Toward detailed analysis for oscillation and non-oscillation of nonlinear ordinary differential equations

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The purpose of this research subject is devoted to the investigation of the oscillatory and nonoscillatory behavior of several forms of the nonlinear ordinary differential equations .Our main results obtained are as belows.(1) We have established oscillation and nonoscillation criteria for a class of the even order quasilinear functional differential equation. Moreover, we have studied the oscillatory and nonoscillatory behavior of solutions of the above mentioned equation. (2) We have given sufficient conditions under which the second order nonlinear non-homogeneous differential equation possesses two positive solutions by means of the principal and the non-principal solutions of the second order nonlinear homogeneous differential equation. (3) We concerned with the oscillatory and nonoscillatory behavior of solutions for a class of the even order nonlinear Sturm-Liouville differential equations. (4) Oscillation criteria are obtained for solutions of forced and unforced second order neutral differential equations with positive and negative coefficients. (6) With regard to half-linear retarded functional differential equations, we have obtained a necessary and sufficient conditions that ensures the existence of a slowly varying solution and a regularly varying solution of index 1 at the same time. (7) We have focused on the regularly varying solutions of generalized Thomas-Fermi differential equations, and that provide criteria for the existence of a decreasing slowly varying solution and an increasing regularly varying solution of index 1. (8) We have established some necessary and sufficient conditions on the some coefficient function for the existence of a pair of solutions which are regularly varying indices different from 0 and 1, for a class of the delayed half-linear differential equation.