Functional Equations and Weierstrass Transforms

The purpose of this article is to illustrate the utility of the Weierstrass transform in the study of functional equations (and systems) of the form 1 $${\mathop \sum^N\limits_{k=0}}\alpha_{k}f(x+r_{k})=f_{0}(x)\ \ \ \, x\in\ {\rm R}.$$ One may think of α₀, α₁,…, α_N as given complex numbers, r₀, r₁,…, r_N as given real numbers, ?₀: ? → C as a given function and ? as the unknown.     

Weierstrass integrability of differential equations

The integrability problem consists of finding the class of functions a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit an elementary or Liouvillian first integral. We define Weierstrass integrability and we determine some Weierstrass integrable systems which are Liouvillian integrable. Inside this new class of integrable systems there are non-Liouvillian integrable systems.     

Functional Equations and Weierstrass Transforms II

The Weierstrass transform is examined on the space of Lebesgue measurable function on Rⁿ having at most exponential growth, thereby extending to higher dimensions the one-dimensional consideration of [4]. The resulting theory has utility in the study of certain functional equations of “translation” type; two such applications are presented.     

Functional equations and martingales

Aequationes mathematicae - We consider functional equations (Cauchy’s, Abel’s and some other functional equations) and show that finding the general solution of these equations...     

Functional differential equations

The method of quasilinearization is a well-known technique for obtaining approximate solutions of nonlinear differential equations. In this paper we apply this technique to functional differential problems. It is shown that linear iterations converge to the unique solution and this convergence is superlinear.     

Functional evolution equations

Of concern is the functional evolution equation

     

Weierstrass

Ohne Zusammenfassung     

Functional equations and group substitutions

Motivated by some investigations of Babbage and a method of solving certain functional equations arising in competition problems, we investigate a class of functional equations and prove a local existence and uniqueness theorem for them. The main tools of the proof are the Inverse Function Theorem and the Global Existence and Uniqueness Theorem.     

Functional equations with extrema

Exact solutions of sine Gordon and multiple sine Gordon equations are constructed in terms of solutions of a linear base equation, the Klein Gordon equation and also in terms of nonlinear base equations where the nonlinearity is polynomial in the dependent variable. Further, exact solutions of nonlinear generalizations of the Schrodinger equation and of additional nonlinear generalizations of the Klein Gordon equation are constructed in terms of solutions of linear base equations. Finally, solutions with spherical symmetry, of nonlinear Klein Gordon equations are given.     

Functional equations on semigroups

Summary. We determine the general solution g:S? F g:S\to F of the d'Alembert equation¶¶g(x+y)+g(x+sy)=2g(x)g(y)       (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)g(y)\qquad (x,y\in S) ,¶the general solution g:S? G g:S\to G of the Jensen equation¶¶g(x+y)+g(x+sy)=2g(x)       (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)\qquad (x,y\in S) ,¶and the general solution g:S? H g:S\to H of the quadratic equation¶¶g(x+y)+g(x+sy)=2g(x)+2g(y)       (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)+2g(y)\qquad (x,y\in S) ,¶ where S is a commutative semigroup, F is a quadratically closed commutative field of characteristic different from 2, G is a 2-cancellative abelian group, H is an abelian group uniquely divisible by 2, and s \sigma is an endomorphism of S with s(s(x)) = x \sigma(\sigma(x)) = x .     

Functional equations involving means

In this paper, the functional equation $$ f(px + (1 - p)y) + f((1 - p)x + py) = f(x) + f(y), (x,y \in I) $$ is considered, where 0 < p < 1 is a fixed parameter and f: I → R is an unknown function. The equivalence of this and Jensen’s functional equation is completely characterized in terms of the algebraic properties of the parameter p. As an application, solutions of certain functional equations involving four weighted arithmetic means are also determined.     

Functional equations onA-orthogonal vectors

Summary LetH be a real (complex) Hilbert space with dimH 3,A:H H a continuous selfadjoint operator with dimA(H) > 2; we introduce onH a suitableA-orthogonality relation and study, in the class of the real (complex) functionals defined onH, two conditional functional equations — the Cauchy and the quadratic one — on the restricted domain of theA-orthogonal vectors.In this paper we determine the general solutions of these equations by theorems in which we establish the equivalence between each equation postulated on the whole space and the respective conditional equation.Our investigations have been motivated by incomplete studies on these conditional functional equations made in 1986 and 1966 by H. Drljevié and F. Vajzovié, respectively.     

Functional equation for the crossover in the model of one-dimensional Weierstrass random walks

We consider the problem of one-dimensional symmetric diffusion in the framework of Markov random walks of the Weierstrass type using two-parameter scaling for the transition probability. We construct a solution for the characteristic Lyapunov function as a sum of regular (homogeneous) and singular (nonhomogeneous) solutions and find the conditions for the crossover from normal to anomalous diffusion.