Estimation of Unknown Functions of Iterative Difference Inequalities and Their Applications
We establish some new nonlinear retarded finite difference inequalities. The results that we propose here can be used as tools in the theory of certain new classes of finite difference equations in various difference situations. We also give applications of the inequalities to show the usefulness of our results.
An integral inequality that provides an explicit bound to the unknown function furnishes a handy tool to investigate qualitative properties of solutions of differential and integral equations. One of the best known and widely used inequalities in the study of nonlinear differential equations is Gronwall-Bellman inequality [1, 2], which can be stated as follows. If and are nonnegative continuous functions on an interval satisfying for some constant , then Being an important tool in the study of 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 [3–5]. Some recent works can be found in [6–12] and the references therein. Along with the development of the theory of integral inequalities and the theory of difference equations, more and more attentions are paid to discrete versions of Gronwall-type inequalities; see [13–36] and the references cited therein.
Sugiyama  established the most precise and complete discrete analogue of the Gronwall inequality  in the following form. Let and be nonnegative functions defined for , and suppose that for every . If where is the set of points (), is a given integer, and is a nonnegative constant, then Pachpatte  established some generalized discrete analogue of the Gronwall inequality in the following form. Let be a positive and monotone nondecreasing function on , and let be nonnegative functions on . If satisfies then where
Lemma 1 (see ). Suppose that is a nonnegative constant and , , , , and are nonnegative functions defined on , for all . If satisfies the inequality then where in which and for all .
Lemma 2 (see [14, 18]). Let be a real-valued function defined for , and monotone nondecreasing with respect to for any fixed . Let be a real-valued function defined for such that Let be a solution of such that . Then
Pachpatte [18, 19] also established some difference inequalities of product form as follows. Let be nonnegative functions defined on and let be a nonnegative constant. Let be a nonnegative function defined for , and monotone nondecreasing with respect to for any fixed . If satisfies then where is defined by (7), and is a solution of Let be nonnegative functions defined for and let be a nonnegative constant. Let be a nonnegative function defined for , and monotone nondecreasing with respect to for any fixed . If satisfies then where is defined by (7), and is a solution of the difference equation
Motivated by the results given in [16, 18, 19], in this paper, we discuss new nonlinear finite difference inequalities: Our inequalities can be used as tools in the study of certain classes of finite difference equations. We also present some immediate applications to show the importance of our results to study the various problems in the theory of finite difference equations.
2. Main Results
Throughout this paper, let , . Let and . For function , , we define the operator by . Obviously, the linear difference equation with the initial condition has the solution . For convenience, in the sequel we complementarily define that and .
Theorem 3. Let be a constant, positive functions defined on , a monotone increasing function, and a monotone decreasing function. Let be a nonnegative function defined on such that (i)Suppose . If , then (ii)Suppose . Then
Proof. (i) We apply mean value theorem for differentiation to the function
and then there exists between and such that
Because is monotone increasing and is monotone decreasing and , we see that and . So for all values of between and we have
From (22) and (27), we have
Since , from (26) and (28) we have
Taking in (29) and summing up over from to , we obtain
From (30), we obtain our required estimation (23).
(ii) Now by following the same steps as in the proof of (i) before (29) we have because . Taking in (31) and summing up over from to , we obtain From (32), we obtain our required estimation (24).
Theorem 4. Let be a positive and monotone nondecreasing function defined on and nonnegative functions defined on . If satisfies then where in which and for all .
Proof. Fix , where is chosen arbitrarily, since is a nonnegative and monotone nondecreasing function, from (33), we have Define a function by the right-hand side of (37). Then is a positive and monotone nondecreasing function defined on . We have Using the definitions of the operator and , we obtain Let Then It follows that Adding to both sides of the above inequality we have Put and then , and We see that the inequality Define a function Multiplying by to both sides of (46) we obtain Let , , , and . Because is monotone increasing, is monotone decreasing and ; applying Theorem 3 to (48) we obtain where , are used. Define a function of the right-hand side of (49). Substituting (49) in (43) we obtain Performing the same derivation as in (46)–(49), we obtain from (50) that Define a function of the right-hand side of (51). Substituting (51) in (39) we obtain Using (38), from (52) it follows that Since is arbitrary, from (53), we get the required estimate (35).
Theorem 5. Let be nonnegative functions defined for and a nonnegative constant. Let be a real-valued function defined for , , and monotone nondecreasing with respect to for any fixed . If satisfies (21), then where in which and is a solution of the difference equation where in which and for all .
Proof. We first assume that and define a function by the right-hand side of (21). Then is a positive and monotone nondecreasing function defined on . We have
Using the definitions of the operator and , we obtain
From (61) it follows that the inequality
holds for all . Setting in (62) and substituting , successively, we get
Define a function by
Using (64), the inequality (63) can be written as
Since is positive and monotone nondecreasing for , satisfy the conditions in Theorem 4. Now an application of Theorem 4 to (66) yields
Since is monotone nondecreasing with respect to for any fixed , from (65) and (67), we have
Now as a suitable application of Lemma 2, we obtain
where is a solution of (57). Using (60), (67), and (71), we obtain our required estimation (54).
If is nonnegative, we can carry out the above procedure with instead of where is an arbitrary small number. Letting , we obtain (54).
3. Application to Finite Difference Equations
In this section, we consider the following difference equation: where , , are real-valued functions defined, respectively, on , , , is as defined in Theorem 5, and is a constant. We assume that where , , are as defined in Theorem 5. Using the definitions of the operator , from (72), we see that the inequality holds for all . It follows that From (76), we have Using the conditions (74), we obtain Now an application of Theorem 5 to (78) yields the estimation of the difference equation (72), that is, where in which and is a solution of the difference equation where in which and for all .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was supported by National Natural Science Foundation of China (Project no. 11161018,11171079), the NSF of Guangxi Zhuang Autonomous Region (no. 2012GXNSFAA053009), and the SRF of the Education Department of Guangxi Zhuang Autonomous Region (nos. 201106LX599 and 201106LX591).
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