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On nonlinear discrete weakly singular inequalities and applications to Volterra-type difference equations
Advances in Difference Equations volume 2013, Article number: 239 (2013)
Some nonlinear discrete weakly singular inequalities, which generalize some known results are discussed. Under suitable parameters, prior bounds on solutions to nonlinear Volterra-type difference equations are obtained. Two examples are presented to show the applications of our results in boundedness and uniqueness of solutions of difference equations, respectively.
As an important branch of the Gronwall-Bellman inequality, various weakly singular integral inequalities and their discrete analogues have attracted more and more attention and play a fundamental role in the study of the theory of singular differential equations and integral equations (for example, see [1–8] and ). When many problems such as the behavior, the perturbation and the numerical treatment of the solution for the Volterra type weakly singular integral equation are studied, they often involve some certain integral inequalities and discrete inequalities. Dixon and McKee  investigated the convergence of discretization methods for the Volterra integral and integro-differential equations, using the following inequalities
As for the second kind Abel-Volterra singular integral equation, Beesack  also discussed the inequalities
where , and . It should be noted that the above mentioned results are based on the assumption that the mesh size is uniformly bounded. Unfortunately, such technique leads to weak results, which do not reflect the true order of consistency of the scheme and may not even yield a convergence result at all. To avoid the shortcoming of these results, Norbuy and Stuart [12, 13] presented some new inequalities to describe the numerical method for weakly singular Volterra integral equations, which is based on a variable mesh.
Another purpose of studying weakly singular integral inequalities and their discrete versions is related to the theory of the parabolic equation (for example, see [14–18] and the references therein). Consider the weakly singular integral inequality
and the corresponding discrete inequality of multi-step,
where , , and . Henry  and Slodicka  discussed the linear case of the two inequalities above and obtained the estimate of the solution. Furthermore, Medveď [20, 21] studied the general nonlinear case. However, his results are based on the ‘(q) condition’: (1) ω satisfies ; (2) there exists such that . Later, Yang and Ma  generalized the results to a new case
In this paper, we are concerned with the following weakly singular inequality on a variable mesh
where , , and . Our proposed method can avoid the so-called ‘q-condition,’ and under a new assumption, the more concise results are derived. Moreover, to show the application of the more general inequality to a Volterra-type difference equation, some examples are presented.
In what follows, we denote ℝ by the set of real numbers. Let and . denotes the collection of continuous functions from the set X to the set Y. As usual, the empty sum is taken to be 0.
Lemma 2.1 (Discrete Jensen inequality, see )
Let be nonnegative real numbers, and let be a real number. Then
Lemma 2.2 (Discrete Hölder inequality, see )
Let , () be nonnegative real numbers, and p, q be positive numbers such that (or , ). Then
Definition 2.1 (See )
Let be an ordered parameter group of nonnegative real numbers. The group is said to belong to the first-class distribution and is denoted by if conditions , , and are satisfied; it is said to belong to the second-class distribution and is denoted by if conditions , and are satisfied.
Lemma 2.3 (See )
Let α, β, γ and p be positive constants. Then
where (, ) is the well-known B-function and .
Lemma 2.4 Suppose that is nondecreasing with for . Let , be nonnegative and nondecreasing in n. If is nonnegative such that
where , , is the inverse function of Ω, and M is defined by
Remark 2.1 Martyniuk et al.  studied the inequality , . Obviously, our result is a more general case of the nonlinear difference inequality.
Lemma 2.5 If , then ; if , then . Furthermore, for sufficiently small , we have
for , where .
Proof By the definition of , . For its proof, see . On one hand, when , it follows from Definition 2.1 that . On the other hand, when , that is, , , we have that
According to the condition , holds, which yields (2.2) holds directly. Thus, when or , we have
Next, we consider the integrated function in the B-function in (2.1).
