A Note on Implicit Functions in Locally Convex Spaces

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

(1.1)

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

(2.1)

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

(2.2)

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

(2.3)

We say that a family is uniformly-bounded if there exists a constant such that

(2.4)

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

(3.1)

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

(3.2)

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

(3.3)

Let . By the assumption (c) there exists such that

(3.4)

for any . Moreover, since and are -osculating at , there are a function and such that

(3.5)

for and . Hence

(3.6)

for and .

Choose such that

(3.7)

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

(3.8)

for . Set we have

(3.9)

for and . This shows that

(3.10)

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

(4.1)

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

(4.2)

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

(4.3)

If , then is clearly a solution of (4.1). Consider the operator defined by

(4.4)

and set for any and . Clearly the operator is continuous at . By the hypothesis made on the operator , there exists such that

(4.5)

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

(4.6)

and set for any and . The operator is continuous at and there exists such that

(4.7)

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.