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

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 𝑑𝑢(𝑡)(𝑡)(𝑡)=𝑑𝑡+sgn𝐴𝑢^𝑡_−𝑡𝐵(𝑠)𝑢(𝑠)𝑑𝑠+𝑓(𝑡),−1≤𝑡≤1 (1.1) in an arbitrary Banach space 𝐸 with unbounded linear operators 𝐴 and 𝐵(𝑡) in 𝐸 with dense domain 𝐷(𝐴)⊂𝐷(𝐵(𝑡)) and 𝐵(𝑡)𝐴⁻¹_𝐸→𝐸≤𝑀,−1≤𝑡≤1. (1.2)

A function 𝑢(𝑡) is called a solution of (1.1) if the following conditions are satisfied:(i)𝑢(𝑡) is continuously differentiable on [−1,1]. The derivatives at the endpoints are understood as the appropriate unilateral derivatives.(ii)The element 𝑢(𝑡) belongs to 𝐷(𝐴) for all 𝑡∈[−1,1], and the functions 𝐴𝑢(𝑡) and 𝐵(𝑡)𝑢(𝑡) are continuous on [−1,1].(iii)𝑢(𝑡) 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 𝐶(𝐸)=𝐶([−1,1],𝐸) of all continuous functions 𝜑(𝑡) defined on [−1,1] with values in 𝐸 equipped with the norm 𝜑_𝐶(𝐸)=max_{−1≤𝑡≤1}𝜑(𝑡)_𝐸. (1.3)

We consider (1.1) under the assumption that the operator −𝐴 generates an analytic semigroup exp{−𝑡𝐴}(𝑡≥0), that is, the following estimates hold: 𝑒^−𝑡𝐴_𝐸→𝐸≤𝑀,𝑡𝐴𝑒^−𝑡𝐴_𝐸→𝐸≤𝑀,0≤𝑡≤1. (1.4)

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 𝑀=𝑐𝑜𝑛𝑠𝑡>0, 𝛿=𝑐𝑜𝑛𝑠𝑡>0, and continuous function 𝑥(𝑡)≥0 satisfies the inequalities 𝑥(𝑡)≤𝛿+𝑀^𝑡₀𝑥(𝑠)𝑑𝑠,0≤𝑡≤𝑇, (1.5) then 𝑥(𝑡)≤𝛿exp[𝑀𝑡],0≤𝑡≤𝑇. (1.6)

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 𝑥_𝑗, 𝑗=0,…,𝑁 is a sequence of real numbers with 𝑥_𝑖≤𝛿+ℎ𝑀^𝑖−1_𝑗=0𝑥_𝑗,𝑖=1,…,𝑁, (1.7) where 𝑀=𝑐𝑜𝑛𝑠𝑡>0 and 𝛿=𝑐𝑜𝑛𝑠𝑡>0, then 𝑥_𝑖≤ℎ𝑀𝑥₀+𝛿exp[𝑀𝑖ℎ],𝑖=1,…,𝑁. (1.8)

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 𝐸. 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 𝑣_𝑖≥0, 𝑎_𝑖≥0, 𝛿_𝑖≥0, 𝑖=−𝑁,…,𝑁+2𝑀 are the sequences of real numbers and the inequalities 𝑣_𝑖≤𝛿_𝑖+ℎ^{|𝑖−𝑀|+𝑀−1}_{𝑗=−|𝑖−𝑀|+𝑀+1}𝑎_𝑗𝑣_𝑗−𝑎_𝑀𝑣_𝑀,𝑖=−𝑁,…,𝑁+2𝑀 (2.1) hold. Then for 𝑣_𝑖 the inequalities 𝑣_𝑀−1≤𝛿_𝑀−1,𝑣_𝑀+1≤𝛿_𝑀+1,𝑣_𝑀≤𝛿_𝑀+ℎ𝑎_𝑀−1𝛿_𝑀−1+𝑎_𝑀+1𝛿_𝑀+1,𝑣_𝑖≤𝛿_𝑖+ℎ^{|𝑖−𝑀|+𝑀−1}_𝑗=𝑀+1𝑎_𝑗𝛿_𝑗+𝑎_2𝑀−𝑗𝛿_2𝑀−𝑗𝐵_{|𝑖−𝑀|+𝑀−1,𝑗},𝑖=−𝑁,…,𝑀−2,𝑀+2,…,𝑁+2𝑀 (2.2)(2.3) are satisfied, where 𝐵_𝑘,𝑗=^𝑘_𝑛=∏_𝑗+11+ℎ𝑎_𝑛+𝑎_2𝑀−𝑛,if𝑗=𝑀+1,…,𝑘−1,1,if𝑗=𝑘. (2.4)

