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

摘要

In this paper, we study the third-order functional dynamic equations with $ gamma$-Laplacian and nonlinearities given by Riemann-Stieltjes integrals egin{equation*} left{ r_{2}left( tight) phi _{gamma _{2}}left( left[ r_{1}left( tight) phi _{gamma _{1}}left( x^{Delta }left( tight) ight) ight] ^{Delta }ight) ight} ^{Delta }+int_{a}^{b}qleft( t,sight) phi _{lpha left( sight) }left( x(gleft( t,sight) )ight) dzeta left( sight) =0, end{equation*} on an above-unbounded time scale $mathbb{T}$, where $phi_{gamma }(u):=leftert uightert^{gamma -1}u$ and $int_{a}^{b}fleft( sight) dzeta left( sight) $ 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.