The Oscillation of the Nonlinear Functional Equation with Variable Coe ffi cients

where ki 1,M 1 is a positive integer, αi is not a positive real number,i = 1, 2, 3 · · · m and m ∑ i=1 αi = 1;P(t),Q(t) : I → R = (0,+∞) are given real valued functions, and I denotes an unbonunded subset of R.x is an unknown real valued function , g(t) : I → I are given functions lim g(t) = ∞, t ∈ I and gi denotes the ith iterate of the function g.i.e. g0(t) = t, gi+1(t) = g(gi(t)), t ∈ I, i = 1, 2, · · · 1n 1994, Golda and Werbowski [1] had studied the second linear functional equation of the form: x(g(t)) = p(t)x(t) + Q(t)x(g2(t)), t t0 (1.2) They proved: Theorem 1.1 All the solution of the functional equation (1.2) is oscillation , if


Introduction
Considering the nonlinear functional equation with variable coefficients: x(g(t)) = P(t)x(t) + Q(t) m i=1 |x(g k i +M (t))| α i sign x(g k i +i (t)) , (1.1) where k i 1, M 1 is a positive integer, α i is not a positive real number,i = 1, 2, 3 • • • m and m i=1 α i = 1;P(t), Q(t) : I → R + = (0, +∞) are given real valued functions, and I denotes an unbonunded subset of R + .x is an unknown real valued function , g(t) : I → I are given functions lim g(t) = ∞, t ∈ I and g i denotes the ith iterate of the function g.i.e.

Journal of Mathematics Research
September , 2009 In this condition, they reached Theorem 1.2 All the sulutions of the equation oscillate,if and Here,λ(A) is the only real roof of equation Obviously, the condition (1.14) had improved the condition (1.4).
By the year of 2000, Zhou Yong, Liu Zheng-rong and Yu Yuan-hong had set up a class of the higher nonlinear functional equation : K+1  (1.16) ¢ www.ccsenet.org/jmrISSN: 1916-9795 then all the solutions of the functional equation (1.15) oscillated.and here λ is the only real roof of the equation λ − 1 , then all the solutions of the equation (1.15) oscillated.
Lin Quan-wen, Wu Ying-zhu and Liao Si-quan also studied the equation (1.15) in [6], they had proved: then all the solutions of the equation (1.15) oscillated and reached the corollary 1.1.This paper is inspired by [4] and [6], we have studied the oscillation of the equation (1.1), we used the method of studying the solution oscillation of the functional equation with variable coefficients then got some new oscillation rules and applied them to the difference equations and educed some main results.Obviously, when M=1 , the equation (2.2) change to the equation (1.14).So our results will improve or expand the results of [1],[2] and [4],[6].Lemma 2 Let 0 ≤ A ≤ K K (K+1) K+1 ,and defining sequence {λ n } ∞ n=0 as follows

Let
The main results of this paper are as follows: Theorem 2.1 if one of the following condition come into existence Where λ(A) is the only one real roof of equation (2.2) in [1, ((K + 1)A)  1.1) we have x(g(t)) ≥ P(t)x(t), By the iteration, we obtain We divide it into two cases: . Substituting (2.5) into (1.1),we obtain We have A ≤ 1 and from the iteration, we obtain Substituting (2.6) into (1.1),we have . By iterating it again, we obtain (2.7) Because βA > 1 ,so there must be an n * , let the equation (2.11) come into existence.And (βA) n * +1 > β (2.8) Substituting (2.7) into (1.1),we obtain: (2.9) ¢ www.ccsenet.org/jmrISSN: 1916-9795 And we can see that it conflict with (2.8).
(b) 0 Substituting (2.5) into (1.1),we obtain: Next, we are going to prove: x Suppose x(g(t)) ≥ λ 1 P(t)x(t), i ≥ 1, from the iteration, we obtain: Substituting (2.11) into (1.1),we will have: . So, from the mathematical induction method, we know that (2.10) exists, from the iteration, we obtain: (2.12) Substituting the (2.12) into (1.1),we reach x(g(t)) ≥ λ [ Let t → ∞ and get the superior limit of (2.13), from the lemma 2, we obtain: And this contradicts with what we had known before.The lemma is proved completely.