Generalized Cauchy difference equations. II

The main result is an improvement of previous results on the equation

for a given function . We find its general solution assuming only continuous differentiability and local nonlinearity of . We also provide new results about the more general equation

for a given function . Previous uniqueness results required strong regularity assumptions on a particular solution . Here we weaken the assumptions on considerably and find all solutions under slightly stronger regularity assumptions on .

     

Generalized stability of the Cauchy and the Jensen functional equations on spheres

We study a generalized stability problem for Cauchy and Jensen functional equations satisfied for all pairs of vectors x,y from a linear space such that γ(x)=γ(y) or γ(x+y)=γ(x−y) with a given function γ.     

Generalized Cauchy Spaces

In this paper a unified theory of Cauchy spaces is presented including the classical cases of filter and sequence Cauchy spaces. To by-pass a lattice-theoretical barrier the notion of Urysohn modification of a functor is introduced. Employing this notion for many types of generalized Cauchy spaces a completion method is given.     

Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator

This paper is devoted to proving the existence and uniqueness of solutions to Cauchy type problems for fractional differential equations with composite fractional derivative operator on a finite interval of the real axis in spaces of summable functions. An approach based on the equivalence of the nonlinear Cauchy type problem to a nonlinear Volterra integral equation of the second kind and applying a variant of the Banach’s fixed point theorem to prove uniqueness and existence of the solution is presented. The Cauchy type problems for integro-differential equations of Volterra type with composite fractional derivative operator, which contain the generalized Mittag-Leffler function in the kernel, are considered. Using the method of successive approximation, and the Laplace transform method, explicit solutions of the open problem proposed by Srivastava and Tomovski (2009) [11] are established in terms of the multinomial Mittag-Leffler function.     

Generalized solutions to Cauchy problems

This paper continues previous attempts to find a convenient mathematical setting in which linear and nonlinear Cauchy problems have a unique global solution, that reduces to a classical solution when the latter exists.With 1 Figure     

Conditional Cauchy equations and applications

Summary. The purpose of this paper is to develop a new perspective concerning conditional Cauchy equations - namely, that of exploiting a natural algebraic structure on the set of solutions of the equation. We then show how this approach leads to applications in both group and loop theory.     

Generalized poisson distribution on groups

The article deals with the arithmetic of distributions on groups. Let X be a locally compact separable abelian metric group, \(e\left( F \right) = e^{ - F\left( X \right)} \left( {\varepsilon _o + F + \frac{{F*2 - }}{{2!}} + ...} \right)\) the generalized Poisson distribution associated with a completely finite measure F, I₀ the class of distributions without indecomposable or idempotent divisors. Some conditions under which the generalized Poisson distributions belong and do not belong to the class I₀ are derived. Some other topics related to the class I₀ are considered.     

Cauchy problem for integrable discrete equations on quad-graphs

Initial value problems for the integrable discrete equations on quad-graphs are investigated. We give a geometric criterion of when such a problem is well-posed. In the basic example of the discrete KdV equation an effective integration scheme based on the matrix factorization problem is proposed and the interaction of the solutions with the localized defects in the regular square lattice are discussed in details. The examples of kinks and solitons on various quad-graphs, including quasiperiodic tilings, are presented.Dedicated to S. P. Novikov on his 65 birthdayOn leave from Landau Institute for Theoretical Physics, Chernogolovka, Russia.     

Cauchy problem for fractional diffusion equations

We consider an evolution equation with the regularized fractional derivative of an order α∈(0,1) with respect to the time variable, and a uniformly elliptic operator with variable coefficients acting in the spatial variables. Such equations describe diffusion on inhomogeneous fractals. A fundamental solution of the Cauchy problem is constructed and investigated.     

The Cauchy problem for Kirchhoff equations

Conferenza tenuta il 28 settembre 1992