Abstract and Applied Analysis

Volume 2012 (2012), Article ID 370946, 18 pages

doi:10.1155/2012/370946

Research Article

A Note on Coincidence Degree Theory

Ali Sırma and Sebaheddin Ṣevgin

Department of Mathematics, Yüzüncü Yıl University, 65080 Van, Turkey

Received 13 March 2012; Accepted 17 June 2012

Academic Editor: Yong Zhou

Copyright © 2012 Ali Sırma and Sebaheddin Ṣevgin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The background of definition of coincidence degree is explained, and some of its basic properties are given.

1. Introduction

Gaines and Mawhin introduced coincidence degree theory in 1970s in analyzing functional and differential equations [1, 2]. Mawhin has continued studies on this theory later on and has made so important contributions on this subject since then this theory is also known as Mahwin's coincidence degree theory. Coincidence theory is very powerful technique especially in existence of solutions problems in nonlinear equations. It has especially so broad applications in the existence of periodic solutions of nonlinear differential equations so that many researchers have used it for their investigations (see [3–32] and references therein). The main goal in the coincidence degree theory is to search the existence of a solutions of the operator equation

in some bounded and open set in some Banach space for being a linear operator and nonlinear operator using Leray-Schauder degree theory. As it is known that, in finite dimensional case, for , , and  , the degree of on with respect to , is well defined. But unfortunately this is not the case in infinite dimension for (see [33], page 172). Luckily, in an arbitrary Banach space , Leray and Schauder proved that for open, bounded set, compact operator and for   the degree of compact perturbation of identity in with respect to , is well defined [34]. One of the main useful properties of degree theory is that if then has at least one solution in . In particular if we take and then the compact operator has at least one fixed point in . In [1], Gaines and Mawhin studied existence of a solution of an operator equation (1.1) defined on a Banach space in an open bounded set using the Leray-Schauder degree theory. But since the operator is not compact in general the need to define a compact operator such that its set of fixed points in would be equal to a solution set of (1.1) in aroused. In [1], the compact operator is given and the coincidence degree for the couple in is defined by .

The aim of this paper is to make an effort to understand the theoretical background of the definition of coincidence degree which has similar properties with the Leray-Schauder degree for an operator couple satisfying some special conditions, to analyze the dependence of coincidence degree to the components of the compact operator and in this way to prepare good resource for one who wants to study and to improve the coincidence degree theory.

The paper is basically prepared using [1]. In this study, we tried to explain the theory that was given densely in [1]. Besides we give proofs of some results that their proofs not given in [1]. Namely, we give proofs of Lemmas 2.1, 2.2, and 3.19 and Theorems 3.3, 4.1, and 4.2. We state and prove Lemma 3.17 which is essential for Proposition 3.18. In Proposition 3.6 we show that the operator is an isomorphism and explain important details, and, in Proposition 3.20, we show that is an automorphism and make necessary explanations. Also in each proof we tried to make important contributions to make the proofs much more understandable and so that it can be improved by interested researchers.

In summary, in Section 2, some preliminaries which are used in the definition of coincidence degree are used. In Section 3, definition of coincidence degree for some linear perturbations of Fredholm mappings on normed spaces is given. In Section 4, some basic properties of coincidence degree are given.

2. Algebraic Preliminaries

In this section, we will give some facts that will be used throughout the paper.

Let and be two vector spaces, the domain of operator , is a linear subspace of , and is a linear operator. Assume that the operators and linear projection operators such that the chain

is exact, that is, and . Let us define the restriction of to as .

Now, let us give the following lemma about .

Lemma 2.1. is an algebraic isomorphism.

Proof. Firstly, let us show that is one-to-one mapping. For this let us take , so that there exists such that . Since is a projection operator we get . Therefore, , so that we obtain that . This means that is one-to-one.

Now let us show that is onto. Since is a projection operator, we can write the vector space as direct sums . From the exactness of the chain above, we get . Take , so that there exists with . Since , there exists unique elements and such that we can write . From here, we can obtain . This means that . So we get and and . So the result follows.

Now, let us define . It is clear that is one-to-one, onto, and .

Lemma 2.2. (1) On , we have . (2) On , we have .

Proof. (1) Take . Therefore, .

(2) Since , then we have , so we obtain . So that in order to prove (2), we need to show the equality . If we can have , then the result follows. Take . Since and is a vector subspace of , we have .

Since , then ; therefore, we have . From here, we obtain . So using (1), the result follows.

