Some new Opial type dynamic inequalities via convex functions and applications

In this paper, we prove some new Opial-type dynamic inequalities on time scales. Our results are obtained in frame of convexity property and by using the chain rule and Jensen and Hölder inequalities. For illustration purpose, we obtain some particular Opial-type inequalities reported in the literature.


Introduction
The theory of convex analysis has become one of the most significant fields of mathematics due to its widespread usefulness in diverse fields of pure and applied sciences. The concept of convexity has been utilized in several directions using innovative techniques to study and unify different problems. Consequently, many new inequalities s associated with convex functions have been derived by many researchers [6][7][8]10].
On other direction, integral inequalities on time scale have been a topic of debate amongst interested researchers. Due to their numerous application potentials, several variants have been established by many authors; see for instance [1,2]. One of the most attractive inequalities that engaged many researchers is the Opial inequality [11]. During the last years, it has been realized that the Opial inequality and its generalizations play a fundamental role in establishing the existence-uniqueness and stability of initial and boundary value problems for various types of differential equations [12,14]. For the sake of completeness, we review some relevant results of Opial inequalities in the context of time scales calculus.
In [3], the authors proved some dynamic inequalities of Opial type on time scales. One of the results states that: If : [0, a] ∩ T → R is delta differentiable with (0) = 0, then Further, it was proved that, if η and ζ are positive rd-continuous functions on [0, b] T , ζ is non-increasing and : In [9], the authors replaced ζ σ with ζ and proved an inequality similar to (2) where ζ is a positive function on [a, b] T , : On the other hand, the authors in [15,16] proved that, if ζ is a positive and non-increasing where : In [13], the author generalized (5) and proved some new dynamic inequalities with two weight functions η and ζ . In particular, it was proved that, if η and ζ are non-negative functions on [a, b] where The objective of this paper is to prove some new dynamic inequalities of Opial type on time scales by using the convexity property and Jensen inequality. Our results in particular cases yield some of the recent results reported on Opial-type inequalities. The paper adheres to the following plan. In Sect. 2, we present some essential preliminaries on time scales as well as some fundamental inequalities. In Sect. 3, we prove the main results of the paper. For illustration, we derive some particular cases of the main results in Sect. 4.

Preliminaries on time scales
In this section, we assemble some definitions and concepts on the theory of time scales calculus. Further, some basic inequalities in the context of time scales are stated. For more details, we refer the reader to the two pioneering monographs [4,5].
A time scale T is an arbitrary nonempty closed subset of the real numbers R. We assume throughout that T has the topology that it inherits from the standard topology on the real numbers R. We define the forward jump operator σ : T → T by σ (θ ) := inf{s ∈ T : s > θ }, for θ ∈ T. The mapping μ : T → [0, ∞) defined by μ(θ ) := σ (θ )θ is called the graininess of T.
A point θ ∈ T is said to be right-dense and right-scattered, if σ (θ ) = θ and σ (θ ) > θ , respectively. A function η : T → R is called rd-continuous provided it is continuous at all right-dense points in T and its left-sided limits exist (finite) at all left-dense points in T. The set of all such rd-continuous functions is denoted by C rd (T). We define η σ := η • σ and define the time scale interval [a, b] Fix θ ∈ T and let η : T → R. Define η (θ ) to be the number (if it exists) with the property that given any > 0 there is a neighborhood s of θ with In this case, we say η (θ ) is the (delta) derivative of η at θ and that η is (delta) differentiable at θ . We will make use of the following product and quotient rules for the derivative of the product ηζ and the quotient η/ζ (where ζ ζ σ = 0, here ζ σ := ζ • σ ) of two differentiable function η and ζ : For a, b ∈ T and a delta differentiable function η, the Cauchy integral of η is defined by The integration by parts formula on time scales is given by The chain rule formula (see [4, Theorem 1.87]) for appropriate functions η and ζ is given as However, we may define another chain rule by where a, b ∈ T and η, ζ ∈ C rd (I, R), p > 1 and 1/p + 1/q = 1. The particular case when p = q = 2 yields the Cauchy-Schwarz inequality, On the other hand, Jensen's inequality on time scales [4,Theorem 6.17] is given by where

Main results
In this section, we state and prove the main results. Throughout this paper we assume that the appropriate functions are delta differentiable and the integrals in the statements of the theorems are assumed to exist.

Theorem 3.1 Let T be a time scale with d, τ ∈ T and assume F is non-negative and in-
This implies that and Applying the chain rule (9), we see that Replacing w(θ ) with |ζ (θ )| and using (15), we have Now using (14), we get (note F is a convex function) which is (13). The proof is complete.

Theorem 3.2 Let T be a time scale with τ , ∈ T and assume that F is non-negative and increasing on
This implies that and Applying the chain rule (9), we see that Replacing w(θ ) with |ζ (θ )| and using (18), we have Now using (17), we get (note F is non-negative and convex) which is (16). The proof is complete.
If we assume that there exists τ ∈ (d, ) so that then we have the following two results when ζ (d) = 0 = ζ ( ).

Theorem 3.3 Let (19) be satisfied and T be a time scale with d, ∈ T. Assume that F is non-negative and increasing on [0, ∞). If F is convex and
Proof It follows that Thus, using Theorems 3.1, 3.2 and (19), we obtain which is the desired inequality (20). The proof is complete.
Proof Since h is a convex function, then using Jensen's inequality (12), we have and since d η(θ ) θ = 1 and h is increasing we obtain Finally, using the increasing behavior of F and substituting (22) into (20), we have which is (21). The proof is complete.

Some applications
In this section, we use the main results of Sect. 3 to obtain Opial-type inequalities.

Remark 4.1 If ν = 1, then inequality (27) becomes the Olech inequality
Following the same arguments of the proofs of Theorem 4.1 and Theorem 4.2, one can prove the following theorems.

Theorem 4.3
Let T be a time scale with τ , ∈ T and > 1.
Similar to Theorem 4.5 and Theorem 4.6, one can prove the following theorems.