Abstract
In this article, some Gronwall-type integral inequalities with impulses on time scales are investigated. Our results extend some known dynamic inequalities on time scales, unify and extend some continuous inequalities and their corresponding discrete analogues. Some applications of the main results are given in the end of this article.
Highlights
The theory of time scales, which has recently received a lot of attention, was initiated by Hilger [1] in his Ph.D. thesis in 1988 to contain both difference and differential calculus in a consistent way
A few papers have studied the theory of integral inequalities on time scales
We study some Gronwall-type integral inequalities on time scales, which extend some known dynamic inequalities on time scales, unify and extend some continuous inequalities and their corresponding discrete analogues
Summary
The theory of time scales, which has recently received a lot of attention, was initiated by Hilger [1] in his Ph.D. thesis in 1988 to contain both difference and differential calculus in a consistent way. Since many authors have investigated the dynamic equations, the calculus of variations and the optimal control problem on time scales (see [2-11]). It is helpful in our result to study dynamic systems and optimal control problem on time scales. Define PClrd(T, R)(PCrrd(T, R)) = {x : T → R|x, PClrd(T, R)(PCrrd(T, R)) = {x : T → R|x is continuous at right-dense point t ∈ T \ , x exists right limit at t Î Λ or left-dense point. ||x||PC = max sup ||x(t+)||, sup ||x(t−)|| ., x is left (right) continuous and exists right t∈T t∈T (left) limit at t Î Λ}. The exponential function ep on time scale plays a very important role for discussing dynamic equations on time scales. Define the generalized exponential function as follows:.
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