A Generalized Nonlinear Sum-Difference Inequality of Product Form
We establish a generalized nonlinear discrete inequality of product form, which includes both nonconstant terms outside the sums and composite functions of nonlinear function and unknown function without assumption of monotonicity. Upper bound estimations of unknown functions are given by technique of change of variable, amplification method, difference and summation, inverse function, and the dialectical relationship between constants and variables. Using our result we can solve both the discrete inequality in Pachpatte (1995). Our result can be used as tools in the study of difference equations of product form.
Being an important tool in the study of existence, uniqueness, boundedness, stability, and other qualitative properties of solutions of differential equations and integral equations, various generalizations of Gronwall inequalities [1, 2] and their applications have attracted great interests of many mathematicians (such as [3–6]). Some recent works can be found, for example, in [7–10] and some references therein. Along with the development of the theory of integral inequalities and the theory of difference equations, more attention is paid to some discrete versions of Gronwall-Bellman type inequalities (such as [3, 4, 11–13]). Some recent works can be found, for example, in [14–24] and some references therein.
Pachpatte  obtained the explicit bound to the unknown function of the following sum-difference inequality: Pachpatte  obtained the estimation of the unknown function of the following inequality: Then, the estimation can be used to study the boundedness, asymptotic behavior, and slow growth of the solutions of the sum-difference equation: However, the bound given on such inequalities in [3, 4] is not directly applicable in the study of certain sum-difference equations. It is desirable to establish new inequalities of the above type, which can be used more effectively in the study of certain classes of sum-difference equations of product form.
In this paper, we establish a new integral inequality of product form where,, may not be monotone. For,, we employ a technique of monotonization to construct two functions; the second possesses stronger monotonicity than the first. We can demonstrate that inequalities (1) and (2), considered in [3, 4], respectively, can also be solved with our result. Finally, we expound that we can give estimation of solutions of a class of sum-difference equations of product form.
2. Main Result and Proof
In this section, we proceed to solve the discrete inequality (4) and present explicit bounds on the embedded unknown function. Let,, and . For function, its difference is defined by. Obviously, the linear difference equationwith the initial conditionhas the solution. For convenience, in the sequel we complementarily define that.
First of all, we monotonize some given functions,,in the sum; let whereandare all nondecreasing in and satisfy Let whereis nondecreasing in andis also nondecreasing inand satisfies where,denote the inverse function of,, respectively.
Theorem 1. Let,be nonnegative and given functions on. Suppose thatis a nonnegative and unknown function. Then, the discrete inequality (4) gives where,,are defined by (9), (10), and (11), respectively,,,denote the inverse functions of,,, respectively, andis the largest natural number such that
Proof. Using (5), (6), (7), and (8), we observe that whereis chosen arbitrarily. Letdenote the function on the right-hand side of (15), namely, which is a nonnegative and nondecreasing function onwith. Then (4) is equivalent to Using the difference formula and the monotonicity ofand, from (16) and (17), we observe that for all. From (19), we have On the other hand, by the mean value theorem for integrals, for arbitrarily given integers,, there existsin the open intervalsuch that whereis defined by (9). From (20) and (21), we have for all. By settingin (22) and substitutingsuccessively, we obtain Letdenote the function on the right-hand side of (23); namely, Then,is a nonnegative and nondecreasing function on, and (23) is equivalent to From (24), we obtain From (26), we have Once again, performing the same procedure as in (21), (22), and (23), (27) gives whereis defined by (10). Letdenote the function on the right-hand side of (28); namely, Then,is a nonnegative and nondecreasing function on, and (28) is equivalent to From (29) and (30), we obtain for all. From (31), we have Once again, performing the same procedure as in (21), (22), and (23), (32) gives Using (17), (25), and (30), from (33) we have As, (34) yields Since, andis chosen arbitrarily in (35), the estimation (12) is derived. This completes the proof of Theorem 1.
We consider a sum-difference equation of product form From (36), we have Let,,,, andin (37); then (37) is the inequality of the form (4). Applying our result we get the estimation of solution of the sum-difference equations of product form (36).
This research was supported by National Natural Science Foundation of China (Project no. 11161018) and Guangxi Natural Science Foundation (2012GXNSFAA053009).
B. G. Pachpatte, Integral and Finite Difference Inequalities and Applications, vol. 205, Elsevier Science, Amsterdam, The Netherlands, 2006.View at: MathSciNet