Global monotonicity and oscillation for second order differential equation

Oscillatory properties of the second order nonlinear equation

On the pullback equation

We discuss the existence of a diffeomorphism such that

φ^*(g)=f

Well-posedness of second order evolution equation on discrete time

We characterize the well-posedness for second order discrete evolution equations in unconditional martingale difference spaces by means of Fourier multipliers and R-boundedness properties of the resolvent operator which defines the equation. Applications to semilinear problems are given.     

A second order splitting method for the Cahn-Hilliard equation

Summary A semi-discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary, fourth order in space, Cahn-Hilliard equation. Optimal order error bounds are derived in various norms for an implementation which uses mass lumping. The continuous problem has an energy based Lyapunov functional. It is proved that this property holds for the discrete problem.Research partially supported by NSF Grant DMS-8896141     

The mixed problem for a nonlinear degenerate elliptic equation of second order

The present paper deals with the mixed boundary value problem for a nonlinear elliptic equation with degenerate rank 0. We first give the formulation of the problem and estimates of solutions of the problem, and then prove the uniqueness and existence of solutions of the above problem for the nonlinear elliptic equation by the extremum principle and the method of parameter extension. The complex method is used to discuss the corresponding problem for degenerate elliptic complex equation of first order and then that of second order.     

Oscillation theorems for a second order delay differential equation

A generalized version is proved of the following inequality, arising in a study of invertible measure preserving transformations: $\text{(∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}_{\mathrm{i}}{}^{\mathrm{n}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}\text{(∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}_{\mathrm{i}}{}^{\mathrm{m}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>m</mtext>}}\text{? (∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}^{\mathrm{mn}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>mn</mtext>}}\text{(∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}_{\mathrm{i}}\text{),}\text{where}\text{x}{}_{\mathrm{i}}\text{? 0, i = 1, 2,…, N,}\text{and}\text{(m ? 1)(n ? 1) > 0}$.     

Zeros of solutions of certain second order linear differential equation

In this paper, we investigate the exponent of convergence of the zero-sequence of solutions of the differential equation

(1.3)     

Analytic solutions of a second order iterative functional differential equation

This paper is concerned with an iterative functional differential equation

c₁x(z)+c₂x^′(z)+c₃x^″(z)=x(az+bx^′(z))

with the delay depends on the argument of the unknown function and the state derivative. By reducing the equation with the Schröder transformation to another functional differential equation without iteration of the unknown function, we give existence of its local analytic solutions which extend the known results in related literature.