Existence of bounded monotonic solutions of second order nonlinear differential equations

In this thesis we discuss the existence of bounded monotonic solutions of the second order nonlinear differential equation pth xt fx't ' =qtg xt ,t≥a. .;We prove that all solutions of the differential equation are divided into four subclasses: Ab = {x ∈ A : limt→infinity | x(t)| → ℓ < infinity}; Ainfinity = { x ∈ A : limt →infinity |x(t)| → infinity}; Bb = {x ∈ B : limt →infinityx(t) →ℓ ≠ 0}; B0 = {x ∈ B : limt →infinityx(t) → 0}, where A and B are two classes of solutions of the differential equation defined as A = {x(·) : there exists a b ≥ a such that x(t)x'(t) > 0, t ∈ [b, alpha]} and B = {x(·) : x(t)x'(t) < 0, t ∈ [a, infinity)}.;The main results of the thesis are the following four theorems: (1) the equation has a positive Ab solution if and only if J1 -infinity; (3) the equation has a positive Bb solution if and only if J4 > -infinity; (4) the equation has negative Bb solution if and only if J3 < infinity, where J 1, J2, J3, and J4 are four integrals of functions p(t) and q(t).;The results obtained in this thesis have generalized and improved some analogous ones existing in the literature.