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.
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