Now, let us define the canonic surjection as

Here, is the quotient space of under the equivalence relation . Thus, . It is clear that the canonic surjection operator is linear and .

Proposition 2.3. If there exists an one-to-one operator , then

will be equivalent to

Here, the operator is defined as .

Proof. Since , then for we have and . From here, it is seen that

Now, let us consider another projection operator couple that will make the chain

exact, and let us search the relation of this operator couple with .

From Lemma 2.2, since , and then we have . So for any , we have . Therefore, we can write . Since the projection operator behaves on as an identity operator, we have . As a result, the equality

follows.

Lemma 2.4. The following relations hold.

(i) ,   (ii) .

Proof. (i) Using , , and (2.7), the result

follows.

(ii) Again using (2.7) and (i), we obtain

In a similar manner, the equality can be obtained.

3. Definition of Coincidence Degree for Some Linear Perturbations of Fredholm Mappings on Normed Spaces

In this section, definition of coincidence degree for some linear perturbations of Fredholm mappings on normed spaces is given.

Let and be two real norm spaces, an open, bounded subset of and an closure of . Let us assume that the operators

satisfy the following conditions:

(i)is linear and is an closed subset of ,(ii) and are finite dimensional spaces and ,(iii)the operator is continuous and is bounded,(iv)the operator is compact on .

Definition 3.1. The operator which satisfies the conditions (i) and (ii) will be called as Fredholm operator of index zero.

Definition 3.2. The operator which satisfies the conditions (iii) and (iv) will be called -compact operator.

It is clear that if we take and the operator reduced to zero operator and the operator turns to an identity operator then -compactness of on reduced to usual compactness for operators.

Theorem 3.3. Let be a Banach space. If the operator is a Fredholm operator of index zero then there exist continuous projections and such that the chain

will be exact.

Proof. Assume that is finite dimensional, and the set is a basis for . Define the vector subspaces as . Since is finite dimensional so is a closed subspace of (see [35, Theorem  2.4-3] and . Let be a basis of such that . Now, let us define the linear operators which satisfy the conditions

Therefore, the operator defined by

is a continuous projection operator (see [36, Remark  2.1.19]).

Now, let us prove the existence of continuous projection operators on with . We know that there exists a subspace such that (see [37, Proposition  I]). The projection operator defined on with the rule

satisfies the relations and . Since is finite dimensional so is , therefore it is closed in . Since is a Banach space, and are closed subsets of , therefore the projection operator is continuous (see [38, Theorem  6.12.6]).

Moreover, the canonical surjection is continuous with the quotient topology on . Now, let us state two theorems that will be used in the proof of following proposition.

Theorem 3.4 (see [35]). Assume and are normed spaces and the operator is linear. Therefore,

(a)If is bounded and then is compact.(b)If then is continuous and compact.

Theorem 3.5 (see [35], Lemma  8.3-2). Let be normed space, be a linear compact operator, and a linear bounded (continuous) operator. So the operators and are also compact.

The following proposition states that the condition (iv) does not depend on the choice of the projection operators and .

Proposition 3.6. Assume that the conditions (i), (ii), (iii) are all satisfied. If the condition (iv) is satisfied for the projection operator couple that makes the chain exact then for any projection operator couple that makes the chain exact is satisfied.

Proof. Let us denote the restriction of to with , and let us show that the linear operator is one-to-one and onto. Since and , then we have . Therefore is one-to-one. To show surjection, let us take an arbitrary element . So there exists such that holds. Since the space can be written as there exist unique elements and such that the relation is satisfied. Since we have

then the surjectivity of follows.

Since we have , then is a finite dimensional linear subspace of . Similarly, is also a finite dimensional subspace of . Therefore, since we have , then is also a finite dimensional subspace of .

Now, let us show that for an arbitrary the relation holds. Since we can write , then

is obtained.

Let denote the restriction of the operator to the finite dimensional space . Using the results obtained until here in this proof and using the equality ,

is achieved. Now, let us explain the operator is compact. Since the operator is compact and is continuous, then the operator is compact. Since , then the operator is compact. From the same reason, the operator is also compact. Since the operators , , and are all continuous, the compactness of follows.

Proposition 3.7. The element is a solution of the operator equation (1.1) if and only if it satisfies

In other words, the set of solutions of (1.1) is equal to the set of fixed points of the operator defined by

Here, is any isomorphism.

Proof. Clear from Proposition 2.3.

Remark 3.8. Note that since , , and , then by definition . That is, any fixed points of , if they exist, should be in the set . Therefore, if (1.1) has a solution in , then the solution should be in the set .

