Generalizations of Wendroff Integral Inequalities and Their Discrete Analogues

Abstract

Generalizations of Wendroff type integral inequalities with four dependent limits and their discrete analogues are obtained. In applications, these results are used to establish the stability estimates for the solution of the Goursat problem.

1. Introduction

Integral inequalities play a significant role in the theory of ordinary and partial differential equations. They are useful to investigate some properties of the solutions of equations, such as existence, uniqueness, and stability, see for instance [1–6].

Most scientific and technical problems can be solved by using mathematical modelling and new numerical methods. This is based on the mathematical description of real processes and the subsequent solving of the appropriate mathematical problems on the computer. The mathematical models of many scientific and technical problems lead to already known or new problems of partial differential equations. In most of the cases it is difficult to find the exact solutions of the problems for partial differential equations. For this reason discrete methods play a significant role, especially due to the increasing role of mathematical methods of solving problems in various areas of science and engineering. A well-known and widely applied method of approximate solutions for problems of 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 partial differential equations. The investigation of stability and convergence of these difference schemes is based on the discrete analogues of integral inequalities, see for instance [1, 7–9].

Due to various motivations, several generalizations and applications of Wendroff-type integral inequality have been obtained and used extensively, see for instance [10–12]. In [12], the following generalizations of Wendroff-type integral inequality in two independent variables are obtained.

Theorem 1.1. Assume that 𝑢(𝑥,𝑦)≥0 and 𝑎(𝑥,𝑦)≥0 are continuous functions on 0≤𝑥≤𝐿, 0≤𝑦≤𝑀 and the inequalities 𝑢𝑥,𝑦≤𝐶+^𝑥₀^𝑦₀𝑎(𝑠,𝑡)𝑢(𝑠,𝑡)𝑑𝑡𝑑𝑠,0≤𝑥≤𝐿,0≤𝑦≤𝑀 (1.1) hold, where 𝐶=const≥0. Then, for 𝑢(𝑥,𝑦) the inequalities: 𝑢𝑥,𝑦≤𝐶exp^𝑥₀^𝑦₀𝑎(𝑠,𝑡)𝑑𝑡𝑑𝑠,0≤𝑥≤𝐿,0≤𝑦≤𝑀 (1.2) are satisfied.

Theorem 1.2. Assume that 𝑢(𝑥,𝑦)≥0 is a continuous function on 0≤𝑥≤𝐿, 0≤𝑦≤𝑀 and the inequalities: 𝑢𝑥,𝑦≤𝑓𝑥,𝑦+^𝑥₀𝑎(𝑠)𝑢𝑠,𝑦𝑑𝑠+^𝑦₀𝑏(𝑡)𝑢(𝑥,𝑡)𝑑𝑡,0≤𝑥≤𝐿,0≤𝑦≤𝑀 (1.3) hold, where 𝑓(𝑥,𝑦)>0 is a continuous function on 0≤𝑥≤𝐿, 0≤𝑦≤𝑀 and increasing with respect to each variable, 𝑎(𝑥)≥0 and 𝑏(𝑦)≥0 are integrable functions on 0≤𝑥≤𝐿 and 0≤𝑦≤𝑀, respectively. Then, for 𝑢(𝑥,𝑦) the inequalities: 𝑢𝑥,𝑦≤𝑓𝑥,𝑦exp^𝑥₀𝑎(𝑠)𝑑𝑠+^𝑦₀𝑏(𝑡)𝑑𝑡+^𝑥₀^𝑦₀𝑎(𝑠)𝑏(𝑡)𝑑𝑡𝑑𝑠,0≤𝑥≤𝐿,0≤𝑦≤𝑀 (1.4) are satisfied.

In this paper, generalizations of Wendroff-type integral inequalities in two independent variables with four dependent limits and their discrete analogues are obtained. In applications, these results are used to obtain the stability estimates of solutions for the Goursat problem.

2. Wendroff-type Integral Inequalities with Four Dependent Limits and Their Discrete Analogues

First of all, let us give the discrete analogue of the Gronwall-type integral inequality with two dependent limits. We will need this result in the remaining part of the paper.

Theorem 2.1. Assume that 𝑣_𝑖≥0, 𝑎_𝑖≥0, 𝛿_𝑖≥0, 𝑖=−𝑁,…,𝑁 are the sequences of real numbers and the inequalities: 𝑣_𝑖≤𝛿_𝑖+ℎ^|𝑖|−1_{|𝑖|𝑗=−+1𝑗≠0}𝑎_𝑗𝑣_𝑗,𝑖=−𝑁,…,𝑁 (2.1) hold. Then, for 𝑣_𝑖 the inequalities: 𝑣_𝑖≤𝛿_𝑖+expℎ^|𝑖|−1_{|𝑖|𝑗=−+1𝑗≠0}𝑎_𝑗ℎ^|𝑖|−1_{|𝑖|𝑗=−+1𝑗≠0}𝑎_𝑗𝛿_𝑗exp−ℎ^|𝑗|_{𝑛=−|𝑗|𝑛≠0}𝑎_𝑛,𝑖=−𝑁,…,𝑁 (2.2) are satisfied.

