- Open Access
Generalized integral inequalities for discontinuous functions with two independent variables and their applications
© Mi; licensee Springer 2014
- Received: 29 October 2013
- Accepted: 8 May 2014
- Published: 15 December 2014
This paper investigates integral inequalities for discontinuous functions with two independent variables involving two nonlinear terms. We do not require that is in the class ℘ or the class ȷ in Gallo and Piccirllo’s paper (Nonlinear Stud. 19:115-126, 2012). My main results can be applied to generalize Borysenko and Iovane’s results (Nonlinear Anal., Theory Methods Appl. 66:2190-2230, 2007) and to give results similar to Gallo-Piccirllo’s. Examples to show the bounds of solutions of a partial differential equation with impulsive terms are also given, which cannot be estimated by Gallo and Piccirllo’s results.
- integral inequalities
- discontinuous functions
Here is an unknown nonnegative continuous function with the exception of the points where there is a finite jump: , .
with two independent variables involving two nonlinear terms and where we do not restrict and to the class ℘ or the class ȷ. Moreover, () has a more general form. We also show that many integral inequalities for discontinuous functions such as (1.3)-(1.6) can be reduced to the form of (1.8). Finally, our main result is applied to an estimation of the bounds of the solutions of a partial differential equation with impulsive terms.
for , and , and let denote the first-order partial derivative of with respect to x and .
Consider (1.8) and assume that
(H1) is defined on Ω and ; is a nonnegative constant for any positive integer i;
(H2) () are continuous and nonnegative functions on and satisfy a certain condition: () if , for arbitrary ;
(H3) and are continuous and nonnegative functions on and are positive on such that is nondecreasing;
(H4) is continuous and nonnegative on Ω;
(H5) is nonnegative and continuous on Ω with the exception of the points where there is a finite jump: , . Here if , , , and , ;
(H6) and () are continuously differentiable and nondecreasing such that on and on .
Let for and where is a given positive constant. Clearly, is strictly increasing so its inverse is well defined, continuous, and increasing in its corresponding domain.
The proof is given in Section 3.
Remark 2.1 If satisfies for , then i in Theorem 2.1 can be any nonzero integer. Reference  pointed out that different choices of in do not affect our results for . If , then define and (2.1) is still true.
- (1)If we take , , , , , and , then (1.8) reduces to (1.3). It is easy to check that and . From Theorem 2.1, we know that for
If we let , , and , the above inequality is the same as (1.8).
which looks much more complicated than (1.8).
This completes the proof of Corollary 2.1. □
According to Corollary 2.1, we have the following result.
where , , and are given in Corollary 2.1.
Obviously, for any , is positive and nondecreasing with respect to x and y, () is nonnegative and nondecreasing with respect to x and y for each fixed s and t. They satisfy and .
This means that (2.1) is true for and if replace with .
This implies that (2.1) is true for and if replace by .
This shows that (2.1) is true for if replace by . By induction, we know that (2.1) holds for for any nonnegative integer i. This completes the proof of Theorem 2.1.
where , , , and .
(C1) where , are nonnegative and continuous on Ω, , for , , ;
(C2) where and m are nonnegative constants.
Remark 4.1 From (4.4), we know . Clearly, does not hold for large . Thus, does not belong to the class ℘ in . Again does not hold for large so does not belong to the class ȷ in . Hence, the results in  cannot be applied to the inequality (4.1).
The author is grateful to Dr. Shengfu Deng for many helpful conversations during the course of this work. This research was supported by National Natural Science Foundation of China (No. 11371314), Guangdong Natural Science Foundation (No. S2013010015957), and the Project of Department of Education of Guangdong Province, China (No. 2012KJCX0074).