Proposition 3.9. Assume that the conditions (i)–(iv) hold. Then, the operator is compact on .

Proof. The projection operator is bounded and then is a finite dimensional therefore, from Theorem 3.4 (a), is compact. By assumption (iv), is compact. Beside these the operator is linear isomorphism and , therefore is compact. Since is continuous then is compact. As a result, we obtained the compactness of the operator on a set .

Let denote the boundary of a set .

(v) If , then the Leray-Schauder degree is well defined [34], since this condition by Proposition 3.7 gives us .

Now, let us search how much the degree depends upon the choice of the operators , , and . To show this, we will need the following definition and results.

Let will be the set of all linear isomorphism from to .

Definition 3.10. If there exists a continuous ,

such that for any the operator then the operator is called homotopic in .

Being homotopic is an equivalence relation in the set . Therefore, this equivalence relation divides the set into homotopy classes.

Proposition 3.11. The operators and are homotopic in if and only if .

Proof. Assume that and are homotopic in . From the condition (ii) we know that we have . Let be the operator defined in Definition 3.10, and be bases of the spaces and , respectively. If for any , denotes the determinant of the matrix corresponding to with respect to these bases, then, for any , since is an isomorphism for any . Beside this, since is continuous, then is also continuous with respect to . Using continuity and the fact that for any , we have the number is always positive or negative, that is it has always same sign. In particular and have the same signs, therefore we have

Conversely assume that . With respect to bases of and , let and denote the matrix representations of the operators and , respectively. By assumption and have the same sign. Therefore, they belong to same connected component of the topological group . Since is locally arcwise connected then the corresponding component is also path connected. Therefore, there exists a continuous operator

Therefore, for any , if we take as a family of isomorphisms corresponding to continuous matrices defined from to , then the proof will be completed.

Corollary 3.12. is separated into two homotopy classes.

Therefore, the set of all isomorphisms with the same sign of determinant will be in the same classes. So one class will be with positive determinant and the other one will be with negative determinant.

Note the following: let be any isomorphism from the set . The sign of determinant of the matrix corresponding to depends upon not only the basis chosen for and but also the order of the elements in these basis. If the operators and are homotopic with respect to chosen bases for and , then they are homotopic with respect to any basis chosen for these spaces.

Now, let us fix an orientation on and , and let be a basis for for the chosen orientation.

Definition 3.13. Let the operator be given. If has the same orientation with basis chosen in , then the operator is said to be an orientation preserving transformation. Otherwise, it is said to be an orientation reversing transformation.

Proposition 3.14. If and are oriented, then the operators and are homotopic in if and only if they are both orientation preserving or both orientation reversing transformations.

Proof. Assume that and are, respectively, bases of and with respect to chosen the orientation. The basis on has the same orientation with if and only if the determinant of the matrix defined by

will be positive. Namely, let be the transition matrix from the basis to the basis and be the transition matrix from the basis to the basis ,

then we have and . has the same orientation with if and only if the determinants of the matrixes and have the same sign. This is only possible in the case the determinant of is positive. Therefore, since the determinant of the matrixes and have the same sign, using the relation , we obtain that .

Let us assume that is a matrix related to a basis . In this case if the matrix is the matrix represent the operator with respect to basis , then we have

Therefore, and are obtained. Since and , then , that is . This means that and have the same orientation.

Conversely, if the operators and have the same orientation, then . Therefore, from the Proposition 3.11, and are homotopic.

Lemma 3.15. Let be a vector space and be two projection operators with . Therefore the operator defined by , , is a projection operator with the property if and only if .

Proof. Let and are real numbers and assume that the operator defined is a projection operator with its image is equal to . Since for any we have and for any , then, for any we have . Therefore, we get the relation . In a similar manner, the equality can be shown. So

is obtained. From here, we get the result , that is . The assumption forces the fact that .

Conversely, if , then

is obtained. Therefore, is a projection operator. Since is a vector space and , then we have . Now, let us take an arbitrary element . Therefore,

and from here we obtain and . So the result follows.

Lemma 3.16. If and are projection operators onto , and , then .

Proof. In the case , the proof is clear. Assume that . Since by Lemma 3.15, is a projection operator and . Since and , then the relation

is obtained.

Lemma 3.17. Let be a vector space, two projection operators with , then for , the operator defined by is a projection operator with .

Proof. First of all, let us show that is a projection operator. Since

then for any there exist unique elements , , , and such that and hold. Therefore,

is obtained.

