Existence and iterative approaches of positive solutions for forced first order differential equations

Abstract

This paper deals with the forced first order nonlinear neutral delay differential equations d/dt [x(t) ± x(t - τ)] + f(t, x(g(t))) = h(t), t ≥ t0 where τ > 0, f ∈ C([t0,+∞) × R, R), g ∈ C([t0,+ ∞), R) with U lim t→∞ g(t) = +∞ and h ∈ C([t0, +∞),R). Based on the Banach fixed point theorem and new techniques, we establish the existence of uncountably many bounded positive solutions for the equations, suggest Mann iterative methods to approximate these positive solutions and obtain several error estimates between the sequences generated by the iterative methods and these positive solutions.