- Research Article
- Open Access
A Note on Implicit Functions in Locally Convex Spaces
© M. Tavernise and A. Trombetta. 2009
- Received: 27 February 2009
- Accepted: 19 October 2009
- Published: 9 November 2009
- Banach Space
- Linear Operator
- Open Subset
- Real Line
- Product Space
Implicit function theorems are an important tool in nonlinear analysis. They have significant applications in the theory of nonlinear integral equations. One of the most important results is the classic Hildebrandt-Graves theorem. The main assumption in all its formulations is some differentiability requirement. Applying this theorem to various types of Hammerstein integral equations in Banach spaces, it turned out that the hypothesis of existence and continuity of the derivative of the operators related to the studied equation is too restrictive. In  it is introduced an interesting linearization property for parameter dependent operators in Banach spaces. Moreover, it is proved a generalization of the Hildebrandt-Graves theorem which implies easily the second averaging theorem of Bogoljubov for ordinary differential equations on the real line.
The notion of osculating operators has been considered from different points of view (see [2, 3]). In this note we reformulate the definition of osculating operators. Our setting is a locally convex topological vector space. Moreover, we present a new implicit function theorem and, as an example of application, we study the solutions of an Hammerstein equation containing a parameter.
is a neighborhood base at . Observe that is a basic calibration for if and only if for each there is such that for and . Given , the family of all maxima of finite subfamily of is a basic calibration.
Now we prove our main result.
for and . Therefore, for each such that , the operator from into is a contraction in the sense of .
for each such that . Then, by [7, Theorem ], when , the operator has a unique fixed point , which is obviously a solution of (3.1).
As an example of application of our main result, we study the stability of the solutions of an operator equation with respect to a parameter.
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