#### 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 [5β8] 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 [12β14]. 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 ββis 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 ββis 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 ββimplies 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 , , ,ββ 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 ββis 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 ,ββ,
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 ,ββ defined bywhere ,ββ 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 , .

Assumption (). *
The function satisfies the conditions: **(1) is continuous and it is nondecreasing with
respect to the both variables,(2) for and the maximal solution of the Cauchy problemis for .*

Assumption (). *
There is such that Assumption ββis satisfied and for , ,
we havewhere .*

Theorem 3.4. *Suppose that Assumptions ββandββ ββare satisfied and *

(1)* is a solution of (1.4), (1.5) and is of class on ,*(2)*, and and there is such that** Under these
assumptions there is a solution of (2.20), (2.21) and there is such that **where is the restriction of to the set *

*Proof. *The
existence of follows from Theorem 3.3. Let , be defined by the relationsFrom Lemma 3.1 and from
assumption (1) of the theorem it follows that there are such thatand , Write and For we put Let be a solution of the difference
problemWe prove thatLet be defined by where We prove that the difference functional
inequalityis satisfied. It follows from
Assumption and from (3.21) that This completes the proof of (3.27).

Since on ,
it follows from Theorem 2.3
that on In a similar way we prove that on The above estimates imply (3.25). Consider the
Cauchy problemIt follows from Assumption that there is such that for ββthe maximal solution of (3.29) is defined on andSince is convex function then we have the difference
inequalitywhere .
Since satisfies (3.24), the above relations imply the
estimateIt follows from (3.30) that
condition (3.20) is satisfied with .
This completes the proof.

Lemma 3.5. *Suppose that Assumption is satisfied
and *

(1)* is a solution of (1.4), (1.5) and is of class on ,*(2)*, and and there is such that*(3)*there exists such that estimatesare satisfied for , and ,*(4)*there is such that** Under these
assumptions there is a solution of (2.20), (2.21), and **where **and is such that **on for *

*Proof. *It
follows that the solution of the difference problem satisfies the condition: for .
Moreover we havewhere 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 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 we put Consider the differential
integral equation with the initial boundary
condition

The function is the solution of the above problem. Let us denote by 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 andwhereand , , The numbers can be called average errors of the difference method for fixed . We put and we have the values of the above defined errors which are shown in Table 1.

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 .

*Example 4.2. *Let and Consider the differential
equation with deviated variables with the initial boundary
conditions where

The function is the solution of the above problem. Let us denote by 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 be defined by (4.4) with , , . We put and we have the values of the above defined errors which are shown in Table 2.

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 .

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.