Abstract
Some new nonlinear impulsive differential inequalities and integral inequalities with integral jump conditions for discontinuous functions are established using the method of successive iteration. These jump conditions at a discontinuous point are related to the integral conditions of the past state, which can be used in the qualitative analysis of the solutions to certain nonlinear impulsive differential systems.
Highlights
Impulsive differential equations, that is, differential equations involving impulse effect, appear as a natural description of observed evolution phenomena of several real world problems
Many processes studied in applied sciences are represented by impulsive differential equations
The situation is quite different in many physical phenomena that have a sudden change in their states such as mechanical systems with impact, biological systems such as heart beats, blood flows, population dynamics theoretical physics, pharmacokinetics, mathematical economy, chemical technology, electric technology, metallurgy, ecology, industrial robotics, biotechnology processes, and so on
Summary
That is, differential equations involving impulse effect, appear as a natural description of observed evolution phenomena of several real world problems. In spite of the importance of impulsive differential equations, the development of the theory of impulsive differential equations has been quite slow due to special features possessed by impulsive differential equations in general, such as pulse phenomena, confluence, and loss of autonomy Among these results, differential inequalities and integral inequalities with impulsive effects play increasingly important roles in the study of quantitative properties of solutions of impulsive differential systems. Under different jump conditions, we will study the upper-bound estimation of the nonlinear inequality m (t) ≤ p(t)m(t) + q(t)mα(t). T ≥ t , m (t) ≤ p(t)m(t) + q(t)mα(t), < α < , tk –σk m –α tk+ ≤ ck m –α(s) ds + bk, tk –τk where ck, bk, σk, τk are defined as in Theorem . Tn –σn m –α tn+ ≤ m –α(tn) + cn m –α(s) ds + bn tn –τn m –α(t ) Ei + Hi Ej e s q(ν)e s ν bn tn –τn t
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