Eigenvalues of discrete Sturm-Liouville problems with nonlinear eigenparameter dependent boundary conditions

We consider the discrete right definite Sturm-Liouville problems with nonlinear eigenparameter dependent boundary conditions

,

where T > 1 is an integer and λ is the spectrum parameter. We obtain the existence of the eigenvalues, the oscillation properties of the eigenfunctions and the interlacing results of the eigenvalues of the above problem with the eigenvalues of the Dirichlet problem and the Neumann problem.     

Lie symmetry analysis and explicit formulas for solutions of some third-order difference equations

Abstract

Lie group theory is applied to rational difference equations of the form

where (a_n)n∈?₀, (b_n)n∈?₀ are non-zero real sequences. Consequently, new symmetries are derived and exact solutions, in unified manner, are constructed. Based on some constraints in the expression of the symmetry generators, we split these solutions into different categories. This work generalises a recent result by Ibrahim [9].     

On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems

In this paper, we obtain a new sufficient condition on the existence of homoclinic solutions of a class of discrete nonlinear periodic systems by using critical point theory in combination with periodic approximations. We prove that it is also necessary in some special cases.     

Existence and stability of nontrivial periodic solutions of periodically forced discrete dynamical Systems

The local existence and local asymptotic stability of nontrivial p-periodic solutions of p-periodically forced discrete systems are proven using Liapunov-Schmidt methods. The periodic solutions bifurcate transcritically from the trivial solution at the critical value n=n^cr of the bifurcation parameter with a typical exchange of stability. If the trivial solution loses (gains) stability as n is increased through n^cr , then the periodic solutions on the nontrivial bifurcating branch are locally asymptotically stable if and only if they correspond to n>n^cr (n n^cr ).     

Eigenvalue comparisons for a class of boundary value problems of second order difference equations

We consider the boundary value problem for second order difference equation

     

On an alternative cauchy equation in two unknown functions. Some classes of solutions

Summary In this paper we consider the alternative Cauchy functional equationg(xy) g(x)g(y) impliesf(xy) = f(x)f(y) wheref, g are functions from a topological group (X, ·) into a group (S,·). First we prove that, ifS is a Hausdorff topological group andX satisfies some weak additional hypotheses, then (f, g) is a continuous solution if and only if eitherf org is a homomorphism. Then we describe a more general class of solutions forX =R ⁿ .Partially supported by M.U.R.S.T. Research funds (40%)Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

On Cauchy differences of all orders

Summary This paper deals with the problem of characterizing higher order Cauchy differences of mappings on groups and semigroups. Symmetric, first order Cauchy differencesf(x + y)–f(x)–f(y) for mapsf between groups were characterized by Jessen, Karpf, and Thorup [8] through the use of first partial Cauchy differences. Our results are similar and extend their result to higher order differences. Our results also extend those of Heuvers [6] for mappings between vector spaces over the rationals.     

On approximate solutions of the Pexider equation

Summary LetX be an abelian (topological) group andY a normed space. In this paper the following functional inequality is considered: {ie143-1} This inequality is a similar generalization of the Pexider equation as J. Tabor's generalization of the Cauchy equation (cf. [3], [4]). The solutions of our inequality have similar properties as the solutions of the Pexider equation. Continuity and related properties of the solutions are investigated as well.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

The concept of spectral dichotomy for linear difference equations II

This paper is a continuation of a previous one (J. Math. Anal. Appl. 185 (1994), 275–287) in which the concept of spectral dichotomy has been introduced. This new notion of dichotomy has proved to be useful since it allows to apply the well known theory of linear operators to study dynamic properties of nonautonomous linear difference equations. In the present paper we extend our result on the equivalence of the spectral dichotomy and the well known exponential dichotomy to the class of linear differenc equations whose right-hand sides are not necessarily invertible. We furthermore investigate equations on the set of positive integers for which we establish necessary and sufficient conditions for exponential and unifrom stability.     

On the mutual noncompatibility of homogeneous analytic non-power means

Summary Homogeneous symmetric meansµ and , defined on ₊ ⁿ and ₊ ⁿ⁺¹ , respectively, are calledcompatible if the value of remains unchanged upon replacing n of its arguments by theirµ-mean. Power means (of a common exponent) are a model example, which turns out to be unique, given analyticity of at least one of the two means considered. This is proved by fixing all but one argument in both and , which leads to a functional equation with two unknown functions, involving their mutual superpositions. The equation is solved in the class of analytic functions by comparing the power series coefficients.     

Some propositions related to a dilation theorem of W. Benz

Summary This paper begins with another proof of a theorem of W. Benz [2] concerning dilations in normed linear spaces. Our proof motivates several questions which are addressed thereafter. For instance it is shown that, ifI is an open interval in ,: I ⁿ , is continuously differentiable and there exista ₁,...,a _n I such that {(a ₁,...,(a _n )} is linearly independent, then {(t): t I} contains a Hamel basis for ⁿ over .     

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]).