- Agarwal RP: On an integral inequality in n independent variables. J. Math. Anal. Appl. 1982, 85: 192–196. 10.1016/0022-247X(82)90034-8MathSciNetView ArticleMATHGoogle Scholar
- Agarwal RP, Deng SF, Zhang WN: Generalization of a retard Gronwall-like inequality and its applications. Appl. Math. Comput. 2005, 165: 599–612. 10.1016/j.amc.2004.04.067MathSciNetView ArticleMATHGoogle Scholar
- Bihari IA: Generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Acta Math. Hung. 1956, 7: 81–94. 10.1007/BF02022967MathSciNetView ArticleMATHGoogle Scholar
- Choi SK, Deng SF, Koo NJ, Zhang WN: Nonlinear integral inequalities of Bihari-type without class H . Math. Inequal. Appl. 2005, 8: 643–654.MathSciNetMATHGoogle Scholar
- Cheung W, Iovane G: Some new nonlinear inequalities and applications to boundary value problems. Nonlinear Anal. 2006, 64: 2112–2128. 10.1016/j.na.2005.08.009MathSciNetView ArticleMATHGoogle Scholar
- Rakhmatullina LF: On application of solvability conditions of Chaplygin problems of boundedness and solution stability of differential equations. Izv. Vysš. Učebn. Zaved., Mat. 1959, 2: 198–201.Google Scholar
- Sun YG: On retarded integral inequalities and their applications. J. Math. Anal. Appl. 2005, 301: 265–275. 10.1016/j.jmaa.2004.07.020MathSciNetView ArticleMATHGoogle Scholar
- Wu Y, Li XP, Deng SF: Nonlinear delay discrete inequalities and their applications to Volterra type difference equations. Adv. Differ. Equ. 2010., 2010: Article ID 795145Google Scholar
- Zheng KL, Zhong SM: Nonlinear sum-difference inequalities with two variables. Int. J. Appl. Math. Comput. Sci. 2010, 6: 31–36.Google Scholar
- Wang WS: A generalized retarded Gronwall-like inequality in two variable and applications to BVP. Appl. Math. Comput. 2007, 191: 144–154. 10.1016/j.amc.2007.02.099MathSciNetView ArticleMATHGoogle Scholar
- Zhang WN, Deng SF: Projected Gronwall-Bellman’s inequality for integrable functions. Math. Comput. Model. 2001, 34: 394–402.MathSciNetView ArticleMATHGoogle Scholar
- Angela G, Anna MP: On some generalizations Bellman-Bihari result for integro-functional inequalities for discontinuous functions and their applications. Bound. Value Probl. 2009., 2009: Article ID 808124Google Scholar
- Borysenko SD: About asymptotical stability on linear approximation of the systems with impulse influence. Ukr. Mat. Zh. 1982, 35: 144–150.Google Scholar
- Borysenko SD: Integro-sum inequalities for functions of many independent variables. Differ. Equ. 1989, 25: 1634–1641.MathSciNetGoogle Scholar
- Borysenko SD, Ciarletta M, Iovane G: Integro-sum inequalities and motion stability of systems with impulse perturbations. Nonlinear Anal. 2005, 62: 417–428. 10.1016/j.na.2005.03.032MathSciNetView ArticleMATHGoogle Scholar
- Borysenko SD, Iovane G: About some new integral inequalities of Wendroff type for discontinuous functions. Nonlinear Anal. 2007, 66: 2190–2203. 10.1016/j.na.2006.03.008MathSciNetView ArticleMATHGoogle Scholar
- Borysenko SD, Iovane G, Giordano P: Investigations of properties motion for essential nonlinear systems perturbed by impulses on some hypersurfaces. Nonlinear Anal. 2005, 62: 345–363. 10.1016/j.na.2005.03.031MathSciNetView ArticleMATHGoogle Scholar
- Borysenko SD, Toscano S: Impulsive differential system: the problem of stability and practical stability. Nonlinear Anal. 2009, 71: e1843-e1849. 10.1016/j.na.2009.02.084MathSciNetView ArticleMATHGoogle Scholar
- Gallo A, Piccirllo AM: On some generalizations Bellman-Bihari result for integrofunctionam inequalities for discontinuous functions and theri applications. Bound. Value Probl. 2009., 2009: Article ID 808124Google Scholar
- Gallo A, Piccirllo AM: About new analogies of Gronwall-Bellman-Bihari type inequalities for discontinuous functions and estimated solution for impulsive differential systems. Nonlinear Anal. 2007, 67: 1550–1568. 10.1016/j.na.2006.07.038MathSciNetView ArticleMATHGoogle Scholar
- Iovane G: Some new integral inequalities of Bellman-Bihari type with delay for discontinuous functions. Nonlinear Anal. 2007, 66: 498–508. 10.1016/j.na.2005.11.043MathSciNetView ArticleMATHGoogle Scholar
- Iovane G: On Gronwall-Bellman-Bihari type integral inequalities in several variables with retardation for discontinuous functions. Math. Inequal. Appl. 2008, 11: 599–606.MathSciNetMATHGoogle Scholar
- Mitropolskiy YA, Iovane G, Borysenko SD: About a generalization of Bellman-Bihari type inequalities for discontinuous functions and theri applications. Nonlinear Anal. 2007, 66: 2140–2165. 10.1016/j.na.2006.03.006MathSciNetView ArticleMATHGoogle Scholar
- Gallo A, Piccirillo AM: Multidimensional impulse inequalities and general Bihari-type inequalities for discontinuous functions with delay. Nonlinear Stud. 2012, 19: 115–126.MathSciNetMATHGoogle Scholar
- Samoilenko AM, Perestjuk NA: Stability of the solutions of differential equations with impulsive action. Appl. Math. Comput. 2005, 165: 599–612. 10.1016/j.amc.2004.04.067MathSciNetView ArticleGoogle Scholar
- Samoilenko AM, Perestyuk NA: Differential Equations with Impulse Effect. Visha Shkola, Kyiv; 1987.Google Scholar
- Samoilenko AM, Perestyuk NA: Stability of the solutions of differential equations with impluse action. Appl. Math. Comput. 2005, 195: 599–613.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.