Abstract
Some new nonlinear impulsive differential and integral inequalities with nonlocal integral jump conditions are presented in this paper. Using the method of mathematical induction, we obtain a new upper bound estimation of certain differential and integral inequalities; these inequalities have both nonlocal integral jump and weakly singular kernels. Finally, we give two examples of these inequalities in estimating solutions of certain equations with Riemann–Liouville fractional integral conditions.
Highlights
As is well known, impulsive differential and impulsive integral inequalities play a fundamental part in the study of theory of impulsive equations
In [12], Thiramanus and Tariboon developed the impulsive inequalities with the following integral jump conditions: tk –σk m tk+ ≤ dkm(tk) + ck m(s) ds + bk, k = 1, 2, . . . , tk –τk where 0 ≤ σk ≤ τk ≤ tk – tk–1
We extend the theories of linear impulsive system to nonlinear impulsive inequalities with nonlocal jump conditions
Summary
Impulsive differential and impulsive integral inequalities play a fundamental part in the study of theory of impulsive equations (see [1,2,3,4]). In [12], Thiramanus and Tariboon developed the impulsive inequalities with the following integral jump conditions: tk –σk m tk+ ≤ dkm(tk) + ck m(s) ds + bk, k = 1, 2, . We note that a weak singular kernel is involved in the nonlocal jump conditions They gave the estimation of m(t) as follows. These results play fundamental roles in the global existence, uniqueness, stability and other properties of various linear impulsive differential and integral equations. We extend the theories of linear impulsive system to nonlinear impulsive inequalities with nonlocal jump conditions. M (t) ≤ p(t)m(t) + q(t)mα(t), t = tk, with different nonlocal jump conditions, we give the upper bound estimation of the inequality, and an estimation of solutions of certain nonlinear equations is involved. We have the following inequality: m(t)1–α ≤ m1–α(t0)
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