Abstract
In this paper, the sharp Hille-type oscillation criteria are proposed for a class of second-order nonlinear functional dynamic equations on an arbitrary time scale, by using the technique of Riccati transformation and integral averaging method. The obtained results demonstrate an improvement in Hille-type compared with the results reported in the literature. Some examples are provided to illustrate the significance of the obtained results.
Highlights
The theory of time scales, which has recently received a lot of interest, was proposed by Stefan Hilger in 1988 in order to unite continuous and discrete analysis; see [1]
The theory was introduced in reality to amalgamate continuous and discrete analyses, which are the basic stones in dynamical systems
The theory of differential equations is one of these theories that can be explored and analyzed by means of time scales to their wide implications in real-word systems and processes
Summary
The theory of time scales, which has recently received a lot of interest, was proposed by Stefan Hilger in 1988 in order to unite continuous and discrete analysis; see [1]. We are worthy of considering the q−difference equations when T = qN0 := {qλ : λ ∈ N0 for q > 1}, which has important applications in quantum theory (see [15]), and various types of time scale such as T = hN, T = N2, and T = Tn, where Tn is the set of the harmonic numbers, can be considered. This work is concerned about the behavior of the oscillatory solutions to the quasilinear functional dynamic equation of second-order b(ξ) x∆(ξ) β−1x∆(ξ)
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