Oscillatory Behavior And The Existence Of Nonoscillatory Solutions Of Several Classes Of Differential Equations With Deviating Arguments

The content of this paper will be mainly divided into four parts. In the part of introduction in the first chapter, we introduce the development of oscillation theory of ordinary differential equations and functional differential equations briefly. On the other hand, we introduce the main study and innovations of this paper. In the second part of this paper, we study the properties of solutions of a class of first order nonlinear differential inequalities with deviating arguments x′(t)+a(t)f(x(t))+p(t)g(x(t))h(x(t-Ï„₁(t)),x(t-Ï„₂(t)),Λ,x(t-Ï„_n(t)))≤0　　　(1)　x′(t)+a(t)f(x(t))+p(t)g(x(t))h(x(t-Ï„₁(t)),x(t-Ï„₂(t)),Λ,x(t-Ï„n(t)))≥0　　　(2) and the corresponding differential equations　x′(t)+a(t)f(x(t))+p(t)g(x(t))h(x(t-Ï„₁(t)),x(t-Ï„₂(t)),Λ,x(t-Ï„_n(t)))=0　　　(3)　By introducing a new transformation, sufficient conditions for no eventually positive solutions of first order nonlinear delay differential inequalities (1), (or no eventually negative solutions of first order nonlinear differential inequalities (2)) and oscillation of equations (3) are obtained. At the same time, this method could also be used in advance differential inequalities and equations. As corollaries to our results, the corre-sponding results in [11] are extended. In the third part of this paper, we study the existence of nonoscillatory solutions of the corresponding equations of the second part of this paper　x′(t)+a(t)f(x(t))+p(t)g(x(t))h(x(t-Ï„₁ (t)),x(t-Ï„₂(t)),Λ,x(t-Ï„_n(t)))=0　　　　　　(4)　By using schauder equilibrium point theorem, sufficient conditions for nonoscillatory solutions of the differential equations (4) are obtained. At the same time, our results are extended to advance differential equations. The results we obtained are new.　In the fourth part of this paper, we consider the second order strongly superlinear delay differential equations of the form By introducing a new transformation, the necessary and sufficient conditions are obtained for equations (5) to have oscillation. As corollaries to our results, the corresp-onding results in [26,28] are improved. All the theoremes and corollaries obtained in this paper are new, and they improved or extended the corresponding results of current document.