Generalized real analysis and its applications

In this paper there are stressed some of the advantages of a generalized real analysis (called pseudo-analysis) based on some real operations which are taken instead of the usual addition and product of reals. Namely, there are covered with one theory and so with unified methods many problems (usually nonlinear) from many fields (system theory, optimization, control theory, differential equations, difference equations, etc.). There are presented some important real aggregation functions as triangular norms and triangular conorms and a real semiring with pseudo-operations. First there is presented how these operations occur as basic operations in the theory of fuzzy logics and fuzzy sets and there is shown a generalization of the utility theory represented by hybrid probabilistic–possibilistic measure. The real semirings serve as a base for pseudo-additive measures, pseudo-integrals, pseudo-convolutions which form the pseudo-analysis. There are presented some of the applications by large deviation principle, nonlinear Hamilton–Jacobi equation, cumulative prospect theory.