=max𝑖||ξ€ΎβˆΆ1β‰€π‘–β‰€π‘˜.(2.1)We define a mesh on Ξ© in the following way. Suppose that (β„Ž0,β„Žξ…ž), β„Žβ€²=(β„Ž1,…,β„Žπ‘›), stand for steps of the mesh. For (π‘Ÿ,π‘š)βˆˆβ„€1+𝑛 where π‘š=(π‘š1,…,π‘šπ‘›), we define nodal points as follows:𝑑(π‘Ÿ)=π‘Ÿβ„Ž0,π‘₯(π‘š)=ξ€·π‘₯(π‘š1)1,…,π‘₯(π‘šπ‘›)𝑛=ξ€·π‘š1β„Ž1,…,π‘šπ‘›β„Žπ‘›ξ€Έ.(2.2)Let us denote by 𝐻 the set of all β„Ž=(β„Ž0,β„Žξ…ž) such that there are 𝐾0βˆˆβ„€ and 𝐾=(𝐾1,…,𝐾𝑛)βˆˆβ„€π‘› satisfying the conditions: 𝐾0β„Ž0=𝑑0 and (𝐾1β„Ž1,…,πΎπ‘›β„Žπ‘›)=𝑑. Setβ„β„Ž1+𝑛=𝑑(π‘Ÿ),π‘₯(π‘š)ξ€ΈβˆΆ(π‘Ÿ,π‘š)βˆˆβ„€1+𝑛,π·β„Ž=π·βˆ©β„β„Ž1+𝑛,πΈβ„Ž=πΈβˆ©β„β„Ž1+𝑛,𝐸0.β„Ž=𝐸0βˆ©β„β„Ž1+𝑛,πœ•0πΈβ„Ž=πœ•0πΈβˆ©β„β„Ž1+𝑛,Ξ©β„Ž=πΈβ„Žβˆͺ𝐸0.β„Žβˆͺπœ•0πΈβ„Ž.(2.3)Let 𝑁0βˆˆβ„• be defined by the relations: 𝑁0β„Ž0β‰€π‘Ž<(𝑁0+1)β„Ž0 andπΈξ…žβ„Ž=𝑑(π‘Ÿ),π‘₯(π‘š)ξ€ΈβˆˆπΈβ„ŽβˆΆ0β‰€π‘Ÿβ‰€π‘0ξ€Ύβˆ’1.(2.4)For functions π‘€βˆΆπ·β„Žβ†’β„π‘˜ and π‘§βˆΆΞ©β„Žβ†’β„π‘˜ we write 𝑀(π‘Ÿ,π‘š)=𝑀(𝑑(π‘Ÿ),π‘₯(π‘š)) on π·β„Ž and 𝑧(π‘Ÿ,π‘š)=𝑧(𝑑(π‘Ÿ),π‘₯(π‘š)) on Ξ©β„Ž. We need a discrete version of the operator (𝑑,π‘₯)→𝑧(𝑑,π‘₯). For a function π‘§βˆΆΞ©β„Žβ†’β„π‘˜ and for a point (𝑑(π‘Ÿ),π‘₯(π‘š))βˆˆπΈβ„Ž we define a function 𝑧[π‘Ÿ,π‘š]βˆΆπ·β„Žβ†’β„π‘˜ by𝑧[π‘Ÿ,π‘š]𝑑(𝜏,𝑦)=𝑧(π‘Ÿ)+𝜏,π‘₯(π‘š)ξ€Έ+𝑦,(𝜏,𝑦)βˆˆπ·β„Ž.(2.5)Solutions of difference equations corresponding to (1.4), (1.5) are functions defined on the mesh. On the other hand (1.4) contains the functional variable 𝑧(𝑑,π‘₯) which is an element of the space 𝐢(𝐷,β„π‘˜). Then we need an interpolating operator π‘‡β„ŽβˆΆπΉ(π·β„Ž,β„π‘˜)→𝐢(𝐷,β„π‘˜). We define π‘‡β„Ž in the following way. Let us denote by (πœ—1,…,πœ—π‘›) the family of sets defined byπœ—π‘–={0,1}if𝑑𝑖>0,πœ—π‘–={0}if𝑑𝑖=0,1≀𝑖≀𝑛.(2.6)Set 𝜐=(𝜐1,…,πœπ‘›)βˆˆβ„€π‘› and πœπ‘–=0 if 𝑑𝑖=0, πœπ‘–=1 if 𝑑𝑖>0 where 1≀𝑖≀𝑛. WriteΞ”+=ξ€½ξ€·πœ†πœ†=1,…,πœ†π‘›ξ€ΈβˆΆπœ†π‘–βˆˆπœ—π‘–ξ€Ύfor1≀𝑖≀𝑛.(2.7)Set 𝑒𝑖=(0,…,0,1,0,…,0)βˆˆβ„π‘› with 1 standing on the 𝑖th place.

Let π‘€βˆˆπΉ(π·β„Ž,β„π‘˜) and (𝑑,π‘₯)∈𝐷. Suppose that 𝑑0>0. There exists (𝑑(π‘Ÿ),π‘₯(π‘š))βˆˆπ·β„Ž such that (𝑑(π‘Ÿ+1),π‘₯(π‘š+𝜐))βˆˆπ·β„Ž and 𝑑(π‘Ÿ)≀𝑑≀𝑑(π‘Ÿ+1), π‘₯(π‘š)≀π‘₯≀π‘₯(π‘š+𝜐). Writeπ‘‡β„Žξ‚€[𝑀](𝑑,π‘₯)=1βˆ’π‘‘βˆ’π‘‘(π‘Ÿ)β„Ž0ξ‚ξ“πœ†βˆˆΞ”+𝑀(π‘Ÿ,π‘š+πœ†)ξ‚€π‘₯βˆ’π‘₯(π‘š)β„Žξ…žξ‚πœ†ξ‚€1βˆ’π‘₯βˆ’π‘₯(π‘š)β„Žξ…žξ‚1βˆ’πœ†+π‘‘βˆ’π‘‘(π‘Ÿ)β„Ž0ξ“πœ†βˆˆΞ”+𝑀(π‘Ÿ+1,π‘š+πœ†)ξ‚€π‘₯βˆ’π‘₯(π‘š)β„Žξ…žξ‚πœ†ξ‚€1βˆ’π‘₯βˆ’π‘₯(π‘š)β„Žξ…žξ‚1βˆ’πœ†,(2.8) whereξ‚€π‘₯βˆ’π‘₯(π‘š)β„Žξ…žξ‚πœ†=𝑛𝑖=1ξ‚€π‘₯π‘–βˆ’π‘₯(π‘šπ‘–)π‘–β„Žπ‘–ξ‚πœ†π‘–,ξ‚€1βˆ’π‘₯βˆ’π‘₯(π‘š)β„Žξ…žξ‚1βˆ’πœ†=𝑛𝑖=1ξ‚€π‘₯1βˆ’π‘–βˆ’π‘₯(π‘šπ‘–)π‘–β„Žπ‘–ξ‚1βˆ’πœ†π‘–(2.9) and we take 00=1 in the above formulas. If 𝑑0=0 then we putπ‘‡β„Žξ“[𝑀](𝑑,π‘₯)=πœ†βˆˆΞ”+𝑀(π‘Ÿ,π‘š+πœ†)ξ‚€π‘₯βˆ’π‘₯(π‘š)β„Žξ…žξ‚πœ†ξ‚€1βˆ’π‘₯βˆ’π‘₯(π‘š)β„Žξ…žξ‚1βˆ’πœ†.(2.10)Then we have defined π‘‡β„Ž[𝑀] on 𝐷. It is easy to see that π‘‡β„Ž[𝑀]∈𝐢(𝐷,β„π‘˜). The above interpolating operator has been first proposed in [10, Chapter 5].

