Paul W. Spikes

Asymptotic behavior of solutions of a second order nonlinear differential equation

where n, y, Y : [to , co) + R, f, g : R + R, a(t) > 0, q(t) > 0, g(i) > 0, and a, q, r, f, and g are continuous. Equations of this type have been studied .by many authors and an excellent survey of known results can be found in Ref. [16]. More recent contributions include Refs. (l-8, 10-14, IS]. Tne approach we take in studying the asymptotic behavior of solutions of Eq. (1) is new. By examining the quotient r(t)/q(t) as t + 03 we are able to obtain boundedness and other behavioral results without requiring that the forcing term, Y(Z), be “small” in some sense. In fact, our results will allow r(t) to become unbounded as t co. This is a significant departure from previous studies of Eq. (1). I n addition, we will be able to relax some of the conditions that other authors place on the functions in Eq. (1 jFinally, in our discussion of the asymptotic behavior of solutions of Eq. (l), we will include a class of solutions, the Z-type sol.utions (see Section 3 for the definition), not considered previously. First, we begin with some new continuability and boundedness results.

Oscillation Theorems for Perturbed Nonlinear Differential Equations

1. INTRODUCTION In this paper we study the oscillatory behavior of the solutions of the perturbed second-order nonlinear differential equation (a(t) x’)’ f Q(t, x) = qt, x, x’). (*) The problem of determining oscillation criteria for second-order nonlinear equations has received a great deal of attention in the twenty years following the publication of the now-classic paper by Atkinson [I]. This has been especially true for the special cases of (*)

Monotone properties of certain classes of solutions of second-order difference equations

The authors consider the difference equations (*)Δ(anΔxn)=qnxn+1 and (**)Δ(anΔxn)=qnf(xn+1) where an > 0, qn > 0, and f: R→R is continuous with uf(u) > 0 for u ≠ 0. They obtain necessary and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (*) and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (**). Sufficient conditions for the existence of these types of nonoscillatory solutions are also presented. Some examples illustrating the results and suggestions for further research are included.

Boundedness and convergence to zero of solutions of a forced second-order nonlinear differential equation

Abstract Sufficient conditions for continuability, boundedness, and convergence to zero of solutions of ( a ( t ) x ′)′ + h ( t , x , x ′) + q ( t ) f ( x ) g ( x ′) = e ( t , x , x ′) are given.

Oscillatory and asymptotic behavior of solutions of nonlinear neutral-type difference equations

The authors consider the higher-order nonlinear neutral delay difference equation and obtain results on the asymptotic behavior of solutions when ( p n ) is allowed to oscillate about the bifurcation value –1. We also consider the case where the sequence { p n } has arbitrarily large zeros. Examples illustrating the results are included, and suggestions for further research are indicated.

On the asymptotic behavior of solutions of a second order nonlinear neutral delay differential equation

Qualitative properties of the solutions of first and higher order neutral delay differential equations, i.e., equations in which the highest order derivative of the unknown function appears both with and without delays, have been studied by several authors in recent years. For discussions of the existence and uniqueness of solutions and some applications of these equations, in addition to how the behavior of their solutions differs from the behavior of solutions of delay equations, the reader is referred to [l-19]. In this paper we study asymptotic properties of the solutions of the second order nonlinear neutral delay differential equation

Asymptotic and oscillatory behavior of solutions of first order nonlinear neutral delay differential equations

The authors consider the first order nonlinear neutral delay differential equation (E) [y(t) + P(t) y(t − τ)]′ − Q(t) ƒ(y(t − σ)) = 0, where P, Q, and ƒ are continuous, Q(t) ≧ 0, τ ≧ 0, σ ≧ 0, and uƒ(u) > 0 if u ≠ 0. They give sufficient conditions for all nonoscillatory solutions of (E) to converge to zero as t → ∞. Two oscillation theorems for equation (E) are also proved.

On existence of positive solutions and oscillations of neutral difference equations of odd order

A criteria for the existence of a positive solution of the odd order neutral difference equation where qn ≥ 0, is established. Results are also obtained for the oscillation of solutions of the above equation when m = I and f is either sublinear or superlinear.

Comparison and nonoscillation results for perturbed nonlinear differential equations

SummaryNonoscillation theorems for perturbed second order nonlinear differential equations are obtained. A nonlinear Picone type identity is introduced to obtain some Sturm-Picone type comparison theorems for nonlinear equations.

BOUNDEDNESS AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF A FORCED DIFFERENCE EQUATION

The authors consider the nonlinear difference equation where Ay, Y,+I Y,, {P,}, {q,}, and {rn} are real sequences, and uf(u) > 0 for u # 0. Sufficient conditions for boundedness and convergence to zero of certain types of solutions axe given. Examples illustrating the results are also included.