SOME GRONWALL – BELLMAN TYPE INEQUALITIES ON TIME SCALES FOR VOLTERRA-FREDHOLM DYNAMIC INTEGRAL EQUATIONS

Abstract In this paper, we prove several new explicit estimations for the solutions of some classes of nonlinear dynamic inequalities of Gronwall–Bellman–Pachpatte type on time scales. Our results formulate some integral and discrete inequalities discussed in the literature as special cases and extend some known dynamic inequalities on time scales. The inequalities given here can be used in the analysis of the qualitative properties of certain classes of dynamic equations on time scales. Some examples are presented to demonstrate the applications of our results.


Introduction
In various situations, we are interested in knowing qualitative properties of solutions without explicit knowledge of the solution process.One of the best known and widely used inequalities in the study of qualitative properties of solutions of nonlinear differential equations can be stated as follows: The inequality given in Theorem 1.1, was discovered by Thomas Gronwall [1] in 1919.In the recent years, these inequalities have been greatly enriched by the recognition of their potential and intrinsic worth in many applications of the applied sciences, (see [2][3][4][5][6][7][8][9][10][11][12]).In 1943, Richard Bellman in [13], proved the fundamental inequality (see Theorem 1.2) named Gronwall-Bellman's inequality as a generalization for Gronwall's inequality and plays a very important role in studying stability and asymptotic behaviour of solutions of linear differential-difference equations.
Theorem 1.2.Let u and f be continuous and nonnegative functions defined on [α, β], and let c be nonnegative constant.Then the inequality implies that Bellman in [14] proved and made use of the following variant of the inequality given by himself in Theorem 1. Gollwitzer [15] gave the following generalization of the Gronwall-Bellman inequality: A fairly general version of Theorem 1.2 is given in the following theorem by Pachpatte [16]: In [17], Pachpatte established also the following inequality: for all t ∈ [a, b] ⊆ R. Kender et al. [18] established the following further generalizations of the inequality (1.5) proved by of Pachpatte in [17] where he replaced the linear term of the unknown function ω by nonlinear term ω p in both sides of the inequality as following It is well known that, the dynamic inequalities play an important role in the development of the qualitative theory of dynamic equations on time scales.The study of dynamic equations on time scales which goes back to Stefan Hilger [19] becomes an area of mathematics and recently has received a lot of attention.The general idea is to prove a result for a dynamic equation or a dynamic inequality where the domain of the unknown function is a so called time scale T, which may be an arbitrary closed subset of the real numbers R see [20,21].The purpose of the theory of time scales is to unify continuous and discrete analysis.The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus (see [22]), i.e, when T = R, T = N and T = q N0 = {q t : t ∈ N 0 } where q > 1.
The book on the subject of time scales by Bohner and Peterson [23] summarizes and organizes much of time scale calculus.
During the past decade a number of dynamic inequalities has been established by some authors which are motivated by some applications, for example, we refer the reader to [23][24][25][26] for contributions, and the references cited therein.
In this paper, we present some new nonlinear dynamic inequalities on an arbitrarily time scale T, these dynamic inequalities unify and extend the inequalities presented in [17] and [18].Our main results will be proved by employing some useful inequalities which will be presented in Section 2. The paper is organized as in the following: In Section 2, some basic concepts of the calculus on time scales and useful lemmas are introduced.In Section 3, we state and prove the main results.In Section 4, we present several applications to study some qualitative properties of the solutions of certain dynamic equations.

Basic Results and Lemmas on Time Scales
In this section, we present some background on time scales.A time scale T is an arbitrary nonempty closed subset of the real numbers.The time scales calculus was initiated by Hilger in his PhD thesis in order to unify discrete and continuous analysis [19].We assume throughout that T has the topology that it inherits from the standard topology on the real numbers R. For t ∈ T, first we define the forward jump operator σ : T → T by: and second, the backward jump operator ρ : T :→ T by: In this definition, we put inf ∅ = sup T and sup ∅ = inf T, where ∅ is the empty set.A point t ∈ T with inf T < t < sup T, is said to be left-dense if ρ(t) = t and is right-dense if σ(t) = t, points that are simultaneously right-dense and left-dense are said to be dense, is left-scattered if ρ(t) < t and right-scattered if σ(t) > t, points that are simultaneously right-scattered and left-scattered are said to be isolated.A function g : T → R is said to be right-dense continuous (rd-continuous) provided g is continuous at right-dense points and at left-dense points in T, left-sided limits exist and are finite.The set of all such rd-continuous functions is denoted by C rd (T).A function f : T → R is said to be left-dense continuous (ld-continuous) provided f is continuous at left-dense points and at right-dense points in T, right-sided limits exist and are finite.The set of all such ld-continuous functions is denoted by C ld (T).
Given a time scale T, we introduce the sets T κ , T κ , and T κ κ as follows.If T has a left-scattered maximum t 1 , then Open intervals and half-closed interval are defined similarly.Let f : T → R be a real-valued function on a time scale T. Then for all t ∈ Tκ, we define f ∆ (t) to be the number (if it exists) with the property that given any ε > 0 there is a neighborhood U of t such that For f : T → R, we define the function A time scale T is said to be regular if the following two conditions are satisfied simultaneously: (1) σ(ρ(t)) = t and (2) ρ(σ(t)) = t, ∀t ∈ T. The product and quotient rules for the derivative of the product f g and the quotient f /g (where gg σ = 0, here g σ = g • σ ) of two differentiable functions f and g are given as the following: and f g