Proof. By putting 𝑖=𝑀−1,𝑀+1,𝑀 directly in (2.1), we obtain the inequalities (2.2), correspondingly. Let us prove (2.3). We denote 𝑦_𝑖=ℎ^{|𝑖−𝑀|+𝑀−1}_{𝑗=−|𝑖−𝑀|+𝑀+1}𝑎_𝑗𝑣_𝑗−𝑎_𝑀𝑣_𝑀,𝑖=−𝑁,…,𝑁+2𝑀. (2.5) Then (2.1) gets the form 𝑣_𝑖≤𝛿_𝑖+𝑦_𝑖,𝑖=−𝑁,…,𝑁+2𝑀. (2.6) Moreover, we have 𝑦_2𝑀−𝑖=ℎ^{|𝑀−𝑖|+𝑀−1}_{𝑗=−|𝑀−𝑖|+𝑀+1}𝑎_𝑗𝑣_𝑗−𝑎_𝑀𝑣_𝑀=𝑦_𝑖,𝑖=−𝑁,…,𝑁+2𝑀. (2.7) Then, using (2.5)–(2.7) for 𝑖=𝑀+1,…,𝑁+2M−1, we obtain 𝑦_𝑖+1−𝑦_𝑖=ℎ^𝑖_{𝑗=2𝑀−𝑖}𝑎_𝑗𝑣_𝑗−𝑎_𝑀𝑣_𝑀−ℎ^𝑖−1_{𝑗=2𝑀−𝑖+1}𝑎_𝑗𝑣_𝑗−𝑎_𝑀𝑣_𝑀=ℎ𝑎_𝑖𝑣_𝑖+𝑎_2𝑀−𝑖𝑣_2𝑀−𝑖≤ℎ𝑎_𝑖𝑦_𝑖+𝛿_𝑖+ℎ𝑎_2𝑀−𝑖𝑦_2𝑀−𝑖+𝛿_2𝑀−𝑖=ℎ𝑎_𝑖+𝑎_2𝑀−𝑖𝑦_𝑖+ℎ𝑎_𝑖𝛿_𝑖+𝑎_2𝑀−𝑖𝛿_2𝑀−𝑖. (2.8) So, 𝑦_𝑖+1≤1+ℎ𝑎_𝑖+𝑎_2𝑀−𝑖𝑦_𝑖+ℎ𝑎_𝑖𝛿_𝑖+𝑎_2𝑀−𝑖𝛿_2𝑀−𝑖,𝑖=𝑀+1,…,𝑁+2𝑀−1. (2.9) Then by induction we can prove that 𝑦_𝑖≤^{𝑖−𝑀−1}_𝑛=11+ℎ𝑎_𝑀+𝑛+𝑎_𝑀−𝑛𝑦_𝑀+1+^𝑖−1_𝑗=𝑀+1ℎ𝑎_𝑗𝛿_𝑗+𝑎_2𝑀−𝑗𝛿_2𝑀−𝑗𝐵_{𝑖−1,𝑗} (2.10) hold for 𝑖=𝑀+2,…,𝑁+2𝑀. Since 𝑦_𝑀+1=0, using (2.6), we obtain (2.3) for 𝑖=𝑀+2,…,𝑁+2𝑀.
Let us prove (2.3) for 𝑖=−𝑁,…,𝑀−2. Using (2.5)–(2.7) for 𝑖=−𝑁+1,…,𝑀−1, we have 𝑦_𝑖−1−𝑦_𝑖=ℎ^2𝑀−𝑖_𝑗=𝑖𝑎_𝑗𝑣_𝑗−𝑎_𝑀𝑣_𝑀−ℎ^{2𝑀−𝑖−1}_𝑗=𝑖+1𝑎_𝑗𝑣_𝑗−𝑎_𝑀𝑣_𝑀=ℎ𝑎_𝑖𝑣_𝑖+𝑎_2𝑀−𝑖𝑣_2𝑀−𝑖≤ℎ𝑎_𝑖𝑦_𝑖+𝛿_𝑖+ℎ𝑎_2𝑀−𝑖𝑦_2𝑀−𝑖+𝛿_2𝑀−𝑖=ℎ𝑎_𝑖+𝑎_2𝑀−𝑖𝑦_𝑖+ℎ𝑎_𝑖𝛿_𝑖+𝑎_2𝑀−𝑖𝛿_2𝑀−𝑖. (2.11) So, 𝑦_𝑖−1≤1+ℎ𝑎_𝑖+𝑎_2𝑀−𝑖𝑦_𝑖+ℎ𝑎_𝑖𝛿_𝑖+𝑎_2𝑀−𝑖𝛿_2𝑀−𝑖,𝑖=−𝑁+1,…,𝑀−1. (2.12) Then by induction we can prove that 𝑦_𝑖≤^{𝑀−𝑖−1}_𝑛=11+ℎ𝑎_𝑀+𝑛+𝑎_𝑀−𝑛𝑦_𝑀−1+^{2𝑀−𝑖−1}_𝑗=𝑀+1ℎ𝑎_𝑗𝛿_𝑗+𝑎_2𝑀−𝑗𝛿_2𝑀−𝑗𝐵_{2𝑀−𝑖−1,𝑗} (2.13) hold for 𝑖=−𝑁,…,𝑀−2. Since 𝑦_𝑀−1=0, using (2.6), we obtain (2.3) for 𝑖=−𝑁,…,𝑀−2. The proof of Theorem 2.1 is complete.

By putting 𝑀=0, 𝛿_𝑖≡const, 𝑎_𝑖≡const, 𝑖=−𝑁,…,𝑁, and using the inequality 1+𝑥<exp[𝑥] for 𝑥>0 in the Theorem 2.1, we get the following result.

Theorem 2.2. Assume that 𝑣_𝑖≥0,𝑖=−𝑁,…,𝑁 is the sequence of real numbers and the inequalities 𝑣_𝑖≤𝛿+𝐿ℎ^|𝑖|−1_{𝑗=−|𝑖|+1}𝑣_𝑗−𝑣₀,𝑖=−𝑁,…,𝑁 (2.14) hold. Then for 𝑣_𝑖 the inequalities 𝑣₀≤𝛿exp[2𝐿ℎ],𝑣_𝑖≤𝛿exp[2𝐿ℎ(|𝑖|−1)],𝑖=−𝑁,…,−1,1,…,𝑁 (2.15) are satisfied.

By putting 𝑁ℎ=1,2𝑀ℎ=𝑇 and passing to limit ℎ→0 in the Theorem 2.1, we obtain the following generalization of Gronwall’s integral inequality with two dependent limits.

