Infinitely many nonoscillatory solutions for second order nonlinear neutral delay differential equations

In this paper we consider the second order nonlinear neutral delay differential equation [a(t)(x(t)+b(t)x(t−τ))′]′+[h(t,x(h1(t)),x(h2(t)),…,x(hk(t)))]′+f(t,x(f1(t)),x(f2(t)),…,x(fk(t)))=g(t),t≥t0, where τ>0,a,b,g∈C([t0,+∞),R)τ>0,a,b,g∈C([t0,+∞),R) with a(t)>0a(t)>0 for t≥t0t≥t0, h∈C1([t0,+∞)×Rk,R)h∈C1([t0,+∞)×Rk,R), f∈C([t0,+∞)×Rk,R)f∈C([t0,+∞)×Rk,R), hl∈C1([t0,+∞),R)hl∈C1([t0,+∞),R) and fl∈C([t0,+∞),R)fl∈C([t0,+∞),R) with limt→+∞hl(t)=limt→+∞fl(t)=+∞,l=1,…,k. Under suitable conditions, by making use of the Banach fixed point theorem, we show the existence of infinitely many nonoscillatory solutions, which are uncountable, for the above equation, suggest several Mann type iterative approximation sequences with errors for these nonoscillatory solutions and establish some error estimates between the approximate solutions and the nonoscillatory solutions. Five nontrivial examples are given to illustrate the advantages of our results.