Global Solvability and Mann Iteration Method with Error for a Third Order Nonlinear Neutral Delay Differential Equation (communicated by Agacik Zafer)

This paper intends to investigate the existence of uncountably many bounded positive solutions of a third order nonlinear neutral delay differential equation d dt { r1(t) d dt [ r2(t) d dt ( x(t)− f(t, x(t− σ)) )]} + d dt [ r1(t) d dt g(t, x(p(t))) ] + d dt h(t, x(q(t))) = l(t, x(η(t))), t ≥ t0 in the following bounded closed and convex set Ω(a, b) = { x(t) ∈ C([t0,+∞),R) : a(t) ≤ x(t) ≤ b(t), ∀t ≥ t0 } , where σ > 0, r1, r2, a, b ∈ C([t0,+∞),R), f, g, h, l ∈ C([t0,+∞)×R,R), p, q, η ∈ C([t0,+∞), [t0,+∞)). By using the Krasnoselskii fixed point theorem, the Schauder fixed point theorem, the Sadovskii fixed point theorem and the Banach contraction principle, four existence results of uncountably many bounded positive solutions of the differential equation are established. Moreover, a perturbed Mann iteration method with error is constructed for approximating the solution of the third order differential equation, and the convergence and stability of the iterative sequence generated by the algorithm are discussed.