Contributions to the theory of functional equations

This thesis has in it several aspects of functional equations. It includes finding the general solutions, investigating the stability, discussing operator equations, and studying conditional equations. There are five main chapters. Four of them are all related to "quadratic", while the fifth one handles "cosine". The approach in every chapter is completely different.; For the general solutions of functional equations, we content ourselves with two equations: the quadratic equation and the cosine equation. Regarding the quadratic equation, we obtain the general solutions on free groups. Applying this result, in principle one can solve it on any group if its generators and defining relations are known. For the cosine equation, we completely describe all its continuous solutions on compact connected groups G. Applying the tools in the theory of semisimple Lie groups, we prove that every non-classical solution factors through a direct factor of G which is isomorphic to SU(2). On SU(2) we prove that the normalized character of its 2-dimensional irreducible representation is the unique continuous non-classical solution.; On the stability of functional equations, we take up the Pexider-quadratic equation on groups. We first improve some results by Jung and Sahoo. As our main results in this topic, we prove that the quadratic equation is stable on amenable (locally compact) groups.; The motivation of studying our operator equations comes from a problem on the representability of quasi-quadratic functionals by sesquilinear ones. In this part, we make some contributions to solving a problem posed by Molnar, which basically asks whether every Jordan *-derivation pair ( E, F) on standard operator algebras A is inner. We solve it when E = F or A is unital. For the general case, we prove that E and F both are real linear. Several other results of independent interest are also obtained along the way.; Our study on conditional functional equations solves an open problem posed by Ratz. It is concerned about characterizing orthogonality spaces with non-zero even orthogonally additive mappings. Our proof is constructive.