On spectral synthesis in varieties containing the solutions of inhomogeneous linear functional equations

As a continuation of our previous work [2] the aim of the recent paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral synthesis in translation invariant closed linear subspaces of additive/multiadditive functions containing the restrictions of the solutions to finitely generated fields. The idea is based on the fundamental work of [5]. Using spectral analysis in some related varieties we can prove the existence of special solutions (automorphisms) of the functional equation but spectral synthesis allows us to describe the entire space of solutions on a large class of finitely generated fields. It is spanned by the so-called exponential monomials which can be given in terms of automorphisms of \({\mathbb C}\) and differential operators. We apply the general theory to some inhomogeneous problems motivated by quadrature rules of approximate integration [8], see also [7, 9].     

A system of two inhomogeneous linear functional equations

Given real nonzero coefficients a, A, b, B and additive functions $\phi,\psi : {\mathbb {R}}\to {\mathbb {R}}$ , necessary and sufficient conditions for existence and uniqueness of additive solutions of the system ξ(ax)?Aξ(x)=?(x), ξ(bx)?Bξ(x)=ψ(x) are presented. According to the various possibilities concerning the arithmetic nature of the four coefficients and the algebraic relationships between them, the additive solution(s) of the system are given explicitly for each case.     

On some properties of solutions of second-order linear functional differential equations

The properties of solutions of the equationu″(t) =p ₁(t)u(τ₁(t)) +p ₂(t)u′(τ₂(t)) are investigated wherep _i :a, + ∞[→R (i=1,2) are locally summable functions τ₁ :a, + ∞[→R is a measurable function, and τ₂ :a, + ∞[→R is a nondecreasing locally absolutely continuous function. Moreover, τ _i (t) ≥t (i = 1,2),p ₁(t)≥0,p ₂ ² (t) ≤ (4 - ɛ)τ ₂ ^′ (t)p ₁(t), ɛ =const > 0 and . In particular, it is proved that solutions whose derivatives are square integrable on [α,+∞] form a one-dimensional linear space and for any such solution to vanish at infinity it is necessary and sufficient that .     

On the uniqueness of solutions of spectral equations

We prove uniqueness of the viscosity solutions of the Dirichlet problem of the spectral equation where is the vector whose components are eigenvalues of a matrix associated with the unknown function u.     

Existence of nontrivial solutions of linear functional equations

In this paper we study the functional equation $$\sum_{i=1}^n a_i f(b_i x+c_i h)=0 \quad (x, h \in \mathbb{C})$$ where a _i , b _i , c _i are fixed complex numbers and \({f \colon \mathbb{C} \to \mathbb{C}}\) is the unknown function. We show, that if there is i such that \({b_i / c_i \neq b_j /c_j}\) holds for any \({1 \leq j \leq n,\ j \neq i}\) , the functional equation has a nonconstant solution if and only if there are field automorphisms \({\phi_1, \ldots, \phi_k}\) of \({\mathbb{C}}\) such that \({\phi_1 \cdots \phi_k}\) is a solution of the equation.     

On the algebraic properties of solutions of inhomogeneous hypergeometric equations

Generalized hypergeometric differential equations of arbitrary order are considered. Necessary and sufficient conditions for the algebraic independence of solutions of collections of such equations, as well as of their values at algebraic points, are obtained.     

On holomorphic solutions of some inhomogeneous linear differential equations in a banach space over a non-archimedean field

Let A be a closed linear operator on a Banach space $ \mathfrak{B} $ \mathfrak{B} over the field Ω of complex p-adic numbers having an inverse operator defined on the whole $ \mathfrak{B} $ \mathfrak{B} , and f be a locally holomorphic at 0 $ \mathfrak{B} $ \mathfrak{B} -valued vector function. The problem of existence and uniqueness of a locally holomorphic at 0 solution of the differential equation y ^(m) − Ay = f is considered in this paper. In particular, it is shown that this problem is solvable under the condition $ \mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{\left\| {A^{ - n} } \right\|}} $ \mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{\left\| {A^{ - n} } \right\|}} = 0. It is proved also that if the vector-function f is entire, then there exists a unique entire solution of this equation. Moreover, the necessary and sufficient conditions for the Cauchy problem for such an equation to be correctly posed in the class of locally holomorphic functions are presented.     

On the identification of inhomogeneous parameters in dynamic linear partial differential equations

The task of identifying inhomogeneous (position-dependent) coefficients of linear dynamic partial differential equations on the basis of a finite collection of points of the solution has practical importance and is the subject of many published analyses, some of which are described herein. The purpose of the present paper is to present new developments on a simple yet appealing method due to the hydrologist B. Sagar. The technique exploits the viewpoint that the coefficient values of the partial differential equation at any point x are uniquely determined by the solution values in a small neighborhood of x. The identification algorithm which results from these considerations is extremely simple, and yet, in view of technical considerations and experimental evidence set forth here, it seems effective. In particular, we have been able to derive error bounds, which the authors believe is a new feature in the literature of identification of partial differential systems.     

Convergent solutions of linear functional difference equations in phase space

By using weighted summable dichotomies and Schauder's fixed point theorem, we prove the existence of convergent solutions of linear functional difference equations. We apply our result to Volterra difference equations with infinite delay.