Proof. The proof of (2.2) for 𝑖=−1,0,1 follows directly from (2.1). Let us prove (2.2) for 𝑖=−𝑁,…,−2,2,…,𝑁. We denote 𝑦_𝑖=ℎ^|𝑖|−1_{|𝑖|𝑗=−+1𝑗≠0}𝑎_𝑗𝑣_𝑗,𝑖=−𝑁,…,𝑁. (2.3) Then, (2.1) gets the form 𝑣_𝑖≤𝛿_𝑖+𝑦_𝑖,𝑖=−𝑁,…,𝑁. (2.4) Moreover, we have 𝑦_−𝑖=𝑦_𝑖,𝑖=−𝑁,…,𝑁. (2.5) Then, using (2.3)–(2.5) for 𝑖=1,…,𝑁−1, we obtain 𝑦_𝑖+1−𝑦_𝑖=ℎ𝑎_𝑖𝑣_𝑖+𝑎_−𝑖𝑣_−𝑖≤ℎ𝑎_𝑖𝑦_𝑖+𝛿_𝑖+ℎ𝑎_−𝑖𝑦_−𝑖+𝛿_−𝑖=ℎ𝑎_𝑖+𝑎_−𝑖𝑦_𝑖+ℎ𝑎_𝑖𝛿_𝑖+𝑎_−𝑖𝛿_−𝑖. (2.6) So, 𝑦_𝑖+1≤1+ℎ𝑎_𝑖+𝑎_−𝑖𝑦_𝑖+ℎ𝑎_𝑖𝛿_𝑖+𝑎_−𝑖𝛿_−𝑖,𝑖=1,…,𝑁−1. (2.7) Then by induction, we can prove that 𝑦_𝑖≤^𝑖−1_𝑗=11+ℎ𝑎_𝑗+𝑎_−𝑗𝑦₁+^𝑖−2_𝑗=1ℎ𝑎_𝑗𝛿_𝑗+𝑎_−𝑗𝛿_−𝑗^𝑖−1_𝑛=𝑗+11+ℎ𝑎_𝑛+𝑎_−𝑛+ℎ𝑎_𝑖−1𝛿_𝑖−1+𝑎_−𝑖+1𝛿_−𝑖+1, (2.8) for 𝑖=2,…,𝑁. Since 𝑦₁=0, using (2.4) and the inequality 1+𝑥<exp[𝑥], 𝑥>0, we obtain 𝑣_𝑖≤𝛿_𝑖+ℎ^𝑖−2_𝑗=1𝑎_𝑗𝛿_𝑗+𝑎_−𝑗𝛿_−𝑗expℎ^𝑖−1_𝑛=𝑗+1𝑎_𝑛+𝑎_−𝑛+ℎ𝑎_𝑖−1𝛿_𝑖−1+𝑎_−𝑖+1𝛿_−𝑖+1=𝛿_𝑖+expℎ^𝑖−1_𝑛=1𝑎_𝑛+𝑎_−𝑛ℎ^𝑖−2_𝑗=1𝑎_𝑗𝛿_𝑗+𝑎_−𝑗𝛿_−𝑗exp−ℎ^𝑗_𝑛=1𝑎_𝑛+𝑎_−𝑛+ℎ𝑎_𝑖−1𝛿_𝑖−1+𝑎_−𝑖+1𝛿_−𝑖+1=𝛿_𝑖+expℎ^𝑖−1_𝑗=1𝑎_𝑗+𝑎_−𝑗ℎ^𝑖−1_𝑗=1𝑎_𝑗𝛿_𝑗+𝑎_−𝑗𝛿_−𝑗exp−ℎ^𝑗_𝑛=1𝑎_𝑛+𝑎_−𝑛=𝛿_𝑖+expℎ^𝑖−1_{𝑗=−𝑖+1𝑗≠0}𝑎_𝑗ℎ^𝑖−1_𝑗=1𝑎_𝑗𝛿_𝑗+𝑎_−𝑗𝛿_−𝑗exp−ℎ^𝑗_{𝑛=−𝑗𝑛≠0}𝑎_𝑛=𝛿_𝑖+expℎ^𝑖−1_{𝑗=−𝑖+1𝑗≠0}𝑎_𝑗ℎ^𝑖−1_{𝑗=−𝑖+1𝑗≠0}𝑎_𝑗𝛿_𝑗exp−ℎ^|𝑗|_{𝑗𝑛=−𝑛≠0}𝑎_𝑛. (2.9) So, we proved (2.2) for 𝑖=2,…,𝑁.
Let us prove (2.2) for 𝑖=−𝑁,…,−2. Using (2.3)–(2.5) for 𝑖=−𝑁+1,…,−1, we have 𝑦_𝑖−1−𝑦_𝑖=ℎ𝑎_𝑖𝑣_𝑖+𝑎_−𝑖𝑣_−𝑖≤ℎ𝑎_𝑖𝑦_𝑖+𝛿_𝑖+ℎ𝑎_−𝑖𝑦_−𝑖+𝛿_−𝑖=ℎ𝑎_𝑖+𝑎_−𝑖𝑦_𝑖+ℎ𝑎_𝑖𝛿_𝑖+𝑎_−𝑖𝛿_−𝑖. (2.10) So, 𝑦_𝑖−1≤1+ℎ𝑎_𝑖+𝑎_−𝑖𝑦_𝑖+ℎ𝑎_𝑖𝛿_𝑖+𝑎_−𝑖𝛿_−𝑖,𝑖=−𝑁+1,…,−1. (2.11) Then by induction, we can prove that 𝑦_𝑖≤^−𝑖−1_𝑛=11+ℎ𝑎_𝑛+𝑎_−𝑛𝑦₋₁+^−𝑖−2_𝑗=1ℎ𝑎_𝑗𝛿_𝑗+𝑎_−𝑗𝛿_−𝑗^−𝑖−1_𝑛=𝑗+11+ℎ𝑎_𝑛+𝑎_−𝑛+ℎ𝑎_−𝑖−1𝛿_−𝑖−1+𝑎_𝑖+1𝛿_𝑖+1, (2.12) for 𝑖=−𝑁,…,−2. Since 𝑦₋₁=0, using (2.4) and the inequality 1+𝑥<exp[𝑥], 𝑥>0, we obtain 𝑣_𝑖≤𝛿_𝑖+ℎ^−𝑖−2_𝑗=1𝑎_𝑗𝛿_𝑗+𝑎_−𝑗𝛿_−𝑗expℎ^−𝑖−1_𝑛=𝑗+1𝑎_𝑛+𝑎_−𝑛+ℎ𝑎_−𝑖−1𝛿_−𝑖−1+𝑎_𝑖+1𝛿_𝑖+1=𝛿_𝑖+expℎ^−𝑖−1_𝑛=1𝑎_𝑛+𝑎_−𝑛ℎ^−𝑖−2_𝑗=1𝑎_𝑗𝛿_𝑗+𝑎_−𝑗𝛿_−𝑗exp−ℎ^𝑗_𝑛=1𝑎_𝑛+𝑎_−𝑛+ℎ𝑎_−𝑖−1𝛿_−𝑖−1+𝑎_𝑖+1𝛿_𝑖+1=𝛿_𝑖+expℎ^−𝑖−1_𝑗=1𝑎_𝑗+𝑎_−𝑗ℎ^−𝑖−1_𝑗=1𝑎_𝑗𝛿_𝑗+𝑎_−𝑗𝛿_−𝑗exp−ℎ^𝑗_𝑛=1𝑎_𝑛+𝑎_−𝑛=𝛿_𝑖+expℎ^−𝑖−1_{𝑗=𝑖+1𝑗≠0}𝑎_𝑗ℎ^−𝑖−1_𝑗=1𝑎_𝑗𝛿_𝑗+𝑎_−𝑗𝛿_−𝑗exp−ℎ^𝑗_{𝑛=−𝑗𝑛≠0}𝑎_𝑛=𝛿_𝑖+expℎ^−𝑖−1_{𝑗=𝑖+1𝑗≠0}𝑎_𝑗ℎ^−𝑖−1_{𝑗=𝑖+1𝑗≠0}𝑎_𝑗𝛿_𝑗exp−ℎ^|𝑗|_{𝑗𝑛=−𝑛≠0}𝑎_𝑛. (2.13) So, we proved (2.2) for 𝑖=−𝑁,…,−2.

