Weighted dynamic inequalities of Opial-type on time scales

In this paper, we will state and prove some weighted dynamic inequalities of Opial-type involving integrals of powers of a function and of its derivative on time scales which not only extend some results in the literature but also improve some of them. The main results will be proved by using some algebraic inequalities, the Hölder inequality and a simple consequence of Keller’s chain rule on time scales. As special cases of the obtained dynamic inequalities, we will get some continuous and discrete inequalities.


Introduction
In 1960, the Polish Mathematician Opial [36] proved an inequality involving integrals of functions and their derivatives; Since the publication of the above result in 1960, numerous papers with new evidence, different speculations, and augmentations have showed up in the literature. Inequalities which involve integrals of functions and their derivatives are of great importance in mathematics with applications in the theory of differential equations, approximations and probability [1-4, 7, 17, 18, 21, 22, 28, 29, 34].
As a generalization of (1.1), Beesack [10] proved that: If x is an absolutely continuous function on [a, b] where r is a positive and continuous function with Yang [44] simplified the Beesack proof and extended the inequality (1.2) as follows: If x is an absolutely continuous function on (a, b) with x(a) = 0, then b a q(t) where r is a positive and continuous function with b a dt r(t) < ∞ and q is a positive, bounded, and nonincreasing function on [a, b].
Recently, the theory of time scales, which has been initiated by Stefen Hilger in his Ph.D. thesis [30] in order to unify discrete and continuous analysis, has gained a lot of attention. During the previous decade, an impressive number of dynamic imbalances have been given by numerous creators who were inspired by certain applications (see [5, 6, 9, 12, 13, 16, 19, 20, 23-27, 31, 35, 37, 39, 41]). The general thought is to demonstrate a result for a dynamic inequality where the domain of the unknown function is a so-called time scale T, which is an arbitrary nonempty closed subset of real numbers. The three best-known time scales are T = R, T = Z and T = q Z = {q z : z ∈ Z} ∪ {0} where q > 1. The books [14] and [15] organize and summarize much of time scales calculus.
In [11], Bohner and Kaymakçalan introduced a dynamic Opial inequality which extended the continuous version inequality (1.1) to a general time scale and studied if Dynamic Opial's inequalities on time scales got a lot of consideration and numerous papers have been composed; see [11,33,38,40,42,43] and the references cited therein. Also in [11] the authors extended the inequality (1.3) of Yang and proved that: If r and q are positive rd-continuous functions on [a, b] T , b a t r(t) < ∞, q is nonincreasing and x : Karpuz et al. [33] established the same inequality as in (1.5) by replacing q σ with q of the form For p ≥ 1, Karpuz and Özkan [32] proved that: If y : [a, τ ] ∩ T → R + is delta differentiable with y(a) = 0 and y does not change sign in (a, τ ) T , then we have where p, q are positive real numbers such that p ≥ 1, and r, s are nonnegative rd-continuous functions on (a, τ ) T such that τ a r -1 p+q-1 (t) t < ∞. In the same paper, the authors proved that: If y : p, q are positive real numbers such that p ≥ 1, and r, s are nonnegative rd-continuous functions on (τ , b) T such that For p ≤ 1, Karpuz and Özkan [32] proved that: If y : [a, τ ]∩T → R + is delta differentiable with y(a) = 0 and y does not change sign in (a, τ ) T , then we have where K 3 (a, τ , p, q) = K 1 (a, τ , p, q) 2 2p-1 , p, q are positive real numbers such that p ≤ 1, p + q > 1 and r, s are nonnegative rdcontinuous functions on (a, τ ) T such that τ a r Also, in the same paper, the authors proved that: If y : [τ , b] ∩ T → R + is delta differentiable with y(b) = 0 and y does not change sign in (τ , b) T , then we have p, q are positive real numbers such that p ≤ 1, p + q > 1 and r, s are nonnegative rd- In this article, motivated by the above inequalities, we will explore some dynamic Opialtype inequalities on time scales, which generalize inequalities (1.7)-(1.12). After each result, we will study the special cases when T = R and T = N to obtain some continuous and discrete results.

