One-parameter semigroups for linear evolution equations

No abstract. October 10, 2000     

On solvability of functional equations and system of functional equations arising in dynamic programming

The purpose of this paper is to study solvability of two classes of functional equations and a class of system of functional equations arising in dynamic programming of multistage decision processes. By using fixed point theorems, a few existence and uniqueness theorems of solutions and iterative approximation for solving these classes of functional equations are established. Under certain conditions, some existence theorems of coincidence solutions for the class of system of functional equations are shown. Some examples are given to demonstrate the advantage of our results than existing ones in the literature.     

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 a system of quasilinear parabolic functional differential equations

Summary We consider initial boundary value problems for a system of second order quasilinear parabolic equations where also the main part contains functional dependence on the unknown function. This system is of type, considered in [6], [7] by U. Hornung, W. J?ger and A. Mikelic.     

On a system of functional equations in Information Theory

Aequationes mathematicae -     

Convolution equations and nonlinear functional equations

The survey is devoted to applications of nonlinear integral equations to linear convolution equations, their discrete analogues, and also the connection of these equations with problems of radiative transfer, in particular, with the Ambartsumyan equations.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 22, pp. 175–244, 1984.     

Quartic functional equations

In this paper, we solve a new functional equation

f(2x+y)+f(2x−y)=4f(x+y)+4f(x−y)+24f(x)−6f(y)     

Dynamical system solutions for homogeneous linear functional equations of higher order

We present the solution of a large class of homogeneous linear functional equations of higher order by using ideas from dynamical systems. A particularly simple example from this class is the functional equation $$f(x) = \frac{1}{2}f \left(\frac{x}{2}\right) + \frac{1}{2}f \left(\frac{x+1}{2}\right), \quad 0 < x < 1.$$ Equations such as these have found important applications in wavelet theory by Hilberdink (Aequa Math 61(1–2):179–189, 2001) where they are called dilation equations and are usually solved by Fourier methods by Daubechies (Comm Pure Appl Math 41(7):909–996, 1988) or iteration methods of Daubechies (SIAM J Math Anal 22(5):1388–1410, 1991). A recent result of Góra (Ergod Theory Dyn Syst 29(5):1549–1583, 2009) allows us to represent the solution as an infinite series that is determined by the dynamics of a map that is defined by the functional equation. In this problem the interplay between dynamical systems and solutions of functional equations is brought into sharp focus.     

Integro-differential equations for the characteristic functional that are generated by a system of random integral equations

We consider a boundary value problem for a special system of integro-differential equations with variational derivatives. We establish the relationship between this problem and a system of integral equations with a power-law nonlinearity whose kernels and right-hand sides are random functions. We study the solvability of the boundary value problem. Special cases and examples are considered.