By putting 𝑁ℎ=1 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 2.2. Assume that 𝑣(𝑡)≥0, 𝛿(𝑡)≥0 are continuous functions on [−1,1], 𝑎(𝑡)≥0 is an integrable function on [−1,1], and the inequalities 𝑣(𝑡)≤𝛿(𝑡)+sgn(𝑡)^𝑡_−𝑡𝑎(𝑠)𝑣(𝑠)𝑑𝑠,−1≤𝑡≤1 (2.14) hold. Then for 𝑣(𝑡), the inequalities: 𝑣(𝑡)≤𝛿(𝑡)+expsgn(𝑡)^𝑡_−𝑡𝑎(𝑠)𝑑𝑠sgn(𝑡)^𝑡_−𝑡𝑎(𝑠)𝛿(𝑠)exp−sgn(𝑠)^𝑠_−𝑠𝑎(𝜏)𝑑𝜏𝑑𝑠,−1≤𝑡≤1 (2.15) are satisfied.

Now, we consider the generalizations of Wendroff inequality for integrals in two independent variables with four dependent limits and their discrete analogues.

Theorem 2.3. Assume that 𝑣^𝑘_𝑛≥0, 𝑎^𝑘_𝑛≥0, 𝑛=−𝑁,…,𝑁, 𝑘=−𝐾,…,𝐾 are sequences of real numbers, and the inequalities: 𝑣^𝑘_𝑛≤𝐶+ℎ₁ℎ₂^|𝑛|−1_{|𝑛|𝑠=−+1𝑠≠0}^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎^𝜏_𝑠𝑣^𝜏_𝑠,𝑛=−𝑁,…,𝑁,𝑘=−𝐾,…,𝐾 (2.16) hold, where 𝐶=const>0, ℎ₁=const>0, ℎ₂=const>0. Then for 𝑣^𝑘_𝑛, the following inequalities: 𝑣^𝑘_𝑛≤𝐶expℎ₁ℎ₂^|𝑛|−1_{|𝑛|𝑠=−+1𝑠≠0}^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎^𝜏_𝑠,𝑛=−𝑁,…,𝑁,𝑘=−𝐾,…,𝐾. (2.17) are satisfied.