For 𝑀,π‘€βˆˆπΉ(π·β„Ž,β„π‘˜) we write 𝑀≀𝑀 if 𝑀(π‘Ÿ,π‘š)≀𝑀(π‘Ÿ,π‘š) where (𝑑(π‘Ÿ),π‘₯(π‘š))βˆˆπ·β„Ž. In a similar way we define the relation 𝑀≀𝑀 for 𝑀,π‘€βˆˆπΆ(𝐷,β„π‘˜) and the relation 𝑧≀𝑧 for 𝑧,π‘§βˆˆπΉ(Ξ©β„Ž,β„π‘˜) and for 𝑧,π‘§βˆˆπΆ(Ξ©,β„π‘˜).

We formulate an implicit difference scheme for (1.4), (1.5). For π‘₯,π‘¦βˆˆβ„π‘› we write π‘₯⋄𝑦=(π‘₯1𝑦1,…,π‘₯𝑛𝑦𝑛)βˆˆβ„π‘›.

Assumption (𝐻[𝑓]). The function 𝑓=(𝑓1,…,π‘“π‘˜)βˆΆΞ£β†’β„π‘˜ of the variables (𝑑,π‘₯,𝑀,π‘ž), π‘ž=(π‘ž1,…,π‘žπ‘›), is continuous and
(1)the partial derivatives (πœ•π‘ž1𝑓𝑖,…,πœ•π‘žπ‘›π‘“π‘–)=πœ•π‘žπ‘“π‘–, 𝑖=1,…,π‘˜, exist on Ξ£ and the functions πœ•π‘žπ‘“π‘–, 𝑖=1,…,π‘˜, are continuous and bounded on Ξ£,(2)there is Μƒπ‘₯∈(βˆ’π‘,𝑏), Μƒπ‘₯=(π‘₯1,…,π‘₯𝑛), such that(π‘₯βˆ’Μƒπ‘₯)β‹„πœ•π‘žπ‘“π‘–(𝑑,π‘₯,𝑀,π‘ž)β‰₯πœƒonΞ£for𝑖=1,…,π‘˜,(2.11)(3)there is πœ€0>0 such that for 0<β„Ž0<πœ€0 and 𝑀,π‘€βˆˆπΆ(𝐷,β„π‘˜), 𝑀≀𝑀, we have𝑀(0,πœƒ)+β„Ž0𝑓(𝑑,π‘₯,𝑀,π‘ž)≀𝑀(0,πœƒ)+β„Ž0𝑓(𝑑,π‘₯,𝑀,π‘ž),(𝑑,π‘₯,π‘ž)βˆˆπΈΓ—β„π‘›.(2.12)

Remark 2.1. The existence theory of classical or generalized solutions to (1.4), (1.5) is based on a method of bicharacteristics. Suppose that π‘§βˆˆπΆ(Ξ©,β„π‘˜), π‘’βˆˆπΆ(Ξ©,ℝ𝑛). Let us denote by 𝑔𝑖𝑔[𝑧,𝑒](β‹…,𝑑,π‘₯)=𝑖.1[𝑧,𝑒](β‹…,𝑑,π‘₯),…,𝑔𝑖.𝑛[𝑧,𝑒](β‹…,𝑑,π‘₯)(2.13)the 𝑖th bicharacteristic of (1.4) corresponding to (𝑧,𝑒). Then 𝑔𝑖[𝑧,𝑒](β‹…,𝑑,π‘₯) is a solution of the Cauchy problemπ‘¦ξ…ž(𝜏)=βˆ’πœ•π‘žπ‘“π‘–ξ€·πœ,𝑦(𝜏),𝑧(𝜏,𝑦(𝜏))ξ€·,π‘’πœ,𝑦(𝜏)ξ€Έξ€Έ,𝑦(𝑑)=π‘₯.(2.14)Assumption (2.11) states that the bicharacteristics satisfy the following monotonicity conditions: If π‘₯π‘—βˆ’ξ‚π‘₯𝑗β‰₯0 the function 𝑔𝑖𝑗[𝑧,𝑒](β‹…,𝑑,π‘₯) is non increasing. If π‘₯π‘—βˆ’ξ‚π‘₯𝑗<0 then 𝑔𝑖𝑗[𝑧,𝑒](β‹…,𝑑,π‘₯) is nondecreasing.
The same property of bicharacteristics is needed in a theorem on the existence and uniqueness of solutions to (1.4), (1.5) see [9]. It is important that our theory of difference methods is consistent with known theorems on the existence of solutions to (1.4), (1.5).

Remark 2.2. Given the function 𝑓𝑓=(1𝑓,…,π‘˜)βˆΆπΈΓ—β„Γ—πΆ(𝐷,β„π‘˜)Γ—β„π‘›β†’β„π‘˜ of the variables (𝑑,π‘₯,𝑝,𝑀,π‘ž). Write 𝑓𝑖𝑓(𝑑,π‘₯,𝑀,π‘ž)=𝑖(𝑑,π‘₯,𝑀𝑖(0,πœƒ),𝑀,π‘ž), 𝑖=1,…,π‘˜, on Ξ£. Then system (1.4) is equivalent to πœ•π‘‘π‘§π‘–ξ‚π‘“(𝑑,π‘₯)=𝑖𝑑,π‘₯,𝑧𝑖(𝑑,π‘₯),𝑧(𝑑,π‘₯),πœ•π‘₯𝑧𝑖(𝑑,π‘₯),𝑖=1,…,π‘˜.(2.15)Note that the dependence of 𝑓 on the classical variable 𝑧(𝑑,π‘₯) is distinguished in (2.15). Suppose that
(1)𝑓 is nondecreasing with respect to the functional variable,(2)there exists the derivative πœ•π‘ξ‚π‘“=(πœ•π‘ξ‚π‘“1,…,πœ•π‘ξ‚π‘“π‘˜) and πœ•π‘ξ‚π‘“π‘–(𝑑,π‘₯,𝑝,𝑀,π‘ž)β‰₯𝐿 for 𝑖=1,…,π‘˜ and 1+πΏβ„Ž0β‰₯0. Then the monotonicity condition (3) of Assumption (𝐻[𝑓])  is satisfied.

Let us denote by 𝐻⋆ the set of all β„Ž=(β„Ž0,β„Žξ…ž)∈𝐻 such thatβ„Žπ‘–ξ€½π‘<minπ‘–βˆ’ξ‚π‘₯𝑖,π‘₯𝑖+𝑏𝑖,𝑖=1,…,𝑛.(2.16)Suppose that πœ”βˆΆΞ©β„Žβ†’β„. We apply difference operators 𝛿=(𝛿1,…,𝛿𝑛) given byifπ‘₯𝑗≀π‘₯(π‘šπ‘—)𝑗<𝑏𝑗thenπ›Ώπ‘—πœ”(π‘Ÿ,π‘š)=1β„Žπ‘—ξ€Ίπœ”(π‘Ÿ,π‘š+𝑒𝑗)βˆ’πœ”(π‘Ÿ,π‘š)ξ€»,ifβˆ’π‘π‘—<π‘₯(π‘šπ‘—)𝑗<π‘₯𝑗thenπ›Ώπ‘—πœ”(π‘Ÿ,π‘š)=1β„Žπ‘—ξ€Ίπœ”(π‘Ÿ,π‘š)βˆ’πœ”(π‘Ÿ,π‘šβˆ’π‘’π‘—)ξ€»,(2.17)and we put 𝑗=1,…,𝑛 in (2.17). Let 𝛿0 be defined by𝛿0πœ”(π‘Ÿ,π‘š)=1β„Ž0ξ€Ίπœ”(π‘Ÿ+1,π‘š)βˆ’πœ”(π‘Ÿ,π‘š)ξ€»(2.18)and 𝛿0𝑧=(𝛿0𝑧1,…,𝛿0π‘§π‘˜). Writeπ”½β„Ž[𝑧](π‘Ÿ,π‘š)=𝑓1𝑑(π‘Ÿ),π‘₯(π‘š),π‘‡β„Žπ‘§[π‘Ÿ,π‘š],𝛿𝑧1(π‘Ÿ+1,π‘š)ξ€Έ,…,π‘“π‘˜ξ€·π‘‘(π‘Ÿ),π‘₯(π‘š),π‘‡β„Žπ‘§[π‘Ÿ,π‘š],π›Ώπ‘§π‘˜(π‘Ÿ+1,π‘š)ξ€Έξ€Έ.(2.19) Given πœ‘β„ŽβˆΆπΈ0.β„Žβˆͺπœ•0πΈβ„Žβ†’β„π‘˜, we consider the functional difference equation𝛿0𝑧(π‘Ÿ,π‘š)=π”½β„Ž[𝑧](π‘Ÿ,π‘š)(2.20)with the initial boundary condition𝑧(π‘Ÿ,π‘š)=πœ‘β„Ž(π‘Ÿ,π‘š)on𝐸0.β„Žβˆͺπœ•0πΈβ„Ž.(2.21)

