A system of functional equations related to plurality functions. A method for the construction of the solutions

Summary In the first part of the present paper we prove some necessary conditions satisfied by the solutions of a system of functional equations related to Plurality Functions. In the second part we describe a geometric-combinatorial procedure for the construction of the solutions of that system. This procedure yields all possible solutions.     

Construction of systems of sets related to the plurality functions

In the present paper we provide a way of construction of all m-elements consistent systems. These kinds of families of sets are the parameters determining the solutions of some functional equation, which express the consistency condition appearing in characterizing the plurality functions. First, we formulate the idea of extending p-elements family to such m-tuples families which are m-elements consistent systems. Then we study some of their properties and we use them in the constructing of extending p-elements families to m-elements consistent systems and all such systems. Finally, we deal with m-elements consistent systems which satisfy an additional condition.     

On the regularity of solutions to a parabolic system related to Maxwell's equations

The goal of this paper is to establish Hölder estimates for the solutions of a certain parabolic system related to Maxwell's equations arising in a quasi-stationary electromagnetic field.     

Extremal solutions to a system of n nonlinear differential equations and regularly varying functions

The strongly increasing and strongly decreasing solutions to a system of n nonlinear first order equations are here studied, under the assumption that both the coefficients and the nonlinearities are regularly varying functions. We establish conditions under which such solutions exist and are (all) regularly varying functions, we derive their index of regular variation and establish asymptotic representations. Several applications of the main results are given, involving n‐th order nonlinear differential equations, equations with a generalized ?‐Laplacian, and nonlinear partial differential systems.     

On functional equations related to a dihedral group

We propose a method for regularization of two three-element functional equations generated by a special case of dihedral group of rotations. These results are applied to the moment problem for entire functions of exponential type and for evaluation of biorthogonally dual systems of analytic functions.     

Attractivity properties of solutions to a system of strongly nonlinear parabolic functional differential equations

In this paper, weak solutions of a system of strongly nonlinear parabolic functional differential equations are considered, and some properties of the solutions are shown for t → ∞.     

Holomorphic solutions to functional differential equations

The pantograph equation is perhaps one of the most heavily studied class of functional differential equations owing to its numerous applications in mathematical physics, biology, and problems arising in industry. This equation is characterized by a linear functional argument. Heard (1973) [10] considered a generalization of this equation that included a nonlinear functional argument. His work focussed on the asymptotic behaviour of solutions for a real variable x as x→∞. In this paper, we revisit Heard's equation, but study it in the complex plane. Using results from complex dynamics we show that any nonconstant solution that is holomorphic at the origin must have the unit circle as a natural boundary. We consider solutions that are holomorphic on the Julia set of the nonlinear argument. We show that the solutions are either constant or have a singularity at the origin. There is a special case of Heard's equation that includes only the derivative and the functional term. For this case we construct solutions to the equation and illustrate the general results using classical complex analysis.     

Remarks on solutions to a generalization of the radical functional equations

During the \(16{\text {th}}\) International Conference on Functional Equations and Inequalities a talk was given concerning the stability of the so-called radical functional equation \(f(\sqrt{x^2+y^2\,}\,)=f(x)+f(y)\). The second author’s question about the general solution of the equation itself was answered later by the first one. Contrary to some assertions in the literature the general solution is not an arbitrary quadratic function, but of the form \(x\mapsto a(x^2)\) with additive a. Here we present far reaching generalizations of this result.