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 [1–4]. For example, Li [5, 6] investigated some new discrete inequalities in two independent variables which provide explicit bounds on unknown functions. Cheung [7–9] 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 [10–13] established some nonlinear retarded difference inequalities and gave some applications of the obtained inequalities to the estimation of finite difference equations. Pachpatte [14–16] obtained certain inequalities arising in the theory of differential equations and provided new estimates for these types of inequalities. Ma et al. [17–19] 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, 17–19, 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. Iffor all , thenfor all , where is defined in Theorem 3, , 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 . Due to the property of , inequality (66) becomesFixing any numbers , , and with , , and , define a positive function byThen , and (70) can restated byAlong the lines of proof of Theorem 3, we obtain From the above inequality and Theorem 3, we observe thatNow using inequalities (74) in (72), we get Taking , , and in the above inequality, since , , and are arbitrary, we get the required inequality.

Corollary 10. Let , , , , , and be as defined in Theorem 3, and let , be as defined in Theorem 9. If for all , then for all , where is defined in Theorem 9.

Proof. The proof is followed by an argument similar to that in the proof of Corollary 4 and Theorem 9 with suitable modification. We omit the details here.

Theorem 11. Let be as defined in Theorem 3, let , , , , and , be as defined in Theorem 5, and let , be as defined in Theorem 9. If for all , then for all , where , are defined in Theorem 5, , , 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 Theorems 5 and 9 with suitable modification. We omit the details here.

3. Applications

In this section, we will show that our results are useful in establishing the boundedness and uniqueness of solutions to certain partial finite difference and sum-difference equations. Consider the partial finite difference equations with the initial boundary conditions:with

The following two results deal with the boundedness on the solutions of problem (82).

Proposition 12. Let be real-valued function such thatwhere and are real-valued nonnegative functions defined for and is nondecreasing in each variables. Suppose that is a strictly increasing function with and is a nondecreasing continuous function with for . If is any solution of problem (82) with the condition (83), thenfor all , where is defined in Theorem 3, , denote the inverse functions of , , and is chosen so that

Proof. It is easy to see that the solution of problem (82) satisfies equivalent sum-difference equation:From (84) and (87), we have Now, a suitable application of inequality given in Theorem 3 to the above yields the desired result (85).

Proposition 13. Let be real-valued function such that where and are real-valued nonnegative functions defined for , is nondecreasing in each variables, and is real-valued function which satisfies the condition for and , be real-valued function. Suppose that is a strictly increasing function with and is a nondecreasing continuous function with for . If is any solution of problem (82) with condition (83), then for all , where is defined in Theorem 3, , 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 3, using Theorem 9 with suitable modification. We omit the details here.

The following theorem deal with the uniqueness on the solutions of problem (82).

Proposition 14. Let be real-valued function such thatwhere is real-valued nonnegative function on . Then problem (82) with condition (83) has at most one positive solution on .

Proof. Let and be two solutions of problem (82) on . By (94), we obtain An application of Corollary 4 with suitable modification to function in the last inequality yields for . Hence on .

Finally, we investigate the continuous dependence of the solutions of problem (82) on the function and the boundary data , , and . For this we consider the following variation of problem (82).with

Proposition 15. Consider problem (82) and problem (97). Assume thatandwhere and . Then for all . Hence, depends continuously on , , , and .

Proof. Let and be solutions of problem (82) and problem (97), respectively. Then satisfies (87) and satisfies the corresponding equation: Hence, we obtain As an application of Corollary 4 with suitable modification, function in the last inequality yields by assumptions (99)-(101) for all . When restricted to any compact sublattice that is bounded, for some for all in this compact sublattice. Hence, depends continuously on , , , and .

4. Conclusions

By using the inequality analyze technique, some new discrete Gronwall-Bellman type inequalities with three independent variables have been obtained. Our main theorems generalize and extend some existing results in the literature. Furthermore, the boundedness and uniqueness of solutions to certain partial finite difference and sum-difference equations have been established by applying our obtained results.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally in this article. They read and approved the final manuscript.