Existence Of Uncountably Many Bounded Nonoscillatory Solutions And Iterative Approximations For Fifth Order Nonlinear Neutral Delay Difference Equation

This paper studies the following fifth order nonlinear neutral delay difference equation deeply,â–³4(anâ–³(xn+bnxn-Ï„))+â–³4f(sf1,xf2n,…,xfkn)+â–³3h(xh1n,xh2n,…,xhkn)+â–³2g(xg1n,xg2n,…,xgkn)+â–³p(xp1n,xp2n,…,xpkn)+q(xq1n,xq2n,…,xqkn)=rn,n≥n0, where Ï„,k∈N,n0∈N0,a,b,r:Nn0â†'R,with an>0for n∈Nn0,f,h,g,p,g:Nn0×Rkâ†'R and f1,h1,g1,p1,q1:Nn0â†'Z wifh limnâ†'∞f1n=limnâ†'∞h1m=limnâ†'∞gln=limnâ†'∞p1n=limq1n=+∞,l∈{1,2,3,…,k}.The first chapter introduces the development process of linear and nonlinear difference equations simply,and states the importance of the study for the above fifth order nonlinear neutral delay difference equation,its form is more general,and includes many cases of the study in[1-34].The second chapter gives some related definitions and two important lemmas.The third chapter with the aid of the Banach fixed point theorem and Mann iterative algorithms,respectively under the condition of bn=-1,0≤bn≤b,b≤bn≤0,b≤bn≤b,|bn|≤b,b<-1and bn=1show that the existence,iterative approximations and the problem of error estimates of the uncountably many bounded nonoscillatory solutions of this difference equtation in the subset A(N,M)of Banach spaces.The text presents seven theorems,and in each proof of theorem,one mapping is constructed,and then the mapping which are used in the process of proof are proved to satisfy the related conditions of the theorems.Finally,according to the corresponding fixed point theorems and iterative schemes, the conditions to ensure the existence of the uncountably many bounded nonoscillatory solutions of the difference equation are obtained.The fourth chapter,seven nontrivial examples are given to illustrate the results present in this paper,at the same time,seven examples explain that the results obtained in this paper are more general than the others.