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.