Proof. The proof of (2.17) for 𝑛=−1,0,1, 𝑘=−𝐾,…,𝐾 and 𝑘=−1,0,1, 𝑛=−𝑁,…,𝑁 follows directly from (2.16). Let us prove (2.17) for 𝑛=−𝑁,…,−2,2,…,𝑁, 𝑘=−𝐾,…,−2,2,…,𝐾. We denote 𝜔^𝑘_𝑛=𝐶+ℎ₁ℎ₂^|𝑛|−1_{|𝑛|𝑠=−+1𝑠≠0}^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎^𝜏_𝑠𝑣^𝜏_𝑠,𝑛=−𝑁,…,𝑁,𝑘=−𝐾,…,𝐾. (2.18) Then, (2.16) gets the form: 𝑣^𝑘_𝑛≤𝜔^𝑘_𝑛,𝑛=−𝑁,…,𝑁,𝑘=−𝐾,…,𝐾. (2.19) Furthermore, we have 𝜔^𝑘_𝑛=𝜔^𝑘_−𝑛=𝜔_𝑛^−𝑘=𝜔^−𝑘_−𝑛,𝑛=−𝑁,…,𝑁,𝑘=−𝐾,…,𝐾. (2.20) From (2.18) for 𝑛=−𝑁,…,𝑁, 𝑘=1,…,𝐾−1, we have 𝜔_𝑛^𝑘+1−𝜔^𝑘_𝑛=ℎ₁ℎ₂^|𝑛|−1_{|𝑛|𝑠=−+1𝑠≠0}𝑎^𝑘_𝑠𝑣^𝑘_𝑠+𝑎_𝑠^−𝑘𝑣_𝑠^−𝑘≥0. (2.21) So, 𝜔^𝑘_𝑛≤𝜔_𝑛^𝑘+1,𝑛=−𝑁,…,𝑁,𝑘=1,…,𝐾−1. (2.22) Then using (2.18)–(2.22) for 𝑛=1,…,𝑁−1, 𝑘=−𝐾,…,−2,2,…,𝐾, we obtain 𝜔^𝑘_𝑛+1−𝜔^𝑘_𝑛=ℎ₁ℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎^𝜏_𝑛𝑣^𝜏_𝑛+𝑎^𝜏_−𝑛𝑣^𝜏_−𝑛≤ℎ₁ℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎^𝜏_𝑛𝜔^𝜏_𝑛+𝑎^𝜏_−𝑛𝜔^𝜏_−𝑛=ℎ₁ℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝜔^𝜏_𝑛𝑎^𝜏_𝑛+𝑎^𝜏_−𝑛=ℎ₁ℎ₂^|𝑘|−1_𝜏=1𝜔^𝜏_𝑛𝑎^𝜏_𝑛+𝑎^𝜏_−𝑛+ℎ₁ℎ₂^|𝑘|−1_𝜏=1𝜔_𝑛^−𝜏𝑎_𝑛^−𝜏+𝑎^−𝜏_−𝑛=ℎ₁ℎ₂^|𝑘|−1_𝜏=1𝜔^𝜏_𝑛𝑎^𝜏_𝑛+𝑎^𝜏_−𝑛+𝑎_𝑛^−𝜏+𝑎^−𝜏_−𝑛≤ℎ₁ℎ₂𝜔_𝑛^{|𝑘||𝑘|−1}_𝜏=1𝑎^𝜏_𝑛+𝑎^𝜏_−𝑛+𝑎_𝑛^−𝜏+𝑎^−𝜏_−𝑛=ℎ₁ℎ₂𝜔^𝑘_𝑛^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎^𝜏_𝑛+𝑎^𝜏_−𝑛. (2.23) So, 𝜔^𝑘_𝑛+1≤𝜔^𝑘_𝑛1+ℎ₁ℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎^𝜏_𝑛+𝑎^𝜏_−𝑛,𝑛=1,…,𝑁−1,𝑘=−𝐾,…,−2,2,…,𝐾. (2.24) Then by induction, we can prove that 𝜔^𝑘_𝑛≤𝜔^𝑘₁^𝑛−1_𝑠=11+ℎ₁ℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎^𝜏_𝑠+𝑎^𝜏_−𝑠,𝑛=2,…,𝑁,𝑘=−𝐾,…,−2,2,…,𝐾. (2.25) Since 𝜔^𝑘₁=𝐶, 𝑘=−𝐾,…,𝐾, using (2.19) and the inequality 1+𝑥<exp[𝑥], 𝑥>0, we obtain 𝑣^𝑘_𝑛≤𝜔^𝑘_𝑛≤𝐶expℎ₁ℎ₂^𝑛−1_𝑠=1^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎^𝜏_𝑠+𝑎^𝜏_−𝑠=𝐶expℎ₁ℎ₂^𝑛−1_{𝑠=−𝑛+1𝑠≠0}^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎^𝜏_𝑠. (2.26) So, we proved (2.17) for 𝑛=2,…,𝑁, 𝑘=−𝐾,…,−2,2,…,𝐾.
Using (2.18)–(2.22) for 𝑛=−𝑁+1,…,−1, 𝑘=−𝐾,…,−2,2,…,𝐾, we obtain 𝜔^𝑘_𝑛−1−𝜔^𝑘_𝑛=ℎ₁ℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎^𝜏_𝑛𝑣^𝜏_𝑛+𝑎^𝜏_−𝑛𝑣^𝜏_−𝑛≤ℎ₁ℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎^𝜏_𝑛𝜔^𝜏_𝑛+𝑎^𝜏_−𝑛𝜔^𝜏_−𝑛=ℎ₁ℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝜔^𝜏_𝑛𝑎^𝜏_𝑛+𝑎^𝜏_−𝑛=ℎ₁ℎ₂^|𝑘|−1_𝜏=1𝜔^𝜏_𝑛𝑎^𝜏_𝑛+𝑎^𝜏_−𝑛+ℎ₁ℎ₂^|𝑘|−1_𝜏=1𝜔_𝑛^−𝜏𝑎_𝑛^−𝜏+𝑎^−𝜏_−𝑛=ℎ₁ℎ₂^|𝑘|−1_𝜏=1𝜔^𝜏_𝑛𝑎^𝜏_𝑛+𝑎^𝜏_−𝑛+𝑎_𝑛^−𝜏+𝑎^−𝜏_−𝑛≤ℎ₁ℎ₂𝜔_𝑛^{|𝑘||𝑘|−1}_𝜏=1𝑎^𝜏_𝑛+𝑎^𝜏_−𝑛+𝑎_𝑛^−𝜏+𝑎^−𝜏_−𝑛=ℎ₁ℎ₂𝜔^𝑘_𝑛^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎^𝜏_𝑛+𝑎^𝜏_−𝑛. (2.27) So, 𝜔^𝑘_𝑛−1≤𝜔^𝑘_𝑛1+ℎ₁ℎ∑₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎^𝜏_𝑛+𝑎^𝜏_−𝑛,𝑛=−𝑁+1,…,−1,𝑘=−𝐾,…,−2,2,…,𝐾. (2.28) Then by induction, we can prove that 𝜔^𝑘_𝑛≤𝜔^𝑘₋₁^−𝑛−1_𝑠=11+ℎ₁ℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎^𝜏_𝑠+𝑎^𝜏_−𝑠,𝑛=−𝑁,…,−2,𝑘=−𝐾,…,−2,2,…,𝐾. (2.29) Since 𝜔^𝑘₋₁=𝐶, 𝑘=−𝐾,…,𝐾 using (2.19) and the inequality 1+𝑥<exp[𝑥], 𝑥>0, we obtain 𝑣^𝑘_𝑛≤𝜔^𝑘_𝑛≤𝐶expℎ₁ℎ₂^−𝑛−1_𝑠=1^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎^𝜏_𝑠+𝑎^𝜏_−𝑠=𝐶expℎ₁ℎ₂^−𝑛−1_{𝑠=𝑛+1𝑠≠0}^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎^𝜏_𝑠. (2.30) So, we proved (2.17) for 𝑛=−𝑁,…,−2, 𝑘=−𝐾,…,−2,2,…,𝐾.