Denote for , where and . If , then is symmetric about . In fact, because of , we get
Moreover, we can obtain the zero-point of as follows
Therefore, the function is decreasing on the interval while increasing sharply on the interval . So, for some given sufficiently small , by the properties of the left-rectangle integral formula, we have
As for the general interval , we can easily obtain the corresponding result (2.1), which is similar to (2.7). We omit the details here. □
3 Main result
To state our result conveniently, we fist introduce the following function
Thus, we have
Denote , where .
Theorem 3.1 Suppose that , are nonnegative functions for . Let , , . If is nonnegative function such that (1.5), then for some sufficiently small :
, letting , , we have(3.1)
for () or (), where
and is the largest integer number such that,
, letting , , we have(3.3)
for () or (), where
and is the largest integer number such that
Proof By the definition of , obviously, is nonnegative and nondecreasing, that is, . It follows from (1.5) that
where is the variable time step.
Next, for convenience, we take the indices , . Denote that if , let and ; if , let , and let . Then holds for .
Using Lemma 2.2 with indices , in (3.5), we have
By Lemma 2.1, the inequality above can be rewritten as
By Lemma 2.5,
where , we get
and is given in Lemma 2.3 for .
Let . Then . It follows from (3.9) that
According to Lemma 2.4, we have
for , where M is the largest integer number such that
For , , . By the definitions of Ω and , we can compute that(3.12)
Observe the second formula in (3.13). To ensure that
we may take for and for , respectively, where is the largest integer number such that
Since , substituting it into (3.13), we can get (3.1).
For , , . Similarly to the computation above, we have(3.14)
Observe the second formula in (3.15). To ensure that
we take for and for , respectively, where is the largest integer number such that
Because , substituting it into (3.15), we can get (3.3). This completes the proof. □
For the case that , let , we obtain from (1.5)
and derive the estimation of the upper bound as follows.
Theorem 3.2 Let , be defined as in Theorem 3.1. If is nonnegative function such that (3.16), then for some sufficiently small :
, , , we have(3.17)
for , where , are defined in Theorem 3.1;
, , , we have(3.18)
for , where , are defined in Theorem 3.1.
Proof In the proof of Theorem 3.1, before we apply Lemma 2.4, it is independent of the comparison of μ and λ. Hence, taking in (3.9), we have
We denote and get
By Lemma 2.4 and the definitions of Ω and for , we have the following result
Finally, considering two situations for and using paremeters α, β, γ to denote , , and in (3.21), we can obtain the estimations, respectively. We omit the details here. □
Remark 3.2 Although Medveď [20, 21] investigated the more general nonlinear case, his result is under the assumption that ‘ satisfies the condition (q).’ In our result, the ‘(q) condition’ is eliminated.
Letting and in Theorem 3.1, we have the following corollary.
Corollary 3.1 Suppose that , are nonnegative functions for , and is nondecreasing such that
then for some sufficiently small :
when , , , we have(3.23)
for , where , are defined in Theorem 3.1;
when , , , we have(3.24)
for , where , are defined in Theorem 3.1.
Remark 3.3 Inequality (3.22) is the extension of the well-known Ou-Iang-type inequality. Clearly, our inequality enriches the results for such an inequality.
In this section, we apply our results to discuss the boundedness and uniqueness of solutions of a Volterra-type difference equation with a weakly singular kernel.
Example 1 Suppose that satisfies the equation
for . Then we get
Letting changes (4.2) into
From (4.3), we can see that
Obviously, . Letting , , we have
Using Corollary 3.1, we get
which implies that in (4.1) is upper bounded.
Example 2 Consider the linear weakly singular difference equation
where , ϵ is an arbitrary positive number, and or . From (4.5) and (4.6), we get
which is the form of inequality (3.16). Applying Theorem 3.2, we have
for . If , let , and we obtain the uniqueness of the solution of equation (4.5).
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The authors are very grateful to both reviewers for carefully reading this paper and their comments. The authors also thank Professor Shengfu Deng for his valuable discussion. 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 authors declare that they have no competing interests.
All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.