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
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.
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.
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.
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.
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
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 .
when and .
Now we prove our main result.
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 .
for and .
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).
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.
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 .
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.
- 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.MathSciNetGoogle Scholar
- 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.MathSciNetMATHGoogle Scholar
- 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.2000.7327MathSciNetView ArticleMATHGoogle Scholar
- Kramar E: Invariant subspaces for some operators on locally convex spaces. Commentationes Mathematicae Universitatis Carolinae 1997,38(4):635–644.MathSciNetMATHGoogle Scholar
- Moore RT: Banach algebras of operators on locally convex spaces. Bulletin of the American Mathematical Society 1969, 75: 68–73. 10.1090/S0002-9904-1969-12147-6MathSciNetView ArticleMATHGoogle Scholar
- 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.Google Scholar
- Tarafdar E: An approach to fixed-point theorems on uniform spaces. Transactions of the American Mathematical Society 1974, 191: 209–225.MathSciNetView ArticleMATHGoogle Scholar
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