LAUSR

AbstractWe investigate oscillation of second-order half-linear variable delay damped dynamic equations [a(t)|xΔ(t)|λ−1xΔ(t)]Δ+b(t)|xΔ(t)|λ−1xΔ(t)+p(t)|x(δ(t))|λ−1x(δ(t))=0$$ \bigl[a(t) \bigl\vert x^{\Delta }(t) \bigr\vert ^{\lambda -1}x^{\Delta }(t) \bigr]^{\Delta }+b(t) \bigl\vert x ^{\Delta }(t) \bigr\vert ^{\lambda -1}x^{\Delta }(t)+p(t) \bigl\vert x \bigl(\delta (t) \bigr) \bigr\vert ^{\lambda -1}x \bigl(\delta (t) \bigr)=0 $$ on a time scale T$\mathbb{T}$. By using the generalized Riccati transformation and the inequality technique, we establish some new oscillation criteria for the equations under the condition ∫t0∞[a−1(s)e−b/a(s,t0)]1/λΔs<∞.$$ \int ^{\infty }_{t_{0}} \bigl[a^{-1}(s)e_{{-b/a}}(s,t_{0}) \bigr]^{1/\lambda } \Delta s< \infty. $$ These results deal with some cases not covered by existing results in the literature.