Eigenvalue comparisons for second order difference equations with Neumann boundary conditions

The boundary value problem for second order difference equation

     

Eigenvalue comparisons for boundary value problems for second order difference equations

We consider the eigenvalue problems for boundary value problems of second order difference equations

(1)

and

(2)

Comparison results for the eigenvalues of the problem (1) and the problem (2) are established.     

First order difference equations with maxima and nonlinear functional boundary value conditions

This paper is devoted to the existence of solutions for a problem of first order difference equations with maxima and with nonlinear functional boundary value conditions. Such boundary conditions include, among others, initial, periodic, antiperiodic and multipoint boundary value conditions, as particular cases.     

Eigenvalue comparisons for second order difference equations with periodic and antiperiodic boundary conditions

We consider a class of boundary value problems of the second order difference equation $$\Delta(r_{i-1}\Delta y_{i-1})-b_{i}y_{i}+\lambda a_{i}y_{i}=0,\quad 1\le i\le n,\ y_{0}=\alpha y_{n},\ y_{n+1}=\alpha y_{1}.$$ The class of problems considered includes those with antiperiodic, Dirichlet, and periodic boundary conditions. We focus on the structure of eigenvalues of this class of problems and comparisons of all eigenvalues as the coefficients {a _i } _i=1 ⁿ ,{b _i } _i=1 ⁿ , and {r _i } _i=0 ⁿ change.     

Existence of weak solutions of two-point boundary value problems for second-order dynamic equations on time scales

In this paper, by using critical point theory, we establish the existence of weak solutions of the two-point boundary value problem for a second-order dynamic equation on an arbitrary time scale T$\mathbb{T}$, so that the well known case of differential dynamic systems (T=R)$(\mathbb{T}=\mathbb{R})$ and the recently developed case of discrete dynamic systems (T=Z)$(\mathbb{T}=\mathbb{Z})$ are unified. To the best of our knowledge, this is the first time that boundary value problems of dynamic equations on time scales have been dealt with by using critical point theory.     

On well-conditioned boundary value problems for systems of second order difference equations

In this paper well-conditioning of boundary value problems for systems of second order difference equations is studied. First, a sufficient condition for the existence of a unique bounded solution (for large enough number of steps) of an associated homogeneous system is given. Finally, a sufficient condition for well-conditioning, intrinsically related to the problem data is proposed. This work has been partially supported by the “Generalitat Valenciana” grant GV1118/93 and the Spanish D.G.I.C.Y.T. grant PB93-0381.     

Periodic solutions for a class of higher-dimension functional differential equations with impulses

By employing a fixed-point theorem in cones, we establish some criteria for existence of positive periodic solutions of a class of n$n$-dimension periodic functional differential equations with impulses. We also give some applications to several biomathematical models and new results are obtained.     

Initial-value problems for linear partial difference equations

The method presented in [4] for the solution of linear difference equations in a single variable is extended to some equations in two variables. Every linear combination of a given functionf and of its partial differences can be obtained by the discrete convolution product off by a suitable functionl (which depends on the considered linear combination), and we want to solve in a convolutional form difference equations in the whole plane. However, the convolution of two functions may not be possible if their supports contain half straight lines with opposite directions. To avoid this, we take four sets of functions corresponding to the quadrants such thatl belong to every set, every set endowed with the convolution and with the usual addition is a ring, and there is an inverse ofl in each of the four rings. This is attained by taking, for each ring, a set of functions whose supports belong to suitable cones. After choosing such rings, a very natural initial-value first-order Cauchy Problem (in partial differences) is reduced to a convolutional form. This is done either by a direct method or by introducing the forward difference functions _i f(i=1,2) in a general way depending on the shape of the support off so that Laplace-like formulas with initial and final values) hold. Applications to difference equations in the whole plane and to partial differential problems are made.     

Applying carvalho's method to find periodic solutions of difference equations

In this paper we apply the method initially developed in [1] for differential-difference equations, to the case of difference equations, in order to find 2 and 3-periodic solutions of some equations that often appear in the literatures as are for instance the case of Applications 2,5 which are examples of population growth models, and Application 4, which is a standard example of nonlinear higher order scalar difference equation depending on two parameters (see, Kocik and Ladas [3]).     

Fredholm boundary value problems for perturbed systems of dynamic equations on time scales

This paper offers conditions ensuring the existence of solutions of linear boundary value problems for systems of dynamic equations on time scales. Utilizing a method of Moore–Penrose pseudo‐inverse matrices leads to an analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditions (defined by a linear vector functional) not coinciding with the number of unknowns of a system of dynamic equations. As an example of an application of the presented results, the problem of bifurcation of solutions of boundary value problems for systems of dynamic equations on time scales with a small parameter is considered.     

Positive solutions for boundary value problems of second order difference equations and their computation

We consider the following two classes of second order boundary value problems for difference equation: