Abstract
In this paper, we establish some new integral inequalities with mixed nonlinearities for discontinuous functions, which provide a handy tool in deriving the explicit bounds for the solutions of impulsive differential equations and differential-integral equations with impulsive conditions.
Highlights
1 Introduction In recent years, the theory of impulsive differential systems has been attracting the attention of many mathematicians, and the interest in the subject is still growing
One effective method for investigating the properties of solutions to impulsive differential systems is related to the integral inequalities for discontinuous functions
Motivated by [3, 8, 17, 18], in this paper, we investigate some new integro-sum inequality with mixed nonlinearities under the condition p > 0, q > 0 (p = q): t t s xp(t) ≤ a(t) + f1(s)xq(s) ds + f2(s) g1(τ )xp(τ ) dτ ds t0
Summary
The theory of impulsive differential systems has been attracting the attention of many mathematicians, and the interest in the subject is still growing. 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: α(t) xm(t) ≤ c(t) + 2. In 2016, Zheng et al [18] considered the following nonlinear integro-sum inequality under the condition p > q > 0: xp(t). We discuss some nonlinear integro-sum inequality with positive and negative coefficients under the condition 0 < q < p < r:. Lemma 2.1 ([19]) Assume that the following conditions for t ≥ t0 hold: (i) x0 is a nonnegative constant, (ii).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