Theorem 2.4. Assume that 𝑣^𝑘_𝑛≥0, 𝛿^𝑘_𝑛>0, 𝑎_𝑛≥0, 𝑏_𝑘≥0, 𝑛=−𝑁,…,𝑁, 𝑘=−𝐾,…,𝐾 are sequences of real numbers, and the inequalities: 𝑣^𝑘_𝑛≤𝛿^𝑘_𝑛+ℎ₁^|𝑛|−1_{|𝑛|𝑠=−+1𝑠≠0}𝑎_𝑠𝑣^𝑘_𝑠+ℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑏_𝜏𝑣^𝜏_𝑛𝛿,𝑛=−𝑁,…,𝑁,𝑘=−𝐾,…,𝐾,_𝑘𝑛≤𝛿^𝑘_𝑛+1𝛿,𝑛=−𝑁,…,𝑁−1,𝑘=−𝐾,…,𝐾,_𝑘𝑛≤𝛿_𝑛^𝑘+1,𝑛=−𝑁,…,𝑁,𝑘=−𝐾,…,𝐾−1 (2.31)(2.32) hold, where ℎ₁=const>0, ℎ₂=const>0. Then for 𝑣^𝑘_𝑛, 𝑛=−𝑁,…,𝑁, 𝑘=−𝐾,…,𝐾, the inequalities: 𝑣^𝑘_𝑛≤𝛿^𝑘_𝑛expℎ₁^|𝑛|−1_{|𝑛|𝑠=−+1𝑠≠0}𝑎_𝑠+ℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑏_𝜏+ℎ₁ℎ₂^|𝑛|−1_{|𝑛|𝑠=−+1𝑠≠0}^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎_𝑠𝑏_𝜏 (2.33) are satisfied.

Proof. We denote 𝑢^𝑘_𝑛=𝑣^𝑘_𝑛/𝛿^𝑘_𝑛, 𝑛=−𝑁,…,𝑁, 𝑘=−𝐾,…,𝐾. Then, (2.31) takes the form 𝑢^𝑘_𝑛≤1+ℎ₁^|𝑛|−1_{|𝑛|𝑠=−+1𝑠≠0}𝑎_𝑠𝑢^𝑘_𝑠+ℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑏_𝜏𝑢^𝜏_𝑛,𝑛=−𝑁,…,𝑁,𝑘=−𝐾,…,𝐾. (2.34) Next, by denoting 𝑇^𝑘_𝑛=1+ℎ₁^|𝑛|−1_{|𝑛|𝑠=−+1𝑠≠0}𝑎_𝑠𝑢^𝑘_𝑠,𝑛=−𝑁,…,𝑁,𝑘=−𝐾,…,𝐾, (2.35) we have 𝑢^𝑘_𝑛≤𝑇^𝑘_𝑛+ℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑏_𝜏𝑢^𝜏_𝑛,𝑛=−𝑁,…,𝑁,𝑘=−𝐾,…,𝐾. (2.36) By using Theorem 2.1, we obtain 𝑢^𝑘_𝑛≤𝑇^𝑘_𝑛+expℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑏_𝜏ℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑏_𝜏𝑇^𝜏_𝑛exp−ℎ₂^|𝜏|_{|𝜏|𝑗=−𝑗≠0}𝑏_𝑗. (2.37) Inserting (2.35) yields 𝑢^𝑘_𝑛≤1+ℎ₁^|𝑛|−1_{|𝑛|𝑠=−+1𝑠≠0}𝑎_𝑠𝑢^𝑘_𝑠+expℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑏_𝜏ℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑏_𝜏exp−ℎ₂^|𝜏|_{|𝜏|𝑗=−𝑗≠0}𝑏_𝑗+expℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑏_𝜏ℎ₁ℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}^|𝑛|−1_{|𝑛|𝑠=−+1𝑠≠0}𝑎_𝑠𝑏_𝜏𝑢^𝜏_𝑠exp−ℎ₂^|𝜏|_{|𝜏|𝑗=−𝑗≠0}𝑏_𝑗≤expℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑏_𝜏+ℎ₁^|𝑛|−1_{|𝑛|𝑠=−+1𝑠≠0}𝑎_𝑠𝑢^𝑘_𝑠+expℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑏_𝜏ℎ₁ℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}^|𝑛|−1_{|𝑛|𝑠=−+1𝑠≠0}𝑎_𝑠𝑏_𝜏𝑢^𝜏_𝑠exp−ℎ₂^|𝜏|_{|𝜏|𝑗=−𝑗≠0}𝑏_𝑗. (2.38) Now, by denoting 𝑤^𝑘_𝑛=𝑢^𝑘_𝑛exp−ℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑏_𝜏,𝑛=−𝑁,…,𝑁,𝑘=−𝐾,…,𝐾, (2.39) we have 𝑤^𝑘_𝑛≤1+ℎ₁^|𝑛|−1_{|𝑛|𝑠=−+1𝑠≠0}𝑎_𝑠𝑤^𝑘_𝑠+ℎ₁ℎ₂^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}^|𝑛|−1_{|𝑛|𝑠=−+1𝑠≠0}𝑎_𝑠𝑏_𝜏𝑤^𝜏_𝑠. (2.40) Let us denote the right-hand side of (2.40) by 𝑅^𝑘_𝑛. Then, (2.40) gets the form 𝑤^𝑘_𝑛≤𝑅^𝑘_𝑛,𝑛=−𝑁,…,𝑁,𝑘=−𝐾,…,𝐾. (2.41) Then by using induction, we can prove that 𝑅^𝑘_𝑛exp−ℎ₁^|𝑛|−1_{|𝑛|𝑠=−+1𝑠≠0}𝑎_𝑠≤1+ℎ₁ℎ₂^|𝑛|−1_{|𝑛|𝑠=−+1𝑠≠0}^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎_𝑠𝑏_𝜏𝑅^𝜏_𝑠exp−ℎ₁^|𝑠|−1_{|𝑠|𝑗=−+1𝑗≠0}𝑎_𝑗. (2.42) By using Theorem 2.3, we obtain 𝑅^𝑘_𝑛exp−ℎ₁^|𝑛|−1_{|𝑛|𝑠=−+1𝑠≠0}𝑎_𝑠≤expℎ₁ℎ₂^|𝑛|−1_{|𝑛|𝑠=−+1𝑠≠0}^|𝑘|−1_{|𝑘|𝜏=−+1𝜏≠0}𝑎_𝑠𝑏_𝜏,𝑛=−𝑁,…,𝑁,𝑘=−𝐾,…,𝐾. (2.43) Finally, by combining (2.39), (2.41), and (2.43), we finish the proof of (2.33).