Theorem 2.3. Assume that 𝑣(𝑡)≥0, 𝛿(𝑡)≥0 are the continuous functions on [−1,1+𝑇] and 𝑎(𝑡)≥0 is an integrable function on [−1,1+𝑇] and the inequalities 𝑣(𝑡)(𝑡)𝑇≤𝛿+sgnt−2^𝑡_𝑇−𝑡𝑎(𝑠)𝑣(𝑠)𝑑𝑠,−1≤𝑡≤1+𝑇 (2.16) hold. Then for 𝑣(𝑡) the inequalities 𝑣(𝑡)≤𝛿(𝑡)+^𝑡_𝑇/2(𝑎(𝑠)𝛿(𝑠)+𝑎(𝑇−𝑠)𝛿(𝑇−𝑠))exp^𝑡_𝑠(𝑎(𝜏)(𝑇+𝑎𝑇−𝜏))𝑑𝜏𝑑𝑠,2𝑣(𝑡)(𝑡)+≤𝑡≤1+𝑇,≤𝛿^𝑇−𝑡_𝑇/2(𝑎(𝑠)𝛿(𝑠)+𝑎(𝑇−𝑠)𝛿(𝑇−𝑠))exp_𝑠^𝑇−𝑡(𝑎(𝜏)(𝑇+𝑎𝑇−𝜏))𝑑𝜏𝑑𝑠,−1≤𝑡<2 (2.17) are satisfied.

Finally, by putting 𝛿(𝑡)≡const, 𝑎(𝑡)≡const, −1≤𝑡≤1, and 𝑇=0 in the Theorem 2.3, we get the following result.

Theorem 2.4. Assume that 𝑣(𝑡)≥0 is a continuous function on [−1,1] and the inequalities 𝑣(𝑡)≤𝐶+𝐿sgn(𝑡)^𝑡_−𝑡𝑣(𝑠)𝑑𝑠,−1≤𝑡≤1 (2.18) hold, where 𝐶=𝑐𝑜𝑛𝑠𝑡≥0 and 𝐿=𝑐𝑜𝑛𝑠𝑡≥0. Then for 𝑣(𝑡) the inequalities 𝑣(𝑡)≤𝐶exp(2𝐿|𝑡|),−1≤𝑡≤1 (2.19) 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 𝐸.

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

Theorem 3.1. Suppose that 𝐹(𝑡)∈𝐶([−1,1],𝐸), 𝐾(𝑡,𝑠)∈𝐶([−1,1],𝐸). Then there is a unique solution of the integral equation 𝑧(𝑡)=𝐹(𝑡)+sgn(𝑡)^𝑡_−𝑡𝐾(𝑡,𝑠)𝑧(𝑠)𝑑𝑠,−1≤𝑡≤1. (3.1)

Proof. The proof of this theorem is based on a fixed-point theorem. It is easy to see that the operator 𝐵𝑧(𝑡)=𝐹(𝑡)+sgn(𝑡)^𝑡_−𝑡𝐾(𝑡,𝑠)𝑧(𝑠)𝑑𝑠,−1≤𝑡≤1 (3.2) maps 𝐶([−1,1],𝐸) into 𝐶([−1,1],𝐸). By using a special value of 𝜆 in the norm ‖𝑣‖_{𝐶∗([−1,1],𝐸)}=max_{−1≤𝑡≤1}𝑒^{−𝜆|𝑡|}‖𝑣(𝑡)‖_𝐸, (3.3) we can prove that 𝐴 is the contracting operator on 𝐶^∗([−1,1],𝐸). Indeed, we have 𝑒^{−𝜆|𝑡|}‖𝐵𝑧(𝑡)−𝐵𝑢(𝑡)‖_𝐸≤_−|𝑡|^|𝑡|‖𝐾(𝑡,𝑠)‖_𝐸→𝐸𝑒^{−𝜆(|𝑡|−|𝑠|)}𝑒^{−𝜆|𝑠|}‖𝑧(𝑠)−𝑢(𝑠)‖_𝐸𝑑𝑠≤max_{−1≤𝑠,𝑡≤1}‖𝐾(𝑡,𝑠)‖_{𝐸→𝐸−|𝑡|}^|𝑡|𝑒^{−𝜆(|𝑡|−|𝑠|)}‖𝑧−𝑢‖_{𝐶∗([−1,1],𝐸)}𝑑𝑠=2max_{−1≤𝑠,𝑡≤1}‖𝐾(𝑡,𝑠)‖_𝐸→𝐸0^|𝑡|𝑒^{−𝜆(|𝑡|−𝑠)}𝑑𝑠‖𝑧−𝑢‖_{𝐶∗([−1,1],𝐸)}=2max_{−1≤𝑠,𝑡≤1}‖𝐾(𝑡,𝑠)‖_𝐸→𝐸‖𝑧−𝑢‖_{𝐶∗([−1,1],𝐸)}1−𝑒^{−𝜆|𝑡|}𝜆≤‖𝑧−𝑢‖_{𝐶∗([−1,1],𝐸)}21−𝑒^−𝜆𝜆max_{−1≤𝑠,𝑡≤1}‖𝐾(𝑡,𝑠)‖_𝐸→𝐸 (3.4) for any 𝑡∈[−1,1]. So, ‖𝐵𝑧−𝐵𝑢‖_{𝐶∗([−1,1],𝐸)}≤‖𝑧−𝑢‖_{𝐶∗([−1,1],𝐸)}𝛼_𝜆, (3.5) where 𝛼_𝜆=(2(1−𝑒^−𝜆)/𝜆)max_{−1≤𝑠,𝑡≤1}‖𝐾(𝑡,𝑠)‖_𝐸→𝐸 and 𝛼_𝜆→0 when 𝜆→∞. Finally, we note that the norms ‖𝑣‖_{𝐶∗([−1,1],𝐸)}=max_{−1≤𝑡≤1}𝑒^{−𝜆|𝑡|}‖𝑣(𝑡)‖_𝐸,‖𝑣‖_{𝐶([−1,1],𝐸)}=max_{−1≤𝑡≤1}‖𝑣(𝑡)‖_𝐸 (3.6) are equivalent in 𝐶([−1,1],𝐸). The proof of Theorem 3.1 is complete.

