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Journal of Function Spaces
Volume 2018, Article ID 8320236, 13 pages
https://doi.org/10.1155/2018/8320236
Research Article

Some New Discrete Gronwall-Bellman Type Inequalities with Three Independent Variables and Applications

1College of Information and Management Science, Henan Agricultural University, Zhengzhou 450002, China
2Faculty of General Education, Zhengzhou Technology and Business University, Zhengzhou 450026, China

Correspondence should be addressed to Weihua Liu; moc.qq@977782969

Received 3 May 2018; Accepted 3 July 2018; Published 1 August 2018

Academic Editor: Hugo Leiva

Copyright © 2018 Weihua Liu and Haisong Huang. 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

Some new discrete Gronwall-Bellman type inequalities with three independent variables which generalize some existing results and can be used as handy tools in the study of qualitative and quantitative properties of solutions of certain classes of difference equation are presented. As applications, some difference equations with the initial boundary conditions are also considered.

1. Introduction

Finite difference inequalities which exhibit explicit bounds on unknown functions in general provide a very useful and important tool in the development of the theory of finite difference equations. Due to the fact that finite difference inequalities have been applied in various branches of finite difference equations, many such inequalities with in one or two variables have been established [14]. For example, Li [5, 6] investigated some new discrete inequalities in two independent variables which provide explicit bounds on unknown functions. Cheung [79] generalized some existing discrete Gronwall-Bellman-type inequalities to more general situations and applied them to study the boundedness, uniqueness, and continuous dependence of solutions of certain discrete boundary value problem for difference equations.

Wang [1013] established some nonlinear retarded difference inequalities and gave some applications of the obtained inequalities to the estimation of finite difference equations. Pachpatte [1416] obtained certain inequalities arising in the theory of differential equations and provided new estimates for these types of inequalities. Ma et al. [1719] obtained some power nonlinear Volterra-Fredholm type discrete inequalities. However, to the best of the authors' knowledge, few papers are only published on discrete inequalities with three independent variables. Hussain et al. [20] discussed some generalized Volterra-Fredholm type nonlinear discrete inequalities involving four iterated infinite sums and applied these results to new explicit bounds for solutions of certain difference equations.

In 1979, Pachpatte and Singare [21] gave discrete Gronwall type inequalities in three independent variables as follows.

Theorem 1. Let and be real-valued nonnegative functions defined for for which the inequality holds for , where ; is the operator defined by are real-valued functions defined on . Then for .

In 1980, Singare and Pachpatte [22] established generalization of discrete Gronwall type inequalities in three independent variables in Theorem 1 in the following.

Theorem 2. Let and be real-valued nonnegative functions defined for , and let be positive nondecreasing in the three variables, defined for for which the inequality holds for . Then for .

Motivated by the results mentioned above, the aim of this paper is to establish some new and more general discrete Gronwall-Bellman type inequalities involving functions of three independent variables, which generalize and extend some existing results in [8, 1719, 21, 22] and can be readily used as handy and powerful tools in the analysis of certain classes of partial finite difference and sum-difference equations. We also give some applications to convey the importance of these new inequalities to study various properties of the solutions of a class of initial value problems involving difference equations.

2. Main Results

In this section, we present some new and more general discrete Gronwall-Bellman type inequalities in three independent variables which might be useful tools in the analysis of certain classes of partial finite difference and sum-difference equations.

In what follows, , and are three fixed lattices of integral points in , where ,. Let , , and denote the sublattice of for any . We also use the following notions of the operators , , , and , and .

Theorem 3. Let , , , and be real-valued nonnegative functions defined for and and be nondecreasing in each variables. Suppose that is a strictly increasing function with and is a nondecreasing continuous function with for . Iffor all , thenfor all , where , denote the inverse functions of , and is chosen so that

Proof. It suffices to consider the case , since the case can be arrived at by continuity argument. Let us first assume that . Fixing any numbers , , and with , and , define a positive function byThen , and (5) can restated byIt is easy observe that is a nondecreasing function in each variables defined for andFrom (11) we haveFrom (12) we observe thatFrom (13) we obtainNow keeping , fixed in (14), setting , and substituting in (14) we obtain the estimate for such thatFrom (15) we observe thatKeeping , fixed in (16), setting , and substituting in (16) we obtain the estimate for such thatFrom the definition of and (17) we obtainAgain keeping , fixed in (18), setting , and substituting in (18), we obtainNow using inequality (19) in (10), we getTaking , , and in inequality (20), since , , and are arbitrary, we get the required inequality.

Corollary 4. Let , , , , and be as defined in Theorem 3. If for all , then for all .

Proof. Suppose first that . Taking , we have and so , in particular, it is defined everywhere on . Hence, by Theorem 3, for all . Finally, by continuity, this should also hold for the case .

Theorem 5. Let , , , and , be real-valued nonnegative functions defined for and and be nondecreasing in each variables. Suppose that are strictly increasing functions with and and is a nondecreasing continuous function with for . Iffor all , let , thenfor all , where , , denote the inverse functions of , , and is chosen so that

Proof. It suffices to consider the case , since the case can be arrived at by continuity argument. Let us first assume that . Fixing any numbers , , and with , and , define a positive function byThen , and (5) can restated byIt is easy observe that is a nondecreasing function in each variable defined for andFrom (31) we haveFrom (32) we observe thatFrom (33) we obtain Along the lines of proof of Theorem 3, we obtainNow, define function by Then and (35) can be restated asIt is easy observe that is a nondecreasing function in each variable defined for andFrom (38) we haveFrom (39) we observe thatFrom (40) we obtain Along the lines of proof of Theorem 3, we obtainNow using inequalities (37) and (42) in (30), we getTaking , , and in inequality (43), since , , and are arbitrary, we get the required inequality.

Corollary 6. Let , , , , and , , be as defined in Theorem 5. Iffor all , let , then for all , where where , denote the inverse functions of , and is chosen so that

Proof. It suffices to consider the case , since the case can be arrived at by continuity argument. Let us first assume that . Fixing any numbers , , and with , , and , define a positive function byThen , and (44) can restated byAlong the lines of proof of Theorem 5, we obtain From the above inequality and Theorem 3, , , we observe thatNow, define a function by Then , and (51) can be restated asAlong the lines of proof of Theorem 5, we obtain From the above inequality and Theorem 3, we observe thatNow using inequalities (53) and (55) in (49), we get Taking , , and in the above inequality, since , , and are arbitrary, we get the required inequality.

For the special case ( is constant), Corollary 6 gives the following retarded discrete inequality for nonlinear functions.

Corollary 7. Let , , and , be as defined in Theorem 5, and let be as defined in Theorem 3. Suppose that are constants. If for all , then for all , where is defined in Corollary 6, , denote the inverse functions of , , and is chosen so that

Proof. The proof is followed by an argument similar to that in the proof of Corollary 6 with suitable modification. We omit the details here.

For the special case and ( are constants), Theorem 5 gives the following retarded discrete inequality for nonlinear functions.

Corollary 8. Let , , and , be as defined in Theorem 5, and let be as defined in Theorem 3. Suppose that are constants. If for all , then for all , where , denote the inverse functions of , , and is chosen so that

Proof. The proof is followed by an argument similar to that in the proof of Theorem 5 with suitable modification. We omit the details here.

Theorems 3 and 5 can easily be applied to generate other useful discrete inequalities in more effective tools of the study of certain partial finite difference and sum-difference equations. For example, we have the following results.

Theorem 9. Let , , , , , and be as defined in Theorem 3, and let be real-valued function which satisfies the following condition:for and , where be real-valued function. If