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The integral-differential equation of the parabolic type in a Banach space is considered. The unique solvability of this equation is established. The stability estimates for the solution of this equation are obtained. The difference scheme approximately solving this equation is presented. The stability estimates for the solution of this difference scheme are obtained.

We consider the integral-differential equation

A function

The element

A solution of (

We consider (

Integral inequalities play a significant role in the theory of differential and integral-differential equations. They are useful to investigate some properties of the solutions of equations, such as existence, uniqueness and stability, see for instance [

Mathematical modelling of real-life phenomena is widely used in various applied fields of science. This is based on the mathematical description of real-life processes and the subsequent solving of the appropriate mathematical problems on the computer. The mathematical models of many real-life problems lead to already known or new differential and integral-differential equations. In most of the cases it is difficult to find the exact solutions of the differential and integral-differential equations. For this reason discrete methods play a significant role, especially with the appearance of highly efficient computers. A well-known and widely applied method of approximate solutions for differential and integral-differential equations is the method of difference schemes. Modern computers allow us to implement highly accurate difference schemes. Hence, the task is to construct and investigate highly accurate difference schemes for various types of differential and integral-differential equations. The investigation of stability and convergence of these difference schemes is based on the discrete analogues of integral inequalities.

Gronwall in 1919 showed the following result [

If

A number of different generalizations of Gronwall’s integral inequality with one and two dependent limits have been obtained, see for instance [

In numerical analysis literature, see for instance [

If

In the current paper, we will derive the discrete analogue of generalization of the Gronwall’s integral inequality. It is used to obtain the generalization of Gronwall’s integral inequality with two dependent limits. We will consider the applications of these inequalities to the integral-differential equation (

First of all, let us obtain the theorems on the Gronwall’s type integral inequalities with two dependent limits and their discrete analogues. We will use these results in the remaining part of the paper.

Assume that

By putting

Let us prove (

By putting

Assume that

By putting

Assume that

Finally, by putting

Assume that

Now, we consider the application of the generalizations of Gronwall’s integral inequality with two dependent limits and their discrete analogues to the integral-differential equation (

First of all, let us give one theorem that will be needed below.

Suppose that

The proof of this theorem is based on a fixed-point theorem. It is easy to see that the operator

Suppose that assumptions (

The proof of the existence and uniqueness of the solution of (

First, we note that the solution of (

Let us now prove (

Now, let

From (

Note that it does not hold, generally speaking

Suppose that assumptions (

First, we rewrite (

Now, we consider the Rothe difference scheme for approximate solutions of (

Suppose that the requirements of the Theorem

By induction we can prove that the initial value problem

Since

In a similar way, we can prove that the initial value problem

Now, the proof of this theorem is based on the Theorem

Note that it does not hold, generally speaking

This approach and theory of difference schemes of [

Suppose that the requirements of the Theorem

Suppose that the requirements of the Theorem

Stability estimates could be also proved for the more general Pade difference schemes of the high order of accuracy, see [

In this paper, the integral-differential equation of the parabolic type with two dependent limits in a Banach space is studied. The unique solvability of this equation is established. The stability estimates for the solution of this equation are obtained. The Rothe difference scheme approximately solving this equation is presented. The stability estimates for the solution of this difference scheme are obtained.