By putting 𝑁ℎ₁=𝐿, 𝐾ℎ₂=𝑀 and passing to limit as ℎ₁→0, ℎ₂→0 in Theorems 2.3 and 2.4, we obtain the following two theorems about the generalizations of Wendroff integral inequality with four dependent limits.

Theorem 2.5. Assume that 𝑣(𝑥,𝑦)≥0 and 𝑏(𝑥,𝑦)≥0 are continuous functions on −𝐿≤𝑥≤𝐿, −𝑀≤𝑦≤𝑀 and the inequalities 𝑣𝑥,𝑦≤𝐶+sgn𝑥𝑦^𝑥_−𝑥^𝑦_−𝑦𝑣(𝑠,𝑡)𝑏(𝑠,𝑡)𝑑𝑡𝑑𝑠,−𝐿≤𝑥≤𝐿,−𝑀≤𝑦≤𝑀 (2.44) hold, where 𝐶=const≥0. Then for 𝑣(𝑥,𝑦) the inequalities 𝑣𝑥,𝑦≤𝐶expsgn𝑥𝑦^𝑥_−𝑥^𝑦_−𝑦𝑏(𝑠,𝑡)𝑑𝑡𝑑𝑠,−𝐿≤𝑥≤𝐿,−𝑀≤𝑦≤𝑀 (2.45) are satisfied.

Theorem 2.6. Assume that 𝑣(𝑥,𝑦)≥0 is a continuous function on −𝐿≤𝑥≤𝐿, −𝑀≤𝑦≤𝑀, and the inequalities: 𝑣𝑥,𝑦≤𝑓𝑥,𝑦+sgn(𝑥)^𝑥_−𝑥𝑣𝑎(𝑠)𝑦𝑠,𝑦𝑑𝑠+sgn^𝑦_−𝑦𝑣(𝑥,𝑡)𝑏(𝑡)𝑑𝑡,−𝐿≤𝑥≤𝐿,−𝑀≤𝑦≤𝑀 (2.46) hold, where 𝑓(𝑥,𝑦)>0 is a continuous function on −𝐿≤𝑥≤𝐿, −𝑀≤𝑦≤𝑀 and increasing with respect to each variable, 𝑎(𝑥)≥0 and 𝑏(𝑦)≥0 are integrable functions on −𝐿≤𝑥≤𝐿 and −𝑀≤𝑦≤𝑀, respectively. Then for 𝑣(𝑥,𝑦), the inequalities 𝑣𝑥,𝑦≤𝑓𝑥,𝑦expsgn(𝑥)∫^𝑥_−𝑥𝑎(𝑠)𝑑𝑠+sgn𝑦∫^𝑦_−𝑦𝑏(𝑡)𝑑𝑡+sgn𝑥𝑦∫^𝑥_−𝑥∫^𝑦_−𝑦𝑎(𝑠)𝑏(𝑡)𝑑𝑠𝑑𝑡 (2.47) are satisfied.

Finally, we formulate (without proofs) the generalizations of Wendroff-type inequalities for the integrals in three independent variables with six dependent limits.

