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A Note on Implicit Functions in Locally Convex Spaces
Fixed Point Theory and Applications volume 2009, Article number: 707406 (2009)
Abstract
An implicit function theorem in locally convex spaces is proved. As an application we study the stability, with respect to a parameter , of the solutions of the Hammerstein equation
in a locally convex space.
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.
Let and
be Banach spaces,
an open subset of the real line
or of the complex plane
,
an open subset of the product space
and
the space of all continuous linear operators from
into
. An operator
and an operator function
are called osculating at
if there exists a function
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ1_HTML.gif)
when 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
Before providing the main results, we need to introduce some basic facts about locally convex topological vector spaces. We give these definitions following [4–6]. Let be a Hausdorff locally convex topological vector space over the field
, where
or
. A family of continuous seminorms
which induces the topology of
is called a calibration for
. Denote by
the set of all calibrations for
. A basic calibration for
is
such that the collection of all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ2_HTML.gif)
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.
A linear operator on
is called
-bounded if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ3_HTML.gif)
Denote by the space of all continuous linear operators on
and by
the space of all
-bounded linear operators
on
. We have
. Moreover, the space
is a unital normed algebra with respect to the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ4_HTML.gif)
We say that a family is uniformly
-bounded if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ5_HTML.gif)
for any .
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
Let be an open subset of the real line
or of the complex plane
. Consider the product space
of
and
provided with the product topology. Let
be an open subset of
and
. Consider a nonlinear operator
and the related equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ6_HTML.gif)
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
.
We say that an operator and an operator
are called
-osculating at
if there exist a function
and
such that
and for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ7_HTML.gif)
when and
.
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.
Let and
be
-osculating at
. Consider the operator
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ8_HTML.gif)
Let . By the assumption (c) there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ9_HTML.gif)
for any . Moreover, since
and
are
-osculating at
, there are a function
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ10_HTML.gif)
for and
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ11_HTML.gif)
for and
.
Choose such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ12_HTML.gif)
for and
. Therefore, for each
such that
, the operator
from
into
is a contraction in the sense of [7].
Since is continuous at
, we may further find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ13_HTML.gif)
for . Set
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ14_HTML.gif)
for and
. This shows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ15_HTML.gif)
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.
Consider in the Hammerstein equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ16_HTML.gif)
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.
Let be
-bounded. Suppose that for each
there exists
such that the operator
satisfies the Lipschitz condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ17_HTML.gif)
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.
Since the linear operator is
-bounded, we can find a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ18_HTML.gif)
If , then
is clearly a solution of (4.1). Consider the operator
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ19_HTML.gif)
and set for any
and
. Clearly the operator
is continuous at
. By the hypothesis made on the operator
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ20_HTML.gif)
for any ; when
and
, the operators
and
are
-osculating at
. Moreover, for each
, we have
and
for any
and
. Then the result follows by Theorem 3.1. Now assume that
is a solution of (4.1) for some
. Let
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ21_HTML.gif)
and set for any
and
. The operator
is continuous at
and there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F707406/MediaObjects/13663_2009_Article_1169_Equ22_HTML.gif)
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.
References
Zabreiko PP, Kolesov JuS, Krasnosel'skij MA: Implicit functions and the averaging principle of N. N. Bogoljubov and N. M. Krylov. Doklady Akademii Nauk SSSR 1969,184(3):526–529.
Trombetta A: An implicit function theorem in complete -normed spaces. Atti del Seminario Matematico e Fisico dell'Università di Modena 2000,48(2):527–533.
Trombetta A: -osculating operators in a space of continuous functions and applications. Journal of Mathematical Analysis and Applications 2001,256(1):304–311. 10.1006/jmaa.2000.7327
Kramar E: Invariant subspaces for some operators on locally convex spaces. Commentationes Mathematicae Universitatis Carolinae 1997,38(4):635–644.
Moore RT: Banach algebras of operators on locally convex spaces. Bulletin of the American Mathematical Society 1969, 75: 68–73. 10.1090/S0002-9904-1969-12147-6
Narici L, Beckenstein E: Topological Vector Spaces, Monographs and Textbooks in Pure and Applied Mathematics. Volume 95. Marcel Dekker, New York, NY, USA; 1985:xii+408.
Tarafdar E: An approach to fixed-point theorems on uniform spaces. Transactions of the American Mathematical Society 1974, 191: 209–225.
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Tavernise, M., Trombetta, A. A Note on Implicit Functions in Locally Convex Spaces. Fixed Point Theory Appl 2009, 707406 (2009). https://doi.org/10.1155/2009/707406
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DOI: https://doi.org/10.1155/2009/707406