# A Note on Coincidence Degree Theory

Authors: Ali Sırma; Sebaheddin Ṣevgin
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

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


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## Body

## 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
(1.1)Lx=Nx
in some bounded and open set Ω in some Banach space for L being a linear operator and N nonlinear operator using Leray-Schauder degree theory. As it is known that, in finite dimensional case, for Ω⊂ℝn, f∈C(Ω¯), and  p∈ℝn∖f(∂Ω),  the degree of f on Ω with respect to p, d(f,Ω,p) is well defined. But unfortunately this is not the case in infinite dimension for f∈C(Ω¯) (see [33], page 172). Luckily, in an arbitrary Banach space X, Leray and Schauder proved that for Ω∈X open, bounded set, M:Ω¯→X compact operator and for  p∈X∖(I-M)(∂Ω)  the degree of compact perturbation of identity I-M in Ω with respect to p, d(I-M,Ω,p) is well defined [34]. One of the main useful properties of degree theory is that if d(I-M,Ω,p)≠0 then (I-M)x=p has at least one solution in Ω. In particular if we take p=0 and d(I-M,Ω,p)≠0 then the compact operator M has at least one fixed point in Ω. In [1], Gaines and Mawhin studied existence of a solution of an operator equation (1.1) defined on a Banach space X in an open bounded set Ω using the Leray-Schauder degree theory. But since the operator I-(L-N) is not compact in general the need to define a compact operator M such that its set of fixed points in Ω would be equal to a solution set of (1.1) in Ω aroused. In [1], the compact operator M is given and the coincidence degree for the couple (L,N) in Ω is defined by d[(L,N),Ω]=d(I-M,Ω,0).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(L,N) satisfying some special conditions, to analyze the dependence of coincidence degree to the components of the compact operator M 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 ΠQ is an isomorphism and explain important details, and, in Proposition 3.20, we show that A 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 Section2, 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.LetX and Z be two vector spaces, the domain of operator L, DomL is a linear subspace of X, and L:DomL→Z is a linear operator. Assume that the operators P:X→X and Q:Z→Z linear projection operators such that the chain
(2.1)X→PDomL→LZ→QZ
is exact, that is, ImP=kerL and ImL=kerQ. Let us define the restriction of L to DomL∩kerP as LP:DomL∩kerP→ImL.Now, let us give the following lemma aboutLP.Lemma 2.1.
L
P is an algebraic isomorphism.Proof.
Firstly, let us show thatLP is one-to-one mapping. For this let us take x∈kerLP⊂kerL=ImP, so that there exists y∈DomP such that x=Py. Since P is a projection operator we get x=Py=P2y=P(Py)=Px=0. Therefore, x=0, so that we obtain that kerLP={0}. This means that LP is one-to-one.
Now let us show thatLP is onto. Since P:X→X is a projection operator, we can write the vector space X as direct sums X=kerP⊕ImP. From the exactness of the chain above, we get X=kerP⊕kerL. Take z∈ImL, so that there exists x∈DomL⊂X with Lx=z. Since X=kerP⊕kerL, there exists unique elements e∈kerP and f∈kerL such that we can write x=e+f. From here, we can obtain z=Lx=L(e+f)=Le+Lf=Le+0=Le. This means that e∈DomL. So we get e∈DomL and e∈kerP and LPe=z. So the result follows.Now, let us defineKP:=LP-1. It is clear that KP:ImL⊂Z→DomL∩kerP is one-to-one, onto, and PKP=0.Lemma 2.2.
(1) OnImL, we have LKP=I. (2) On DomL, we have KPL=I-P.Proof.
(1) Takex∈ImL. Therefore, LKPx=L(KP(x))=LP(KP(x))=Ix.
(2) SinceImP=kerL, then we have LP=0, so we obtain KPL=KPL(I-P). So that in order to prove (2), we need to show the equality KPL(I-P)=KPLP(I-P). If we can have Im(I-P)⊆Dom(LP)=DomL∩kerP, then the result follows. Take x∈DomL. Since P(x)∈kerL⊂DomL and DomL is a vector subspace of X, we have (x-Px)∈DomL.
SinceP(x-Px)=Px-P2x=Px-Px=0, then (x-Px)∈kerP; therefore, we have (x-Px)∈DomL∩kerP. From here, we obtain Im(I-P)⊂DomL∩kerP. So using (1), the result KPL(I-P)=KPLP(I-P)=I-P follows.Now, let us define the canonic surjectionΠ as
(2.2)Π:Z→CokerLz↦z+ImL.
Here, CokerL=Z/ImL is the quotient space of Z under the equivalence relation z~z'⇔z-z'∈ImL. Thus, CokerL={z¯=z+ImL:z∈Z}. It is clear that the canonic surjection operator Π is linear and kerΠ=kerQ.Proposition 2.3.
If there exists an one-to-one operatorΛ:CokerL→kerL, then
(2.3)Lx=y,y∈Z
will be equivalent to
(2.4)(I-P)x=(ΛΠ+KP,Q)y.
Here, the operator KP,Q:Z→X is defined as KP,Q=KP(I-Q).Proof.
SinceImL=kerQ=kerΠ, then for y∈ImL we have Qy=0 and ΛΠy=ΛImL=Λ0¯=0. From here, it is seen that
(2.5)Lx=y⟺Lx=y-Qy⟺KPLx=KP(y-Qy)⟺(I-P)x=KP(I-Q)y⟺(I-P)x=(ΛΠ+KP(I-Q))y.Now, let us consider another projection operator couple(P′,Q′) that will make the chain
(2.6)X→P′DomL→LZ→Q′Z
exact, and let us search the relation of this operator couple with (P,Q).From Lemma2.2, since LKP=I, and LKP′=I then we have L(KP-KP′)=0. So for any z∈ImL, we have (KP-KP′)z∈kerL. Therefore, we can write KP-KP′:ImL→kerL=ImP=ImP′. Since the projection operator P behaves on ImP as an identity operator, we have KP-KP′=P(KP-KP′). As a result, the equality
(2.7)KP-KP′=P(KP-KP′)=P′(KP-KP′)
follows.Lemma 2.4.
The following relations hold.(i)PKP′+P′KP=0,   (ii) KP′=(I-P′)KP.Proof.
(i) UsingPKP=0, P′KP′=0, and (2.7), the result
(2.8)P(Kp-KP′)=P′(KP-KP′)-PKP′=P′KP
follows.
(ii) Again using (2.7) and (i), we obtain
(2.9)KP-KP′=P′(KP-KP′)⇒KP-KP′=P′KP-P′KP′⇒KP=P′KP+(-PKP′)⇒KP=(I-PKP′).
In a similar manner, the equality KP′=(I-P′KP) 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.LetX and Z be two real norm spaces, Ω⊂X an open, bounded subset of X and Ω¯ an closure of Ω. Let us assume that the operators
(3.1)L:DomL⊂X→Z,N:Ω¯⊂X→Z
satisfy the following conditions:(i)
Lis linear and ImL is an closed subset of Z,(ii)
kerL and CokerL=Z/ImL are finite dimensional spaces and dimkerL=dimCokerL,(iii)
the operatorN:Ω¯⊂X→Z is continuous and ΠN(Ω) is bounded,(iv)
the operatorKP,QN:Ω¯→Z is compact on Ω¯.Definition 3.1.
The operatorL which satisfies the conditions (i) and (ii) will be called as Fredholm operator of index zero.Definition 3.2.
The operatorN:