Non-Global Higher-Order Differentiable And ?-Lie Higher-Order Differentiable Nonlinear Mappings

In this paper,we study nonlinear higher derivable maps related to idempotents and square zero elements on triangular algebras.The following is the main contents:In Chapter 1,we introduce some common symbols and definitions(for example,triangular algebra and higher derivation)which will be used in this paper.In Chapter 2,we study a class of higher derivable nonlinear maps related to idempotents.Let T=Tri(A,M,B)be a triangular algebra,and ??n?n?N:T?T be a family of maps(without the assumption of additivity and where ?0 is the identity map).If ??n}n?N satisfies(?)for any U,V ? T and at least one of them is an idempotent,then {?n?n?N is an additive higher derivation on T.In Chapter 3,we study nonlinear maps which are ?-Lie higher derivable by Lie product square zero elements on triangular algebras.Let T=Tri(A,M,B)be a 2-torsion free triangular algebra,S={S ?T:S2=0} be the set of square zero elements on T and {?n}n?NN:T?T be a family of maps(without the assumption of additivity and where ?0 is the identity map).If {?n}n?N satisfies(?)(?)for any W,X ?T and[W,X]?S,then ??n}n?N is an additive higher derivation on T.