Positive solutions for quasilinear second order differential equation

It is well known that the Krasnoselskii's fixed point theorem is very very important. It was extensively used for studying the boundary value problems. In this article, the Krasnoselskii's fixed point theorem is extended. The new fixed point theorem is obtained. The second order quasilinear differential equation (Φ (y′))′+a(t)f(t,y,y′)=0,, 0<t<1 subject to mixed boundary condition is studied, where f is a nonnegative continuous function, Φ (v)= |v|^p-2 v, p>1. We show the existence of at least one positive solution by using the new fixed point theorem in cone.     

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.     

The mixed BVP for second order nonlinear ordinary differential equation at resonance

Efficient sufficient conditions are established for the solvability of the mixed problem where in the case where the homogeneous linear problem has nontrivial solutions.     

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.     

On stability of the second order neutral differential equation

There exist a well-developed stability theory for neutral differential equations of the first order and only a few results on functional differential equations of the second order. One of the aims of this paper is to fill this gap. Explicit tests for stability of linear neutral delay differential equations of the second order are obtained.     

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     

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: $\text{(∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}_{\mathrm{i}}{}^{\mathrm{n}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}\text{(∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}_{\mathrm{i}}{}^{\mathrm{m}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>m</mtext>}}\text{? (∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}^{\mathrm{mn}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>mn</mtext>}}\text{(∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}_{\mathrm{i}}\text{),}\text{where}\text{x}{}_{\mathrm{i}}\text{? 0, i = 1, 2,…, N,}\text{and}\text{(m ? 1)(n ? 1) > 0}$.     

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

The value distribution of meromorphic solutions of some second order nonlinear difference equation

In this paper, we investigate some properties of solutions for some nonlinear difference equation, and obtain some estimates of the exponent of convergence of poles and growth of its transcendental meromorphic solutions $f(z)$ and its difference $\Delta f(z)$. Moreover, we study the existence and forms of rational solutions. We also give some examples to support our theoretical discussion.     

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)