Research Article | Open Access

Yange Huang, Wu-Sheng Wang, Yong Huang, "A Class of Volterra-Fredholm Type Weakly Singular Difference Inequalities with Power Functions and Their Applications", *Journal of Applied Mathematics*, vol. 2014, Article ID 826173, 9 pages, 2014. https://doi.org/10.1155/2014/826173

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

**Academic Editor:**Junjie Wei

#### 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).

#### References

- T. H. Gronwall, “Note on the derivatives with respect to a parameter of the solutions of a system of differential equations,”
*Annals of Mathematics*, vol. 20, no. 4, pp. 292–296, 1919. View at: Publisher Site | Google Scholar | MathSciNet - R. Bellman, “The stability of solutions of linear differential equations,”
*Duke Mathematical Journal*, vol. 10, pp. 643–647, 1943. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. S. Mitrinović, J. E. Pečarić, and A. M. Fink,
*Inequalities Involving Functions and Their Integrals and Derivatives*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991. View at: Publisher Site | MathSciNet - D. Bainov and P. Simeonov,
*Integral Inequalities and Applications*, Kluwer Academic, Dordrecht, The Netherlands, 1992. View at: Publisher Site | MathSciNet - B. G. Pachpatte,
*Inequalities for Differential and Integral Equations*, vol. 197 of*Mathematics in Science and Engineering*, Academic Press, New York, NY, USA, 1998. View at: MathSciNet - R. P. Agarwal, S. Deng, and W. Zhang, “Generalization of a retarded Gronwall-like inequality and its applications,”
*Applied Mathematics and Computation*, vol. 165, no. 3, pp. 599–612, 2005. View at: Publisher Site | Google Scholar | MathSciNet - W. Cheung, “Some new nonlinear inequalities and applications to boundary value problems,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 64, no. 9, pp. 2112–2128, 2006. View at: Publisher Site | Google Scholar | MathSciNet - W. S. Wang, “A generalized retarded Gronwall-like inequality in two variables and applications to BVP,”
*Applied Mathematics and Computation*, vol. 191, no. 1, pp. 144–154, 2007. View at: Publisher Site | Google Scholar | MathSciNet - A. Abdeldaim and M. Yakout, “On some new integral inequalities of Gronwall-Bellman-Pachpatte type,”
*Applied Mathematics and Computation*, vol. 217, no. 20, pp. 7887–7899, 2011. View at: Publisher Site | Google Scholar | MathSciNet - Y. S. Lu, W. S. Wang, X. L. Zhou, and Y. Huang, “Generalized nonlinear Volterra-Fredholm type integral inequality with two variables,”
*Journal of Applied Mathematics*, vol. 2014, Article ID 359280, 14 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - K. Cheng and C. Guo, “New explicit bounds on Gamidov type integral inequalities for functions in two variables and their applications,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 539701, 9 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - D. Henry,
*Geometric Theory of Semilinear Parabolic Equations*, Springer, New York, NY, USA, 1981. View at: MathSciNet - M. Medveď, “A new approach to an analysis of Henry type integral inequalities and their Bihari type versions,”
*Journal of Mathematical Analysis and Applications*, vol. 214, no. 2, pp. 349–366, 1997. View at: Publisher Site | Google Scholar | MathSciNet - M. Medveď, “Nonlinear singular integral inequalities for functions in two and n independent variables,”
*Journal of Inequalities and Applications*, vol. 5, no. 3, pp. 287–308, 2000. View at: Publisher Site | Google Scholar | MathSciNet - Q. H. Ma and E. H. Yang, “Estimates on solutions of some weakly singular Volterra integral inequalities,”
*Acta Mathematicae Applicatae Sinica*, vol. 25, no. 3, pp. 505–515, 2002. View at: Google Scholar | MathSciNet - Y. Wu and S. Deng, “Generalization on some weakly singular Volterra integral inequalities,”
*Journal of Sichuan University (Natural Science Edition)*, vol. 41, no. 3, pp. 472–479, 2004. View at: Google Scholar - K. M. Furati and N. Tatar, “Behavior of solutions for a weighted Cauchy-type fractional differential problem,”
*Journal of Fractional Calculus*, vol. 28, pp. 23–42, 2005. View at: Google Scholar | Zentralblatt MATH | MathSciNet - H. Ye, J. Gao, and Y. Ding, “A generalized Gronwall inequality and its application to a fractional differential equation,”
*Journal of Mathematical Analysis and Applications*, vol. 328, no. 2, pp. 1075–1081, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. Cheung, Q. H. Ma, and S. Tseng, “Some new nonlinear weakly singular integral inequalities of Wendroff type with applications,”
