Nonoscillatory solutions to forced higher-order nonlinear neutral dynamic equations on time scales

Abstract

By employing Kranoselskii's fixed point theorem, we obtain sufficient conditions for the existence of nonoscillatory solutions of the forced higher-order nonlinear neutral dynamic equation \[ [x(t)+p(t)x(\tau(t))]^{\nabla^m }+\sum^{k}_{i=1}p_{i}(t)f_{i}(x(\tau_{i}(t)))=q(t) \] on a time scale, where $p_{i}(t)$, $f_{i}(t)$ and $q(t)$ may be oscillatory. Then we establish sufficient and necessary conditions for the existence of nonoscillatory solutions to the equation $[x(t)+p(t)x(\tau(t))]^{\nabla^m }+F(t,x(\delta(t)))=q(t)$. Finally, we deal with dynamic equation \[ [x(t)+p(t)x(\tau(t))]^{\nabla^{m-1} \Delta}+\sum^{k}_{i=1}p_{i}(t)f_{i}(x(\tau_{i}(t)))=q(t) \] with mixed $\nabla$ and $\Delta$ derivatives. In particular, some interesting examples are included to illustrate the versatility of our results.