On Stability Of Coarse Isometries And Functional Equations Of Banach Spaces

In this dissertation,we mainly study stability and weak stability of non-surjective coarse isometries of Banach spaces,especially Lp(1<p<∞)spaces and finite dimensional normed linear spaces,and generalized stability of a class of non-surjective functional equations in uniformly convex spaces.The main results are as follows.Theorem 1.Let f:Lp→)Lp be a standard coarse isometry.(Ⅰ)If f is pointwise weakly stable at each point of a fundamental set,then there is a linear isometry U:Lp→ Lp such that(?),x∈Lp;(Ⅱ)If f is uniformly weakly stable on the unit sphere,then there is a linear isometry U:Lp→Lp such that‖f(x)-Ux‖=o(‖x‖),‖x‖→ ∞.Theorem 2.Let f:Lp→Lp be a standard coarse isometry.If U:Lp→ Lp is a linear isometry and P:Lp→U(Lp)is a projection with ‖P‖=1,Then the following statements are equivalent.(Ⅰ)‖f(x)-Ux‖=o(‖x‖),‖x‖→ ∞(Ⅱ)‖Pf(x)-Ux‖=o(‖x‖),‖x‖→ ∞.Theorem 3.Let X,Y be finite dimensional normed linear spaces with dim X=dim Y,and let f:X→Y be a standard coarse isometry.If f satisfies one of the following conditions.(Ⅰ)f is pointwise weakly stable at each point of a fundamental set;(Ⅱ)f satisfies integral convergence condition whereεf(t)=sup{|‖f(x)-f(y)‖-‖x-y‖|:x,y∈X‖x-y‖≤},t≥ 0.Then there is a surjective linear isometry U:X→Y such that‖f(x)-Ux‖=o(‖x‖),‖x‖→∞.Theorem 4.Let(G,+)be an abelian group,and let Y be a uniformly convex space of power type p.If a standard map f:G→Y satisfies d(u,f(G))≤δ‖u‖r for each u∈Y and|‖f(x)-f(y)‖-‖f(x-y)‖|≤ε,x,y ∈ G,where δ,ε≥ 0,0<r<1.Then there is an additive map A:G→ Y such that‖f(x)-Ax‖=o(‖f(x)‖),‖f(x)‖→∞.