Comparison theorems for second order nonlinear difference equations

New oscillation results are obtained for the second order nonlinear difference equation

Δ(rnf(Δxn−1))+g(n,xn)=0,     

Second order linear difference equations

We provide the explicit solution of a general second order linear difference equation via the computation of its associated Green function. This Green function is completely characterized and we obtain a closed expression for it using functions of two–variables, that we have called Chebyshev functions due to its intimate relation with the usual one–variable Chebyshev polynomials. In fact, we show that Chebyshev functions become Chebyshev polynomials if constant coefficients are considered.     

A difference method for solving parabolic equations of order 2n

In this paper, a finite difference method is used to approximate for the solution of the parabolic partial differential equation of order 2n and error of the method is determined. The resulting system is solved by efficient implicit iterations. In some numerical examples, MAPLE modules are designed for the purpose of testing and using the method.     

A note on temporal regularity of second order difference equations

This work deals with the study of maximal ?^p‐regularity of a pair (A,B) of bounded linear operators on a complex Banach space associated to the second order difference equation u_{n + 2} = Bu_{n + 1} + Au_n + f_n, where f is a given sequence on . We obtain results of characterization based on spectral analysis of the discrete sine family, which is the resolvent family of this equation.     

Nonlinear equations for the recurrence coefficients of discrete orthogonal polynomials

We study polynomials orthogonal on a uniform grid. We show that each weight function gives two potentials and each potential leads to a structure relation (lowering operator). These results are applied to derive second order difference equations satisfied by the orthogonal polynomials and nonlinear difference equations satisfied by the recursion coefficients in the three-term recurrence relations.     

New explicit global asymptotic stability criteria for higher order difference equations

New explicit sufficient conditions for the asymptotic stability of the zero solution of higher order difference equations are obtained. These criteria can be applied to autonomous and nonautonomous equations. The celebrated Clark asymptotic stability criterion is improved. Also, applications to models from mathematical biology and macroeconomics are given.     

Homoclinic orbits and subharmonics for nonlinear second order difference equations

We show the existence of a nontrivial homoclinic orbit and subharmonic solutions for a class of second order difference equations by applying the “Mountain Pass” theorem relying on Ekeland’s variational principle and the diagonal method, and the homoclinic orbit as the limit of the subharmonics. A completely new way is provided for dealing with the existence of solutions for difference equations.     

Nonoscillation and oscillation of second order half-linear differential equations

We study the oscillation problems for the second order half-linear differential equation ^′[p(t)Φ(x^′)]+q(t)Φ(x)=0, where Φ(u)=|u|^r−1u with r>0, 1/p and q are locally integrable on R₊; p>0, q?0 a.e. on R₊, and . We establish new criteria for this equation to be nonoscillatory and oscillatory, respectively. When p≡1, our results are complete extensions of work by Huang [C. Huang, Oscillation and nonoscillation for second order linear differential equations, J. Math. Anal. Appl. 210 (1997) 712-723] and by Wong [J.S.W. Wong, Remarks on a paper of C. Huang, J. Math. Anal. Appl. 291 (2004) 180-188] on linear equations to the half-linear case for all r>0. These results provide corrections to the wrongly established results in [J. Jiang, Oscillation and nonoscillation for second order quasilinear differential equations, Math. Sci. Res. Hot-Line 4 (6) (2000) 39-47] on nonoscillation when 0<r<1 and on oscillation when r>1. The approach in this paper can also be used to fully extend Elbert's criteria on linear equations to half-linear equations which will cover and improve a partial extension by Yang [X. Yang, Oscillation/nonoscillation criteria for quasilinear differential equations, J. Math. Anal. Appl. 298 (2004) 363-373].     

Nonlinear boundary value problems for second order differential equations with causal operators

In this paper we deal with second order differential equations with causal operators. To obtain sufficient conditions for existence of solutions we use a monotone iterative method. We investigate both differential equations and differential inequalities. An example illustrates the results obtained.     

On solutions of second and fourth order elliptic equations with power-type nonlinearities

We study second and fourth order semilinear elliptic equations with a power-type nonlinearity depending on a power p$p$ and a parameter λ>0$\lambda >0$. For both equations we consider Dirichlet boundary conditions in the unit ball B⊂Rⁿ$B\subset {\mathbb{R}}^n$. Regularity of solutions strictly depends on the power p$p$ and the parameter λ$\lambda$. We are particularly interested in the radial solutions of these two problems and many of our proofs are based on an ordinary differential equation approach.     

A study on a second order finite difference scheme for fractional advection–diffusion equations

Second order finite difference schemes for fractional advection–diffusion equations are considered in this paper. We note that, when studying these schemes, advection terms with coefficients having the same sign as those of diffusion terms need additional estimates. In this paper, by comparing generating functions of the corresponding discretization matrices, we find that sufficiently strong diffusion can dominate the effects of advection. As a result, convergence and stability of schemes are obtained in this situation.     

Periodic solutions of some second order difference equations

In this paper, using critical point theory, some sufficient conditions are obtained for the existence of periodic solutions of a class of second order difference equation.     

An existence result for a class of second order evolution equations of mixed type

We give an existence and uniqueness result for a linear abstract evolution equation of second order with some coefficient in front of the second temporal derivative which may degenerate to zero and change sign.