NONOSCILLATORY SOLUTIONS TO FORCED HIGHER-ORDER NONLINEAR NEUTRAL DYNAMIC EQUATIONS ON TIME SCALES

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 ( τ ( t ) ) ] ∇ m + ∑ i = 1 k p i ( t ) f i ( x ( τ i ( t ) ) ) = q ( t ) on a time scale, where pi(t), fi(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 ( τ ( t ) ) ] ∇ m + F ( t , x ( δ ( t ) ) ) = q ( t ) . Finally, we deal with dynamic equation [ x ( t ) + p ( t ) x ( τ ( t ) ) ] ∇ m − 1 Δ + ∑ i = 1 k p i ( t ) f ( x ( τ i ( t ) ) ) i = q ( t ) with mixed ∇ and Δ derivatives. In particular, some interesting examples are included to illustrate the versatility of our results.