The above problem is considered as an implicit difference method for (1.4), (1.5). It is important that the difference expressions (𝛿1𝑧𝑖,…,𝛿𝑛𝑧𝑖), 1β‰€π‘–β‰€π‘˜, are calculated at the point (𝑑(π‘Ÿ+1),π‘₯(π‘š)) and the functional variable π‘‡β„Žπ‘§[π‘Ÿ,π‘š] appears in a classical sense.

We prove a theorem on implicit difference inequalities corresponding to (2.20), (2.21). Note that results on implicit difference methods presented in [18] are not applicable to (2.20), (2.21).

Theorem 2.3. Suppose that Assumption (𝐻[𝑓])  is satisfied and
(1)β„Žβˆˆπ»β‹†, β„Ž0<πœ€0 and the functions 𝑒,π‘£βˆΆΞ©β„Žβ†’β„π‘˜ satisfy the difference functional inequality𝛿0𝑒(π‘Ÿ,π‘š)βˆ’π”½β„Ž[𝑒](π‘Ÿ,π‘š)≀𝛿0𝑣(π‘Ÿ,π‘š)βˆ’π”½β„Ž[𝑣](π‘Ÿ,π‘š)onπΈξ…žβ„Ž,(2.22)(2)the initial boundary estimate 𝑒(π‘Ÿ,π‘š)≀𝑣(π‘Ÿ,π‘š) holds on 𝐸0.β„Žβˆͺπœ•0πΈβ„Ž. Then 𝑒(π‘Ÿ,π‘š)≀𝑣(π‘Ÿ,π‘š)onπΈβ„Ž.(2.23)

Proof. We prove (2.23) by induction on π‘Ÿ. It follows from assumption (2) that estimate (2.23) is satisfied for π‘Ÿ=0 and (𝑑(0),π‘₯(π‘š))βˆˆπΈβ„Ž. Assume that 𝑒(𝑗,π‘š)≀𝑣(𝑗,π‘š) for (𝑑(𝑗),π‘₯(π‘š))βˆˆπΈβ„Žβˆ©([0,𝑑(π‘Ÿ)]×ℝ𝑛). We prove that 𝑒(π‘Ÿ+1,π‘š)≀𝑣(π‘Ÿ+1,π‘š) for (𝑑(π‘Ÿ+1,π‘š),π‘₯(π‘š))βˆˆπΈβ„Ž. Writeπ‘ˆπ‘–(π‘Ÿ,π‘š)=𝑒𝑖(π‘Ÿ,π‘š)+β„Ž0𝑓𝑖𝑑(π‘Ÿ),π‘₯(π‘š),π‘‡β„Žπ‘’[π‘Ÿ,π‘š],𝛿𝑒𝑖(π‘Ÿ+1,π‘š)ξ€Έβˆ’π‘£π‘–(π‘Ÿ,π‘š)βˆ’β„Ž0𝑓𝑖𝑑(π‘Ÿ),π‘₯(π‘š),π‘‡β„Žπ‘£[π‘Ÿ,π‘š],𝛿𝑒𝑖(π‘Ÿ+1,π‘š)ξ€Έ,𝑖=1,…,π‘˜.(2.24) It follows from (2.22) thatξ€·π‘’π‘–βˆ’π‘£π‘–ξ€Έ(π‘Ÿ+1,π‘š)β‰€π‘ˆπ‘–(π‘Ÿ,π‘š)+β„Ž0𝑓𝑖𝑑(π‘Ÿ),π‘₯(π‘š),π‘‡β„Žπ‘£[π‘Ÿ,π‘š],𝛿𝑒𝑖(π‘Ÿ+1,π‘š)ξ€Έβˆ’π‘“π‘–ξ€·π‘‘(π‘Ÿ),π‘₯(π‘š),π‘‡β„Žπ‘£[π‘Ÿ,π‘š],𝛿𝑣𝑖(π‘Ÿ+1,π‘š),ξ€Έξ€»(2.25) where 𝑖=1,…,π‘˜. The monotonicity condition (3) of Assumption (𝐻[𝑓])  implies the inequalities π‘ˆπ‘–(π‘Ÿ,π‘š)≀0 for (𝑑(π‘Ÿ),π‘₯(π‘š))βˆˆπΈβ„Ž, 𝑖=1,…,π‘˜. Then we haveξ€·π‘’π‘–βˆ’π‘£π‘–ξ€Έ(π‘Ÿ+1,π‘š)β‰€β„Ž0𝑛𝑗=1ξ€œ10πœ•π‘žπ‘—π‘“π‘–ξ€·π‘„π‘–(π‘Ÿ,π‘š)ξ€Έ[𝑣,𝜏]π‘‘πœπ›Ώπ‘—ξ€·π‘’π‘–βˆ’π‘£π‘–ξ€Έ(π‘Ÿ+1,π‘š),(2.26)where 𝑖=1,…,π‘˜ and𝑄𝑖(π‘Ÿ,π‘š)𝑑[𝑣,𝜏]=(π‘Ÿ),π‘₯(π‘š),π‘‡β„Žπ‘£[π‘Ÿ,π‘š],𝛿𝑣𝑖(π‘Ÿ+1,π‘š)𝑒+πœπ›Ώπ‘–βˆ’π‘£π‘–ξ€Έ(π‘Ÿ+1,π‘š)ξ€Έ.(2.27)WriteΞ“+(π‘š)=ξ€½π‘—βˆˆ{1,…,𝑛}∢π‘₯(π‘šπ‘—)π‘—βˆˆξ€Ίξ‚π‘₯𝑗,𝑏𝑗,Ξ“βˆ’(π‘š)={1,…,𝑛}⧡Γ+(π‘š).(2.28)It follows from (2.11), (2.17) thatξ€·π‘’π‘–βˆ’π‘£π‘–ξ€Έ(π‘Ÿ+1,π‘š)1+β„Ž0𝑛𝑗=11β„Žπ‘—ξ€œ10||πœ•π‘žπ‘—π‘“π‘–ξ€·π‘„π‘–(π‘Ÿ,π‘š)ξ€Έξ‚„[𝑣,𝜏]|π‘‘πœβ‰€β„Ž0ξ“π‘—βˆˆΞ“+(π‘š)1β„Žπ‘—ξ€œ10πœ•π‘žπ‘—π‘“π‘–ξ€·π‘„π‘–(π‘Ÿ,π‘š)𝑒[𝑣,𝜏]π‘‘πœπ‘–βˆ’π‘£π‘–ξ€Έ(π‘Ÿ+1,π‘š+𝑒𝑗)βˆ’β„Ž0ξ“π‘—βˆˆΞ“βˆ’(π‘š)1β„Žπ‘—ξ€œ10πœ•π‘žπ‘—π‘“π‘–ξ€·π‘„π‘–(π‘Ÿ,π‘š)𝑒[𝑣,𝜏]π‘‘πœπ‘–βˆ’π‘£π‘–ξ€Έ(π‘Ÿ+1,π‘šβˆ’π‘’π‘—),𝑖=1,…,π‘˜.(2.29) We define ξ‚π‘šβˆˆβ„€π‘› and πœ‡βˆˆβ„•, 1β‰€πœ‡β‰€π‘˜, as follows:ξ€·π‘’πœ‡βˆ’π‘£πœ‡ξ€Έ(π‘Ÿ+1,ξ‚π‘š)=max1β‰€π‘–β‰€π‘˜π‘’maxξ€½ξ€·π‘–βˆ’π‘£π‘–ξ€Έ(π‘Ÿ+1,π‘š)βˆΆξ€·π‘‘(π‘Ÿ+1),π‘₯(π‘š)ξ€ΈβˆˆΞ©β„Žξ€Ύ.(2.30)If (𝑑(π‘Ÿ+1),π‘₯(ξ‚π‘š))βˆˆπœ•0πΈβ„Ž then assumption (2) implies that (π‘’πœ‡βˆ’π‘£πœ‡)(π‘Ÿ+1,ξ‚π‘š)≀0. Let us consider the case when (𝑑(π‘Ÿ+1),π‘₯(ξ‚π‘š))βˆˆπΈβ„Ž. Then we have from (2.29) thatξ€·π‘’πœ‡βˆ’π‘£πœ‡ξ€Έ(π‘Ÿ+1,ξ‚π‘š)ξ‚Έ1+β„Ž0𝑛𝑗=11β„Žπ‘—ξ€œ10||πœ•π‘žπ‘—π‘“π‘–(𝑄𝑖(π‘Ÿ,ξ‚π‘š)||ξ‚Ή[𝑣,𝜏])π‘‘πœβ‰€β„Ž0ξ€·π‘’πœ‡βˆ’π‘£πœ‡ξ€Έ(π‘Ÿ+1,ξ‚π‘š)ξ‚Έξ“π‘—βˆˆΞ“+(ξ‚π‘š)1β„Žπ‘—ξ€œ10πœ•π‘žπ‘—π‘“π‘–ξ€·π‘„π‘–(π‘Ÿ,ξ‚π‘š)ξ€Έβˆ’ξ“[𝑣,𝜏]π‘‘πœπ‘—βˆˆΞ“βˆ’(ξ‚π‘š)1β„Žπ‘—ξ€œ10πœ•π‘žπ‘—π‘“π‘–ξ€·π‘„(π‘Ÿ,ξ‚π‘šξ€Έπ‘–ξ€Έξ‚Ή.[𝑣,𝜏]π‘‘πœ(2.31) It follows that (π‘’πœ‡βˆ’π‘£πœ‡)(π‘Ÿ+1,ξ‚π‘š)≀0. The the proof of (2.23) is completed by induction.

