The Application Of Difference Equations Of Frequency Measure Of Sequence

There are many conclusions about the oscillatory and convergence for the difference e-quations. But the classical concepts of oscillation and convergence can not fully describe the oscillatory and convergent properties of a real sequence, thus cannot accurately depict oscillation and convergence of solutions. In such a case, it becomes inevitable that the es-tablishment of the concept of frequency measures of sequences and the frequent oscillation and the frequent convergence of the solutions could be described correctly by utilizing the properties of the frequency measures. The frequency measure is one of the basic tools for studying sequences and discrete systems, in turn, sequences and difference equations are the main research objects of the frequency measure theories.Firstly, by employing the concept and the properties of frequency measures, we discuss frequent oscillation of degree ω of the solutions of the following difference equation:We establish some new frequently oscillatory criteria of the solutions of the difference equation, and give the example to illustrate our results.Secondly, frequent convergence of solutions are discussed for the following nonlinear dif-ference equations, which may change signs: xn+2=1-xn2, n=0,1,2,…We present that for any ρ,σ∈[0,1] which is different from (5^1/2-1)/2, the solution X={xn} determined by ρ and a has two frequent limits0and1of the same degree0.5.Finally, we use the properties of frequency measures and the definition of frequent stability to discuss the frequent stability of the following Fibonacci System: xn+1=xn+xn+xn-1, n=1,2,3,…and we get some corresponding conclusions.In this paper, our results generalize the previous oscillation criteria in literature. Mean-while, we draw some conclusions about the frequent convergence and frequent stability of the difference equations, and thus improve some known results before.