Abstract and Applied Analysis

Volume 2012 (2012), Article ID 178084, 18 pages

doi:10.1155/2012/178084

Research Article

A Note on the Stability of the Integral-Differential Equation of the Parabolic Type in a Banach Space

Maksat Ashyraliyev

Department of Mathematics and Computer Sciences, Bahcesehir University, Besiktas, 34353 Istanbul, Turkey

Received 20 May 2012; Accepted 15 June 2012

Academic Editor: Ravshan Ashurov

Copyright © 2012 Maksat Ashyraliyev. 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.

Abstract.

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.

1. Introduction

We consider the integral-differential equation

in an arbitrary Banach space with unbounded linear operators and in with dense domain and

A function is called a solution of (1.1) if the following conditions are satisfied:

(i) is continuously differentiable on . The derivatives at the endpoints are understood as the appropriate unilateral derivatives.(ii)The element belongs to for all , and the functions and are continuous on .(iii) satisfies (1.1).

A solution of (1.1) defined in this manner will from now on be referred to as a solution of (1.1) in the space of all continuous functions defined on with values in equipped with the norm

We consider (1.1) under the assumption that the operator generates an analytic semigroup , that is, the following estimates hold:

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 [1–11].

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 [12].

Lemma 1.1. If , , and continuous function satisfies the inequalities

then

A number of different generalizations of Gronwall’s integral inequality with one and two dependent limits have been obtained, see for instance [13, 14].

In numerical analysis literature, see for instance [15, 16], one can find the following discrete analogue of Lemma 1.1.

Lemma 1.2. If , is a sequence of real numbers with

where and , then

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 (1.1) of the parabolic type with two dependent limits in a Banach space . 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.

2. Gronwall’s Type Integral Inequality with Two Dependent Limits and Its Discrete Analogue

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.

Theorem 2.1. Assume that , , , are the sequences of real numbers and the inequalities

hold. Then for the inequalities

are satisfied, where

Proof. By putting directly in (2.1), we obtain the inequalities (2.2), correspondingly. Let us prove (2.3). We denote

Then (2.1) gets the form

Moreover, we have

Then, using (2.5)–(2.7) for , we obtain

So,

Then by induction we can prove that

hold for . Since , using (2.6), we obtain (2.3) for .

Let us prove (2.3) for . Using (2.5)–(2.7) for , we have

So,

Then by induction we can prove that

hold for . Since , using (2.6), we obtain (2.3) for . The proof of Theorem 2.1 is complete.

By putting , , , , and using the inequality for in the Theorem 2.1, we get the following result.

Theorem 2.2. Assume that is the sequence of real numbers and the inequalities

hold. Then for the inequalities

are satisfied.

By putting and passing to limit in the Theorem 2.1, we obtain the following generalization of Gronwall’s integral inequality with two dependent limits.

Theorem 2.3. Assume that , are the continuous functions on and is an integrable function on and the inequalities

hold. Then for the inequalities

are satisfied.

Finally, by putting , , , and in the Theorem 2.3, we get the following result.

Theorem 2.4. Assume that is a continuous function on and the inequalities

hold, where and . Then for the inequalities

are satisfied.

3. The Integral-Differential Equation of the Parabolic Type

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 (1.1) of the parabolic type with two dependent limits in a Banach space .

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

Theorem 3.1. Suppose that , . Then there is a unique solution of the integral equation

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

maps into . By using a special value of in the norm

we can prove that is the contracting operator on . Indeed, we have

for any . So,

where and when . Finally, we note that the norms

are equivalent in . The proof of Theorem 3.1 is complete.

Theorem 3.2. Suppose that assumptions (1.2) and (1.4) for the operators and hold. Assume that is continuously differentiable on function. Then there is a unique solution of (1.1) and stability inequality

holds, where does not depend on and .

Proof. The proof of the existence and uniqueness of the solution of (1.1) is based on the following formula:

and the Theorem 3.1.

First, we note that the solution of (1.1) satisfies . Indeed, assume that is the solution of (1.1) with . Then

and from the continuity of , and at we get

This leads to , and it follows that .

Let us now prove (3.8). First, we consider the case when . It is well known that the Cauchy problem

for differential equations in an arbitrary Banach space with positive operator has the unique solution

for smooth . By putting

we have

Since

we obtain (3.8) for .

Now, let . Then we consider the problem

for differential equations in an arbitrary Banach space with positive operator , which has the unique solution

By putting

we have

Since

we obtain (3.8) for .

From (3.8) it follows that

Applying the triangle inequality and assumptions (1.2) and (1.4), we get

Then, using the Theorem 2.4, we have

So,

By applying the triangle inequality in (1.1) and assumptions (1.2) and (1.4), we obtain

So,

Then using (3.24), we have

So, stability inequality (3.7) holds with . The proof of Theorem 3.2 is complete.

Note that it does not hold, generally speaking

in an arbitrary Banach space for the general strong positive operator , see [17, Section 1.5, Chapter 1]. Nevertheless, we can establish the following theorem.

Theorem 3.3. Suppose that assumptions (1.4) for the operator hold and

Assume that is a continuous on function. Then there is a unique solution of (1.1) and stability inequality

holds, where does not depend on and . Here the fractional spaces , consisting of all for which the following norms are finite:

Proof. First, we rewrite (3.8) as

The proof of the existence and uniqueness of the solution of (1.1) is based on the formula (3.32) and an analogue of the Theorem 3.1. Let us prove (3.30). From (3.32) it follows that

Applying the triangle inequality, the definition of the norm of the space and assumptions (1.4) and (3.29), we obtain

By [17, Chapter 1, Theorem 4.1], we obtain

So,

for . Then, using the Theorem 2.4, we have

So,

Then, using the triangle inequality in (1.1) yields

Combining last two inequalities, we obtain (3.30) with . The proof of Theorem 3.3 is complete.

Now, we consider the Rothe difference scheme for approximate solutions of (1.1).

Theorem 3.4. Suppose that the requirements of the Theorem 3.2 are satisfied. Then for the solution of difference scheme (3.40), the following stability inequalities

hold, where does not depend on , .

Proof. By induction we can prove that the initial value problem

for difference equations in an arbitrary Banach space with positive operator has a unique solution

where . By putting , we obtain

Using

we have

Furthermore,

Putting (3.46)-(3.47) in (3.44), we get

for .

Since

applying estimates (1.4) gives

Then, applying the triangle inequality and the estimate (1.2) in (3.48), we obtain

So, for ,

holds, where .

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

for difference equations in an arbitrary Banach space with positive operator has a unique solution and inequalities

hold for .

Now, the proof of this theorem is based on the Theorem 2.2 and the inequalities (3.52) and (3.54). The proof of Theorem 3.4 is complete.

Note that it does not hold, generally speaking

in the arbitrary Banach space for the general strong positive operator , see [17, 18].

This approach and theory of difference schemes of [17] permit us to obtain the following two theorems on stability estimates for the solution of difference scheme (3.40).

Theorem 3.5. Suppose that the requirements of the Theorem 3.2 are satisfied. Then for the solution of difference scheme (3.40) the following stability inequalities

hold, where does not depend on , .

Theorem 3.6. Suppose that the requirements of the Theorem 3.3 are satisfied. Then for the solution of difference scheme (3.40), the following stability inequalities

hold, where does not depend on , . Here the fractional spaces , consisting of all for which the following norms are finite:

Stability estimates could be also proved for the more general Pade difference schemes of the high order of accuracy, see [17, 19].

4. Conclusion

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.

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