Existence and iterative approximations of nonoscillatory solutions of higher order nonlinear neutral delay differential equations

This paper studies a few existence results of nonoscillatory solutions for the following higher order nonlinear neutral delay differential equation:dndtnx(t)+cx(t-τ)+(-1)n-i+1didtih(t,x(h1(t)),x(h2(t)),…,x(hk(t)))+(-1)n+1f(t,x(f1(t)),x(f2(t)),…,x(fk(t)))=g(t),t⩾t0,where n   is a positive integer, 0⩽i⩽n-10⩽i⩽n-1, c∈R⧹{-1}c∈R⧹{-1}, τ>0τ>0, h∈Ci([t0,+∞)×Rk,R)h∈Ci([t0,+∞)×Rk,R), f∈C([t0,+∞)×Rk,R)f∈C([t0,+∞)×Rk,R), g∈C([t0,+∞),R)g∈C([t0,+∞),R), hl∈Ci([t0,+∞),R)hl∈Ci([t0,+∞),R) and fl∈C([t0,+∞),R)fl∈C([t0,+∞),R) withlimt→+∞hl(t)=limt→+∞fl(t)=+∞,l=1,…,kconstructs 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. In addition sufficient conditions for the existence of infinitely many nonoscillatory solutions for the above equation are given. Three nontrivial examples are given to illustrate the advantages of our results.