g(σ(t)) .
A function F : T → R is called a delta antiderivative of f : T → R provided that F ∆ (t) = f (t) holds for all t ∈ T κ , and the delta integral of f is defined by We will frequently use the following useful relations between calculus on time scales T and differential calculus on R, difference calculus on Z, and quantum calculus on q Z .Note that It can be shown (see [23]) that if g ∈ C rd (T), then the Cauchy integral G(t) := t t0 g(s)∆s exists, t 0 ∈ T, and satisfies Now, we will give the definition of the generalized exponential function and its derivatives.We say that p : T → R is regressive provided 1 + µ(t)p(t) = 0 for all t ∈ T κ , we define the set of all regressive and rd-continuous functions.We define the set + of all positively regressive elements of by + = {p ∈ : 1 + µ(t)p(t) > 0, ∀t ∈ T}.The set of all regressive functions on a time scale T forms an Abelian group under the addition ⊕ defined by p ⊕ q = p + q + µpq.If p ∈ , then we can define the exponential function by where ξ h (z) is the cylinder transformation, which is defined by If p ∈ , then e p (t, s) is real-valued and nonzero on T. If p ∈ + , then e p (t, t 0 ) is always positive.Note that a(s)ds ; (2.6) 1 + (q − 1)sa(s) . (2.8) In the following, we present the basic lemmas that will be needed in the proof of our main results.

Lemma 2.1 ( [27]
).If p, q ∈ and a, b, c ∈ T, then 1. e p (t, t) = 1 and e 0 (t, s) = 1; 2. e p (σ(t), s) = (1 + µ(t)p(t))e p (t, s); Lemma 2.2 (See [27]).If p ∈ and fix t ∈ T, then the exponential function e p (t, t 0 ) is the unique solution of the following initial value problem: (2.9) Lemma 2.3 (See [27]).Let t 0 ∈ T κ and k : T × T κ → R be continuous at (t, t), where t > t 0 and t ∈ T κ .Assume that If k ∆ denotes the derivative of k with respect to the first variable, then where Now we are ready to state and prove our main results, which give us the time scales version of the inequalities proved in [17] and [18].

Main results
In this section, we will state and prove the main results and investigate some dynamic Gronwall-Bellman inequalities on time scales.
, c be delta-differentiable on T with c ∆ (t) ≥ 0, and p ≥ 1 be a constant.If where and ) where m 1 , m 2 are defined as in Lemma 2.5, and e m1g (t, a) is a solution of the initial value problem (2.9) in Lemma 2.2 when p(t) replaced by m 1 g.
Proof.Define a function z 1 (t) by then, we get that and from (3.6), and using (3.7), we have: Therefore, using Lemma 2.5, from (3.9), we get that where A(t) is as defined in (4.6).Now an application of Lemma 2.3 to (3.10) yields As a special case of Theorem 3.1, if T = Z and using the relations (2.4) and (2.7), we obtain the following discrete result.
Corollary 3.1.Let T = Z and assume that ω, g, c and f are nonnegative sequences defined for t ∈ N 0 , then the inequality , where and, where and where η(t) is define as in (3.18).Using Lemma 2.5 in (3.24), the inequality (3.24) can be written as, where A 1 (t) is define as in (3.17).Now, using Lemma 2.4 in (3.25) yields that As a special case of Theorem 3.2, if T = Z and using the relations (2.4) and (2.7), we get the following discrete result.
Corollary 3.2.Let T = Z and assume that ω, g, k(t, s), ∆k(t, s), c and f are nonnegative sequences defined for t ∈ N 0 , then the inequality , where where ∆k(t, s) = k(t + 1, s) − k(t, s), and , Theorem 3.3.Let ω and c be defined as in Theorem 3.2, where where where m 1 , m 2 are as defined in Lemma 2.
By using Lemma 2.5 on z 1 p 3 (t) and z 3 (s) in (3.37), we have Therefore, using Lemma (2.4) in (3.38) we get that As a special case of Theorem 3.3, if T = Z and using the relations (2.4) and (2.7), we obtain the following discrete result.
Corollary 3.3.Let T = Z and assume that ω, k 1 (t, s), k 2 (t, s) 2 , ∆k 1 (t, s), ∆k 2 (t, s), c and f are nonnegative sequence defined for t ∈ N 0 , then the inequality , where and where where Theorem 3.4.Let ω and c be defined as in Theorem 3. where where where m 1 , m 2 are as defined in Lemma 2.5, and e m1E3+E4 (t, a) is a solution of the initial value problem (2.9) in Lemma 2.2, when p(t) replaced by Proof.The proof is similar to the proof of Theorem 3.3.As a special case of Theorem 3.4, if T = Z and using the relations (2.4) and (2.7), we obtain the following discrete result.

Applications
In this section, we present some applications of Theorem 3.1 and Theorem 3.3 to obtain the explicit estimates on the solutions of certain dynamic equations, and also prove the uniqueness and global existence of solutions for a class of nonlinear dynamic integral equations.
Consider the following dynamic integral equation on time scale where where c, f , g ∈ C rd ([a, b] T k , R + ) and r, l are given constants.
for all t ∈ [a, b] T k , where M, A are defined as in the following: and where m 1 , m 2 are defined as in Lemma 2.5.
Proof.Let ω be a solution of the dynamical system (4.1).

Acknowledgment
The author thanks the referees very much for their careful comments and valuable suggestions on this paper.

Remark 3 . 5 .
By taking T = R in Theorem 3.4 and using the relation (2.3), it is easy to observe that the inequality obtained in Theorem 3.3 reduces to the inequality obtained by Kender et al. in [18, Theorem 2.4].