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.
Proof. Define a function by for all . Then, from (8), we have Applying Lemma 3 to (38), we obtain for all . Substituting (39) into (37), we obtain for all , where , , and , are defined by (34), (35), and (36), respectively. Since , , and are nonnegative and nondecreasing in both and and by (34), (35), and (36), , , and are also nonnegative and nondecreasing in both and . Using Theorem 2, from (40), we obtain for all . Substituting (41) into (38), we get our required estimation (33) of unknown function in (8).
3. Difference Inequality with Weakly Singular
For the reader’s convenience, we present some necessary Lemmas.
Lemma 5 (discrete Jensen inequality [47]). Let be nonnegative real numbers, a real number, and a natural number. Then,
Lemma 6 (discrete Hölder inequality [48]). Let be nonnegative real numbers and , positive numbers such that . Then,
Lemma 7 (see [15, 49]). Let , , and . If , , and , then where and is the well-known -function.
Now, we consider the weakly singular difference inequality (9).
Theorem 8. Let , , , , and . Assume that , , , , , , , and are nonnegative functions on and , , and are nondecreasing. If and satisfies (9), then where and , , and , are arbitrary constants.
Proof. Applying Lemma 6 with , to (8), we obtain that for all , , where is used. Applying Lemma 5 to (48), we have for all , . By discrete Jensen inequality (42) with , , from (49), we obtain that Applying Theorem 4 to (50), we have This is our required estimation (46) of unknown function in (9).
4. Applications
In this section, we apply our results to discuss the boundedness of solutions of an iterative difference equation with a weakly singular kernel.
Example 9. Suppose that satisfies the difference equation where , , , , , , , , and are nonnegative functions on , and , , and are nondecreasing. From (52), we have Let , , and , are arbitrary constants, and If Applying Theorem 8 to (53), we obtain the estimation of the solutions of difference equation (52)
Conflict of Interests
The authors declare that they have no conflict of interests.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (Project no. 11161018), the Guangxi Natural Science Foundation of China (Projects nos. 2012GXNSFAA053009, 2013GXNSFAA019022), and the Scientific Research Foundation of the Education Department of Guangxi Autonomous Region (no. 2013YB243).