On Weakly Singular Versions of Discrete Nonlinear Inequalities and Applications

and Applied Analysis 3 that is, 0 > n 1 − 1 > n 2 − 1 > −1. (18) On the other hand, the zero-point of f󸀠(s) can be obtained as follows: s 0 = n 1 − 1 n 1 + n 2 − 2 < 1 2 . (19) Therefore, the function f(s) is decreasing on the interval (0, s 0 ] while increasing sharply on the interval [s 0 , 1). Consequently, for some given sufficiently small τ k , by the properties of the left-rectangle integral formula, we have


Introduction
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][2][3][4][5].Starting from the basic form,

𝑢 (𝑛) ≤ 𝑎 (𝑛) +
discussed originally by Pachpatte in [4], 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][9][10].For example, Dixon and McKee [7] investigated the convergence of discretization methods for the Volterra integral and integrodifferential equations using the following inequality: and Beesack [6] also discussed the inequality, for the second kind Abel-Volterra singular integral equations.Henry [9] presented a linear inequality to investigate some qualitative properties for a parabolic equation.In particular, to avoid the shortcoming of analysis, Medved [11][12][13] used a new method to discuss some nonlinear weakly singular integral inequalities and difference inequalities.Following Medved's work, Ma and Yang [14] improved his method to discuss a more general nonlinear weakly singular integral inequality, As for other new weakly singular inequalities, recent work can be found, for example, in [16][17][18][19][20][21][22][23][24][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.
Remark 7. When  = 1 and  = 1, the inequality was discussed by Medved [12] which is the special case of our result.Moreover, his result holds under the assumption "() satisfies the condition ();" that is, " then for  ≥ 0.
Clearly, let  = 1 and  = 1 in (31), and we can get the same formula.
Remark 10.Let ] = 2 and  = 1; we can get the interesting Henry's version of the Ou-Iang-Pachpatte type difference inequality [26].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 [27] and here we present the discrete version of their result.Furthermore, the result in [27] is established for the cases when the ordered parameter group [, , ] obeys distribution I or II (for details, see [27]) which makes the application of inequality more inconvenient.Clearly, our result is based on the concise assumption to overcome this weakness.
Theorem 13.Under assumptions ( 3 ) and ( 4 ), suppose that   ,   are nonnegative for  ∈ N. If   is nonnegative such that (8), then for 0 ≤  ≤  2 , where Ω and Ω −1 are defined as in Theorem 6, and  2 is the largest integer number such that Proof.By the definition of ã , we have