We first obtain some sufficient conditions for an isometric mapping defined on the unit sphere(or ball)of aβ-normed space(0＜β■1)can bc extended to be a linear isometry on the whole space.Secondly,in aβ-normed space E,we study the extension problem of(λ,Ψ,2)-isometry.The main result says that positively ho- mogeneous mapping V_0:B_1(E)→B_1(E)is(1,Ψ,2)-isometry if and only if‖V_0x‖■‖x‖,■x∈B_1(E),hence this result generalizes the corresponding result in Zhang L.