# A Note on Implicit Functions in Locally Convex Spaces

- Marianna Tavernise
^{1}Email author and - Alessandro Trombetta
^{1}

**2009**:707406

**DOI: **10.1155/2009/707406

© M. Tavernise and A. Trombetta. 2009

**Received: **27 February 2009

**Accepted: **19 October 2009

**Published: **9 November 2009

## Abstract

## 1. Introduction

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 [1] 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.

*osculating*at if there exists a function such that and

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.

## 2. Preliminaries

*calibration*for . Denote by the set of all calibrations for . A

*basic calibration*for is such that the collection of all

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.

In the following we will assume that is a complete Hausdorff locally convex topological vector space and that is a basic calibration for .

## 3. Main Result

Assume that is a solution of the above equation. A fundamental problem in nonlinear analysis is to study solutions of (3.1) for close to .

*-osculating*at if there exist a function and such that and for any

Now we prove our main result.

Theorem 3.1.

Suppose that and satisfy the following conditions:

(a) is a solution of (3.1) and the operator is continuous at ;

(b)there exists an operator function such that and are -osculating at ;

(c)the linear operator is invertible and for each . Moreover the family is uniformly -bounded.

Then there are , and such that, for each with , (3.1) has a unique solution .

Proof.

for and . Therefore, for each such that , the operator from into is a contraction in the sense of [7].

for each
such that
. Then, by [7, Theorem
], when
, the operator
has a unique fixed point
, which is obviously a solution of (3.1)*.*

## 4. An Application

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.

containing a parameter . In our case is a continuous linear operator on and is the so-called superposition operator. We have the following theorem.

Theorem 4.1.

for any and , where . If is a solution of (4.1) for , then there exist and such that, for each with , (4.1) has a unique solution .

Proof.

for any , when and . So the operators and are -osculating at . Further, assuming for some , we can find such that for any and . As before, the proof is completed by appealing to Theorem 3.1.

## Authors’ Affiliations

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## Copyright

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