Existence for positive solutions of second-order neutral nonlinear differential equations

We consider the nonlinear neutral differential equations. This work contains some sufficient conditions for the existence of a positive solution which is bounded with exponential functions. The case when the solution converges to zero is also treated.     

On the oscillation of certain second-order nonlinear differential equations

In this paper, oscillation criteria for the nonlinear second-order ordinary differential equation

     

On the solutions of certain linear nonhomogeneous second-order differential equations

The system Nω=(N-α)ω+y, N= bN+aωω^T, N(t)?R^m×m, ω(t)?R^m which originally arose from a model for the pathological behavior of neural networks, is studied. Similar equations can arise in a variety of applications. It is shown that if N(0) is positive definite, then solutions exist for all time. Equilibrium points are determined. N is found to be singular at the equilibrium points, making the analysis of the asymptotic properties of the system non-trivial. The asymptotic behavior when y = 0 is completely described. Some results are proven on the asymptotic behavior of N and ω when y≠0     

On certain properties of solutions of second-order linear differential equations

For a second-order linear differential equation with coefficients in a differential fieldL of meromorphic functions, a classical result of C. L. Siegel states that if the logarithmic derivative of every nontrivial solution is transcendental overL, then no nontrivial solution can satisfy a first-order algebraic differential equation overL. In Part 1, we consider a situation for the equation (*)w¨+A(z)w=0 where Siegel's condition is violated, namely where the logarithmic derivative of a nontrivial solution belongs toL. In this case, we obtain a representation for any solution of (*) which satisfies a first-order algebraic differential equation overL, and we illustrate our result with examples for various choices ofA(z) andL. In Part 2, we treat the question of determining when all solutions of (*) satisfy first-order algebraic differential equations overL, and in Part 3, we obtain a refinement of Siegel's classical theorem on Bessel's equation.This research was supported in part by the National Science Foundation (MCS-8002269).     

On the almost periodic solutions of certain second-order abstract differential equations

In a Hilbert space, certain solutions of second-order nonhomogeneous linear differential equations with a Stepanov almost periodic forcing function are shown to be almost periodic.     

Monotone solutions of second-order nonlinear differential equations

Necessary and sufficient conditions are obtained for existence of monotone solutions of a nonlinear differential equation. As applications, several existence criteria and comparison theorems are derived.     

Positive solutions for second-order nonlinear differential equations

We prove the existence of positive solutions to the scalar equation y^″(x)+F(x,y,y^′)=0$y^{″}(x)+F(x,y,y^{\prime })=0$. Applications to semilinear elliptic equations in exterior domains are considered.     

Oscillation of solutions of second-order nonlinear self-adjoint differential equations

We are concerned with the oscillation problem for the nonlinear self-adjoint differential equation (a(t)x′)′+b(t)g(x)=0. Here g(x) satisfied the signum condition xg(x)>0 if x≠0, but is not imposed such monotonicity as superlinear or sublinear. We show that certain growth conditions on g(x) play an essential role in a decision whether all nontrivial solutions are oscillatory or not. Our main theorems extend recent results in a serious of papers and are best possible for the oscillation of solutions in a sense. To accomplish our results, we use Sturm's comparison method and phase plane analysis of systems of Liénard type. We also explain an analogy between our results and an oscillation criterion of Kneser-Hille type for linear differential equations.     

Existence for nonoscillatory solutions of second-order nonlinear differential equations

In this paper, the existence of nonoscillatory solutions of the second-order nonlinear neutral differential equation

     

二阶微分方程的亚纯及代数元素解

利用待定系数的方法研究了一类二阶线性齐次亚纯系数复微分方程的亚纯解及代数元素解的存在性.     

Asymptotic behavior of solutions of second-order nonlinear impulsive differential equations

In this work, the asymptotic behavior of all solutions of second-order nonlinear ordinary differential equations with impulses is investigated. By impulsive differential inequality and Riccati transformation, sufficient conditions of asymptotic behavior of all solutions of second-order nonlinear ordinary differential equations with impulses are obtained. An example is also inserted to illustrate the impulsive effect.     

On the nonexistence of entire solutions of certain type of nonlinear differential equations

By utilizing Nevanlinna's value distribution theory of meromorphic functions, it is shown that the following type of nonlinear differential equations:

fⁿ(z)+P_n−3(f)=p₁e^α₁z+p₂e^α₂z