Actions of Join Spaces of Continuous Functions on Hypergroups of Second-Order Linear Differential Operators

One from classical constructions of algebraic binary hyperstructures (semihypergroups and hypergroups) from ordered algebraic systems is based on a certain Lemma on principal ends generated by products of pairs of elements – shortly termed as Ends-Lemma. First version of this auxiliary lemma was obtained in the monography [3]. The mentioned lemma is applied not only in case of hyperstructures created from ordered or quasi-ordered semigroups and groups but also in constructions based centralizers of various transformations (in partricular endomorphism monoids of mono-unary algebras of various functions or operators). Moreover constructions of this type yield the way to obtain semihypergroups and hypergroups of linear differential operators – ordinary differential operators of a given order n or partial differential operators and mentioned constructions allow also to create hyperstructures of integral operators of Fredholm or Volterra-type – cf. [2], [5]. In the present contribution we construct actions of commutative transposition hypergroups i.e. join spaces created from rings of continuous and smooth functions of a given class on semihypergroups or hypergroups of second order linear ordinary differential operators. These constructed structures are in fact discrete dynamical systems with a phase (additive) hypergroups of continuous and smooth functions and phase set formed by the above mentioned differential operators. As a suitable synonyma we can use a multiautomaton without output function, where the transition function or next state funcion satisfies so called Generalized Mixed Asociativity Condition (GMAC) – see the complete contribution. It is to be noted that the systematic investigation of transformations of ordinary differential equations using algebraic tools and methods is going back into fifties which has been initiated by Professor Otakar Bor̊uvka and his school, [7]. In this connection recall from the paper [6] of one of his outstanding successors – Professor Frantǐsek Neuman, to whom we dedicate our contribution: ”Algebraic, topological and geometrical tools together with the methods of the theory of dynamical systems and functional equations make it possible to deal with problems concerning global properties of solutions by contrast to the previous local investigations and isolated results”. The main part (4) of the contribution contains the isomorphism theorem between actions (discrete dynamical systems – called in the last Bor̊uvka’s paper [1] as algebraic spaces with operators) of an additive join space of real functions of the given class C on the phase space formed by second-order linear ordinary differential operators and actions of the mentioned functional join space of hypergroups of transformation operators which are restrictions on intervals of the real line of Laplace-transform images of the Volterra-type integral operators with difference kernels. Notice that the just mentioned classical Laplace transform, which is a part of curricula of mathematical subjects – in particular at faculties of electrical engineering of universities of technology, enables the construction of an embedding of the above mentioned Volterra integral operators in the hypergroup of certain complex transformations generalizing affine transformations of vector function spaces. These facts serve as motivating factors of presented constructions.