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Some New Nonlinear Weakly Singular Inequalities and Applications to Volterra-Type Difference Equation
Some new nonlinear weakly singular difference inequalities are discussed, which generalize some known weakly singular inequalities and can be used in the analysis of nonlinear Volterra-type difference equations with weakly singular kernel. An application to the upper bound of solutions of a nonlinear difference equation is also presented.
The discrete version of the well-known Gronwall-Bellman inequality is an important tool in the development of the theory of difference equations as well as the analysis of the numerical schemes of differential equations. A great deal of interest has been given to these inequalities, and many results on their generalizations have been found; for example, see [1–4]. Among them, one of the fundamental cases is Pachpatte’s result  for the difference inequality:
In particular, due to the study of the behavior and numerical solutions for the singular integral equations, some discrete weakly singular integral inequalities also have drawn more and more attention [5–7]. Dixon and McKee  investigated the convergence of discretization methods for the Volterra integral and integrodifferential equations, by using the following inequality:
Henry  presented a linear integral inequality with weakly kernel: to investigate some qualitative properties for a parabolic equation. The corresponding discrete version was discussed by Slodika . But he studied the case , that is, the case of constant differences. Furthermore, the first formulation of the inequality with a nonlinearity and nonconstant was studied in , in which the general nonlinear discrete case as follows: was considered. However, his results are based on the so-called “ condition”: (1) satisfies ; (2) there exists such that . Recently, a new nonlinear difference inequality: was discussed by Yang et al. . For other new weakly singular inequalities, lots of work can be found, for example, in [12–22] and references therein.
In this paper, we investigate the new nonlinear weakly singular inequality: where , , , , and . Compared to the existing result, our result does not need the so-called “ condition” proposed in  and can be used to obtain pointwise explicit bounds on solutions for a class of more general weakly singular inequalities of Volterra type. Finally, we also present an application to Volterra-type difference equation with weakly singular kernel.
Let be the set of real numbers, , and . denotes the collection of continuous functions from the set to the set . As usual, the empty sum is taken to be .
Lemma 1 (Discrete Jensen inequality, ). Let be nonnegative real numbers, and let be a real number. Then,
Lemma 2 (Discrete Hölder inequality, ). Let , be nonnegative real numbers, and let , be positive numbers such that (or ). Then,
Furthermore, take ; then, one gets the discrete Cauchy-Schwarz inequality.
Lemma 3. Suppose that is nondecreasing. Let , be nonnegative and nondecreasing in . If is nonnegative such that Then, where ,, is the inverse function of , and is defined by
3. Main Results
Assume that (A1) , are nonnegative functions for , respectively; (A2) is nondecreasing and .
Define and , where is the variable time step.
Theorem 4. Under assumptions () and (), if is nonnegative such that (6), then(1)for , letting and , one has for , where ,, is the inverse function of , and is the largest integer number such that (2)for , letting and , one has for , where , and is the largest integer number such that
Proof. By definition of and assumption (), is nonnegative and nondecreasing and . It follows from (6) that (1)If , using Lemma 2 with the indices , for (18), we get By Lemma 1, the inequality above yields Consider that where and is the well-known -function. Thus, we have Let , , and . Obviously, , are nondecreasing for and satisfies the assumption (). Equation (22) can be rewritten as which is similar to inequality (9). Using Lemma 3, from (23), we have for , where is the largest integer number such that Therefore, by , (12) holds for .(2)If , applying Cauchy-Schwarz inequality for (18), that is, , we get By Lemma 1, the inequality above yields Because where ,, it follows from (27) that Let , , and . Similarly, , also are nondecreasing for and also satisfies the assumption (). Equation (29) can be rewritten as which also is similar to inequality (9). Using Lemma 3, from (30), we have for , and is the largest integer number such that Clearly, by , (15) also holds for .
Remark 5. Here, we note that the most significant work in the study of weakly singular inequalities is Medve’s method, originally presented in the paper  and also applied in the paper . But his result holds under the assumption “ satisfies the condition (q),” that is, “, where is a continuous, nonnegative function.” In our result, the condition (q) is eliminated.
Corollary 6. Under assumptions () and (), let , . If is nonnegative such that then(1)if , let and , and one gets for , where is defined as in Theorem 4;(2)if , let , and one gets for , where is defined as in Theorem 4
Proof. Let , then and . From (33), we have
Clearly, satisfies the assumption . According to the definition of in Theorem 4, for , letting , we have
It can be seen easily from (38) that . Substituting (37) and (38) into (12), we get
In view of , we can obtain (34). For the case that , in fact, and are the same as (37) and (38), respectively. So, it follows from (37), (38), and (15) that for .
Remark 8. Letting and , we can get the interesting Henry version of the Ou-Iang-Pachpatte-type difference inequality . Thus, our result is a more general discrete analogue for such inequality.
Corollary 9. Under assumptions () and (), if is nonnegative such that then(1)if , let and , and one gets for , where is defined as in Theorem 4;(2)if , let , and one gets for , where is defined as in Theorem 4.
In this section, we apply our results to discuss the upper bound of solution of a Volterra type difference equation with weakly singular kernel.
Consider the following the inequality:
Thus, we can take and ; then, . Moreover,
According to Theorem 4, we obtain for , which indicates that we get the upper bound of .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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). The work is also supported by the Applied Basic Research Project of Sichuan Province (no. 2013JY0096).
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