Constantin’s inequality for nabla and diamond-alpha derivative

Ayşe Feza Güvenilir,Neslihan Nesliye Pelen,Billur Kaymakçalan

doi:10.1186/s13660-015-0681-9

Ayşe Feza Güvenilir, Neslihan Nesliye Pelen + Show 1 more

Open Access

https://doi.org/10.1186/s13660-015-0681-9

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Abstract

Calculus for dynamic equations on time scales, which offers a unification of discrete and continuous systems, is a recently developed theory. Our aim is to investigate Constantin’s inequality on time scales that is an important tool used in determining some properties of various dynamic equations such as global existence, uniqueness and stability. In this paper, Constantin’s inequality is investigated in particular for nabla and diamond-alpha derivatives.

Highlights

To study the boundedness of solutions for some nonautonomous second order linear differential equations, Ou-Iang [ ] used a nonlinear integral inequality
∀t ∈ [, T], where f, g, h ∈ C(R+, R+ ) and w belongs to the class of continuous nondecreasing functions on R+ such that w(r) > if r > and satisfies ds w(s)
The class of real rd-continuous functions defined on a time scale T is denoted by Crd(T, R)

Summary

Introduction

To study the boundedness of solutions for some nonautonomous second order linear differential equations, Ou-Iang [ ] used a nonlinear integral inequality. ∀t ∈ [ , T], where f , g, h ∈ C(R+ , R+ ) and w belongs to the class of continuous nondecreasing functions on R+ such that w(r) > if r > and satisfies. For the same function define the ∇-derivative of f at t ∈ Tκ , denoted by f ∇ (t), for all >. The class of real rd-continuous functions defined on a time scale T is denoted by Crd(T, R). The class of real ld-continuous functions defined on a time scale T is denoted by Cld(T, R). By [ ], if a function h(t) : T → R is continuous, it is diamond-alpha integrable, and the fundamental theorem of calculus is not true for α-derivative.

Also assume that the function x satisfies r ds

Then x

Then we have