Stability For Several Types Of Equations On Some Abstract Spaces

The Hyer-Ulam stability of functional equations is an important topic in nonlinear analysis.The stability problem of functional equations originated from a question of Ulam,concerning the stability of group homomorphisms.Hyers was able to give a partial solution to Ulam's question that was the first significant breakthrough and step toward more solutions in this area.The stability phenomenon attracted several mathematicians worldwide who began to be stimulated to investigate the stability problems of functional equations.In this thesis,we obtain the Hyers-Ulam stability of operatorial equations on some abstract spaces.The emphases in this thesis are summarized as follows.1.Using the direct and the fixed point methods,we prove the Hyers-Ulam stability of the mixed additive and quadratic functional equation in complete non-Archimedean normed spaces.Meantime,sufficient conditions for the stability of the AQ equation are given.2.We introduce a notion of non-Archimedean metric space endowed with the non-Archimedean Pompeiu-Hausdorff metric.Then,we prove the Hyers-Ulam stability for additive,quadratic and cubic set-valued functional equations in the framework of complete non-Archimedean metric spaces.We indeed present an interdisciplinary relation between the theory of set-valued mappings,the theory of non-Archimedean spaces and the stability theory of functional equations.3.Using the weakly Picard operator technique,we establish some abstract Hyers-Ulam stability results for operatorial equations in noncommutative metric spaces.As an application,results for the existence,uniqueness and stability of the solution of the fractional differential equation are given.4.We consider some abstract Hyers-Ulam stability results of the initial value problem of fractional differential equations in quaternionic analysis,and provide a sufficient condition for associated pairs with the Dirac operator.Sufficient conditions for the existence of solutions of the initial value problem are given by the application of the method of associated spaces.Moreover,we obtain some Hyers-Ulam stability results for fractional differential equations.5.Using Kronecker products and vector operator technique,we present the general exact solutions of non-homogenous linear matrix fractional differential equations and such coupled linear matrix fractional differential equations.And some Hyers-Ulam stability results for these linear matrix fractional differential equations are obtained.