Bounded solutions of delay dynamic equations on time scales

Abstract

In this paper we discuss the asymptotic behavior of solutions of a delay dynamic equation $$y^{\Delta}(t)=f(t,y(\tau(t)))$$ where $f\colon\mathbb{T}\times\mathbb{R}\rightarrow\mathbb{R}$, \tau\colon\T\rightarrow \T$ is a delay function and $\mathbb{T}$ is a time scale. We formulate a principle which gives the guarantee that the graph of at least one solution of above mentioned equation stays in the prescribed domain. This principle uses the idea of the retraction method and is a suitable tool for investigating the asymptotic behavior of solutions of dynamic equations. This is illustrated by an example.In this paper we discuss the asymptotic behavior of solutions of a delay dynamic equation $$y^{\Delta}(t)=f(t,y(\tau(t)))$$ where $f\colon\mathbb{T}\times\mathbb{R}\rightarrow\mathbb{R}$, \tau\colon\T\rightarrow \T$ is a delay function and $\mathbb{T}$ is a time scale. We formulate a principle which gives the guarantee that the graph of at least one solution of above mentioned equation stays in the prescribed domain. This principle uses the idea of the retraction method and is a suitable tool for investigating the asymptotic behavior of solutions of dynamic equations. This is illustrated by an example.