3. Implicit Difference Schemes

We define 𝑁=(𝑁1,…,𝑁𝑛)βˆˆπ‘π‘› by the relations:𝑁1β„Ž1,…,π‘π‘›β„Žπ‘›ξ€Έ<𝑏1,…,𝑏𝑛≀𝑁1ξ€Έβ„Ž+11𝑁,…,π‘›ξ€Έβ„Ž+1𝑛(3.1)and we assume that (𝑁𝑖+1)β„Žπ‘–=𝑏𝑖 if 𝑑𝑖=0. For π‘€βˆˆπΆ(𝐷,β„π‘˜) we write‖𝑀‖𝐷‖‖‖‖=max𝑀(𝑑,π‘₯)βˆžξ€ΎβˆΆ(𝑑,π‘₯)∈𝐷.(3.2)In a similar way we define the norm in the space 𝐹(π·β„Ž,β„π‘˜) : if π‘€βˆΆπ·β„Žβ†’β„π‘˜ thenβ€–π‘€β€–π·β„Žξ€½β€–β€–π‘€=max(π‘Ÿ,π‘š)β€–β€–βˆžβˆΆξ€·π‘‘(π‘Ÿ),π‘₯(π‘š)ξ€Έβˆˆπ·β„Žξ€Ύ.(3.3)The following properties of the operator π‘‡β„Ž are important in our considerations.

Lemma 3.1. Suppose that π‘€βˆΆπ·β†’β„π‘˜ is of class 𝐢1 and π‘€β„Ž is the restriction of 𝑀 to the set π·β„Ž. Let 𝐢 be such a constant that β€–πœ•π‘‘π‘€β€–π·, β€–πœ•π‘₯𝑖𝑀‖𝐷≀𝐢 for 1≀𝑖≀𝑛. Then β€–π‘‡β„Ž[π‘€β„Ž]βˆ’π‘€β€–π·β‰€ξ‚πΆβ€–β„Žβ€– where β€–β„Žβ€–=β„Ž0+β„Ž1+β‹―+β„Žπ‘›.

Lemma 3.2. Suppose that π‘€βˆΆπ·β†’β„π‘˜ is of class 𝐢2 and π‘€β„Ž is the restriction of 𝑀 to the set π·β„Ž. Let 𝐢 be such a constant that β€–πœ•π‘‘π‘‘π‘€β€–π·, β€–πœ•π‘‘π‘₯𝑖𝑀‖𝐷, β€–πœ•π‘₯𝑖π‘₯𝑗𝑀‖𝐷≀𝐢,   𝑖,𝑗=1,…,𝑛. Then β€–π‘‡β„Ž[π‘€β„Ž]βˆ’π‘€β€–π·β‰€ξ‚πΆβ€–β„Žβ€–2.

The above lemmas are consequences of [10, Lemma 3.19 and Theorem 5.27].

We first prove a theorem on the existence and uniqueness of solutions to (2.20), (2.21).

Theorem 3.3. If Assumption (𝐻[𝑓])  is satisfied and πœ‘β„ŽβˆˆπΉ(𝐸0.β„Žβˆͺπœ•0πΈβ„Ž,β„π‘˜) then there exists exactly one solution π‘’β„Ž=(π‘’β„Ž.1,…,π‘’β„Ž.π‘˜)βˆΆΞ©β„Žβ†’β„π‘˜ of difference functional problem (2.20), (2.21).

