Unbounded Positive Solutions Of A Second Order Differential Equation

Recent years,functional differential equations has become a very powerful tool for studying a rich kind of problems arising in many different fields of sciences. Functional differential equations provide a large mathematics framework for discussing many interesting problems in many fields.In this paper, we study the second order nonlinear neutral delay differential equation [x(t)+a(t)x(t-τ)]"+[h(t,x(h₁(t)),x(h₂(t)),...x(h_k(t)))]'+f(t,x(f₁(t)),x(f₂(t)),...x(f_x(t)))=b(t),t≥t_o,whereτ>0,a,b∈C([t_o,+∞),R),h∈C'([t_o,+∞)×R^k,R),h₁∈C'([t_o,+∞),R),f₁∈C([t_o,+∞),R)且有lim h₁(t)=lim f₁(t)=+∞,l-1,2...k.The functional differential equation in this paper includes a number of known functional differential equations as special case, the contraction mapping principle and some new techniques are employed to study the existence of solutions for the above equation.Utilizing the Banach fixed point theory, the existence results of uncountably many positive solutions which are unbounded of the equation are shown. We suggest several Mann type iterative schemes with errors and discuss the error estimates between the unbounded positive solutions and sequences generated by the Mann iterative schemes, we also discuss the convergence of the iterative schemes.At last, four nontrivial examples are given to illustrate the results presented in this paper.The results presented in this paper extend, improve and unify several known results in the literature.