*Journal of Inequalities and Applications*, vol. 2008, Article ID 909156, 12 pages, 2008. View at: Publisher Site | Google Scholar | MathSciNet - Q. Ma and J. Pečarić, “Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations,”
*Journal of Mathematical Analysis and Applications*, vol. 341, no. 2, pp. 894–905, 2008. View at: Publisher Site | Google Scholar | MathSciNet - S. Deng and C. Prather, “Generalization of an impulsive nonlinear singular Gronwall-Bihari inequality with delay,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 9, no. 34, 11 pages, 2008. View at: Google Scholar | MathSciNet - Y. Wu, “A new type of weakly singular Volterra integral inequalities,”
*Acta Mathematicae Applicatae Sinica*, vol. 31, no. 4, pp. 584–591, 2008. View at: Google Scholar | MathSciNet - S. Mazouzi and N.-E. Tatar, “New bounds for solutions of a singular integro-differential inequality,”
*Mathematical Inequalities & Applications*, vol. 13, no. 2, pp. 427–435, 2010. View at: Publisher Site | Google Scholar | MathSciNet - H. Wang and K. Zheng, “Some nonlinear weakly singular integral inequalities with two variables and applications,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 345701, 12 pages, 2010. View at: Publisher Site | Google Scholar | MathSciNet - H. Ye and J. Gao, “Henry-Gronwall type retarded integral inequalities and their applications to fractional differential equations with delay,”
*Applied Mathematics and Computation*, vol. 218, no. 8, pp. 4152–4160, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Q. H. Ma and E. H. Yang, “Bounds on solutions to some nonlinear Volterra integral inequalities with weakly singular kernels,”
*Annals of Differential Equations*, vol. 27, no. 3, pp. 283–292, 2011. View at: Google Scholar | MathSciNet - K. Zheng, “Bounds on some new weakly singular Wendroff-type integral inequalities and applications,”
*Journal of Inequalities and Applications*, vol. 2013, article 159, 2013. View at: Publisher Site | Google Scholar | MathSciNet - K. Cheng, C. Guo, and M. Tang, “Some nonlinear Gronwall-Bellman-GAMidov integral inequalities and their weakly singular analogues with applications,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 562691, 9 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - T. E. Hull and W. A. J. Luxemburg, “Numerical methods and existence theorems for ordinary differential equations,”
*Numerische Mathematik*, vol. 2, pp. 30–41, 1960. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. Willett and J. S. W. Wong, “On the discrete analogues of some generalizations of Gronwall's inequality,”
*Monatshefte für Mathematik*, vol. 69, no. 4, pp. 362–367, 1965. View at: Publisher Site | Google Scholar | MathSciNet - S. Sugiyama, “On the stability problems of difference equations,”
*Bulletin of Science and Engineering Research Laboratory, Waseda University*, vol. 45, pp. 140–144, 1969. View at: Google Scholar | MathSciNet - B. G. Pachpatte and S. G. Deo, “Stability of discrete time systems with retarded argument,”
*Utilitas Mathematica*, vol. 4, pp. 15–33, 1973. View at: Google Scholar - B. G. Pachpatte, “Finite difference inequalities and discrete time control systems,”
*Indian Journal of Pure and Applied Mathematics*, vol. 9, no. 12, pp. 1282–1290, 1978. View at: Google Scholar | MathSciNet - P. Y. H. Pang and R. P. Agarwal, “On an integral inequality and its discrete analogue,”
*Journal of Mathematical Analysis and Applications*, vol. 194, no. 2, pp. 569–577, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. G. Pachpatte, “On some new inequalities related to certain inequalities in the theory of differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 189, no. 1, pp. 128–144, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. G. Pachpatte, “A note on certain integral inequality,”
*Tamkang Journal of Mathematics*, vol. 33, no. 4, pp. 353–358, 2002. View at: Google Scholar | MathSciNet - W. S. Cheung and J. Ren, “Discrete non-linear inequalities and applications to boundary value problems,”
*Journal of Mathematical Analysis and Applications*, vol. 319, no. 2, pp. 708–724, 2006. View at: Publisher Site | Google Scholar | MathSciNet - B. G. Pachpatte,
*Integral and Finite Difference Inequalities and Applications*, vol. 205 of*North-Holland Mathematics Studies*, Elsevier Science, Amsterdam, The Netherlands, 2006. - F. C. Jiang and F. W. Meng, “Explicit bounds on some new nonlinear integral inequalities with delay,”
*Journal of Computational and Applied Mathematics*, vol. 205, no. 1, pp. 479–486, 2007. View at: Publisher Site |