Proof. Suppose that 0β‰€π‘Ÿβ‰€π‘0βˆ’1 is fixed and that the solution π‘’β„Ž of problem (2.20), (2.21) is given on the set Ξ©β„Žβˆ©([βˆ’π‘‘0,𝑑(π‘Ÿ)]×ℝ𝑛). We prove that the vectors π‘’β„Ž(π‘Ÿ+1,π‘š),β€‰β€‰βˆ’π‘β‰€π‘šβ‰€π‘, exist and that they are unique. It is sufficient to show that there exists exactly one solution of the system of equations1β„Ž0𝑧𝑖(π‘Ÿ+1,π‘š)βˆ’π‘’(π‘Ÿ,π‘š)β„Ž.𝑖=𝑓𝑖𝑑(π‘Ÿ),π‘₯(π‘š),𝑇(π‘’β„Ž)[π‘Ÿ,π‘š],𝛿𝑧𝑖(π‘Ÿ+1,π‘š)ξ€Έ,(3.4)where βˆ’π‘β‰€π‘šβ‰€π‘,𝑖=1,…,π‘˜, with the initial boundary condition (2.21). There exists π‘„β„Ž>0 such thatπ‘„β„Žβ‰₯β„Ž0ξ‚Έξ“π‘—βˆˆΞ“+(π‘š)1β„Žπ‘—πœ•π‘žπ‘—π‘“π‘–ξ€·π‘‘(π‘Ÿ),π‘₯(π‘š),π‘‡β„Žξ€·π‘’β„Žξ€Έ[π‘Ÿ,π‘š]ξ€Έβˆ’ξ“,π‘žπ‘—βˆˆΞ“βˆ’(π‘š)1β„Žπ‘—πœ•π‘žπ‘—π‘“π‘–ξ€·π‘‘(π‘Ÿ),π‘₯(π‘š),π‘‡β„Žξ€·π‘’β„Žξ€Έ[π‘Ÿ,π‘š]ξ€Έξ‚Ή,π‘ž,(3.5) where βˆ’π‘β‰€π‘šβ‰€π‘,𝑖=1,…,π‘˜. It is clear that system (3.4) is equivalent to the following one:𝑧𝑖(π‘Ÿ+1,π‘š)=1π‘„β„Žξ‚ƒπ‘„+1β„Žπ‘§π‘–(π‘Ÿ+1,π‘š)+𝑒(π‘Ÿ,π‘š)β„Ž.𝑖+β„Ž0𝑓𝑖𝑑(π‘Ÿ),π‘₯(π‘š),π‘‡β„Žξ€·π‘’β„Žξ€Έ[π‘Ÿ,π‘š],𝛿𝑧𝑖(π‘Ÿ+1,π‘š),ξ‚ξ‚„βˆ’π‘β‰€π‘šβ‰€π‘,𝑖=1,…,π‘˜(3.6) Write π‘†β„Ž={π‘₯(π‘š)∢π‘₯(π‘š)∈[βˆ’π‘,𝑐]}. Elements of the space 𝐹(π‘†β„Ž,β„π‘˜) are denoted by πœ‰, πœ‰. For πœ‰βˆΆπ‘†β„Žβ†’β„π‘˜, πœ‰=(πœ‰1,…,πœ‰π‘˜), we write πœ‰(π‘š)=πœ‰(π‘₯(π‘š)) andπ›Ώπœ‰π‘–(π‘š)=𝛿1πœ‰π‘–(π‘š),…,π›Ώπ‘›πœ‰π‘–(π‘š)𝛿,1β‰€π‘–β‰€π‘˜,π‘—πœ‰π‘–(π‘š)=1β„Žπ‘—ξ€Ίπœ‰(π‘š+𝑒𝑗𝑖𝑗)βˆ’πœ‰π‘–(π‘š)ξ€»ifπ‘₯(π‘šπ‘—)π‘—βˆˆξ€Ίξ‚π‘₯𝑗,𝑏𝑗,π›Ώπ‘—πœ‰π‘–(π‘š)=1β„Žπ‘—ξ€Ίπœ‰π‘–(π‘š)βˆ’πœ‰(π‘šβˆ’π‘’π‘—)𝑖ifπ‘₯(π‘šπ‘—)π‘—βˆˆξ€·π‘π‘—,π‘₯𝑗,(3.7)where 𝑗=1,…,𝑛. The norm in the space 𝐹(π‘†β„Ž,β„π‘˜) is defined byβ€–πœ‰β€–β‹†ξ€½β€–β€–πœ‰=max(π‘š)β€–β€–βˆžβˆΆπ‘₯(π‘š)βˆˆπ‘†β„Žξ€Ύ.(3.8)Let us consider the setπ‘‹β„Ž=ξ€½ξ€·π‘†πœ‰βˆˆπΉβ„Ž,β„π‘˜ξ€ΈβˆΆπœ‰(π‘š)=πœ‘(π‘Ÿ+1,π‘š)forπ‘₯(π‘š)ξ€Έβˆˆ[βˆ’π‘,𝑐]⧡(βˆ’π‘,𝑏}.(3.9)We consider the operator π‘Šβ„ŽβˆΆπ‘‹β„Žβ†’π‘‹β„Ž,β€‰β€‰π‘Šβ„Ž=(π‘Šβ„Ž.1,…,π‘Šβ„Ž.𝑛) defined byπ‘Šβ„Ž.𝑖[πœ‰](π‘š)=1π‘„β„Žξ€Ίπ‘„+1β„Žπœ‰π‘–(π‘š)+𝑒(π‘Ÿ,π‘š)β„Ž.𝑖+β„Ž0𝑓𝑖𝑑(π‘Ÿ),π‘₯(π‘š)𝑒,π‘‡β„Žξ€Έ[π‘Ÿ,π‘š],π›Ώπœ‰π‘–(π‘š)ξ€Έξ€»,(3.10)where βˆ’π‘β‰€π‘šβ‰€π‘,  𝑖=1,…,π‘˜ andπ‘Šβ„Ž[πœ‰](π‘š)=πœ‘β„Ž(π‘Ÿ+1,π‘š)forπ‘₯(π‘š)∈[βˆ’π‘,𝑐]⧡(βˆ’π‘,𝑏),(3.11)where πœ‰=(πœ‰1,…,πœ‰π‘˜)∈𝐹(π‘†β„Ž,β„π‘˜). We prove thatβ€–β€–π‘Šβ„Ž[πœ‰]βˆ’π‘Šβ„Ž[β€–β€–πœ‰]β‹†β‰€π‘„β„Žπ‘„β„Ž+1β€–πœ‰βˆ’πœ‰β€–β‹†ξ€·π‘†onπΉβ„Ž,β„π‘˜ξ€Έ.(3.12)
It follows from (3.10) that we have for βˆ’π‘β‰€π‘šβ‰€π‘:π‘Šβ„Ž.𝑖[πœ‰](π‘š)βˆ’π‘Šβ„Ž.𝑖[πœ‰](π‘š)=1π‘„β„Žξ‚ƒπ‘„+1β„Žξ€·πœ‰π‘–βˆ’πœ‰π‘–ξ€Έ(π‘š)βˆ’β„Ž0ξ“π‘—βˆˆΞ“+(π‘š)1β„Žπ‘—ξ€œ10πœ•π‘žπ‘—π‘“π‘–ξ€·π‘ƒπ‘–(π‘Ÿ,π‘š)[π‘’β„Žξ€Έξ€·πœ‰,𝜏]π‘‘πœπ‘–βˆ’πœ‰π‘–ξ€Έ(π‘š)+ξ“π‘—βˆˆΞ“βˆ’(π‘š)1β„Žπ‘—ξ€œ10πœ•π‘žπ‘—π‘“π‘–ξ€·π‘ƒπ‘–(π‘Ÿ,π‘š)[π‘’β„Žξ€Έ,𝜏]π‘‘πœ(πœ‰π‘–βˆ’πœ‰π‘–)(π‘š)+β„Ž0ξ“π‘—βˆˆΞ“+(π‘š)1β„Žπ‘—ξ€œ10πœ•π‘žπ‘—π‘“π‘–ξ€·π‘ƒπ‘–(π‘Ÿ,π‘š)[π‘’β„Žξ€Έ,𝜏]π‘‘πœ(πœ‰π‘–βˆ’πœ‰π‘–)(π‘š+𝑒𝑗)βˆ’β„Ž0ξ“π‘—βˆˆΞ“βˆ’(π‘š)1β„Žπ‘—ξ€œ10πœ•π‘žπ‘—π‘“π‘–ξ€·π‘ƒπ‘–(π‘Ÿ,π‘š)[π‘’β„Žξ€Έ,𝜏]π‘‘πœ(πœ‰π‘–βˆ’πœ‰π‘–)(π‘šβˆ’π‘’π‘—)ξ‚„,(3.13) where 𝑖=1,…,π‘˜ and𝑃𝑖(π‘Ÿ,π‘š)ξ€Ίπ‘’β„Žξ€»=𝑑,𝜏(π‘Ÿ),π‘₯(π‘š),π‘‡β„Žξ€·π‘’β„Žξ€Έ[π‘Ÿ,π‘š],π›Ώπœ‰π‘–(π‘š)ξ€·πœ‰+πœπ›Ώπ‘–βˆ’πœ‰π‘–ξ€Έ(π‘š).(3.14)It follows from the above relations and from (3.5) that|||π‘Šβ„Ž.𝑖[πœ‰](π‘š)βˆ’π‘Šβ„Ž.𝑖[πœ‰](π‘š)|||β‰€π‘„β„Žπ‘„β„Ž+1β€–πœ‰βˆ’πœ‰β€–β‹†forβˆ’π‘β‰€π‘šβ‰€π‘,𝑖=1,…,π‘˜.(3.15)According to (3.12) we haveπ‘Šβ„Ž.𝑖[πœ‰](π‘š)βˆ’π‘Šβ„Ž.𝑖[πœ‰](π‘š)=0forπ‘₯(π‘š)∈[βˆ’π‘,𝑐]⧡(βˆ’π‘,𝑏),𝑖=1,…,π‘˜.(3.16)This completes the proof of (3.12).
It follows from the Banach fixed point theorem that there exists exactly one solution πœ‰βˆΆπ‘†β„Žβ†’β„π‘˜ of the equation πœ‰=π‘Šβ„Ž[πœ‰] and consequently, there exists exactly one solution of (3.6), (2.21). Then the vectors π‘’β„Ž(π‘Ÿ+1,π‘š), βˆ’π‘β‰€π‘šβ‰€π‘, exist and they are unique. Then the proof is completed by induction with respect to π‘Ÿ, 0β‰€π‘Ÿβ‰€π‘0.