Theorem 3.2. Suppose that assumptions (1.2) and (1.4) for the operators 𝐴 and 𝐵(𝑡) hold. Assume that 𝑓(𝑡) is continuously differentiable on [−1,1] function. Then there is a unique solution of (1.1) and stability inequality max_{−1≤𝑡≤1}𝑑𝑢(𝑡)𝑑𝑡_𝐸+max_{−1≤𝑡≤1}‖𝐴𝑢(𝑡)‖_𝐸≤𝑀^∗𝑓(0)_𝐸+¹₋₁𝑓^′(𝑠)_𝐸𝑑𝑠 (3.7) holds, where 𝑀^∗ does not depend on 𝑓(𝑡) and 𝑡.

Proof. The proof of the existence and uniqueness of the solution of (1.1) is based on the following formula: 𝑢(𝑡)=sgn(𝑡)𝐴⁻¹𝑓(𝑡)−sgn(𝑡)𝑒^{−|𝑡|𝐴}𝐴⁻¹𝑓(0)−sgn(𝑡)^𝑡₀𝑒^{−(|𝑡|−|𝑠|)𝐴}𝐴⁻¹𝑓^′(𝑠)(𝑡)𝑑𝑠+sgn^𝑡_−𝑡𝐼−𝑒^{−(|𝑡|−|𝑠|)𝐴}𝐴⁻¹𝐵(𝑠)𝑢(𝑠)𝑑𝑠,−1≤𝑡≤1 (3.8) and the Theorem 3.1.
First, we note that the solution of (1.1) satisfies 𝑢(0)=0. Indeed, assume that 𝑢(𝑡) is the solution of (1.1) with 𝐵≡0. Then 𝑢^′(𝑡)(𝑡)(𝑡)𝑢+𝐴𝑢=𝑓,0<𝑡≤1,_′(𝑡)−𝐴𝑢(𝑡)=𝑓(𝑡),−1≤𝑡<0, (3.9) and from the continuity of 𝑓,𝑢′(𝑡), and 𝐴𝑢 at 𝑡=0 we get 𝑢^′(0)(0)(0),𝑢+𝐴𝑢=𝑓_′(0)−𝐴𝑢(0)=𝑓(0). (3.10) This leads to 2𝐴𝑢(0)=0, and it follows that 𝑢(0)=0.
Let us now prove (3.8). First, we consider the case when 0≤𝑡≤1. It is well known that the Cauchy problem 𝑑𝑢(𝑡)(𝑡)(𝑡)𝑢(0)𝑑𝑡+𝐴𝑢=𝐹,0≤𝑡≤1,=0 (3.11) for differential equations in an arbitrary Banach space 𝐸 with positive operator 𝐴 has the unique solution 𝑢(𝑡)=^𝑡₀𝑒^{−(𝑡−𝑠)𝐴}𝐹(𝑠)𝑑𝑠,0≤𝑡≤1 (3.12) for smooth 𝐹(𝑡). By putting 𝐹(𝑡)=^𝑡_−𝑡𝐵(𝑠)𝑢(𝑠)𝑑𝑠+𝑓(𝑡), (3.13) we have 𝑢(𝑡)=^𝑡₀𝑒^{−(𝑡−𝑠)𝐴}𝑓(𝑠)𝑑𝑠+^𝑡₀𝑒^{−(𝑡−𝑠)𝐴𝑠}_−𝑠𝐵(𝜏)𝑢(𝜏)𝑑𝜏𝑑𝑠,0≤𝑡≤1. (3.14) Since ^𝑡₀^𝑠_−𝑠𝑒^{−(𝑡−𝑠)𝐴}𝐵(𝜏)𝑢(𝜏)𝑑𝜏𝑑𝑠=^𝑡₀^𝑡_𝜏𝑒^{−(𝑡−𝑠)𝐴}𝐵(𝜏)𝑢(𝜏)+𝑑𝑠𝑑𝜏_{0−𝑡𝑡−𝜏}𝑒^{−(𝑡−𝑠)𝐴}𝐵(𝜏)𝑢(𝜏)=𝑑𝑠𝑑𝜏_𝑡0𝐼−𝑒^{−(𝑡−𝜏)𝐴}𝐴⁻¹𝐵(𝜏)𝑢(𝜏)+𝑑𝜏_0−𝑡𝐼−𝑒^{−(𝑡+𝜏)𝐴}𝐴⁻¹𝐵(𝜏)𝑢(𝜏)=𝑑𝜏_𝑡−𝑡𝐼−𝑒^{−(𝑡−|𝑠|)𝐴}𝐴⁻¹𝐵(𝑠)𝑢(𝑠)𝑑𝑠,_𝑡0𝑒^{−(𝑡−𝑠)𝐴}𝑓(𝑠)𝑑𝑠=𝐴⁻¹𝑓(𝑡)−𝑒^−𝑡𝐴𝐴⁻¹𝑓(0)−^𝑡₀𝑒^{−(𝑡−𝑠)𝐴}𝐴⁻¹𝑓^′(𝑠)𝑑𝑠, (3.15) we obtain (3.8) for 0≤𝑡≤1.
