Abstract
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.
Highlights
1 Introduction In recent years, there exist a large number of published papers on the theory of time scales which was introduced by Stefan Hilger [1] in his Ph.D. thesis in 1988 in order to unify and extend the difference and differential calculus in a consistent way, for instance, [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] and the references therein
T0 τs τ s, t ∈ I, Liu Journal of Inequalities and Applications (2018) 2018:211 where I = [t0, α] ∩ T, t0 ∈ T, α ∈ T, α > t0, u, f1, f2, f3 are rd-continuous functions defined on I, f1, f2, f3 are nonnegative, w ∈ C(R+, R+) is a nondecreasing function with w(u) > 0 for u > 0, and k is a nonnegative constant
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)
Summary
There exist a large number of published papers on the theory of time scales which was introduced by Stefan Hilger [1] in his Ph.D. thesis in 1988 in order to unify and extend the difference and differential calculus in a consistent way, for instance, [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] and the references therein. T0 where I = [t0, α] ∩ T, t0 ∈ T, α ∈ T, α > t0, u0 is a nonnegative constant, u, f , g, and h are nonnegative rd-continuous functions defined on I. Lemma 2.1 ([5, Theorem 1.16]) Assume that f : T → R is a function and let t ∈ T. Lemma 2.2 ([5, Theorem 1.98]) Assume that ν : T → R is a strictly increasing function and T := ν(T) is a time scale. Lemma 2.5 ([5, Theorem 6.1]) Suppose that y and f are rd-continuous functions and p ∈ R+. Note that w is rd-continuous and B ⊕ C ∈ R+, from Lemma 2.5, (3.16), and (3.22), we obtain w(t) ≤ w(t0)eB⊕C(t, t0) = MeB⊕C(t, t0), t ∈ I. Using Theorem 3.1, we obtain the desired inequality (4.7)
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