HALF-LINEAR VOLTERRA-FREDHOLM TYPE INTEGRAL INEQUALITIES ON TIME SCALES AND THEIR APPLICATIONS∗

The main aim of this paper is to establish some new half-linear Volterra-Fredholm type integral inequalities on time scales. Our results not only extend and complement some known integral inequalities but also provide an effective tool for the study of qualitative properties of solutions of some dynamic equations.


Introduction
In 1988, Stefan Hilger [21] introduced the theory of time scales in order to unify and extend the difference and differential calculus in a consistent way. Since this pioneering work, the theory has been growing up and applied to many different fields of mathematics. As one of the most fundamental objects, dynamic equations on time scales has been extensively investigated in recent years, we refer the reader to the books [8,9] and to the papers [2, 5, 6, 10-12, 15-17, 19, 22, 27, 28, 31-36, 38, 42, 45, 47, 49-52] and the references therein.
In the present paper, we continue our investigation to obtain some new halflinear Volterra-Fredholm type integral inequalities on time scales. Our results not only complement the results established in [43] in the sense that the results can be applied in the cases when 1 < α < β or 0 < β < α < 1, but also furnish a handy tool for the study of qualitative properties of solutions of some Volterra-Fredholm integral equations and dynamic equations.

Preliminaries
Throughout this paper, a knowledge and understanding of time scales and time scale notation is assumed. For an excellent introduction to the calculus on time scales, we refer the reader to [8] and [9] .

List of abbreviations.
In what follows, we always assume that R denotes the set of real numbers, R + = [0, ∞), Z denotes the set of integers, C rd denotes the set of all rd-continuous functions, T is an arbitrary time scale (nonempty closed subset of R), R denotes the set of all regressive and rd-continuous functions, The following lemmas are useful in the proof of the main results of this paper.

Main results
Theorem 3.1.
Suppose that x satisfies .
If we let w(t, s) ≡ 1 in Theorem 3.2 , then we obtain the following corollary. Assume that x, f, g, k, a, b, c, α, β, and M are defined the same as in Theorem 3.2. Suppose that x satisfies If we let k(t) ≡ 0 in Theorem 3.2, then we obtain the following corollary.
Assume that x, f, g, w, a, b, c, α, β, P , F , and G are defined the same as in Theorem 3.2, f is nondecreasing, and λ ≥ 0 is a constant. Suppose that x satisfies where we use the convention that 1 0 = +∞. Proof. From Lemma 2.1 and (3.29), we have w(σ(t), s) a(s)z(s) + M (s)z(σ(s)) ∆s w(σ(t), s) a(s) + M (s) ∆s where P (t) and G(t) are defined as in (3.14) and (3.16). By a similar argument with Theorem 3.2, we get From (3.33) and (3.35), we obtain In view of (3.30) and (3.40), we get Substituting (3.41) into (3.37), we obtain Noting x(t) ≤ z(t), we get the desired inequality (3.31). This completes the proof.
If we let w(t, s) ≡ 1 in Theorem 3.3 , then we obtain the following corollary. Assume that x, f, g, a, b, c, M, α, β, λ, P, F and G are defined the same as in corollary 3.1. Suppose that x satisfies

Applications
In this section, we will present some simple applications for our results.
Example 4.1. Consider the following Volterra-Fredholm type dynamic integral equation on time scales: The following theorem gives an estimate for the solutions of Eq.(4.1).