International Journal of Stochastic Analysis

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 and we denote by the class of all continuous functions from into We will use vectorial inequalities with the understanding that the same inequalities hold between their corresponding components. Writewhere , , for and , , Let andFor a function , , and for a point where is the closure of , we define a function by , Then is the restriction of to the set and this restriction is shifted to the set Write and suppose that and , , are given functions. Let us denote by an unknown function of the variables , Writeand , We consider the system of functional differential equationswith the initial boundary conditionIn 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

Example 1.1. Suppose that the function satisfies the condition: for For a given we put where . Then (1.4) is reduced to the system of differential equations with deviated variables

Example 1.2. For the above we define Then (1.4) is equivalent to the system of differential integral equations

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 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 . It is assumed that

(i)the function of the variables , , is of class with respect to and the functions , are bounded,(ii) satisfies the Perron type estimates with respect to the functional variable The second type conditions concern the mesh. It is required that difference schemes generated by (1.4), (1.5) satisfy the conditionwhere and are steps of the mesh with respect to and 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 and 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 and we denote by the class of all functions defined on and taking values in . Let and be the sets of natural numbers and integers, respectively. For , we putWe define a mesh on in the following way. Suppose that , stand for steps of the mesh. For where , we define nodal points as follows:Let us denote by the set of all such that there are and satisfying the conditions: and SetLet be defined by the relations: andFor functions and we write on and on . We need a discrete version of the operator For a function and for a point we define a function bySolutions 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 which is an element of the space Then we need an interpolating operator We define in the following way. Let us denote by the family of sets defined bySet and if , if where WriteSet with 1 standing on the th place.

Let and Suppose that There exists such that and , Write where and we take in the above formulas. If then we putThen we have defined on It is easy to see that The above interpolating operator has been first proposed in [10, Chapter 5].

For we write if where In a similar way we define the relation for and the relation for and for

We formulate an implicit difference scheme for (1.4), (1.5). For we write

Assumption (). The function of the variables , is continuous and
(1)the partial derivatives , exist on and the functions , are continuous and bounded on (2)there is , such that(3)there is such that for and , we have

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 , Let us denote by the th bicharacteristic of (1.4) corresponding to Then is a solution of the Cauchy problemAssumption (2.11) states that the bicharacteristics satisfy the following monotonicity conditions: If the function is non increasing. If then 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 of the variables Write , on Then system (1.4) is equivalent to Note that the dependence of on the classical variable is distinguished in (2.15). Suppose that
(1) is nondecreasing with respect to the functional variable,(2)there exists the derivative and for and Then the monotonicity condition (3) of Assumption   is satisfied.

Let us denote by the set of all such thatSuppose that We apply difference operators given byand we put in (2.17). Let be defined byand . Write Given we consider the functional difference equationwith the initial boundary condition

The above problem is considered as an implicit difference method for (1.4), (1.5). It is important that the difference expressions , are calculated at the point and the functional variable 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   is satisfied and
(1), and the functions satisfy the difference functional inequality(2)the initial boundary estimate holds on Then

Proof. We prove (2.23) by induction on . It follows from assumption (2) that estimate (2.23) is satisfied for and Assume that for We prove that for Write It follows from (2.22) that where The monotonicity condition (3) of Assumption   implies the inequalities for , Then we havewhere andWriteIt follows from (2.11), (2.17) that We define and , as follows:If then assumption (2) implies that Let us consider the case when Then we have from (2.29) that It follows that . The the proof of (2.23) is completed by induction.

3. Implicit Difference Schemes

We define by the relations:and we assume that if . For we writeIn a similar way we define the norm in the space : if thenThe following properties of the operator are important in our considerations.

Lemma 3.1. Suppose that is of class and is the restriction of to the set . Let be such a constant that , for Then where

Lemma 3.2. Suppose that is of class and is the restriction of to the set . Let be such a constant that , , ,   Then

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   is satisfied and then there exists exactly one solution of difference functional problem (2.20), (2.21).

Proof. Suppose that is fixed and that the solution of problem (2.20), (2.21) is given on the set . We prove that the vectors ,  , exist and that they are unique. It is sufficient to show that there exists exactly one solution of the system of equationswhere with the initial boundary condition (2.21). There exists such that where It is clear that system (3.4) is equivalent to the following one: Write Elements of the space are denoted by , . For , , we write andwhere The norm in the space is defined byLet us consider the setWe consider the operator ,   defined bywhere ,   andwhere . We prove that
It follows from (3.10) that we have for : where andIt follows from the above relations and from (3.5) thatAccording to (3.12) we haveThis completes the proof of (3.12).
It follows from the Banach fixed point theorem that there exists exactly one solution of the equation and consequently, there exists exactly one solution of (3.6), (2.21). Then the vectors , , exist and they are unique. Then the proof is completed by induction with respect to , .