A Class of Volterra-Fredholm Type Weakly Singular Difference Inequalities with Power Functions and Their Applications

Abstract

We discuss a class of Volterra-Fredholm type difference inequalities with weakly singular. The upper bounds of the embedded unknown functions are estimated explicitly by analysis techniques. An application of the obtained inequalities to the estimation of Volterra-Fredholm type difference equations is given.

1. Introduction

Being an important tool in the study of existence, uniqueness, boundedness, stability, invariant manifolds, and other qualitative properties of solutions of differential equations and integral equations, various generalizations of Gronwall inequalities [1, 2] and their applications have attracted great interests of many mathematicians [3–5]. Some recent works can be found in [6–28].

In 1981, Henry [12] discussed the following linear singular integral inequality: In 2007, Ye et al. [18] discussed linear singular integral inequality In 2014, Cheng et al. [28] discussed the following inequalities:

On the other hand, difference inequalities which give explicit bounds on unknown functions provide a very useful and important tool in the study of many qualitative as well as quantitative properties of solutions of nonlinear difference equations. More attentions are paid to some discrete versions of Gronwall-Bellman type inequalities (such as [29–50]).

In 2002, Pachpatte [36] discussed the following difference inequality: In 2010, Ma [45] discussed the following difference inequality with two variables: In 2014, Huang at el. [50] discussed the following linear singular difference inequality:

Motivated by the results given in [6, 11, 28, 36, 45, 49, 50], in this paper, we discuss the following inequalities:

2. Difference Inequalities with Two Variables

Throughout this paper, let , , and . For a function , its first-order difference is defined by . Obviously, the linear difference equation with the initial condition has the solution . For convenience, in the sequel, we complementarily define that .

Lemma 1. Assume that , , , and are nonnegative functions on . If and satisfies the following difference inequality: then

Proof. Since is a constant. Let . From (10), we have Since is nonnegative, we have Let and in (13) and substituting and , successively, we obtain From (14), we have where . Substituting inequality (15) into (13), we get the explicit estimation (11) for .

Theorem 2. Assume that , , , , , and are nonnegative functions on and , , and are nondecreasing in both and . If and satisfies the difference inequality (7), then for all .

Proof. Fixing any arbitrary , from (7), we have for all , where , , and are nondecreasing in both and .
Define a function by the right side of (18); that is, for all . Obviously, we have Using the difference formula and relation (20), from (21), we have where we have used the monotonicity of in . From (22), we observe that On the other hand, by the mean-value theorem for integrals, for arbitrarily given integers , with , , there exists in the open interval such that From (23) and (24), we have Let and in (25), and substituting and , successively, we obtain It implies that Using (20) and (21), from (27), we have for all . Taking and in (28), we have Since , are chosen arbitrarily, we replace and in (29) with and , respectively, and obtain that for all . Applying the result of Lemma 1 to inequality (30), we obtain desired estimation (17).

Lemma 3 (see [39]). Let , , and . Then,

Theorem 4. Assume that , , , , , and are defined as in Theorem 2 and that and . If and satisfies difference inequality (8), then for all , where and are arbitrary constants.