Basics of time scales
Firstly, we recall some essentials of time scales, and some universal symbols that will be used in the present paper. From now on, R and Z are the set of real numbers and the set of integers, respectively.
A time scale T is an arbitrary nonempty closed subset of the set of real numbers R. Throughout the article, we assume that T has the topology that it inherits from the standard topology on R. We define the forward jump operator σ : T → T for any t ∈ T by σ (t) := inf{s ∈ T : s > t}, and the backward jump operator ρ : T → T for any t ∈ T by ρ(t) := sup{s ∈ T : s < t}.
In the preceding two definitions, we set inf ∅ = sup T (i.e., if t is the maximum of T, then σ (t) = t) and sup ∅ = inf T (i.e., if t is the minimum of T, then ρ(t) = t), where ∅ denotes the empty set.
Points that are simultaneously right-dense and left-dense are said to be dense points. Points that are simultaneously rightscattered and left-scattered are said to be isolated points.
The forward graininess function μ : T → [0, ∞) is defined for any t ∈ T by μ(t) := σ (t)-t and the backward graininess function ν : The sets T κ , T κ and T κ κ are introduced as follows: If T has a left-scattered maximum t 1 , We define the open intervals and half-closed intervals similarly.
Assume f : T → R is a function and t ∈ T κ . Then f (t) ∈ R is said to be the delta derivative of f at t if for any ε > 0 there exists a neighborhood U of t such that, for every s ∈ U, we have A function f : T → R is said to be right-dense continuous (rd-continuous) if f is continuous at all right-dense points in T and its left-sided limits exist at all left-dense points in T.
In a similar manner, a function f : T → R is said to be left-dense continuous (ldcontinuous) if f is continuous at all left-dense points in T and its right-sided limits exist at all right-dense points in T.
The delta integration by parts on time scales is given by the following formula: whereas the nabla integration by parts on time scales is given by We will use the following crucial relations between calculus on time scales T and either differential calculus on R or difference calculus on Z. Note that: where and ∇ are the forward and backward difference operators, respectively.

Main results
In this section, we will state and prove our main results.
First, we present the basic theorems that will be needed in the proof of our main results.
(3.1) Theorem 3.2 (Chain rule on time scales [14]) Let f : R → R be continuously differentiable and suppose g :

Theorem 3.3 (Dynamic Hölder inequality [14]) Let a, b ∈ T and f
Also, the main results here will be proved by employing the inequalities (see [8], page 51) Next, we enlist the following assumptions for the proofs of our main results: where K 5 (a, τ , p, q) where Proof , (3.11) Since p ≥ 1, by taking the power p for both sides of (3.11), we have (3.12) Applying the inequality (3.4) on the right-hand side of (3.12), we deduce x a r(t) y (t) p+q t p p+q + 2 p-1 y(a) p .
Since y σ = y + μy , we have Obviously, p ≥ 1. Taking power p for both sides of (3.13) and using the inequality (3.4), we deduce we see that z(a) = 0, and From (3.16), we get (3.17) From (3.14), (3.17) and since s is nonnegative on (a, τ ) T , we have Integrating the above inequality from a to τ , we get By applying Hölder inequality (3.3) with (p + q)/p and (p + q)/q on the right side of integral of the above inequality, we have From (3.1), we obtain Since z (x) ≥ 0 and x ≤ c, we get Substituting (3.19) into (3.18) and since z(a) = 0, we have The above inequality, (3.15) and (3.16) imply that which is the desired inequality (3.6).
(b) The proof follows from (a) by setting r = s. (c) It is noted from the chain rule on time scales (3.2) that (ta) p+q = (p + q) From (3.7) and (3.8) (by taking r(t) = 1) and using (3.20), we get which is the desired inequality (3.9). This completes the proof.