Oscillation And Asymptotic Behavior Of Third-order Nonlinear Difference Equations

The dissertation is divided into three chapters according to the content.The first chapter summarizes the background and the main problems of this research.In the second chaper, we consider the oscillation and asymptotic behavior of the third-order nonlinear neutral delay difference equation whereΔdenotes the forward difference operator defined byΔxn=xn+1-xn for any sequence{xn} of real numbers,0<γ≤1 is a quotient of odd posi-tive integers, (?) andσare nonnegative integers such (?)≤σ, the real sequences and the function f satisfies the follow-ing conditions(h1){cn}n∞=n0,{dn}n∞=n0 are positive sequences of real numbers such that(h2) 0≤pn<1, qn≥0 and has a positive subsequence;(h3) f:R R→R is a continuous function such that uf(u)>0 for u≠0 and f(u)/u≥K>0.By using Riccati transformation and estimates of inequalities, we establish some sufficient conditions which guarantee that every solution{xn} of Eq.(1.1) either applications oscillates or satisfies limn→∞xn=0. Our results solve an open problem posed by Saker. Two examples illustrating the applications of our results are also given. In the third chapter, we study the nonoscillation and the oscillation of a class of third-order nonlinear difference equations where N is the set of natural numbers, a,b∈R\{0}, R is the set of real numbers,{pn} and {qn} are real sequences with pn≠0 and f:N×R3→R.The main results in this chapter are divided into two parts. In the first part, nonoscillatory theorems for Eq.(1.2) with a>0 are established. Oscil-latory theorems for Eq.(1.2) with a<0 are given in the second part. The obtained results extend and improve the ones of Parhi and Panda. At last, three examples are provided to illustrate our results.