On nonlinear discrete weakly singular inequalities and applications to Volterra-type difference equations
© Zheng et al.; licensee Springer 2013
Received: 22 May 2013
Accepted: 23 July 2013
Published: 8 August 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.
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.
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 )
Lemma 2.2 (Discrete Hölder inequality, see )
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 )
where (, ) is the well-known B-function and .
Remark 2.1 Martyniuk et al.  studied the inequality , . Obviously, our result is a more general case of the nonlinear difference inequality.
for , where .
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
Denote , where .
- (1), letting , , we have(3.1)
- (2), letting , , we have(3.3)
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 .
and is given in Lemma 2.3 for .
- (1)For , , . By the definitions of Ω and , we can compute that(3.12)
- (2)For , , . Similarly to the computation above, we have(3.14)
Because , substituting it into (3.15), we can get (3.3). This completes the proof. □
and derive the estimation of the upper bound as follows.
- (1), , , we have(3.17)
- (2), , , we have(3.18)
for , where , are defined in Theorem 3.1.
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.
- (1)when , , , we have(3.23)
- (2)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.
which implies that in (4.1) is upper bounded.
for . If , let , and we obtain the uniqueness of the solution of equation (4.5).
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).
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