Nonlinear integral inequalities with delay for discontinuous functions and their applications

Abstract

This paper investigates integral inequalities with delay for discontinuous functions involving two nonlinear terms. We do not require the classes ℘ and j 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. MSC: 26D15; 26D20

Some new integral inequalities with mixed nonlinearities for discontinuous functions

“…One effective method for investigating the properties of solutions to impulsive differential systems is related to the integral inequalities for discontinuous functions (integro-sum inequalities). Up to now, a lot of integro-sum inequalities (for example, [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein) have been discovered. For example, in 2003, Borysenko [3] considered the following integro-sum inequality: In 2009, Gallo and Piccirillo [8] further discussed the following nonlinear integro-sum inequality: In 2012, Wang et al [17] considered the nonlinear integro-sum inequality as follows: …”