Necessary and sufficient conditions on asymptotic behaviour of solutions of forced neutral delay dynamic equations

In this article, we first study the asymptotic behavior of delay dynamic equations having the following form: [x(t)+A(t)x(α(t))]Δ+B(t)F(x(β(t)))−C(t)G(x(γ(t)))=φ(t)fort∈[t0,∞)T, where TT is a time scale unbounded from above, t0∈Tt0∈T, F,G∈Crd(R,R), A,φ∈Crd([t0,∞)T,R), B,C∈Crd([t0,∞)T,[0,∞)R) and α,β,γα,β,γ are delay functions. Then, we extend our results to equations of the form: [x(t)+A(t)x(α(t))]Δ+B(t)F(x(β(t)))=φ(t)fort∈[t0,∞)T, where B∈Crd[t0,∞)T,R is allowed to oscillate. Our results state necessary and sufficient conditions on all solutions of the equations to be oscillatory or convergent (→0→0) or divergent (→±∞→±∞) depending on various ranges of the coefficient AA. We generalize and improve some of the results for differential and difference equations using the time scale theory. Most of our results for the equations are not, thus far, presented in the literature–not even for the discrete or the continuous cases.