Comparison Criteria for Nonlinear Functional Dynamic Equations of Higher Order

Taher S. Hassan,Said R. Grace

doi:10.1155/2016/6847956

Abstract

We will consider the higher order functional dynamic equations with mixed nonlinearities of the formxnt+∑j=0Npjtϕγjxφjt=0, on an above-unbounded time scaleT, wheren≥2,xi(t)≔ri(t)ϕαixi-1Δ(t), i=1,…,n-1, with x0=x, ϕβ(u)≔uβsgn⁡u, andα[i,j]≔αi⋯αj. The functionφi:T→Tis a rd-continuous function such thatlimt→∞φi(t)=∞forj=0,1,…,N. The results extend and improve some known results in the literature on higher order nonlinear dynamic equations.

Highlights

In this paper, we consider comparison criteria for higher order nonlinear dynamic equation with mixed nonlinearities of the formN x[n] (t) + ∑pj (t) φγj (x (φj (t))) = 0, (1)j=0 on an above-unbounded time scale T, where (i) n i ≥ =2 is an integer, 1, 2, . . . , n − 1, and t∈ x[i] T,(t) fl with ri rn (t=)φ1α,i [(x[i−1])Δ αn = 1,(t)], and x[0] = x;(ii) φβ(u) fl |u|β sgn u for β > 0
The results extend and improve some known results in the literature on higher order nonlinear dynamic equations
Discrete Dynamics in Nature and Society x ∈ Cr1d[Tx, ∞)T for some Tx ≥ t0 such that x[i] ∈ Cr1d[Tx, ∞)T, i = 1, 2, . . . , n − 1, and x(t) satisfies (1) on [Tx, ∞)T, where Crd is the space of right-dense continuous functions

Summary

Introduction

We consider comparison criteria for higher order nonlinear dynamic equation with mixed nonlinearities of the form x[n] (t) + ∑pj (t) φγj (x (φj (t))) = 0, (1) N − 1, and x(t) satisfies (1) on [Tx, ∞)T , where Crd is the space of right-dense continuous functions. Grace and Hassan [17] establish oscillation criteria for more general higher order dynamic equation x[n] (t) + p (t) φγ (xσ (φ (t))) = 0.

Objectives

Results

Conclusion