Now, let −1≤𝑡≤0. Then we consider the problem 𝑑𝑢(𝑡)(𝑡)(𝑡)𝑢(0)𝑑𝑡−𝐴𝑢=𝐹,−1≤𝑡≤0,=0 (3.16) for differential equations in an arbitrary Banach space 𝐸 with positive operator 𝐴, which has the unique solution 𝑢(𝑡)=−⁰_𝑡𝑒^{(𝑡−𝑠)𝐴}𝐹(𝑠)𝑑𝑠,−1≤𝑡≤0. (3.17) By putting 𝐹(𝑡)=^𝑡_−𝑡𝐵(𝑠)𝑢(𝑠)𝑑𝑠+𝑓(𝑡), (3.18) we have 𝑢(𝑡)=−⁰_𝑡𝑒^{(𝑡−𝑠)𝐴}𝑓(𝑠)𝑑𝑠+⁰_𝑡𝑒^{(𝑡−𝑠)𝐴}_𝑠^−𝑠𝐵(𝜏)𝑢(𝜏)𝑑𝜏𝑑𝑠,−1≤𝑡≤0. (3.19) Since ⁰_𝑡𝑠^−𝑠𝑒^{(𝑡−𝑠)𝐴}𝐵(𝜏)𝑢(𝜏)𝑑𝜏𝑑𝑠=₀^−𝑡_𝑡^−𝜏𝑒^{(𝑡−𝑠)𝐴}𝐵(𝜏)𝑢(𝜏)+𝑑𝑠𝑑𝜏_{0𝑡𝜏𝑡}𝑒^{(𝑡−𝑠)𝐴}𝐵(𝜏)𝑢(𝜏)=𝑑𝑠𝑑𝜏_0−𝑡𝐼−𝑒^{(𝑡+𝜏)𝐴}𝐴⁻¹𝐵(𝜏)𝑢(𝜏)+𝑑𝜏_0𝑡𝐼−𝑒^{(𝑡−𝜏)𝐴}𝐴⁻¹𝐵(𝜏)𝑢(𝜏)=𝑑𝜏_𝑡−𝑡𝐼−𝑒^{−(−𝑡−|𝑠|)𝐴}𝐴⁻¹𝐵(𝑠)𝑢(𝑠)𝑑𝑠,_0𝑡𝑒^{(𝑡−𝑠)𝐴}𝑓(𝑠)𝑑𝑠=𝐴⁻¹𝑓(𝑡)−𝑒^𝑡𝐴𝐴⁻¹𝑓(0)+⁰_𝑡𝑒^{−(−𝑡+𝑠)𝐴}𝐴⁻¹𝑓^′(𝑠)𝑑𝑠, (3.20) we obtain (3.8) for −1≤𝑡≤0.
From (3.8) it follows that 𝐴𝑢(𝑡)=sgn(𝑡)𝑓(𝑡)−sgn(𝑡)𝑒^{−|𝑡|𝐴}𝑓(0)−sgn(𝑡)^𝑡₀𝑒^{−(|𝑡|−|𝑠|)𝐴}𝑓^′(𝑠)(𝑡)𝑑𝑠+sgn^𝑡_−𝑡𝐼−𝑒^{−(|𝑡|−|𝑠|)𝐴}𝐵(𝑠)𝑢(𝑠)𝑑𝑠,−1≤𝑡≤1. (3.21) Applying the triangle inequality and assumptions (1.2) and (1.4), we get ‖𝐴𝑢(𝑡)‖_𝐸≤𝑓(𝑡)_𝐸+𝑒^{−|𝑡|𝐴}_𝐸→𝐸𝑓(0)_𝐸+_−|𝑡|^|𝑡|𝑒^{−(|𝑡|−|𝑠|)𝐴}_𝐸→𝐸𝑓^′(𝑠)_𝐸+𝑑𝑠_{−|𝑡||𝑡|}1+𝑒^{−(|𝑡|−|𝑠|)𝐴}_𝐸→𝐸𝐵(𝑠)𝐴⁻¹_𝐸→𝐸‖𝐴𝑢(𝑠)‖_𝐸≤(𝑓(0𝑑𝑠𝑀+1))_𝐸+¹₋₁𝑓^′(𝑠)_𝐸(𝑡)𝑀()𝑑𝑠+sgn𝑀+1^𝑡_−𝑡‖𝐴𝑢(𝑠)‖_𝐸𝑑𝑠,−1≤𝑡≤1. (3.22) Then, using the Theorem 2.4, we have ‖𝐴𝑢(𝑡)‖_𝐸≤(𝑀+1)𝑓(0)_𝐸+¹₋₁𝑓^′(𝑠)_𝐸𝑑𝑠𝑒^{2𝑀(𝑀+1)|𝑡|}≤(𝑀+1)𝑒^{2𝑀(𝑀+1)}𝑓(0)_𝐸+¹₋₁𝑓^′(𝑠)_𝐸𝑑𝑠,−1≤𝑡≤1. (3.23) So, max_{−1≤𝑡≤1}‖𝐴𝑢(𝑡)‖_𝐸≤(𝑀+1)𝑒^{2𝑀(𝑀+1)}𝑓(0)_𝐸+¹₋₁𝑓^′(𝑠)_𝐸𝑑𝑠. (3.24) By applying the triangle inequality in (1.1) and assumptions (1.2) and (1.4), we obtain 𝑑𝑢(𝑡)𝑑𝑡_𝐸≤‖𝐴𝑢(𝑡)‖_𝐸+_−|𝑡|^|𝑡|𝐵(𝑠)𝐴⁻¹_𝐸→𝐸‖𝐴𝑢(𝑠)‖_𝐸𝑑𝑠+𝑓(𝑡)_𝐸≤(2𝑀+1)max_{−1≤𝑡≤1}‖𝐴𝑢(𝑡)‖_𝐸+𝑓(0)_𝐸+¹₋₁𝑓^′(𝑠)_𝐸𝑑𝑠,−1≤𝑡≤1. (3.25) So, max_{−1≤𝑡≤1}𝑑𝑢(𝑡)𝑑𝑡_𝐸≤(2𝑀+1)max_{−1≤𝑡≤1}‖𝐴𝑢(𝑡)‖_𝐸+𝑓(0)_𝐸+¹₋₁𝑓^′(𝑠)_𝐸𝑑𝑠. (3.26) Then using (3.24), we have max_{−1≤𝑡≤1}𝑑𝑢(𝑡)𝑑𝑡_𝐸+max_{−1≤𝑡≤1}‖𝐴𝑢(𝑡)‖_𝐸≤2(𝑀+1)max_{−1≤𝑡≤1}‖𝐴𝑢(𝑡)‖_𝐸+𝑓(0)_𝐸+¹₋₁𝑓^′(𝑠)_𝐸≤2()𝑑𝑠𝑀+1²𝑒^{2𝑀(𝑀+1)}+1𝑓(0)_𝐸+¹₋₁𝑓^′(𝑠)_𝐸𝑑𝑠. (3.27) So, stability inequality (3.7) holds with 𝑀^∗=2(𝑀+1)²𝑒^{2𝑀(𝑀+1)}+1. The proof of Theorem 3.2 is complete.

