On some nonlinear retarded Volterra–Fredholm type integral inequalities on time scales and their applications

In this paper, we establish some new nonlinear retarded Volterra–Fredholm type integral inequalities on time scales. Our results not only generalize and extend some known integral inequalities, but also provide a handy and effective tool for the study of qualitative properties of solutions of some Volterra–Fredholm type dynamic equations.

In 2013, the authors in [22] established and applied the following useful linear Volterra-Fredholm type integral inequality on time scales: where I = [t 0 , α] ∩ T, t 0 ∈ T, α ∈ T, α > t 0 , u 0 is a nonnegative constant, u, f , g, and h are nonnegative rd-continuous functions defined on I.
Very recently, the author in [29] discovered the retarded Volterra-Fredholm type integral inequality on time scales where I = [t 0 , T] ∩ T, t 0 ∈ T, T ∈ T, T > t 0 , α : I → I is continuous and strictly increasing satisfying α(t) ≤ t, α is rd-continuous, u, a, b, f , and g : I → R + are rd-continuous functions and a is nondecreasing.
Inspired by the ideas employed in [22,27,29], here we obtain some new nonlinear Volterra-Fredholm type integral inequalities on time scales. Our results not only generalize and extend the results of [22,27] and some known integral inequalities but also provide a handy and effective tool for the study of qualitative properties of solutions of some complicated Volterra-Fredholm type dynamic equations.

Preliminaries
For an excellent introduction to the calculus on time scales, we refer the reader to [5] and [6].
In what follows, we always assume that R denotes the set of real numbers, R + = [0, ∞), Z denotes the set of integers, and T is an arbitrary time scale (nonempty closed subset of R), R denotes the set of all regressive and rd-continuous functions, The set T κ is defined as follows: If T has a maximum m and m is left-scattered, then T κ = T -{m}. Otherwise T κ = T. The graininess function μ : T → [0, ∞) is defined by μ(t) := σ (t)t, the forward jump operator σ : T → T by σ (t) := inf{s ∈ T : s > t}, and the "circle plus" We give the following lemmas in order to use them in our proofs. One can find details in [5].

Lemma 2.3 ([29])
Let α : I → I be a continuous and strictly increasing function such that α(t) ≤ t, and α is rd-continuous. Assume that f : I → R is an rd-continuous function, then Lemma 2.6 Let m ≥ n ≥ 0, m = 0, and a ≥ 0, then a n ≤ n m k n-m a m + mn m k n (2.5) for any k > 0.
Proof Set F(x) = n m x n-m a m + m-n m x n , x > 0. It is seen that F(x) obtains its minimum at x 0 = a. Hence we get (2.5) holds for any k > 0.

Main results
Then z is nondecreasing on I. From (3.1) and (3.9) we have , and j = i/2 with m = 1 and n = p, q, r for any k 1 , k 2 , k 3 > 0, respectively, we have Now using (3.4) and (3.5) and (3.10) we get Since V (t) is nondecreasing on I, then for t ∈ I, from the above inequality we have Then (3.13) can be restated as Then w(t) is nondecreasing, and from (3.15) and (3.16) we obtain Using Lemma 2.3, taking delta derivative of (3.16), and from (3.17), we have where A(t) and C(t) are defined as in (3.6) and (3.8). From (3.7), we get which yields i.e., Note that w is rd-continuous and B ⊕ C ∈ R + , from Lemma 2.5, (3.16), and (3.22), we obtain From (3.17) and (3.23), we have Using (3.24) on the right-hand side of (3.14) and according to (3.2), we obtain From (3.24) and (3.25), we obtain Noting (3.10), we get the desired inequality (3.3). This completes the proof.
If we take p = q = r = 1, we can get the following corollary.

B(t) = A(t) 1μ(t)A(t)
, (3.33) Proof Denote Using Lemma 2.6, we obtain It is similar to the proof of Theorem 3.1, we get Then, using u l (t) ≤ z(t), we have (3.29). This completes the proof. If there exist positive constants k i (i = 1, 2) such that

Applications
In this section, we will present some simple applications for our results. First, we consider the following Volterra-Fredholm type dynamic integral equation: Using Theorem 3.1, we obtain the desired inequality (4.7).

Conclusions
In this paper, we have established some new retarded Volterra-Fredholm type integral inequalities on time scales, which extend some known inequalities and provide a handy tool for deriving bounds of solutions of retarded dynamic equations on time scales. Unlike some existing results in the literature, the integral inequalities considered in this paper involve the power nonlinearity, which results in difficulties in the estimation on the explicit bounds of unknown function u(t). We establish an inequality to overcome the difficulties, which can be used as a handy tool to solve the similar problems.