Abstract
We study some nonlinear dynamic integral inequalities on time scales by introducing two adjusting parameters, which provide improved bounds on unknown functions. Our results include many existing ones in the literature as special cases and can be used as tools in the qualitative theory of certain classes of dynamic equations on time scales.
Highlights
Following Hilger’s landmark paper 1, there have been plenty of references focused on the theory of time scales in order to unify continuous and discrete analysis, where a time scale is an arbitrary nonempty closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations
The purpose of this paper is to further investigate some sublinear integral inequalities on time scales that have been studied in a recent paper 6
The definitions below merely serve as a preliminary introduction to the time scale calculus; they can be found in the context of a much more robust treatment than is allowed here in the text [15, 16] and the references therein
Summary
Following Hilger’s landmark paper 1 , there have been plenty of references focused on the theory of time scales in order to unify continuous and discrete analysis, where a time scale is an arbitrary nonempty closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations. Many other interesting time scales exist; for example, T qN0 {qt : t ∈ N0} for q > 1 which has important applications in quantum theory , T hN with h > 0, T N2, and T Hn the space of the harmonic numbers. Many authors have extended some continuous and discrete integral inequalities to arbitrary time scales. The purpose of this paper is to further investigate some sublinear integral inequalities on time scales that have been studied in a recent paper 6. We provide improved bounds on unknown functions, which include many existing results in [6, 14] as special cases and can be used as tools in the qualitative theory of certain classes of dynamic equations on time scales
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