Note that it does not hold, generally speaking max_{−1≤𝑡≤1}𝑑𝑢(𝑡)𝑑𝑡_𝐸+max_{−1≤𝑡≤1}‖𝐴𝑢(𝑡)‖_𝐸≤𝑀^∗max_{−1≤𝑡≤1}𝑓(𝑡)_𝐸 (3.28) in an arbitrary Banach space 𝐸 for the general strong positive operator 𝐴, 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 𝐴 hold and 𝐵(𝑡)𝐴⁻¹_{𝐸𝛼→𝐸𝛼}≤𝑀,−1≤𝑡≤1. (3.29) Assume that 𝑓(𝑡) is a continuous on [−1,1] function. Then there is a unique solution of (1.1) and stability inequality max_{−1≤𝑡≤1}𝑑𝑢(𝑡)𝑑𝑡_𝐸𝛼+max_{−1≤𝑡≤1}‖𝐴𝑢(𝑡)‖_𝐸𝛼≤𝑀^∗(𝛼)max_{−1≤𝑡≤1}𝑓(𝑡)_𝐸𝛼 (3.30) holds, where 𝑀^∗(𝛼) does not depend on 𝑓(𝑡) and 𝑡. Here the fractional spaces 𝐸_𝛼=𝐸_𝛼(𝐸,𝐴)(0<𝛼<1), consisting of all 𝑣∈𝐸 for which the following norms are finite: ‖𝑣‖_𝐸𝛼=sup_0<𝑧𝑧^1−𝛼𝐴exp{−𝑧𝐴}𝑣_𝐸. (3.31)

Proof. First, we rewrite (3.8) as 𝑢(𝑡)=^𝑡₀𝑒^{−(|𝑡|−|𝑠|)𝐴}𝑓(𝑠)𝑑𝑠+sgn(𝑡)^𝑡_−𝑡𝐼−𝑒^{−(|𝑡|−|𝑠|)𝐴}𝐴⁻¹𝐵(𝑠)𝑢(𝑠)𝑑𝑠,−1≤𝑡≤1. (3.32) 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 𝐴𝑢(𝑡)=^𝑡₀𝐴𝑒^{−(|𝑡|−|𝑠|)𝐴}𝑓(𝑠)𝑑𝑠+sgn(𝑡)^𝑡_−𝑡𝐼−𝑒^{−(|𝑡|−|𝑠|)𝐴}𝐵(𝑠)𝑢(𝑠)𝑑𝑠,−1≤𝑡≤1. (3.33) Applying the triangle inequality, the definition of the norm of the space 𝐸_𝛼 and assumptions (1.4) and (3.29), we obtain ‖𝐴𝑢(𝑡)‖_𝐸𝛼≤_−|𝑡|^|𝑡|𝐴𝑒^{−(|𝑡|−|𝑠|)𝐴}𝑓(𝑠)𝑑𝑠_𝐸𝛼+_−|𝑡|^|𝑡|1+𝑒^{−(|𝑡|−|𝑠|)𝐴}_𝐸→𝐸𝐵(𝑠)𝐴⁻¹_{𝐸𝛼→𝐸𝛼}‖𝐴𝑢(𝑠)‖_𝐸𝛼≤𝑑𝑠_{−|𝑡||𝑡|}𝐴𝑒^{−(|𝑡|−|𝑠|)𝐴}𝑓(𝑠)𝑑𝑠_𝐸𝛼+sgn(𝑡)𝑀(𝑀+1)^𝑡_−𝑡‖𝐴𝑢(𝑠)‖_𝐸𝛼𝑑𝑠. (3.34) By [17, Chapter 1, Theorem 4.1], we obtain_−|𝑡|^|𝑡|𝐴𝑒^{−(|𝑡|−|𝑠|)𝐴}𝑓(𝑠)𝑑𝑠_𝐸𝛼≤𝑀𝛼(1−𝛼)max_{−1≤𝑡≤1}𝑓(𝑡)_𝐸𝛼. (3.35) So, ‖𝐴𝑢(𝑡)‖_𝐸𝛼≤𝑀𝛼(1−𝛼)max_{−1≤𝑡≤1}𝑓(𝑡)_𝐸𝛼+sgn(𝑡)𝑀(𝑀+1)^𝑡_−𝑡‖𝐴𝑢(𝑠)‖_𝐸𝛼𝑑𝑠 (3.36) for −1≤𝑡≤1. Then, using the Theorem 2.4, we have ‖𝐴𝑢(𝑡)‖_𝐸𝛼≤𝑀𝛼(1−𝛼)max_{−1≤𝑡≤1}𝑓(𝑡)_𝐸𝛼𝑒^{2𝑀(𝑀+1)|𝑡|},−1≤𝑡≤1. (3.37) So, max_{−1≤𝑡≤1}‖𝐴𝑢(𝑡)‖_𝐸𝛼≤𝑀𝛼(1−𝛼)max_{−1≤𝑡≤1}𝑓(𝑡)_𝐸𝛼𝑒^{2𝑀(𝑀+1)}. (3.38) Then, using the triangle inequality in (1.1) yields max_{−1≤𝑡≤1}𝑑𝑢(𝑡)𝑑𝑡_𝐸𝛼≤𝑀(𝑀+1)𝛼(1−𝛼)max_{−1≤𝑡≤1}𝑓(𝑡)_𝐸𝛼𝑒^{2𝑀(𝑀+1)}. (3.39) Combining last two inequalities, we obtain (3.30) with 𝑀^∗(𝛼)=(𝑀(𝑀+2)/𝛼(1−𝛼))𝑒^{2𝑀(𝑀+1)}. The proof of Theorem 3.3 is complete.

