Asymptotic behavior of third order functional dynamic equations with $\gamma$-Laplacian and nonlinearities given by Riemann-Stieltjes integrals

Abstract

In this paper, we study the third-order functional dynamic equations with $ \gamma$-Laplacian and nonlinearities given by Riemann-Stieltjes integrals \begin{equation*} \left\{ r_{2}\left( t\right) \phi _{\gamma _{2}}\left( \left[ r_{1}\left( t\right) \phi _{\gamma _{1}}\left( x^{\Delta }\left( t\right) \right) \right] ^{\Delta }\right) \right\} ^{\Delta }+\int_{a}^{b}q\left( t,s\right) \phi _{\alpha \left( s\right) }\left( x(g\left( t,s\right) )\right) d\zeta \left( s\right) =0, \end{equation*} on an above-unbounded time scale $\mathbb{T}$, where $\phi_{\gamma }(u):=\left\vert u\right\vert^{\gamma -1}u$ and $\int_{a}^{b}f\left( s\right) d\zeta \left( s\right) $ denotes the Riemann-Stieltjes integral of the function $f$ on $[a,b]$ with respect to $\zeta $. Results are obtained for the asymptotic and oscillatory behavior of the solutions. This work extends and improves some known results in the literature on third order nonlinear dynamic equations.

Keywords

• asymptotic behavior
• oscillation
• $\gamma $-laplacian
• nonlinear dynamic equations
• time scales