Abstract
This paper is devoted to the study of Gronwall–Bellman-type inequalities on an arbitrary time scale mathbb{T}. We investigate some new explicit bounds of a certain class of nonlinear retarded dynamic inequalities of Gronwall–Bellman type on time scales. These inequalities extend some known dynamic inequalities on time scales. We also generalize and unify some continuous inequalities and their corresponding discrete analogues. To illustrate the benefits of our work, we present some applications of these results. The main results will be proved by using some analysis techniques and a simple consequence of the Keller’s chain rule on time scales.
Highlights
In 1919, Gronwall [1] discovered a celebrated inequality
A where a, ξ, ζ, and h are nonnegative constants, 0 ≤ (t) ≤ ξ heζh, t ∈ D. This inequality has been very important in the theory of differential equations and difference equations
A implies that t (t) ≤ c exp f (s) ds, t ∈ [a, b]
Summary
In 1919, Gronwall [1] discovered a celebrated inequality. He proved that if uous function defined on the interval D = [a, a + h] and is a contint0 ≤ (t) ≤ ζ (s) + ξ ds, t ∈ D, (1.1)a where a, ξ , ζ , and h are nonnegative constants, 0 ≤ (t) ≤ ξ heζh, t ∈ D.This inequality has been very important in the theory of differential equations and difference equations. He proved that if , f , a, ∈ C(R+, R+) and a is a nondecreasing function, the inequality t (t) ≤ a(t) + f (s) (s) ds, t ∈ R+, (1.3)
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