Now, we consider the Rothe difference scheme for approximate solutions of (1.1). 𝑢_𝑘−𝑢_𝑘−1𝜏+𝐴𝑢_𝑘=^𝑘−1_{𝑖=−𝑘+1}𝐵_𝑖𝑢_𝑖𝜏+𝜑_𝑘𝑢,𝑘=1,…,𝑁,_𝑘−𝑢_𝑘−1𝜏−𝐴𝑢_𝑘−1=−^−𝑘_𝑖=𝑘𝐵_𝑖𝑢_𝑖𝜏+𝜑_𝑘𝐵,𝑘=−𝑁+1,…,0,_𝑘=𝐵𝑡_𝑘,𝑡_𝑘𝑢=𝑘𝜏,𝑘=−𝑁,…,𝑁,₀=0. (3.40)

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 max_{𝑘=−𝑁+1,…,𝑁}𝑢_𝑘−𝑢_𝑘−1𝜏_𝐸+max_{𝑘=−𝑁,…,𝑁}𝐴𝑢_𝑘𝐸≤𝑀^∗𝜑_0𝐸+^𝑁_{𝑘=−𝑁+1}𝜑_𝑘−𝜑_𝑘−1𝐸 (3.41) hold, where 𝑀^∗ does not depend on 𝜑_𝑘, 𝑘=−𝑁,…,𝑁.

