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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.

The 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

We give examples of equations which can be obtained
from (

Suppose that the function

For the above

It is clear
that more complicated differential systems with deviated variables and
differential integral problems can be obtained from (

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 (

the function

We show that there are implicit difference methods for
(

The paper is organized as follows. A theorem on
implicit difference functional inequalities with unknown function of several
variables is proved in Section

We use in the paper general ideas for finite
difference equations which were introduced in [

For any two
sets

Let

For

We formulate an implicit difference scheme for (

The function

the partial derivatives

there is

there is

The
existence theory of classical or generalized solutions to (

The same property of bicharacteristics is needed in a
theorem on the existence and uniqueness of solutions to (

Given the function

there exists the derivative

Let us denote by

The above problem is considered as an implicit
difference method for (

We prove a theorem on implicit difference inequalities
corresponding to (

Suppose that Assumption

the initial boundary estimate

We prove
(

We define

Suppose
that

Suppose
that

The above lemmas
are consequences of [

We first prove a theorem on the existence and
uniqueness of solutions to (

If
Assumption

Suppose
that

It follows from (

It follows from the Banach fixed point theorem that
there exists exactly one solution

The function

There is

Suppose that Assumptions

The
existence of

Since

Suppose that Assumption

there exists

there is

It
follows that the solution

In the result on error estimates we need
estimates for the derivatives of the solution

For

The function

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

Table of errors.

0.25 | 0.30 | 0.35 | 0.40 | 0.45 | 0.50 | |

0.0006 | 0.0007 | 0.0009 | 0.0010 | 0.0012 | 0.0014 |

Note that our equation and the
steps of the mesh do not satisfy condition (

Let

The function

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

Let

Table of errors.

0.25 | 0.30 | 0.35 | 0.40 | 0.45 | 0.5 | |

0.0002 | 0.0003 | 0.0004 | 0.0004 | 0.0005 | 0.0006 |

Note that our equation and the
steps of the mesh do not satisfy condition (

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 (

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