Study on the estimates of Gronwall–Ou-Iang dynamic integral inequalities by means of diamond-α derivatives

By the utilization of diamond-α derivatives, certain new generalizations of Ou-Iang type of dynamic integral inequalities of one independent variable on time scales are examined. The resulting inequalities are significant in the study of various fields of dynamic equations. A few mathematical applications are also presented.


Introduction
Integral inequalities have an extraordinarily advantageous position in strengthening the traditional differential and integral equations hypotheses. Gronwall [10] discussed the integral inequality for some c ≥ 0. Ou-Iang inequalities and their subsequent developments have demonstrated to be valuable devices in the concept of stability, oscillation, and boundedness and in different fields of differential and integral equations. Like Gronwall's inequality, Ou-Iang's inequality is additionally utilized to obtain an explicit bound of unknown functions. An introduction to continuous and discrete OuIang inequalities can be found in [26,27], and in [1,7,8,13,14,24] one finds generalizations to numerous integrals. Pachpatte [19] introduced the following integral inequality: c is a nonnegative constant and v ∈ R + .
It is noteworthy that the dynamic inequalities assume the role of a necessary key in the improvement of the subjective concept of dynamic equations on time scales. Hilger [12] started the advancement of analysis of time scales. The general impression is to show an equation for a dynamic equation or a dynamic inequality where the domain of the unknown function is supposed to be a time scale T. The motivation behind the hypothesis of time scales is to unify continuous and discrete investigation. In the course of recent years, many authors completed an exhaustive examination of the properties and usage of various sorts of these types of inequalities on time scale; see [2, 3, 5, 9, 16-18, 23, 25] and the references therein. Bohner et al. [4] inspected the integral inequality on time scales of the kind Consequently in 2010, Li [15] tested the subsequent nonlinear integral inequality of one independent variable associated with time scales Pachpatte [20] stepped forward to discover the extension of the integral inequality of the form Haidong [11] proposed the generalization of the nonlinear integral inequality as follows: where λ ≥ 0. Despite the fact that diamond-α derivatives cannot be identified as a standard derivative due to the absence of an antiderivative [21], its powerful distinct-alike pattern allows for accessibility in computational situations. Although there has been done much work on the integral inequalities related to the delta derivative or the nabla derivative, yet we do not carry out any significant research of integral inequalities based on diamond-α derivatives on time scales. In view of the work listed above and utilizing a similar setting to Gronwall-Bellman type inequalities, in this paper, we generalize and sum up the accompanying nonlinear integral inequalities of one variable by virtue of diamond alpha derivatives on time scales. The obtained results are useful to investigate the qualitative properties of different issues of certain classes of integral equations and evolution equations.

Basics on diamond-α derivatives and integrals
In what follows, denote R + = [0, ∞). A time scale T is a nonempty closed subset of the real line R. For v ∈ T, the forward and backward jump operators , ς : T → T are defined by (v) = inf(v, ∞) T , and ς(v) = sup(-∞, v) T simultaneously, whereas the forward and backward graininess functions , χ : T → [0, ∞) are defined by (v) = (v)v and χ(v) = vς(v), respectively. We have the set T k = T/(ς(sup T), T] and the set T k = T/[inf T, (inf T)). A function p : T → R is called regressive and defines the set of all regressive and rd-continuous functions if 1 + (v)p(v) = 0 for v ∈ T k . Also a function q : T → R is known as v-regressive provided 1χ(v)q(v) = 0 for v ∈ T k and v are the set of all v-regressive and ld-continuous functions. The delta derivative of the function j : T → R, denoted by j (t) is The function e p (v, v 0 ) = exp( v v 0 ξ (τ ) (p(τ )) τ ) stands for the -exponential function where p ∈ and the cylinder transformation is ε h (z) = 1 h log(1 + zh), log is the principal logarithm function. Analogously, the ∇-exponential function is defined by , which is a combination of the and ∇ exponential functions. C rd (T, R) denotes the class of real rd-continuous functions defined on a time scale T. If j ∈ C rd (T, R) is rd-continuous, i.e. it is continuous at right-dense points and left-sided limits exist at left-dense points in T, then there exists a function For the general primary ideas and background of time scale analysis, we refer to a book by Bohner et al. [6].
Presently, on time scales, we declare the essential lemmas that will be used later in the verifications of the paper.

Results and discussion
The statements of our main results are as follows.
be satisfied for some j > 0. Then there exist fixed constants l, n such that where Proof for v ∈ [r, v 0 ] T , then (1) can be modified as since z (v), z ∇ (v) ≥ 0, from (5) and Lemma 2.1, we deduce is non-decreasing, therefore the above inequality by using z(ς(v)) ≥ z(v) ≥ z( (v)) implies that

z(v) is bounded as it is regulated in [r, q] T and composed of C([r, q] T , R + ) and never will be zero. For a constant
, we multiply (6) by l, hence which by integrating from r to v, using Lemma 3.1 and z(r) = j 2 , leads to Take the right hand side of (7) as and here G α (v) ≥ 0 and by the delta and nabla derivative, ). Therefore (9) yields Put By the definition of Π(v) and from (10), we obtain ). Therefore, from (11), we get Π(ς (v)) , multiplying the last inequality by n, we have ≤ 2nl 1 -2α + 2α 2 2 + 4nl αα 2 2 b 2 (v) which by using (8) gives the estimate where Λ(v) is defined in (3). From (10) and (12), we obtain or equivalently The conclusion in (2) can be obtained by substituting the last inequality in √ z(v) ≤ G(v) and (5).
The following theorem is useful. that y(v), b, d, , r, q are as mentioned in Theorem 3.2 and ψ(v) ∈ C(R + , R + ) is non-decreasing. If
Proof Let v 1 ∈ [r, q] and a non-decreasing function z 1 (v) be defined by (13) can be restated as as , multiplying the previous inequality by l 1 , (17) integrating (17) from r to v, using Lemma 3.1 and z 1 (r) = j 2 , we have where By following the same steps from (8)- (10) to (19) with some alterations, we conclude where . Thus from (20), we acquire ≤ 2l 1 1 -2α + 2α 2 2 + 4l 1 αα 2 2 b 2 (v) + 4l 1 where x 0 is a constant, and H : T k × R × R → R, N : T k × T k × R → R are continuous functions. The boundedness on the solution of (23) can be addressed in the first example.