QUASI-LOCAL CONJUGACY THEOREMS IN BANACH SPACES

Let f : U(x0) (?) E→F be a. C1 map and f'(X0) be the Prechet derivative of /fat X0. In local analysis of nonlinear functional analysis, implicit function theorem, inverse function theorem, local surjectivity theorem, local injectivity theorem, and the local conjugacy theorem are well known. Those theorems are established by using the properties: f'(x0) is double splitting and R(f'(x))∩N(T0 ) = {0} near X0. However, in infinite dimensional Banach spaces, f'(x0) is not always double splitting (i.e., the generalized inverse of f'(xo) does not always exist), but its bounded outer inverse of f'(x0) always exists. Only using the C1 map f and the outer inverse T0# of f'(x0), the authors obtain two quasi-local conjugacy theorems, which imply the local conjugacy theorem if X0 is a locally fine point of f. Hence the quasi-local conjugacy theorems generalize the local conjugacy theorem in Banach spaces.

QUASI-LOCAL CONJUGACY THEOREMS IN BANACH SPACES

Let f: U(xo)() E → F be a C1 map and f'(x0) be the Frechet derivative of f at x0. In local analysis of nonlinear functional analysis, implicit function theorem, inverse function theorem, local surjectivity theorem, local injectivity theorem, and the local conjugacy theorem are well known. Those theorems are established by using the properties: f'(x0) is double splitting and R(f'(x)) ∩ N(T0+) = {0} near x0. However,in infinite dimensional Banach spaces, f'(x0) is not always double splitting (i.e., the generalized inverse of f'(x0) does not always exist), but its bounded outer inverse of f'(x0) always exists.Only using the C1 map f and the outer inverse T0# of f'(x0), the authors obtain two quasi-local conjugacy theorems, which imply the local conjugacy theorem if x0 is a locally fine point of f. Hence the quasi-local conjugacy theorems generalize the local conjugacy theorem in Banach spaces.