Assumption (𝐻[𝜎]). The function 𝜎∢[0,π‘Ž]×ℝ+→ℝ+ satisfies the conditions:
(1)𝜎 is continuous and it is nondecreasing with respect to the both variables,(2)𝜎(𝑑,0)=0 for π‘‘βˆˆ[0,π‘Ž] and the maximal solution of the Cauchy problemπœ‚ξ…ž(𝑑)=𝜎(𝑑,πœ‚(𝑑)),πœ‚(0)=0,(3.17)is Μƒπœ‚(𝑑)=0 for π‘‘βˆˆ[0,π‘Ž].

Assumption (𝐻[𝑓,𝜎]). There is 𝜎∢[0,π‘Ž]×ℝ+→ℝ+ such that Assumption (𝐻[𝜎])  is satisfied and for 𝑀,π‘€βˆˆβ„‚(𝐷,β„π‘˜), 𝑀β‰₯𝑀, we have𝑓𝑖(𝑑,π‘₯,𝑀,π‘ž)βˆ’π‘“π‘–(𝑑,π‘₯,𝑀,π‘ž)β‰€πœŽπ‘‘,β€–π‘€βˆ’π‘€β€–π·ξ€Έ,𝑖=1,…,π‘˜,(3.18)where (𝑑,π‘₯,π‘ž)βˆˆπΈΓ—β„π‘›.

Theorem 3.4. Suppose that Assumptions (𝐻[𝑓])  and   (𝐻[𝑓,𝜎])  are satisfied and
(1)π‘£βˆΆΞ©β†’β„ is a solution of (1.4), (1.5) and 𝑣 is of class 𝐢1 on Ξ©,(2)β„Žβˆˆπ»βˆ—, β„Ž0<πœ€ and πœ‘β„ŽβˆΆπΈ0.β„Žβˆͺπœ•0πΈβ„Žβ†’β„π‘˜ and there is 𝛼0βˆΆπ»βˆ—β†’β„+ such thatβ€–β€–πœ‘(π‘Ÿ,π‘š)βˆ’πœ‘β„Ž(π‘Ÿ,π‘š)β€–β€–βˆžβ‰€π›Ό0(β„Ž)on𝐸0.β„Žβˆͺπœ•0πΈβ„Ž,limβ„Žβ†’0𝛼0(β„Ž)=0.(3.19) Under these assumptions there is a solution π‘’β„ŽβˆΆΞ©β„Žβ†’β„π‘˜ of (2.20), (2.21) and there is π›ΌβˆΆπ»βˆ—β†’β„+ such that β€–β€–ξ€·π‘’β„Žβˆ’π‘£β„Žξ€Έ(π‘Ÿ,π‘š)β€–β€–βˆžβ‰€π›Ό(β„Ž)onπΈβ„Ž,limβ„Žβ†’0𝛼(β„Ž)=0,(3.20)where π‘£β„Ž is the restriction of 𝑣 to the set Ξ©β„Ž.

Proof. The existence of π‘’β„Ž follows from Theorem 3.3. Let Ξ“β„ŽβˆΆπΈξ…žβ„Žβ†’β„π‘˜, Ξ“0.β„ŽβˆΆπΈ0.β„Žβˆͺπœ•0πΈβ„Žβ†’β„π‘˜ be defined by the relations𝛿0π‘£β„Ž(π‘Ÿ,π‘š)=π”½β„Žξ€Ίπ‘£β„Žξ€»(π‘Ÿ,π‘š)+Ξ“β„Ž(π‘Ÿ,π‘š)onπΈξ…žβ„Ž,𝑣(3.21)β„Ž(π‘Ÿ,π‘š)=πœ‘β„Ž(π‘Ÿ+1,π‘š)+Ξ“(π‘Ÿ,π‘š)0.β„Žξ€·π‘‘for(π‘Ÿ),π‘₯(π‘š)ξ€ΈβˆˆπΈ0.β„Žβˆͺπœ•0πΈβ„Ž.(3.22)From Lemma 3.1 and from assumption (1) of the theorem it follows that there are 𝛾,𝛾0βˆΆπ»βˆ—β†’β„+ such thatβ€–β€–Ξ“β„Ž(π‘Ÿ,π‘š)β€–β€–βˆžβ‰€π›Ύ(β„Ž)onπΈξ…žβ„Ž,β€–β€–Ξ“(π‘Ÿ,π‘š)0.β„Žβ€–β€–βˆžβ‰€π›Ύ0(β„Ž)on𝐸0.β„Žβˆͺπœ•0πΈβ„Ž(3.23)and limβ„Žβ†’0𝛾(β„Ž)=0, limβ„Žβ†’0𝛾0(β„Ž)=0. Write 𝐽=[0,π‘Ž] and π½β„Ž={𝑑(π‘Ÿ)∢0β‰€π‘Ÿβ‰€π‘0}. For π›½βˆΆπ½β„Žβ†’β„ we put 𝛽(π‘Ÿ)=𝛽(𝑑(π‘Ÿ)). Let π›½β„ŽβˆΆπ½β„Žβ†’β„+ be a solution of the difference problem𝛽(π‘Ÿ+1)=𝛽(π‘Ÿ)+β„Ž0πœŽξ€·π‘‘(π‘Ÿ),𝛽(π‘Ÿ)ξ€Έ+β„Ž0𝛾(β„Ž),0β‰€π‘Ÿβ‰€π‘0βˆ’1,𝛽(0)=𝛼0(β„Ž).(3.24)We prove thatβ€–β€–ξ€·π‘’β„Žβˆ’π‘£β„Žξ€Έ(π‘Ÿ,π‘š)β€–β€–βˆžβ‰€π›½β„Ž(π‘Ÿ)onπΈβ„Ž.(3.25)Let ξ‚π‘£β„Žξƒ€π‘£=(β„Ž.1𝑣,…,β„Ž.π‘˜)βˆΆΞ©β„Žβ†’β„π‘˜ be defined by𝑣(π‘Ÿ,π‘š)β„Ž.𝑖=𝑣(π‘Ÿ,π‘š)β„Ž.𝑖+π›½β„Ž(0)on𝐸0.β„Ž,𝑣(π‘Ÿ,π‘š)β„Ž.𝑖=𝑣(π‘Ÿ,π‘š)β„Ž.𝑖+π›½β„Ž(𝑖)onπΈβ„Žβˆͺπœ•0πΈβ„Ž,(3.26) where 𝑖=1,…,π‘˜. We prove that the difference functional inequality𝛿0ξ‚π‘£β„Žβ‰₯π”½β„Žξ€Ίξ‚π‘£β„Žξ€»(π‘Ÿ,π‘š),𝑑(π‘Ÿ),π‘₯(π‘š)ξ€ΈβˆˆπΈξ…žβ„Ž,(3.27)is satisfied. It follows from Assumption (𝐻[𝑓,𝜎]) and from (3.21) that𝛿0𝑣(π‘Ÿ,π‘š)β„Ž.𝑖=𝛿0𝑣(π‘Ÿ,π‘š)β„Ž.𝑖+1β„Ž0ξ€·π›½β„Ž(π‘Ÿ+1)βˆ’π›½β„Ž(π‘Ÿ)ξ€Έ=𝑓𝑖𝑑(π‘Ÿ),π‘₯(π‘š),π‘‡β„Žξ€·ξ‚π‘£β„Žξ€Έ[π‘Ÿ,π‘š]ξ„Ÿπ‘£,𝛿(π‘Ÿ+1,π‘š)β„Ž.𝑖+1β„Ž0ξ€·π›½β„Ž(π‘Ÿ+1)βˆ’π›½β„Ž(π‘Ÿ)ξ€Έ+𝑓𝑖𝑑(π‘Ÿ),π‘₯(π‘š),π‘‡β„Žξ€·π‘£β„Žξ€Έ[π‘Ÿ,π‘š],𝛿𝑣(π‘Ÿ+1,π‘š)β„Ž.π‘–ξ€Έβˆ’π‘“π‘–ξ€·π‘‘(π‘Ÿ),π‘₯(π‘š),π‘‡β„Žξ€·ξ‚π‘£β„Žξ€Έ[π‘Ÿ,π‘š],𝛿𝑣(π‘Ÿ+1,π‘š)β„Ž.𝑖β‰₯𝑓𝑖𝑑(π‘Ÿ),π‘₯(π‘š),π‘‡β„Žξ€·ξ‚π‘£β„Žξ€Έ[π‘Ÿ,π‘š]ξ„Ÿπ‘£,𝛿(π‘Ÿ+1,π‘š)β„Ž.π‘–ξ€Έξ€·π‘‘βˆ’πœŽ(π‘Ÿ),π›½β„Ž(π‘Ÿ)ξ€Έ+1β„Ž0ξ€·π›½β„Ž(π‘Ÿ+1)βˆ’π›½β„Ž(π‘Ÿ)ξ€Έ=𝑓𝑖(𝑑(π‘Ÿ),π‘₯(π‘š),π‘‡β„Ž(ξ‚π‘£β„Ž)[π‘Ÿ,π‘š]ξ„Ÿπ‘£,𝛿(π‘Ÿ+1,π‘š)β„Ž.𝑖),𝑖=1,…,π‘˜.(3.28) This completes the proof of (3.27).
Since π‘£β„Ž(π‘Ÿ,π‘š)β‰€ξƒ΄π‘£β„Ž(π‘Ÿ,π‘š) on 𝐸0.β„Žβˆͺπœ•0πΈβ„Ž, it follows from Theorem 2.3 that π‘’β„Ž(π‘Ÿ,π‘š)β‰€π‘£β„Ž(π‘Ÿ,π‘š)+π›½β„Ž(π‘Ÿ) on πΈβ„Ž. In a similar way we prove that π‘£β„Ž(π‘Ÿ,π‘š)βˆ’π›½β„Ž(π‘Ÿ)β‰€π‘’β„Ž(π‘Ÿ,π‘š) on πΈβ„Ž. The above estimates imply (3.25). Consider the Cauchy problemπœ‚ξ…ž(𝑑)=𝜎(𝑑,πœ‚(𝑑))+𝛾(β„Ž),πœ‚(0)=𝛼0(β„Ž).(3.29)It follows from Assumption (𝐻[𝜎]) that there is Μƒπœ€>0 such that for β€–β„Žβ€–β‰€Μƒπœ€β€‰β€‰the maximal solution πœ‚(β‹…,β„Ž) of (3.29) is defined on [0,π‘Ž] andlimβ„Žβ†’0πœ‚(𝑑,β„Ž)=0uniformlyon[0,π‘Ž].(3.30)Since πœ‚(β‹…,β„Ž) is convex function then we have the difference inequalityπœ‚ξ€·π‘‘(π‘Ÿ+1)𝑑,β„Žβ‰₯πœ‚(π‘Ÿ)ξ€Έ,β„Ž+β„Ž0πœŽξ€·π‘‘(π‘Ÿ)𝑑,πœ‚(π‘Ÿ),β„Žξ€Έξ€Έ+β„Ž0𝛾(β„Ž),(3.31)where 0β‰€π‘Ÿβ‰€π‘0βˆ’1. Since π›½β„Ž satisfies (3.24), the above relations imply the estimateπ›½β„Ž(π‘Ÿ)ξ€·π‘‘β‰€πœ‚(π‘Ÿ)ξ€Έ,β„Žβ‰€πœ‚(π‘Ž,β„Ž),0β‰€π‘Ÿβ‰€π‘0.(3.32)It follows from (3.30) that condition (3.20) is satisfied with 𝛼(β„Ž)=πœ‚(π‘Ž,β„Ž). This completes the proof.