Theorem 2.7. Assume that 𝑣(𝑥,𝑦,𝑧)≥0, 𝑏(𝑥,𝑦,𝑧)≥0(−𝑙₁≤𝑥≤𝑙₁,−𝑙₂≤𝑦≤𝑙₂,−𝑙₃≤𝑧≤𝑙₃) are continuous functions, and the inequalities: 𝑣𝑥,𝑦,𝑧≤𝐶+sgn𝑥𝑦𝑧^𝑥_−𝑥^𝑦_−𝑦^𝑧_−𝑧𝑣(𝑠,𝑡,𝜏)𝑏(𝑠,𝑡,𝜏)𝑑𝜏𝑑𝑡𝑑𝑠 (2.48) hold, where 𝐶=const≥0. Then for 𝑣(𝑥,𝑦,𝑧), the inequalities 𝑣𝑥,𝑦,𝑧≤𝐶expsgn𝑥𝑦𝑧^𝑥_−𝑥^𝑦_−𝑦^𝑧_−𝑧𝑏(𝑠,𝑡,𝜏)𝑑𝜏𝑑𝑡𝑑𝑠,−𝑙₁≤𝑥≤𝑙₁,−𝑙₂≤𝑦≤𝑙₂,−𝑙₃≤𝑧≤𝑙₃ (2.49) are satisfied.

Theorem 2.8. Assume that 𝑣(𝑥,𝑦,𝑧)≥0(−𝑙₁≤𝑥≤𝑙₁,−𝑙₂≤𝑦≤𝑙₂,−𝑙₃≤𝑧≤𝑙₃) is a continuous function, and the inequalities: 𝑣𝑥,𝑦,𝑧≤𝛿𝑥,𝑦,𝑧+sgn(𝑥)^𝑥_−𝑥𝑣𝑠,𝑦,𝑧𝑎(𝑠)𝑑𝑠+sgn𝑦^𝑦_−𝑦𝑣()𝑏(𝑡)(𝑧)𝑥,𝑡,𝑧𝑑𝑡+sgn^𝑧_−𝑧𝑣(𝑠,𝑡,𝜏)𝑐(𝜏)𝑑𝜏 (2.50) hold, where 𝑎(𝑥)≥0(−𝑙₁≤𝑥≤𝑙₁), 𝑏(𝑦)≥0(−𝑙₂≤𝑦≤𝑙₂), and 𝑐(𝑧)≥0(−𝑙₃≤𝑧≤𝑙₃) are integrable functions, 𝛿(𝑥,𝑦,𝑧)>0 is continuous and increasing with respect to each variable. Then for 𝑣(𝑥,𝑦,𝑧), the inequalities: 𝑣𝑥,𝑦,𝑧≤𝛿𝑥,𝑦,𝑧exp𝑇𝑥,𝑦,𝑧+³sgn𝑥𝑦𝑧^𝑥_−𝑥^𝑦_−𝑦^𝑧_−𝑧𝑎(𝑠)𝑏(𝑡)𝑐(𝜏)𝑑𝜏𝑑𝑡𝑑𝑠𝑒^{𝑇(𝑥,𝑦,𝑧)}−𝑙₁≤𝑥≤𝑙₁,−𝑙₂≤𝑦≤𝑙₂,−𝑙₃≤𝑧≤𝑙₃ (2.51) are satisfied, where 𝑇𝑥,𝑦,𝑧=sgn(𝑥)^𝑥_−𝑥𝑎(𝑠)𝑑𝑠+sgn𝑦^𝑦_−𝑦𝑏(𝑡)𝑑𝑡+sgn(𝑧)^𝑧_−𝑧𝑐(𝜏)𝑑𝜏+sgn𝑥𝑦^𝑥_−𝑥^𝑦_−𝑦𝑎(𝑠)𝑏(𝑡)𝑑𝑡𝑑𝑠+sgn(𝑥𝑧)^𝑥_−𝑥^𝑧_−𝑧𝑎(𝑠)𝑐(𝜏)𝑑𝜏𝑑𝑠+sgn𝑦𝑧^𝑦_−𝑦^𝑧_−𝑧𝑏(𝑡)𝑐(𝜏)𝑑𝜏𝑑𝑡. (2.52)

3. Applications

In applications, we consider the Goursat problem for hyperbolic equations: 𝑢_𝑥𝑦+𝜕=𝑓𝑥,𝑦𝑎𝑢𝑢+𝜕𝜕𝑦𝑥,𝑦𝑥,𝑦+𝑎−𝑥,𝑦−𝑥,𝑦𝑏𝑢𝑢𝜕𝑥𝑥,𝑦𝑥,𝑦+𝑏𝑥,−𝑦𝑥,−𝑦,−𝑙₁<𝑥<𝑙₁,−𝑙₂<𝑦<𝑙₂,𝑢(𝑥,0)=𝜙(𝑥),−𝑙₁≤𝑥≤𝑙₁,𝑢0,𝑦=𝜓𝑦,−𝑙₂≤𝑦≤𝑙₂. (3.1)

A function 𝑢(𝑥,𝑦) is called a solution of the Goursat problem (3.1) if the following conditions are satisfied:𝑢(𝑥,𝑦) is twice continuously differentiable on the region [−𝑙₁,𝑙₁]×[−𝑙₂,𝑙₂], and the derivatives at the endpoints are understood as the appropriate unilateral derivatives;

Theorem 3.1. Assume that the functions 𝜙(𝑥) and 𝜓(𝑦) are continuously differentiable and 𝜙(0)=𝜓(0). Let 𝑎(𝑥,𝑦), 𝑏(𝑥,𝑦), and 𝑓(𝑥,𝑦) be continuously differentiable functions. Then, there is a unique solution of the problem (3.1) and the stability inequalities: 𝑢𝑥,𝑦≤𝑙₁𝑙₂3𝑓+2𝜙+𝜓+2𝑎𝑙₁𝜙+2𝑏𝑙₂𝜓𝑒^{2𝑎|𝑥|+2𝑏|𝑦|+4𝑎𝑏|𝑥𝑦|},𝑢_𝑥𝑥,𝑦,𝑢_𝑦𝑥,𝑦≤𝑀₁,𝑢_𝑥𝑦𝑥,𝑦≤𝑀₂ (3.2) hold, where 𝑀₁ and 𝑀₂ do not depend on 𝑥 and 𝑦 and 𝑓=max_{|𝑥|≤𝑙1𝑦≤𝑙2}𝑓𝑥,𝑦,𝑎=max_{|𝑥|≤𝑙1𝑦≤𝑙2}𝑎𝑥,𝑦,𝑏=max_{|𝑥|≤𝑙1𝑦≤𝑙2}𝑏𝑥,𝑦,𝜙=max_{|𝑥|≤𝑙1}𝜙(𝑥),𝜓=max_𝑦≤𝑙2𝜓𝑦. (3.3)

