Oscillatory and nonoscillatory properties of solutions of functional differential equations and difference equations

摘要

Oscillation and nonoscillation of solutions of functional differential equations and difference equations are analyzed qualitatively. A qualitative approach is usually concerned with the behavior of solutions of a given equation and does not seek explicit solutions. The dissertation is divided into five chapters. The first chapter is essentially introductory in nature. Its main purpose is to introduce certain well-known basic concepts and to present some result that are not as well-known. In chapter 2 and chapter 3 we present sufficient conditions for oscillation of solutions of neutral differential equations of the form [a(t)[x(t) + p(t)x([tau](t))] [superscript](n-1)] [superscript]\u27 + q(t)f(x([sigma](t))) = 0and [x(t) + p(t)x([tau](t))] [superscript](n) + q[subscript]1(t)f(x([sigma][subscript]1(t))) + q[subscript]2(t)f(x([sigma][subscript]2(t))) = h(t)respectively. Chapter 4 discusses the oscillation, nonoscillation, and the asymptotic behavior of solutions of higher order functional differential equations of the form (r[subscript]2(r[subscript]1 x[superscript]\u27(t))[superscript]\u27)[superscript]\u27 + q(t)f(x([sigma](t))) = h(t)and x[superscript](n)(t) + F(t,x([sigma][subscript]1(t)),...,x([sigma][subscript]m(t))) = h(t).Chapter 5 is devoted the study of oscillatory solutions of neutral type difference equations of the form [delta][a[subscript]n[delta][superscript]m-1(x[subscript]n + p[subscript]nx[subscript][tau][subscript]n)] + q[subscript]nf(x[subscript][sigma][subscript]n) = 0and that of asymptotic behavior for n → [infinity] of solutions of equations of the form [delta][superscript]mx[subscript]n + F(n, x[subscript][sigma][subscript]n) = h[subscript]n.The results obtained here are the discrete analogs of several of those in chapter 1 and chapter 4;A function x(t) : [a,[infinity]) → R is said to be oscillatory if it has a zero on [T,[infinity]) for every T ≥ a; otherwise it is called nonoscillatory. Similarly a sequence \x[subscript]n of real numbers is oscillatory if it is not eventually positive or eventually negative; otherwise it is nonoscillatory.