Recent Developments and Applications on Qualitative Theory of Fractional Equations and Related TopicsView this Special Issue
On Weakly Singular Versions of Discrete Nonlinear Inequalities and Applications
Some new weakly singular versions of discrete nonlinear inequalities are established, which generalize some existing weakly singular inequalities and can be used in the analysis of nonlinear Volterra type difference equations with weakly singular kernels. A few applications to the upper bound and the uniqueness of solutions of nonlinear difference equations are also involved.
Recently, along with the development of the theory of integral inequalities and difference equations, many authors have researched some discrete versions of Gronwall-Bellman type inequalities [1–5]. Starting from the basic form, discussed originally by Pachpatte in , various such new inequalities have been established, which can be used as a powerful tool in the analysis of certain classes of finite difference equations. Among these results, discrete weakly singular integral inequalities also play an important role in the study of the behavior and numerical solutions for singular integral equations [6, 7] and the theory for parabolic equations [8–10]. For example, Dixon and McKee  investigated the convergence of discretization methods for the Volterra integral and integrodifferential equations using the following inequality: and Beesack  also discussed the inequality, for the second kind Abel-Volterra singular integral equations. Henry  presented a linear inequality to investigate some qualitative properties for a parabolic equation. In particular, to avoid the shortcoming of analysis, Medved [11–13] used a new method to discuss some nonlinear weakly singular integral inequalities and difference inequalities. Following Medved’s work, Ma and Yang  improved his method to discuss a more general nonlinear weakly singular integral inequality, and a nonlinear difference inequality , As for other new weakly singular inequalities, recent work can be found, for example, in [16–25] and references therein.
In this paper, we investigate some new nonlinear discrete weakly singular inequalities Compared to the existing result, our result is more concise and can be used to obtain pointwise explicit bounds on solutions of a class of more general weakly singular difference equations of Volterra type. Finally, to illustrate the usefulness of the result, we give some applications to Volterra type difference equation with weakly singular kernels.
For convenience, before giving our main results, we first cite some useful lemmas here.
Lemma 1 (discrete Jensen inequality, see ). Let be nonnegative real numbers and a real number. Then
Lemma 2 (discrete Hölder inequality, see ). Let be nonnegative real numbers, and positive numbers such that (or ). Then
Lemma 3 (see ). Let , and be positive constants. Then where is the well-known -function and .
In what follows, denote to be the set of real numbers. Let and . denotes the collection of continuous functions from the set to the set . As usual, the empty sum is taken to be .
2. Some New Nonlinear Weakly Singular Difference Inequalities
Lemma 4. Suppose that is nondecreasing with for . Let be nonnegative and nondecreasing in . If is nonnegative such that then where , is the inverse function of , and is defined by
Assume that) and such that ;() are nonnegative functions for , respectively;() is nondecreasing and .
Lemma 5. Suppose that satisfies assumption (); then for sufficiently small , one has
Proof. Consider the -function in (15). Consider
and denote for , where and . If , then is symmetric about . According to assumption (),
On the other hand, the zero-point of can be obtained as follows:
Therefore, the function is decreasing on the interval while increasing sharply on the interval . Consequently, for some given sufficiently small , by the properties of the left-rectangle integral formula, we have
where . For the general interval , we can easily obtain the corresponding result (15) by the similar method and omit the details here.
Denote and , where is the variable time step. Next, we first discuss inequality (7) and obtain the following result.
Theorem 6. Under assumptions , , and , if is nonnegative such that (7), then for , where , , is the inverse function of , , and is the largest integer number such that
Proof. Since , according to assumption (), is nonnegative and nondecreasing, and . From (7), we have
Due to assumption (), take suitable indices such that . An application of Lemma 2 yields By Lemma 1, it follows from the inequality above that Considering in which we apply Lemma 5, we have
Let , , and . Obviously, are nondecreasing for and satisfies assumption . Equation (27) can be rewritten as which is similar to inequality (12). Using Lemma 4 to (28), we have for , where , , is the inverse function of , and is the largest integer number such that Therefore, by , (21) holds for .
Remark 7. When and , the inequality was discussed by Medved  which is the special case of our result. Moreover, his result holds under the assumption “ satisfies the condition ;” that is, “, where is a continuous, nonnegative function.” In our result, the condition is eliminated.
Corollary 8. Under assumptions () and (), let , . If is nonnegative such that then for .
Proof. Let ; then or . From (31) we have Denote . Clearly, satisfies assumption . With the definition of in Theorem 6, letting , we have Substituting (34) and (35) into (29), we get In view of , we can obtain (32).
Remark 10. Let and ; we can get the interesting Henry’s version of the Ou-Iang-Pachpatte type difference inequality . Thus, our results are a more general discrete analogue for such inequality.
Remark 11. Ma and Pečarić discussed the continuous case of (2.15) in  and here we present the discrete version of their result. Furthermore, the result in  is established for the cases when the ordered parameter group obeys distribution I or II (for details, see ) which makes the application of inequality more inconvenient. Clearly, our result is based on the concise assumption to overcome this weakness.
Corollary 12. Under assumptions () and (), if is nonnegative such that then for .
Proof. In (7), also satisfies assumption . Thus, we have
Similar to the computation in Corollary 8, estimate (38) holds.
Now, we discuss inequality (8) Since there are two different parameter groups and , assumption is revised as follows: () , and , such that and .
Theorem 13. Under assumptions () and (), suppose that are nonnegative for . If is nonnegative such that (8), then for , where and are defined as in Theorem 6, and is the largest integer number such that
Proof. By the definition of , we have Let which yields directly Using Corollary 12, from the inequality above, we get where . Letting from (47), we get Clearly, inequality (49) is similar to (7). According to Theorem 6, we obtain for .
Remark 14. Our result for inequality (8) is also the discrete analogue of inequality (5). In fact, with the different choice of the parameter groups and in , the complicate results must be presented by four cases, respectively. Apparently, compared to their results, our result is quite simple.
In this section, we apply our results to discuss the upper bound and the uniqueness of solutions of a Volterra type difference equation with certain weakly singular kernels.
Example 15. Consider the following inequality: Obviously, (51) is the special case of inequality (8), and we get Next, we discuss the choice of parameter . By assumption (), from the conditions and , we have . From the conditions and , we have . Thus, we can take ; then , . According to Theorem 13, we obtain Substituting the results above into (41), we can get the upper bound of and omit the details for its complicated formula.
Example 16. Consider the linear weakly singular difference equation where , is an arbitrary positive number, and the parameter group satisfies assumption . From (54) and (55), we get which is the form of inequality (37). Applying Corollary 12, we have for . If , let and we obtain the uniqueness of the solution of (54).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are very grateful to the referees for their helpful comments and valuable suggestions. This work is supported by the Doctoral Program Research Funds of Southwest University of Science and Technology (no. 11zx7129) and the Fundamental Research Funds for the Central Universities (no. skqy201324).
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