Proof. By induction we can prove that the initial value problem 𝑢_𝑘−𝑢_𝑘−1𝜏+𝐴𝑢_𝑘=𝜓_𝑘,𝑘=1,…,𝑁,𝑢₀=0 (3.42) for difference equations in an arbitrary Banach space 𝐸 with positive operator 𝐴 has a unique solution 𝑢_𝑘=^𝑘_𝑖=1𝑅^{𝑘−𝑖+1}𝜓_𝑖𝜏,𝑘=1,…,𝑁, (3.43) where 𝑅=(𝐼+𝜏𝐴)⁻¹. By putting 𝜓_𝑘=∑^𝑘−1_{𝑖=−𝑘+1}𝐵_𝑖𝑢_𝑖𝜏+𝜑_𝑘, we obtain 𝐴𝑢_𝑘=𝐴^𝑘_𝑖=1𝑅^{𝑘−𝑖+1}𝜑_𝑖𝜏+𝐴^𝑘_𝑖=1𝑅^{𝑘−𝑖+1𝑖−1}_{𝑗=−𝑖+1}𝐵_𝑗𝑢_𝑗𝜏²,𝑘=1,…,𝑁. (3.44) Using 𝜏^𝑘_{𝑖=±𝑗+1}𝑅^{𝑘−𝑖+1}=𝜏𝑅+𝑅²+⋯+𝑅^𝑘∓𝑗=𝜏𝑅(𝐼−𝑅)⁻¹𝐼−𝑅^𝑘∓𝑗=𝐴⁻¹𝐼−𝑅^𝑘∓𝑗,𝑘=1,…,𝑁, (3.45) we have 𝐴^𝑘_𝑖=1𝑅^{𝑘−𝑖+1𝑖−1}_{𝑗=−𝑖+1}𝐵_𝑗𝑢_𝑗𝜏²=𝐴⁰_{𝑗=−𝑘+1}𝜏^𝑘_{𝑖=−𝑗+1}𝑅^{𝑘−𝑖+1}𝐵_𝑗𝑢_𝑗𝜏+𝐴^𝑘−1_𝑗=1𝜏^𝑘_𝑖=𝑗+1𝑅^{𝑘−𝑖+1}𝐵_𝑗𝑢_𝑗𝜏=𝐴⁰_{𝑗=−𝑘+1}𝐴⁻¹𝐼−𝑅^𝑘+𝑗𝐵_𝑗𝑢_𝑗𝜏+𝐴^𝑘−1_𝑗=1𝐴⁻¹𝐼−𝑅^𝑘−𝑗𝐵_𝑗𝑢_𝑗𝜏=^𝑘−1_{𝑖=−𝑘+1}𝐼−𝑅^{𝑘−|𝑖|}𝐵_𝑖𝑢_𝑖𝜏. (3.46) Furthermore, 𝐴^𝑘_𝑖=1𝑅^{𝑘−𝑖+1}𝜑_𝑖𝜏=^𝑘_𝑖=1(𝐼−𝑅)𝑅^𝑘−𝑖𝜑_𝑖=^𝑘_𝑖=1𝑅^𝑘−𝑖𝜑_𝑖−^𝑘_𝑖=1𝑅^{𝑘−𝑖+1}𝜑_𝑖=^𝑘+1_𝑖=2𝑅^{𝑘−𝑖+1}𝜑_𝑖−1−^𝑘_𝑖=1𝑅^{𝑘−𝑖+1}𝜑_𝑖=^𝑘_𝑖=1𝑅^{𝑘−𝑖+1}𝜑_𝑖−1−𝑅^𝑘𝜑₀+𝜑_𝑘−^𝑘_𝑖=1𝑅^{𝑘−𝑖+1}𝜑_𝑖=𝜑_𝑘−𝑅^𝑘𝜑₀−^𝑘_𝑖=1𝑅^{𝑘−𝑖+1}𝜑_𝑖−𝜑_𝑖−1. (3.47) Putting (3.46)-(3.47) in (3.44), we get 𝐴𝑢_𝑘=𝜑_𝑘−𝑅^𝑘𝜑₀−^𝑘_𝑖=1𝑅^{𝑘−𝑖+1}𝜑_𝑖−𝜑_𝑖−1+^𝑘−1_{𝑖=−𝑘+1𝑖≠0}𝐼−𝑅^{𝑘−|𝑖|}𝐵_𝑖𝑢_𝑖𝜏 (3.48) for 𝑘=1,…,𝑁.
Since 𝑅^𝑘=(𝐼+𝜏𝐴)^−𝑘=1(𝑘−1)!^∞₀𝑡^𝑘−1𝑒^−𝑡𝑒^{−𝜏𝑡𝐴}𝑑𝑡,𝑘=1,…,𝑁, (3.49) applying estimates (1.4) gives 𝑅^𝑘_𝐸→𝐸≤𝑀(𝑘−1)!^∞₀𝑡^𝑘−1𝑒^−𝑡𝑑𝑡=𝑀,𝑘=1,…,𝑁. (3.50) Then, applying the triangle inequality and the estimate (1.2) in (3.48), we obtain 𝐴𝑢_𝑘𝐸≤𝜑_𝑘𝐸+𝑅^𝑘_𝐸→𝐸𝜑_0𝐸+^𝑘_𝑖=1𝑅^{𝑘−𝑖+1}_𝐸→𝐸𝜑_𝑖−𝜑_𝑖−1𝐸+^𝑘−1_{𝑖=−𝑘+1𝑖≠0}1+𝑅^{𝑘−|𝑖|}_𝐸→𝐸𝐵_𝑖𝐴⁻¹_𝐸→𝐸𝐴𝑢_𝑖𝐸𝜏≤^𝑘_𝑖=1𝜑_𝑖−𝜑_𝑖−1+𝜑_0𝐸+𝑀𝜑_0𝐸+𝑀^𝑘_𝑖=1𝜑_𝑖−𝜑_𝑖−1𝐸+𝑀(𝑀+1)^𝑘−1_{𝑖=−𝑘+1𝑖≠0}𝐴𝑢_𝑖𝐸𝜏≤(𝑀+1)^𝑁_{𝑖=−𝑁+1}𝜑_𝑖−𝜑_𝑖−1𝐸+(𝑀+1)𝜑_0𝐸+𝑀(𝑀+1)^𝑘−1_{𝑖=−𝑘+1𝑖≠0}𝐴𝑢_𝑖𝐸𝜏,𝑘=1,…,𝑁. (3.51) So, for 𝑘=1,…,𝑁, 𝐴𝑢_𝑘𝐸≤𝑀𝜑_0𝐸+^𝑁_{𝑖=−𝑁+1}𝜑_𝑖−𝜑_𝑖−1𝐸+𝑀(𝑀+1)^𝑘−1_{𝑖=−𝑘+1𝑖≠0}𝐴𝑢_𝑖𝐸𝜏 (3.52) holds, where 𝑀=max{𝑀(1+𝜏(𝑀+1)),𝑀+1}.
In a similar way, we can prove that the initial value problem 𝑢_𝑘−𝑢_𝑘−1𝜏−𝐴𝑢_𝑘−1=−^−𝑘_𝑖=𝑘𝐵_𝑖𝑢_𝑖𝜏+𝜑_𝑘,𝑘=−𝑁+1,…,0,𝑢₀=0 (3.53) for difference equations in an arbitrary Banach space 𝐸 with positive operator 𝐴 has a unique solution and inequalities 𝐴𝑢_𝑘𝐸≤𝑀𝜑_0𝐸+^𝑁_{𝑖=−𝑁+1}𝜑_𝑖−𝜑_𝑖−1𝐸+𝑀(𝑀+1)^−𝑘−1_{𝑖=𝑘+1𝑖≠0}𝐴𝑢_𝑖𝐸𝜏 (3.54) hold for 𝑘=−𝑁…,0.
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 max_{𝑘=−𝑁+1,…,𝑁}𝑢_𝑘−𝑢_𝑘−1𝜏_𝐸+max_{𝑘=−𝑁,…,𝑁}𝐴𝑢_𝑘𝐸≤𝑀^∗max_{𝑘=−𝑁,…,𝑁}𝜑_𝑘𝐸 (3.55) in the arbitrary Banach space 𝐸 for the general strong positive operator 𝐴, 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 max_{𝑘=−𝑁+1,…,𝑁}𝑢_𝑘−𝑢_𝑘−1𝜏_𝐸+max_{𝑘=−𝑁,…,𝑁}𝐴𝑢_𝑘𝐸≤𝑀^∗1minln𝜏,1+ln‖𝐴‖_𝐸→𝐸max_{𝑘=−𝑁,…,𝑁}𝜑_𝑘𝐸 (3.56) hold, where 𝑀^∗ does not depend on 𝜑_𝑘, 𝑘=−𝑁,…,𝑁.

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 max_{𝑘=−𝑁+1,…,𝑁}𝑢_𝑘−𝑢_𝑘−1𝜏_𝐸^′_𝛼+max_{𝑘=−𝑁,…,𝑁}𝐴𝑢_{𝑘𝐸′𝛼}≤𝑀^∗(𝛼)max_{𝑘=−𝑁,…,𝑁}𝜑_{𝑘𝐸′𝛼} (3.57) hold, where 𝑀^∗(𝛼) does not depend on 𝜑_𝑘, 𝑘=−𝑁,…,𝑁. Here the fractional spaces 𝐸^′_𝛼=𝐸^′_𝛼(𝐸,𝐴)(0<𝛼<1), consisting of all 𝑣∈𝐸 for which the following norms are finite: ‖𝑣‖_𝐸′𝛼=sup_0<𝑧𝑧^𝛼𝐴(𝑧+𝐴)⁻¹𝑣_𝐸. (3.58)

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.