Lemma 3.5. Suppose that Assumption (𝐻[𝑓]) is satisfied and
(1)π‘£βˆΆΞ©β†’β„ is a solution of (1.4), (1.5) and 𝑣 is of class 𝐢2 on Ξ©,(2)β„Žβˆˆπ»βˆ—, β„Ž0<πœ€ and πœ‘β„ŽβˆΆπΈ0.β„Žβˆͺπœ•0πΈβ„Žβ†’β„π‘˜ and there is 𝛼0βˆΆπ»βˆ—β†’β„+ such thatβ€–β€–πœ‘(π‘Ÿ,π‘š)βˆ’πœ‘β„Ž(π‘Ÿ,π‘š)β€–β€–βˆžβ‰€π›Ό0(β„Ž)on𝐸0.β„Žβˆͺπœ•0πΈβ„Ž,limβ„Žβ†’0𝛼0(β„Ž)=0.(3.33)(3)there exists πΏβˆˆβ„+ such that estimates𝑓𝑖(𝑑,π‘₯,𝑀,π‘ž)βˆ’π‘“π‘–ξ‚ξ‚(𝑑,π‘₯,𝑀,π‘ž)β‰€πΏβ€–π‘€βˆ’π‘€β€–π·,𝑖=1,…,π‘˜,(3.34)are satisfied for (𝑑,π‘₯,π‘ž)βˆˆπΈΓ—β„π‘›, 𝑀,π‘€βˆˆπΆ(𝐷,β„π‘˜) and 𝑀𝑀β‰₯,(4)there is πΆβˆˆβ„+ such thatβ€–β€–πœ•π‘žπ‘“π‘–β€–β€–β‰€(𝑑,π‘₯,𝑀,π‘ž)𝐢onΞ£for𝑖=1,…,π‘˜.(3.35) Under these assumptions there is a solution π‘’β„ŽβˆΆΞ©β„Žβ†’β„π‘˜ of (2.20), (2.21), and β€–β€–ξ€·π‘’β„Žβˆ’π‘£β„Žξ€Έ(π‘Ÿ,π‘š)β€–β€–βˆžβ‰€ξ‚π›Ό(β„Ž)onπΈβ„Ž,(3.36)where 𝛼(β„Ž)=𝛼0(β„Ž)π‘’πΏπ‘Žπ‘’+̃𝛾(β„Ž)πΏπ‘Žβˆ’1𝐿if𝐿>0,𝛼(β„Ž)=𝛼0(β„Ž)+π‘ŽΜƒπ›Ύ(β„Ž)if𝐿=0,̃𝛾(β„Ž)=0.5πΆβ„Ž0(1+𝐢)+πΏπΆβ€–β„Žξ…žβ€–2+0.5πΆξ‚πΆβ€–β„Žβ€–(3.37)and ξ‚πΆβˆˆβ„+ is such that β€–β€–πœ•π‘‘π‘‘β€–β€–π‘£(𝑑,π‘₯)∞,β€–β€–πœ•π‘‘π‘₯𝑖‖‖𝑣(𝑑,π‘₯)∞,β€–β€–πœ•π‘₯𝑖π‘₯𝑗‖‖𝑣(𝑑,π‘₯)βˆžβ‰€ξ‚πΆ(3.38)on Ξ© for 𝑖,𝑖=1,…,𝑛.

Proof. It follows that the solution π›½β„ŽβˆΆπ½β„Žβ†’β„+ of the difference problem𝛽(π‘Ÿ+1)=ξ€·1+πΏβ„Ž0𝛽(π‘Ÿ)+β„Ž0𝛾(β„Ž),0β‰€π‘Ÿβ‰€π‘0π›½βˆ’1,(0)=𝛼0(β„Ž)(3.39) satisfies the condition: π›½β„Ž(π‘Ÿ)≀𝛼(β„Ž) for 0β‰€π‘Ÿβ‰€π‘0. Moreover we haveβ€–β€–Ξ“β„Ž(π‘Ÿ,π‘š)β€–β€–βˆžβ‰€Μƒπ›Ύ(β„Ž)onπΈξ…žβ„Ž,(3.40)where Ξ“β„Ž is given by (3.21). Then we obtain the assertion from Lemma 3.2 and Theorem 3.4.

