Abstract
We discuss a class of new nonlinear weakly singular difference inequality, which is solved by change of variable, discrete Hölder inequality, discrete Jensen inequality, the mean-value theorem for integrals and amplification method, and Gamma function. Explicit bound for the unknown function is given clearly. Moreover, an example is presented to show the usefulness of our results.
1. Introduction
Being an important tool in the study of qualitative properties of solutions of differential equations and integral equations, various generalizations of Gronwall inequalities and their applications have attracted great interests of many mathematicians (such as [1–21]). Gronwall-Bellman inequality [22, 23] can be stated as follows: if and are nonnegative and continuous functions on an interval satisfying for some constant , then In 1981, Henry [2] discussed the following linear singular integral inequality: In 2007, Ye et al. [20] discussed linear singular integral inequality: In 2011, Abdeldaim and Yakout [21] studied a new integral inequality of Gronwall-Bellman-Pachpatte type On the other hand, many physical problems arising in a wide variety of applications are governed by finite difference equations. The theory of difference equations has been developed as a natural discrete analogue of corresponding theory of differential equations. 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 (such as [24–32]). Sugiyama [26] established the most precise and complete discrete analogue of the Gronwall inequality in the following form: For instance, Pachpatte [27] considered the following discrete inequality: In 2006, Cheung and Ren [29] studied Later, Zheng et al. [31] discussed the following discrete inequality: Motivated by the results given in [2, 20, 21, 32], in this paper, we discuss a new linear singular integral inequality where , , , and .
For the reader’s convenience, we present some necessary lemmas.
Lemma 1 (discrete Jensen inequality [28]). Let be nonnegative real numbers, is a real number, and is a natural number. Then
Lemma 2 (discrete Hölder inequality [30]). Let be nonnegative real numbers, and let be positive numbers such that ; then
Lemma 3. Let , , , and . If , then where , is the well-known -function.
Proof. By the definition of integration and the conditions in Lemma 3, we have Using a change of variables and , we have the estimation Since , , , and , from (14) and (15), we have the relation (13).
2. Main Result
In this section, we give the estimation of unknown function in (10). Let . For function , its 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 .
Theorem 4. Suppose that is a constant, , are nonnegative and nondecreasing functions defined on , are nonnegative, nondecreasing, and continuous functions defined on , , , , and . If satisfies (10), then where and , , , is the largest integer number such that
Proof. From (10), we have
Applying Lemma 2 with , to (24), we obtain that
where is used. Applying Lemma 3, we have
By discrete Jensen inequality (11) with , from (26) we obtain that
Again using discrete Jensen inequality (11) with , , from (27) we obtain that
For in (28), applying Lemma 2 with , , we obtain that
here Lemma 3 is used. Substituting (29) into (28), we have
where and are defined by (21) and (22), respectively. Let ; from (30) we have
Since , are nondecreasing functions, from (31) we have
where is chosen arbitrarily.
Let denote the function on the right-hand side of (32), which is a positive and nondecreasing function on . From (32), we have
Using and (33), we obtain
for all .
Let
Then
From (35), we have
It implies that, for all ,
On the other hand, by the mean-value theorem for integrals, for arbitrarily given integers , there exists in the open interval such that
for all , where is defined by (18). From (38) and (39), we have
Taking in (40) and summing up over from to , from (40) we obtain
Let denote the function on the right-hand side of (41), which is a positive and nondecreasing function on . From (41), we have
Using and (42), we obtain
From (43), we have
for all . Again by the mean-value theorem for integrals, for arbitrarily given integers , there exists in the open interval such that
where is defined by (19). From (44) and (45), we have
Taking in (46) and summing up over from to , from (46) we obtain
for all . Let
Then
From (48) and (49), we have
Using the mean-value theorem for integrals, from (50) we have
where is defined by (20). From (36), (42), (49), and (51), we have
Using and (33), from (52) we obtain that
Since is chosen arbitrarily, from (53) we have
This is our required estimation (16) of unknown function in (10).
3. Application
In this section, we apply our results to discuss the boundedness of solutions of an iterative difference equation with a weakly singular kernel.
Example 5. Suppose that satisfies the difference equation where , , , and . Then we have for all . Let , . From (18) to (20) we obtain that Using Theorem 4, we get which is an upper bound of in (55).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research was supported by National Natural Science Foundation of China (Project no. 11161018), the NSF of Guangxi Zhuang Autonomous Region (no. 2012GXNSFAA053009), the high school specialty and curriculum integration project of Guangxi Zhuang Autonomous Region (no. GXTSZY2220), the Science Innovation Project of Department of Education of Guangdong province (Grant 2013KJCX0125), and the NSF of Guangdong Province (no. s2013010013385). The authors are very grateful to the editor and the referees for their careful comments and valuable suggestions on this paper.