Improved Oscillation Results for Functional Nonlinear Dynamic Equations of Second Order

Abstract

In this paper, the functional dynamic equation of second order is studied on an arbitrary time scale under milder restrictions without the assumed conditions in the recent literature. The Nehari, Hille, and Ohriska type oscillation criteria of the equation are investigated. The presented results confirm that the study of the equation in this formula is superior to other previous studies. Some examples are addressed to demonstrate the finding.

Amended Criteria of Oscillation for Nonlinear Functional Dynamic Equations of Second-Order

“…(4) Regarding dynamic equations on time scales, the oscillation criteria here are an important improvement compared to the literature outcomes. In particular, our results improve those reported in [28]; 27) and ( 34) improves ( 12)).…”

“…The object of this paper is to deduce some sharp Hille-type oscillation criteria for (4) in the cases where γ ≥ β and γ ≤ β and for the both cases advanced and delayed dynamic equations. The results which will be proven in this paper present critical improvement to the results in [23,24,28]; for more details, see Section 4.…”

“…Let β ≥ 1. By dint ofB β (ξ) ∞ ξ φ(τ)Q(τ)∆τ ≥ B β (ξ) ∞ σ(ξ) φ(τ)Q(τ)∆τ and β β l β(β−1) (β + 1) β+1 < β β l β 2 (β + 1) β+1 for 0 < l < 1,Theorems 2 and 3 improve ([28], Theorems 4 and 8) (conditions (…”

“…Without loss of generality, we let x(ξ) > 0 and x(η(ξ)) > 0 on [ξ 0 , ∞) T . It is easy to achieve that (see [28], Theorem 1)…”

“…For Hille-type oscillation criteria of second-order dynamic equations, References [23][24][25][26][27] proposed some Hille-type oscillation criteria to special cases of Equation (7). Recently, Hassan et al [28] established the following Hille-type oscillation criterion for (4) and showed that if l > 0 and…”