Dynamic Inequalities On Time Scales by Ravi Agarwal Donal O'Regan & Samir Saker

Author:Ravi Agarwal, Donal O'Regan & Samir Saker
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(3.3.22)

where

(3.3.23)

Proof. Since y Δ (t) does not change sign in , we have

This implies that

Now, since r is nonnegative on , then it follows from the Hölder inequality lder inequality with

that

Then, for a ≤ x ≤ τ, we get (note y(a) = 0) that

(3.3.24)

Since , we have

Applying the inequality ( 3.3.2), we get (note p ≤ 1) that

(3.3.25)

Setting

(3.3.26)

we see that z(a) = 0, and

(3.3.27)

From this, we get that

(3.3.28)

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