Projectors on the generalized eigenspaces for functional differential equations using integrated semigroups

The aim of this article is to derive explicit formulas for the projectors on the generalized eigenspaces associated to some eigenvalues for linear functional differential equations (FDE) by using integrated semigroup theory. The idea is to formulate the FDE as a non-densely defined Cauchy problem and obtain an explicit formula for the integrated solutions of the non-densely defined Cauchy problem, from which we then derive explicit formulas for the projectors on the generalized eigenspaces associated to some eigenvalues. The results are useful in studying bifurcations in some semi-linear problems.     

Asymptotic expansions for singular differential operators with matricial coefficients

This paper presents a detailed analysis of the asymptotic expansion, in terms of Bessel functions, for some eigenfunctions of a singular second-order differential operator with matrix coefficients. In application, we recover the asymptotic behavior of the associated Harish-Chandra function and interesting approximations at infinity of the related spectral function and scattering matrix.     

Point initial value problem for linear functional differential equations in Banach spaces

Summary The problem of existence and uniqueness of solutions defined on the whole real line and satisfying given initial point data for general abstract linear functional differential equations is considered. The equation is not assumed to be of the delay type. The essence of the method presented here consists in the representation of a solution in the form analogous to the variation of constants formula known for linear ordinary differential equations. It is shown that such an approach can be effectively applied to the problem of existence and uniqueness of solutions satisfying an exponential growth estimate, provided that the deviation of the argument is sufficiently small. The proofs are based on the Banach fixed point principle. Detailed comparison and discussion of the hypotheses ensuring the existence and uniqueness of solutions are presented.     

On a class of abstract neutral functional differential equations

By using the theory of semigroups of growth α, we discuss the existence of mild solutions for a class of abstract neutral functional differential equations. A concrete application to partial neutral functional differential equations is considered.     

Theory of fractional functional differential equations

In this paper, the basic theory for the initial value problems for fractional functional differential equations is considered, extending the corresponding theory of ordinary functional differential equations.     

Oscillation constant of second-order non-linear self-adjoint differential equations

Our concern is to solve the oscillation problem for the non-linear self-adjoint differential equation (a(t)x’)’+b(t)g(x)=0, where g(x) satisfies the signum condition xg(x)>0 if x≠0, but is not assumed to be monotone. Sufficient conditions and necessary conditions are given for all non-trivial solutions to be oscillatory. The obtained results show that the number 1/4 is a critical value for this problem. This paper takes a different approach from most of the previous research. Proof is given by means of phase plane analysis of systems of Liénard type. Examples are included to illustrate the relation between our theorems and results which were given by Cecchi, Marini and Villari. Received: January 5, 2001?Published online: June 11, 2002     

Global existence for second order functional semilinear equations

In this paper we study the global existence of solutions for initial value problems for functional semilinear equations, where the linear operator in the differential equation is the infinitesimal generator of a strongly cosine family in a Banach spaceX. Using the Leray-Schauder Alternative, we derive conditions under which a solution exists globally.     

Analytical and numerical investigation of mixed-type functional differential equations

This paper is concerned with the approximate solution of a linear non-autonomous functional differential equation, with both advanced and delayed arguments. We search for a solution x(t), defined for t∈[−1,k], (k∈N), that satisfies this equation almost everywhere on [0,k−1] and assumes specified values on the intervals [−1,0] and (k−1,k]. We provide a discussion of existence and uniqueness theory for the problems under consideration and describe numerical algorithms for their solution, giving an analysis of their convergence.     

Heat kernel expansions on the integers and the Toda lattice hierarchy

We consider the heat equation u _t = Lu where L is a second-order difference operator in a discrete variable n. The fundamental solution has an expansion in terms of the Bessel functions of imaginary argument. The coefficients α _k (n, m) in this expansion are analogs of Hadamard’s coefficients for the (continuous) Schr?dinger operator. We derive an explicit formula for α _k in terms of the wave and the adjoint wave functions of the Toda lattice hierarchy. As a first application of this result, we prove that the values of these coefficients on the diagonals n = m and n = m + 1 define a hierarchy of differential-difference equations which is equivalent to the Toda lattice hierarchy. Using this fact and the correspondence between commutative rings of difference operators and algebraic curves we show that the fundamental solution can be summed up, giving a finite formula involving only two Bessel functions with polynomial coefficients in the time variable t, if and only if the operator L belongs to the family of bispectral operators constructed in [18].      

Existence of global solutions for second order impulsive abstract partial differential equations

In this paper, we study the existence of global solutions for a class of second order impulsive abstract functional differential equations. The results are obtained by using Leray-Schauder’s Alternative fixed point theorem. An application is provided to illustrate the theory.     

Study the stability of solutions of functional differential equations via fixed points

In this paper, we study the stability properties of solutions of a class of functional differential equations with variable delay. By using the fixed point theory under an exponentially weighted metric, we obtain some interesting sufficient conditions ensuring that the zero solution of the equations is stable and asymptotically stable.     

Oscillation of even order partial differential equations with distributed deviating arguments

Some Kamenev-type oscillation criteria are established for a class of boundary value problems associated with even-order partial differential equations with distributed deviating arguments. Our approach is to reduce the high-dimensional oscillation problem to a one-dimensional oscillation one, and the general means developed by Philos and Wong is used as the main tool. The results obtained here extend and improve some known results in the literature.     

Mellin-type differential equations and associated sampling expansions

This paper is devoted to a self-contained approach to Mellin-type differential equations and associated ssampling expansions. Here the first order differential operator is not the normal d/dx but D_M,c=xd/dx+c,c E R being connected with the definition of the Mellin transform. Existence and uniqueness theorems are established for a system of first order Mellin equations and the properties of nth order linear equations are investigated. Then self adjoint Mellin-type second order Sturm-Liouville eigenvalue problems are considered and properties of the eigenvalues, eigenfunctions and Green's functions are derived. As applications. sampling representations for two classes of integral transforms arising from the eigenvalue problem are introduced. In the first class the kernesl are solutions of the problem and in the second they are expressed in terms of green's function.     

Quasilinearization for functional differential equations with retardation and anticipation

The technique of generalised quasilinearization is developed, using the method of lower and upper solutions and the monotone iterative technique (MIT), to prove the existence of unique solutions for functional differential equations (FDE) with retardation and anticipation.     

On solving certain differential equations with variable coefficients

We show how to solve certain types of linear ordinary differential equations with variable coefficients by using Appell polynomials.     

Decay for nonlinear Klein-Gordon equations

We study the asymptotic behavior of the semilinear Klein-Gordon equation with nonlinearity of fractional order. By the aid of a suitable generalization of the weighted Sobolev spaces we define the weighted Sobolev spaces on the upper branch of the unit hyperboloid. In these spaces of fractional order we obtain a weighted Sobolev embedding and a nonlinear estimate. Using these, we establish the decay estimate of the solution for large time provided the power of nonlinearity is greater than a critical value.     

Hyperbolicity of linear partial differential equations with delay

Robust hyperbolicity and stability results for linear partial differential equations with delay will be given and, as an application, the effect of small delays to the asymptotic properties of feedback systems will be analyzed.The author thanks W. Desch (Graz), I. Gyri (Veszprém) and R. Schnaubelt (Halle) for helpful discussions.