Some new dynamic Gronwall–Bellman–Pachpatte type inequalities with delay on time scales and certain applications

The main objective of the present article is to prove some new delay nonlinear dynamic inequalities of Gronwall–Bellman–Pachpatte type on time scales. We introduce very important generalized results with the help of Leibniz integral rule on time scales. For some specific time scales, we further show some relevant inequalities as special cases: integral inequalities and discrete inequalities. Our results can be used as handy tools for the study of qualitative and quantitative properties of solutions of dynamic equations on time scales. Some examples are provided to demonstrate the applications of the results.


Introduction
In 1919 Thomas Gronwall [1] discovered a vital inequality, which can be used as an effective tool in the study of existence, uniqueness, boundedness, stability, and other qualitative properties of solutions of certain nonlinear differential and difference equations. The Gronwall inequality is stated as follows: If u is a continuous function defined on the interval D = [a, a + h] and where a, ξ , ζ , and h are nonnegative constants, then 0 ≤ u(t) ≤ ξ he ζ h , ∀t ∈ D.
In 1943, Richard Bellman [2] proved the fundamental inequality, named Gronwall-Bellman's inequality, as a generalization for Gronwall's inequality. He proved that: If u and f are continuous and nonnegative functions defined on [a, b], and let c be a nonnegative constant, then the inequality implies that As a generalization of (1.1), Bellman himself [3] proved that: If u, f , a, ∈ C(R + , R + ) and a is nondecreasing, then the inequality The discrete version of (1.2) was studied by Pachpatte in [4]. In particular, he proved that: If (n), f (n), γ (n) are nonnegative sequences defined for n ∈ N 0 , and f (n) is nondecreasing for n ∈ N 0 , then 1 + γ (s) , n ∈ N 0 .
In [5], Pachpatte studied the following inequalities: where , a, b, g, h and c ∈ C(R + , R + ), k(t, s) and its partial derivative ∂k(t,s) ∂t are real-valued nonnegative continuous functions for 0 ≤ s ≤ t ≤ ∞, f : R + × R + → R + is a continuous function, and p > 1 is a constant.
In [35, Theorem 6.4, page 256], Bohner and Peterson introduced a dynamic inequality on a time scale T which unifies the continuous version inequality (1.2) and the discrete version inequality (1.3) as follows: If , ζ are right dense continuous functions and γ ≥ 0 is a regressive and right-dense continuous function, then In this paper, motivated by the above-mentioned inequalities, we prove some new delay dynamic inequalities of Gronwall-Bellman-Pachpatte type on time scales. Some special cases of our results contain continuous Gronwall type inequalities and their discrete analogues. We also present some application examples to illustrate our results at the end. The paper is organized as follows: Sect. 2 contains the main results of this paper. In Sect. 3, an application to study some qualitative properties of the solutions of certain retarded dynamic equations are demonstrated. In Sect. 4, we state the conclusion.
Before we arrive at the main results in the next section, we need the following lemmas and essential relations on some time scales such as R, Z, hZ, and q Z . Note that: (1.10) If λ ∈ C rd (T) (see [35]), then the Cauchy integral (τ ) := The function η : T → R is called regressive provided 1 + μ(t)η(t) = 0 for all t ∈ T κ . The set of all positively regressive elements of is + = {η ∈ : 1 + μ(t)η(t) > 0, ∀t ∈ T}. We form an Abelian group under the addition ⊕ by the set of all regressive functions on a time scale T by η ⊕ ζ = η + ζ + μηζ . If η ∈ , then the exponential function is defined by whereξĥ(z) is the cylinder transformation, which is defined bŷ If η ∈ , then e η (τ , s) is real-valued and nonzero on T. If η ∈ + , then e η (τ , τ 0 ) is always positive.
Note that: If η ∈ and fix t 0 ∈ T, then the exponential function e η (t, t 0 ) is the unique solution of the following initial value problem: (1.14) where ς denotes the derivative of ς with respect to the first variable, then implies

Main results
In this section, the authors state and justify the main results and investigate some dynamic Gronwall-Bellman inequalities on time scales.

5)
where m 1 , m 2 are defined as in Lemma 1.5.
Proof Define a function χ 1 (t) by We notice that χ 1 (t) ≥ 0 and nondecreasing on [a, b] T . Since α(a) = a, we get that Then from (2.1), (2.6) and by using the monotonicity of χ 1 (t), we get which implies From (2.6), (2.8) and using Theorem 1.6, we have Therefore, using (2.9) and Lemma 1.5, we get that Thus, from (2.13), we obtain where 1 is defined as in (2.3). Then we get the desired inequality (2.2) by combining (2.12) and (2.14). This completes the proof. As a special case of Theorem 2.1, if we take T = Z and the delay function α(n) = nτ , where τ > 0, and so α(n) = 1 > 0, then, using relations (1.9) and (1.12), we obtain the following completely new discrete result.

Corollary 2.5
Assume that (n), g(n), c(n), and f (n) are nonnegative sequences defined for n ∈ N 0 , with c(n) ≥ 0 for n ∈ N 0 . If (n) satisfies the following delay discrete inequality: Theorem 2.6 Let a, b ∈ T k with a < b, and let , g, c ∈ C rd ([a, b] T , R + ) and α : T → T. Further, assume that α and c are delta-differentiable on T with c (t) ≥ 0, α (t) ≥ 0, α(t) ≤ t, and α(a) = a. Moreover, assume that k(t, s), where m 1 , m 2 are defined as in Lemma 1.5.
Proof Define a function χ 2 (t) by Then from (2.15), (2.19) and by using the monotonicity of χ 2 (t), we obtain Using Lemma 1.5, inequality (2.22) can be rewritten as

Theorem 2.11
Assume that a, b ∈ T k with a < b, and let , α, and c be defined as in Theorem 2.6. Further, suppose that k 1 (t, s), k 2 (t, s), k 1 (t, s), and k 2 (t, s) where m 1 , m 2 are defined as in Lemma 1.5.
Proof Define a function χ 3 (t) by We notice that χ 3 (t) is nonnegative nondecreasing on [a, b] T . Since α(a) = a, we get that By applying Lemma 1.5 to (2.35), we get As a special case of Theorem 2.11, if we take T = Z and the delay function α(n) = nτ , where τ > 0, and so α(n) = 1 > 0, then, using relations (1.9) and (1.12), we obtain the following completely new discrete result.

Applications
In this section, by using Theorem 2.11, we demonstrate the global existence of solutions for a class of nonlinear retarded dynamic integral equations of the form where ϒ ∈ C rd ([a, b] T × R + × R + , R + ). Now, in the following theorem, we obtain the explicit estimates for the solution of (3.1). where , c, h, ∈ C rd ([a, b] T , R + ), c is delta-differentiable on T k with c (t) ≥ 0, k 1 (t, s), k 1 (t, s), k 2 (t, s), k 2 (t, s) ∈ C rd ([a, b] T × [a, b] T , R + ) for a ≤ s ≤ t ≤ b and p ≥ 1 is a constant.