Nonlinear Limit Circle/Limit Point Criteria For Second Order Damping Differential Equations

The limit circle/limit point criterion for differential equations is one of important branches of differential equations.In the field of modern applied mathematics, it has made considerable headway in recent years, because all the structures of its emergence have deep physical background and realistic mathematical models. Many scholars take on the research of this field, they have achieved many good results. With the increasing development of science and technology, there are many problems relating to differential equation derived from lots of real applications and practice, such as whether certain differential equation has a limit circle/limit point solution or not, and whether all of its solutions are of limit circle/limit point case or not. In very resent years, great changes of this field have taken place. Especially, the second order differential equation has been paid more attentions and investigated in various classes by using different methods(see [1]-[30]).The present paper employs Lyapunov function, Schwartz inequality, Gronwall inequality and so on to investigate the nonlinear limit circle/limit point criteria for nonlinear differential equations, the results generalize and improve some known limit circle/limit point criteria in literature.The thesis is divided into three sections according to contents.In Chapter 1, we introduce the main contents of this paper.In Chapter 2, we study the nonlinear limit circle/limit point criteria for superlinear differential equation with damping term(?) , (2.1.1) whereα, r: R₊ (?)R and b: R₊ (?)R₊ are continuous,α', r'∈AC_loc(R₊),α", r"∈L_loc²(R₊),α(t) > 0,r(i) > 0, and k > 0 is a positive integer. we mainly employed Lyapunov function, Schwartz inequality, Gronwall inequality and so on to investigate the limit circle/limit point criteria for superlinear differential equation with damping term, and we shall further generalize and improve the main results of John R.Graef, Miroslav Bartusek and Zuzana Dosla [10], we obtained several new nonlinear limit circle/limit point criteria for superlinear differential equation with damping term at the end of this section.In Chapter 3, we study the nonlinear limit circle/limit point criteria for sublinear differential equation with damping term(a(t)y')' + b(t)y' + r(t)y^γ=0. (3.1.1)whereα, r: R₊ (?) R and b: R₊ (?) R₊ are continuous,α', r'∈AC_loc(R₊),α", r''∈L_loc²(R₊),α(t) > 0,r(t) > 0, and 0 <γ≤1, (?) say (?), M and N are positive integers, we can writeγ= 2k - 1, where k = (?). In this chapter we mainly employed Lyapunov function, Schwartz inequality, Gronwall inequality and so on to investigate the limit circle/limit point criteria for sublinear differential equation with damping term, and we shall further generalize and improve the main results of John R.Graef, Miroslav Bartusek and Zuzana Dosla [10], we obtained several new nonlinear limit circle/limit point criteria for sublinear differential equation with damping term at the end of this section.