Reconstruction of nonlinear integral inequalities associated with time scales calculus

In this paper, we build up some generalizations of nonlinear integral inequalities and recreate the results of some Pachpatte’s inequalities on time scales. We not just settle new estimated bounds of a particular class of nonlinear retarded dynamic inequalities, but additionally determine and unify continuous analogs alongside a subjective time scale T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{T}$\end{document}. We demonstrate applications of the treated inequalities to reflect the benefits of our work. The key effects will be proven by using the analysis procedure and the standard time-scale comparison theorem technique.


Introduction
A dynamic system containing discrete and continuous times is an important tool for modeling real-world problems. It is fair to check if a structure can be given that helps us to integrate all dynamic systems simultaneously to gain some perspective and a superior comprehension of the contrasts between discrete and continuous domains. To counter this, a concept was composed by Hilger [1]. The primary target of dynamic equations on time scales is that they construct a connection between continuous and discrete situations. A while later, this perception was evolved by many researchers [2][3][4].
Linear and nonlinear versions of Pachpatte's inequalities on time scale have been a matter of conversation for quite a while. These inequalities were advanced by means of several authors [17][18][19][20][21][22]. Bohner has planned an assortment of dynamic inequalities, which are basically founded on the inequality of Gronwall. Originally, Bohner et al. [23] unify the continuous-type Gronwall inequality as follows where j is a right-dense continuous function, x ≥ 0 is a regressive right-dense continuous function, and T is a time scale. Bohner et al. [24] further suggested the integral inequality on time scales x(l) ≤ a(l) + p(l) l l 0 b(l 1 )x(l 1 ) + q(l 1 ) l 1 .
After that in 2010, Li [25] considered the nonlinear integral inequality of one independent variable associated with time scales Pachpatte [26] stepped forward to discover the extension of the integral inequality of the form Meng et al. [27] inquired the expansion of the nonlinear integral inequality on time scales as follows: Recently, in 2017, Haidong [28] proved the retarded Volterra-Fredholm integral inequality on time scales where λ ≥ 0. To delineate the hypothetical theorems, it has been demonstrated that the acquired inequalities can be utilized as significant apparatuses in the investigation of specific properties of dynamic equations on time scales.
Moreover, Nasser et al. [29] introduced some new generalizations and rectifications of many known results of Pachpatte kind, consolidating two nonlinear integral terms on time scales. These acquired consequences played a crucial role in reading a few lessons of integral and integro-differential equations.
Often, the previously noted inequalities are not practical directly in the evaluation of certain retarded differential and integral equations. Therefore it is alluring to discover a few new estimates in which the nonretarded term l is changed to the retarded argument ρ(l) in specific circumstances. To overcome this hollow, primarily based on the expertise of the research mentioned, in this text, we are able to seek the nonlinear dynamic inequalities constructed up for the solution of the integral inequalities and unifying some known results in the literature.
At the point when we want to examine certain properties of a differential equation, these types of inequalities have many applications (see [30][31][32]). Around the completion of this paper, we discuss several applications to investigate the uniqueness and global existence of solutions of nonlinear delay dynamic integral equations.
The remaining portions of the document are structured as follows. In Sect. 2, we describe major realities and fundamental lemmas that are key devices for our primary results. Theoretical conversations on nonlinear dynamic Pachpatte's inequalities on general time scales with some finishing remarks are committed in Sect. 3. The final section accomplishes the applications of the abstract results.

Preliminaries on time scales
A time scale T is a nonempty closed subset of the real line R. For l ∈ T, the forward jump operator σ : T → R is defined by σ (l) = inf{n ∈ T : n > l}, the backward jump operator ς : T → R by ς(l) = sup{n ∈ T : n < l} and the graininess function ψ : is the set of all regressive and rd-continuous functions, and + = {y ∈ : 1 + ψ(l)y(l) > 0, l ∈ T}.
On time scales, the reader is supposed to be acquainted with the skills and basic ideas about the analytics given by Bohner [3]. Next, we give some basic lemmas on time scales which will be required in the evidence of the exhibited paper.

Lemma 2.2 ([19])
Let l 0 ∈ T k , and let j : where j be the derivative of j with respect to the first variable. Then Chain Rule 2: Assume that j : T → R is strictly increasing and T * = j(T) is a time scale. Let v : T * → R and j (l), v (j(l)) exist for l ∈ T k . Then

Lemma 2.4 ([32])
Let j ∈ C rd and l ∈ T k . Then Lemma 2.5 ([19]) If j ∈ and l ∈ T, then the exponential function e j (l, l 0 ) is the unique solution of the initial value problem x(l 0 ) = 1.

Results and discussion
Without compromising nonspecific statements, throughout in this task, we denote To demonstrate our elementary results, we first rundown the accompanying suppositions: We now present the principle lemma and theorems. Lemma 3.1 Let a ∈ C rd , l ∈ T k , and let ρ(l) ∈ C rd be a strictly increasing function for l ∈ T. Then Theorem 3.2 Suppose that suppositions (P1)-(P5) with 1 (l) = 2 (l) and the inequality are satisfied. Then where ξ = 1, Λ -1 is the inverse function of Λ, and L 1 is the largest number for all l < L 1 with Proof Fix an arbitrary l * ∈ T 0 for l ∈ [l 0 , l * ] ∩ T and denote by 1 (J(l)) the function on the right side of (1), which is nonnegative and nondecreasing. Therefore and so that by (1) x(l) ≤ J(l), l ∈ T 0 .
Equation (5) by Lemma 2.2 and delta derivative with respect to l imply that l 0 m(l, q 1 ) 1 J(q 1 ) q 1 ξ , since 1 (J(l)) = 2 (J(l)). This inequality becomes where Delta differentiating (9) and utilizing J(l) ≤ W (l) and (8), we derive that Consider ρ(l) It is easy to observe from Lemma 3.1 that By substituting (12) into (11) we have From (10) and (13) we obtain Integrating both sides of (14) from l 0 to l and using W (l 0 ) = -1 1 (b(l * )) and W (l) > 0 yield the estimate also, Define the function R 1-ξ (l) as the right-hand side of (15). Since R(l) is nondecreasing, we have From (16) by delta differentiating R 1-ξ (l) with respect to l we get that which leads to In comparison, for l ∈ [l 0 , T] ∩ T, if σ (l) > l, then . (18) If σ (l) = l, then we have where μ lies between R(l 1 ) and R(l). Together (18) and (19) produce . (20) Inequalities (17) and (20) turn out into From the definition of Λ in (3) by integration (20) from l 0 to l we get Since Λ is increasing and 1-ξ , the last inequality takes the form The conclusion in (2) can be achieved by the arbitrariness of l * , inserting (21) into (16) and (8) simultaneously, integrating the resulting inequality, and taking the benefit of (6) and (7). Explanations are discarded.
Essential comments on Theorem 3.2 are listed underneath.
The desired bound in (23) can be carried out by integrating over [l 0 , l], using (25) and (32), setting l = l * , and simultaneously putting the resultant inequality into (30) and (28). The proof is completed.
We can notice from Theorem 1.90 of [23] and N (l) ≥ 0 that    Proof The proof of Corollary 3.12 is the same that of Theorem 3.9 with appropriate alterations.