Now, let us show that . For this, take an arbitrary element . Therefore,

This means that . Now, take . Since , then there exist unique elements and such that holds. Therefore, we obtain

That is . If , since then and then . So that . If , then . Then,

is obtained. In this case, we get . Since , this gives us . From here, we get that is obtained. So in any case we showed that .

Proposition 3.18. If the assumptions (i)−(v) hold, then Leray-Schauder degree depends on only , , and homotopy class of in .

Proof. Let the operators , , , be the projection operators with the properties , and , two isomorphisms from to in the same homotopy class. From Lemma 3.15 and Lemma 3.16, it is clear that for each the operators

are the projection operators with the property of for each , and . Beside this, from Lemma 3.16, we have . Let the operator be the operator given in Definition 3.1. Using Proposition 3.7, we see that for each the fixed points of the operator

coincide with the solutions of the operator equation (1.1). From the condition (v), we have and

Clearly, we have

Now let us show that is compact on . From the open form

it is clear that is continuous. So, in order to show that the set is relatively compact the only delicate point is the last term. Using the fact that , we obtain the last term as

Therefore, compactness can be proven like in the proof of Proposition 3.9. Using the invariance of Leray-Schauder degree with respect to compact homotopy, we obtain that

that is,

Now, let us indicate how the degree depends on homotopy class of . For this, let us prove the following lemma.

Lemma 3.19. If is any automorphism and

then the relation

is satisfied.

Proof. Since , , and , then

Proposition 3.20. If and

then we have

Proof. In Lemma 3.19, if we take , then

is obtained. Now let us show that the operator is an automorphism on . For this take, , then we have

If we apply the operator to both sides, we get . Since is one-to-one, this result gives . If we substitute this result in (3.40) we obtain that , and therefore is one-to-one. For surjectivity, take . Therefore, there exists unique elements , such that . Now, we are looking for , , such that and . So

Using uniqueness in direct sum and the fact that is an automorphism on , we get and . Therefore, taking ontoness of the operator is proved. Therefore, is an automorphism on . So using the identity and Leray Product Theorem, we have

Therefore the result

is achieved.

Corollary 3.21. Under the assumptions of Proposition 3.18, Leray-Schauder degree only depends upon , , and .

Now, if the orientation on the spaces and is fixed, then we can give the following beautiful and fruitful definition.

Definition 3.22. If the operators and satisfy the conditions (i)–(v) then the coincidence degree of and in defined by

Here, in is an orientation preserving isomorphism.

This definition is supported with all the arguments given in this paper.

4. Basic Properties of Coincidence Degree

In this section, we will see that the coincidence degree satisfies all the basic properties of the Leray-Schauder degree. First, let us consider the simplest case where and . In this situation, and , so that . Therefore and then the assumptions (i) and (ii) are clearly satisfied. Since and , then , and . Thus , and . Therefore, the conditions (iii) and (iv) reduced to the compactness of on . Since and , then the condition (v) in this case means that has no fixed point on . Since , , and , then . Therefore,

That is the coincidence degree of and in this case is nothing but the Leray-Schauder degree of .

Now, we will give the basic properties of coincidence degree.

Theorem 4.1. Assume that the conditions (i) to (v) are satisfied. Then coincidence degree satisfies the following basic properties.

(1)Existence theorem: if , then . (2)Excision property: if is an open set such that , then

(3) Additivity property: if with and are open, bounded, disjoint subsets of , then

(4)Invariance under homotopy property: if the operator

is -compact in and such that for each , , then coincidence degree , is independent of in . In particular

Proof. (1) If , then such that . But in fact, we know that . Also, by Proposition 3.7, . That is .

(2) Assume that is an open set such that , then by Proposition 3.7,  . Therefore, by the excision property of the Leray-Schauder degree, . So, by the definition of coincidence degree, we have .

(3) If with and are open, bounded, disjoint subsets of , then additive property of Leray-Schauder degree we have . So the result follows from the definition of coincidence degree.

(4) Since the operator is -compact for each and for each ,  , then for each the coincidence degree is well defined. Since the operator is -compact on then it is a homotopy of compact operators on . Therefore, by invariance of the Leray-Schauder degree under homotopy property the result follows.

The famous Borsuck theorem for degree theory is also valid for coincidence degree.

Theorem 4.2. If is symmetric with respect to 0 and contains it and if in , then coincidence degree is an odd integer.

Proof. We proved that the operator is compact on . Since a projection operator is linear then it is odd in and is odd in then the operator is odd in . Therefore, the result follows from the validity of Borsuck theorem in the Leray-Schauder degree.

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