- Open Access
Nonlinear integral inequalities with delay for discontinuous functions and their applications
© Mi et al.; licensee Springer. 2013
- Received: 16 May 2013
- Accepted: 26 August 2013
- Published: 11 September 2013
This paper investigates integral inequalities with delay for discontinuous functions involving two nonlinear terms. We do not require the classes ℘ and ȷ in Gallo and Piccirillo’s paper (Bound. Value Probl. 2009:808124, 2009). Our main results can be applied to generalize Gallo and Piccirillo’s results and Iovane’s results (Nonlinear Anal., Theory Methods Appl. 66:498-508, 2007). Examples to show the bounds of solutions of an impulsive differential equation are also given, which can not be estimated by Gallo and Piccirillo’s results.
- integral inequalities
- discontinuous functions
- impulsive differential equations
with a general nonlinear term of u. They assumed that or , where the class ℘ consists of all nonnegative, nondecreasing and continuous functions on such that and for all and , and the class ȷ consists of all positive, nondecreasing and continuous functions on such that and for all and . The classes ℘ and ȷ allow a reduction of to the case of a constant by dividing if is a positive and nondecreasing function. Actually, when we study behaviors of solutions of impulsive differential equations, may not be a nondecreasing function, and w may not satisfy the condition or . For example, does not belong to the class ℘ and ȷ for any and large . Thus, it is interesting to avoid such conditions.
with two nonlinear terms and of u, where we do not restrict and to the class ℘ or the class ȷ. We also show that many integral inequalities for discontinuous functions such as (1.3), (1.4) and (1.6) can be reduced to the form of (1.7). Our main result is applied to estimate the bounds of solutions of an impulsive ordinary differential equation.
Consider (1.7), and assume that
(C1) and are continuous and nondecreasing functions on and are positive on such that is nondecreasing;
(C2) is defined on and ; is a nonnegative constant for any positive integer i;
(C3) and are continuous and nonnegative functions on ;
(C4) and are continuously differentiable and nondecreasing such that and on ;
(C5) For , is nonnegative and piecewise-continuous with the first kind of discontinuities at the points , where i is a nonnegative integer and .
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.
- (2)Take , , , , and . Hence, (1.7) becomes (1.1). It is easy to check that and . From Theorem 2.1, we know that for ,
- (3)Clearly, (1.2) and (1.3) are special cases of (1.7). If on , then (1.6) can be rewritten as
Let and , the inequality above is same as (1.7). Similarly, (1.5) can also be reduced to (1.7).
which looks more complicated than (1.7).
where and its related functions are defined as in Theorem 2.1 by replacing with , .
Remark 2.2 Using the same way, we can change inequality (1.4) into the form of (1.7) with , , , and .
Obviously, is positive and nondecreasing in x, and is nonnegative and nondecreasing in x for each fixed s and . 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 yields 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 , , (), , , for all .
, where , are nonnegative and continuous on ;
, where and m are nonnegative constants.
Remark 4.1 From (4.3), we know that . Clearly, does not hold for large . Thus, does not belong to the class ℘. Again does not hold for large , so does not belong to the class ȷ. Hence, the results in  can not be applied to inequality (4.3).
This work was supported by the Project of Department of Education of Guangdong Province, China (No. 2012KJCX0074), the PhD Start-up Fund of the Natural Science Foundation of Guangdong Province, China (No. S2011040000464), the China Postdoctoral Science Foundation-Special Project (No. 201104077), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry (No. (2012)940), the Natural Fund of Zhanjiang Normal University (No. LZL1101), and the Doctoral Project of Zhanjiang Normal University (No. ZL1109).
- 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 ArticleGoogle Scholar
- Agarwal RP, Deng S, Zhang W: Generalization of a retarded Gronwall-like inequality and its applications. Appl. Math. Comput. 2005, 165: 599–612. 10.1016/j.amc.2004.04.067MathSciNetView ArticleGoogle Scholar
- Bihari I: A 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 ArticleGoogle Scholar
- Borysenko S, Matarazzo G, Pecoraro M: A generalization of Bihari’s lemma for discontinuous functions and its application to the stability problem of differential equations with impulse disturbance. Georgian Math. J. 2006, 13: 229–238.MathSciNetGoogle Scholar
- Cheung W: Some new nonlinear inequalities and applications to boundary value problems. Nonlinear Anal. 2006, 64: 2112–2128. 10.1016/j.na.2005.08.009MathSciNetView ArticleGoogle Scholar
- Choi SK, Deng S, Koo NJ, Zhang W: Nonlinear integral inequalities of Bihari-type without class H . Math. Inequal. Appl. 2005, 8: 643–654.MathSciNetGoogle Scholar
- Pachpatte BG: Integral inequalities of the Bihari type. Math. Inequal. Appl. 2002, 5: 649–657.MathSciNetGoogle Scholar
- Pinto M: Integral inequalities of Bihari-type and applications. Funkc. Ekvacioj 1990, 33: 387–403.Google Scholar
- Banov D, Simeonov P Mathematics and Its Applications 57. In Integral Inequalities and Applications. Kluwer Academic, Dordrecht; 1992.View ArticleGoogle Scholar
- Borysenko S, 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 ArticleGoogle Scholar
- Borysenko SD, Toscano S: Impulsive differential systems: The problem of stability and practical stability. Nonlinear Anal. 2009, 71: e1843-e1849. 10.1016/j.na.2009.02.084MathSciNetView ArticleGoogle Scholar
- Hu SC, Lakshmikantham V, Leela S: Impulsive differential systems and the pulse phenomena. J. Math. Anal. Appl. 1989, 137: 605–612. 10.1016/0022-247X(89)90266-7MathSciNetView ArticleGoogle 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.MathSciNetGoogle Scholar
- Lakshmikantham V, Bainov DD, Simeonov PS Series in Modern Applied Mathematics 6. In Theory of Impulsive Differential Equations. World Scientific, Teaneck; 1989.View ArticleGoogle Scholar
- Mitropolskiy YuA, Samoilenko AM, Perestyuk N: On the problem of substantiation of overoging method for the second equations with impulse effect. Ukr. Mat. Zh. 1977, 29: 750–762.Google Scholar
- Mitropolskiy YuA, Iovane G, Borysenko SD: About a generalization of Bellman-Bihari type inequalities for discontinuous functions and their applications. Nonlinear Anal. 2007, 66: 2140–2165. 10.1016/j.na.2006.03.006MathSciNetView ArticleGoogle Scholar
- Samoilenko AM, Perestyuk N: Differential Equations with Impulse Effect. Visha Shkola, Kyiv; 1987.Google 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 ArticleGoogle Scholar
- Borysenko SD, Iovane G, Giordano P: Investigations of the 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 ArticleGoogle Scholar
- Borysenko, SD: About one integral inequality for piece-wise continuous functions, p. 323. In: Proc. Int. Kravchuk Conf. Kyiv (2004)Google 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 ArticleGoogle Scholar
- Gallo A, Piccirillo AM: 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
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