Existence of uncountably many bounded nonoscillatory solutions and their iterative approximations for second order nonlinear neutral delay difference equations

In this paper, the Banach fixed-point theorem is employed to establish several existence results of uncountably many bounded nonoscillatory solutions for the second order nonlinear neutral delay difference equationΔa(n)Δx(n)+b(n)x(n-τ)+Δh(n,x(h1(n)),x(h2(n)),…,x(hk(n)))+f(n,x(f1(n)),x(f2(n)),…,x(fk(n)))=c(n),n⩾n0,where τ,k∈Nτ,k∈N, n0∈N0n0∈N0, a,b,c:Nn0→Ra,b,c:Nn0→R with a(n)>0a(n)>0 for n∈Nn0n∈Nn0, h,f:Nn0×Rk→Rh,f:Nn0×Rk→R and hl,fl:Nn0→Zhl,fl:Nn0→Z withlimn→∞hl(n)=limn→∞fl(n)=+∞,l∈{1,2,…,k}.A few Mann type iterative approximation schemes with errors are suggested, and the errors estimates between the iterative approximations and the nonoscillatory solutions are discussed. Seven nontrivial examples are given to illustrate the advantages of our results.