The proof of this theorem is based on the formula: 𝑢𝑥,𝑦=𝜙(𝑥)+𝜓𝑦−𝜙(0)−^𝑥_−𝑥𝑎(𝑠,0)𝜙(𝑠)𝑑𝑠−^𝑦_−𝑦𝑏(0,𝑡)𝜓(𝑡)𝑑𝑡+^𝑥₀^𝑦₀𝑓()+𝑠,𝑡𝑑𝑡𝑑𝑠_𝑥−𝑥𝑎𝑠,𝑦𝑢𝑠,𝑦𝑑𝑠+^𝑦_−𝑦𝑏(𝑥,𝑡)𝑢(𝑥,𝑡)𝑑𝑡,−𝑙₁≤𝑥≤𝑙₁,−𝑙₂≤𝑦≤𝑙₂ (3.4) and on the Theorem 2.6.

Now, we consider the difference schemes for approximate solutions of problem (3.1): 𝑢^𝑘_𝑦,𝑛−𝑢^𝑘−1_𝑦,𝑛ℎ₁=𝑎^𝑘_𝑛𝑢^𝑘_𝑛−𝑎_𝑛^𝑘−1𝑢_𝑛^𝑘−1ℎ₂+𝑎^𝑘_−𝑛𝑢^𝑘_−𝑛−𝑎^𝑘−1_−𝑛𝑢^𝑘−1_−𝑛ℎ₂+𝑏^𝑘_𝑛𝑢^𝑘_𝑛−𝑏^𝑘_𝑛−1𝑢^𝑘_𝑛−1ℎ₁+𝑏_𝑛^−𝑘𝑢_𝑛^−𝑘−𝑏^−𝑘_𝑛−1𝑢^−𝑘_𝑛−1ℎ₁+𝑓^𝑘_𝑛𝑎,−𝑁+1≤𝑛≤𝑁,−𝐾+1≤𝑘≤𝐾,_𝑘𝑛=𝑎𝑥_𝑛,𝑦_𝑘,𝑏^𝑘_𝑛=𝑏𝑥_𝑛,𝑦_𝑘,𝑓^𝑘_𝑛=𝑓𝑥_𝑛,𝑦_𝑘𝑥,−𝑁≤𝑛≤𝑁,−𝐾≤𝑘≤𝐾,_𝑛=𝑛ℎ₁,−𝑁≤𝑛≤𝑁,𝑁ℎ₁=𝑙₁,𝑦_𝑘=𝑘ℎ₂,−𝐾≤𝑘≤𝐾,𝐾ℎ₂=𝑙₂,𝑢⁰_𝑛=𝜙𝑥_𝑛,−𝑁≤𝑛≤𝑁,𝑢^𝑘₀=𝜓𝑦_𝑘,−𝐾≤𝑘≤𝐾, (3.5) where 𝑢^𝑘_𝑦,𝑛=(𝑢^𝑘_𝑛−𝑢^𝑘_𝑛−1)/ℎ₂.

Theorem 3.2. For the solution of difference schemes (3.5), the following estimates are satisfied: max_{−𝑁≤𝑛≤𝑁−𝐾≤𝑘≤𝐾}𝑢^𝑘_𝑛≤𝑀max_{−𝑁+1≤𝑛≤𝑁−𝐾+1≤𝑘≤𝐾}𝑓^𝑘_𝑛+max_{−𝑁≤𝑛≤𝑁}𝜙_𝑛+max_{−𝐾≤𝑘≤𝐾}𝜓_𝑘, (3.6) where 𝑀 does not depend on ℎ₁, ℎ₂, 𝑓^𝑘_𝑛, 𝜙_𝑛, 𝜓_𝑘(−𝑁≤𝑛≤𝑁,−𝐾≤𝑘≤𝐾).

The proof of this theorem is based on the following formula: 𝑢^𝑘_𝑛=𝜙_𝑛+𝜓_𝑘−𝜙₀−ℎ₁^𝑛_{𝑠=−𝑛𝑠≠0}𝑎⁰_𝑠𝜙_𝑠−ℎ₂^𝑘_{𝜏=−𝑘𝜏≠0}𝑏^𝜏₀𝜓_𝜏+ℎ₁ℎ₂^𝑛𝑘_𝑠=1𝜏=1𝑓^𝜏_𝑠+ℎ₁^𝑛_{𝑠=−𝑛𝑠≠0}𝑎^𝑘_𝑠𝑢^𝑘_𝑠+ℎ₂^𝑘_{𝜏=−𝑘𝜏≠0}𝑏^𝜏_𝑛𝑢^𝜏_𝑛,−𝑁≤𝑛≤𝑁,−𝐾≤𝑘≤𝐾 (3.7) and on Theorem 2.4.

4. Conclusion

In this paper, generalizations of Wendroff-type inequalities for the integrals in two independent variables with four dependent limits and their discrete analogues are studied. The generalizations of Wendroff-type integral inequalities are used to establish stability estimates for the solution of the Goursat problem. A difference scheme approximately solving the Goursat problem is presented. Bu using the discrete analogues of the generalizations of Wendroff-type integral inequalities, stability estimates for the solution of this difference scheme are established.