On the Diffeomorphisms Between Banach and Hilbert Spaces

Abstract

In this paper, we give some sufficient conditions for f : X → H to be a diffeomorphism, where X is a Banach space and H is a Hilbert space. The proof of the result is based on the mountain pass theorem. Using this result, in the final part of the paper, we prove an existence theorem for some class of integro-differential equations.

On a global diffeomorphism between two Banach spaces and some application

“…Our results extends their counterparts from [7] where the case p = 2 is considered, i.e. diffeomorphism between two Hilbert spaces works.…”

“…diffeomorphism between two Hilbert spaces works. Thus while we use similar general ideas several key details have to be considered in the other way due to the fact that now we do not have a Hilbert space structure which is extensively exploited in [7]. Concerning the abstract tool applied in [7] it is now modified accordingly.…”

“…Thus while we use similar general ideas several key details have to be considered in the other way due to the fact that now we do not have a Hilbert space structure which is extensively exploited in [7]. Concerning the abstract tool applied in [7] it is now modified accordingly. In [7] in the proof of the diffeomorphism theorem, properties of Hilbert space are exploited, the local diffeomorphism and a mountain pass geometry for C 1 functionals is used.…”

Control system defined by some integral operator

“…The problem of the smooth dependence of solutions on the parameter of integral or integro-differential equations was discussed in papers [2,6,7,9,10].…”

“…The novelty of this paper, in contrast to papers [9,10] is, among others, the application of the Bielecki norm in the space of solutions which allows us to weaken one of the assumptions (see Remark 2.3 for details).…”