Remark 3.6. In the result on error estimates we need estimates for the derivatives of the solution 𝑣 of problem (1.4), (1.5). One may obtain them by the method of differential inequalities, see [10, Chapter 5].

4. Numerical Examples

Example 4.1. For 𝑛=2 we put 𝐸=[0,0.5]Γ—[βˆ’1,1]Γ—[βˆ’1,1],𝐸0={0}Γ—[βˆ’1,1]Γ—[βˆ’1,1].(4.1)Consider the differential integral equationπœ•π‘‘ξ€Ίπ‘§(𝑑,π‘₯,𝑦)=arctan2π‘₯πœ•π‘₯𝑧(𝑑,π‘₯,𝑦)+2π‘¦πœ•π‘¦ξ€·π‘§(𝑑,π‘₯,𝑦)βˆ’π‘‘2π‘₯2𝑦2βˆ’π‘₯2βˆ’π‘¦2𝑧(𝑑,π‘₯,𝑦)+𝑑1βˆ’π‘¦2ξ€Έξ€œπ‘₯βˆ’1𝑠𝑧(𝑑,𝑠,𝑦)𝑑𝑠+𝑑1βˆ’π‘₯2ξ€Έξ€œπ‘¦βˆ’1ξ€Ίξ€·π‘₯𝑠𝑧(𝑑,π‘₯,𝑠)𝑑𝑠+𝑧(𝑑,π‘₯,𝑦)4+0.252π‘¦βˆ’1ξ€Έξ€·2βˆ’1ξ€Έξ€»βˆ’4(4.2) with the initial boundary condition𝑧(0,π‘₯,𝑦)=1,(π‘₯,𝑦)∈[βˆ’1,1]Γ—[βˆ’1,1],𝑧(𝑑,βˆ’1,𝑦)=𝑧(𝑑,1,𝑦)=1,(𝑑,𝑦)∈[0,0.5]Γ—[βˆ’1,1],𝑧(𝑑,π‘₯,βˆ’1)=𝑧(𝑑,π‘₯,1)=1,(𝑑,π‘₯)∈[0,0.5]Γ—[βˆ’1,1].(4.3)

The function 𝑣(𝑑,π‘₯,𝑦)=exp[0.25𝑑(π‘₯2βˆ’1)(𝑦2βˆ’1)] is the solution of the above problem. Let us denote by π‘§β„Ž an approximate solution which is obtained by using the implicit difference scheme.

The Newton method is used for solving nonlinear systems generated by the implicit difference scheme. Write π‘š=(π‘š1,π‘š2) andπœ€β„Ž(π‘Ÿ)=1ξ€·2𝑁1βˆ’1ξ€Έξ€·2𝑁2ξ€Έξ“βˆ’1π‘šβˆˆΞ ||π‘§β„Ž(π‘Ÿ,π‘š)βˆ’π‘£(π‘Ÿ,π‘š)||,0β‰€π‘Ÿβ‰€π‘0,(4.4)whereξ€½ξ€·π‘šΞ =π‘š=1,π‘š2ξ€ΈβˆΆβˆˆβ„€2βˆΆβˆ’π‘1+1β‰€π‘š1≀𝑁1βˆ’1,βˆ’π‘1+1β‰€π‘š2≀𝑁2ξ€Ύβˆ’1(4.5)and 𝑁1β„Ž1=1, 𝑁2β„Ž2=1, 𝑁0β„Ž0=0.5. The numbers πœ€β„Ž(π‘Ÿ) can be called average errors of the difference method for fixed 𝑑(π‘Ÿ). We put β„Ž0=β„Ž1=β„Ž2=0.005 and we have the values of the above defined errors which are shown in Table 1.


t(r)0.250.300.350.400.450.50
Ξ΅h(r)0.00060.00070.00090.00100.00120.0014

Note that our equation and the steps of the mesh do not satisfy condition (1.10) which is necessary for the explicit difference method to be convergent. In our numerical example the average errors for the explicit difference method exceeded 102.

Example 4.2. Let 𝑛=2 and 𝐸=[0,0.5]Γ—[βˆ’0.5,0.5]Γ—[βˆ’0.5,0.5],𝐸0={0}Γ—[βˆ’0.5,0.5]Γ—[βˆ’0.5,0.5].(4.6)Consider the differential equation with deviated variablesπœ•π‘‘π‘§(𝑑,π‘₯,𝑦)=2π‘₯πœ•π‘₯𝑧(𝑑,π‘₯,𝑦)+2π‘¦πœ•π‘¦ξ€Ίπ‘§(𝑑,π‘₯,𝑦)+cos2π‘₯πœ•π‘₯𝑧(𝑑,π‘₯,𝑦)βˆ’2π‘¦πœ•π‘¦ξ€·π‘₯𝑧(𝑑,π‘₯,𝑦)βˆ’π‘‘2βˆ’π‘¦2ξ€Έξ€»+𝑧(𝑑,π‘₯,𝑦)𝑧𝑑2ξ€Έ,π‘₯,𝑦+𝑓(𝑑,π‘₯,𝑦)𝑧(𝑑,π‘₯,𝑦)βˆ’1,(4.7) with the initial boundary conditions𝑧(0,π‘₯,𝑦)=1,(π‘₯,𝑦)∈[βˆ’0.5,0.5]Γ—[βˆ’0.5,0.5]𝑧(𝑑,βˆ’0.5,𝑦)=𝑧(𝑑,0.5,𝑦)=1,(𝑑,𝑦)∈[0,0.5]Γ—[βˆ’0.5,0.5],𝑧(𝑑,π‘₯,βˆ’0.5)=𝑧(𝑑,π‘₯,0,5)=1,(𝑑,π‘₯)∈[0,0.5]Γ—[βˆ’0.5,0.5],(4.8) whereξ€·π‘₯𝑓(𝑑,π‘₯,𝑦)=2βˆ’0.25ξ€Έξ€·0.25βˆ’π‘¦2ξ€Έξ€Ί+𝑑8π‘₯2𝑦2βˆ’π‘₯2βˆ’π‘¦2ξ€»βˆ’expξ€½ξ€·0.5𝑑2π‘₯βˆ’π‘‘ξ€Έξ€·2βˆ’0.25ξ€Έξ€·0.25βˆ’π‘¦2.ξ€Έξ€Ύ(4.9)

The function 𝑣(𝑑,π‘₯,𝑦)=exp[[𝑑(π‘₯2βˆ’0.25)(0.25βˆ’π‘¦2)] is the solution of the above problem. Let us denote by π‘§β„Ž an approximate solution which is obtained by using the implicit difference scheme.

The Newton method is used for solving nonlinear systems generated by the implicit difference scheme.

Let πœ€β„Ž be defined by (4.4) with 𝑁1β„Ž1=0.5, 𝑁2β„Ž2=0.5, 𝑁0β„Ž0=0.5. We put β„Ž0=β„Ž1=β„Ž2=0.005 and we have the values of the above defined errors which are shown in Table 2.


t(r)0.250.300.350.400.450.5
Ξ΅h(r)0.00020.00030.00040.00040.00050.0006

Note that our equation and the steps of the mesh do not satisfy condition (1.10) which is necessary for the explicit difference method to be convergent. In our numerical example the average errors for the explicit difference method exceeded 102.

The above examples show that there are implicit difference schemes which are convergent, and the corresponding classical method is not convergent. This is due to the fact that we need assumption (1.10) for explicit difference methods. We do not need this condition in our implicit methods.

Our results show that implicit difference schemes are convergent on all meshes.

References

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Copyright © 2009 Z. Kamont and K. Kropielnicka. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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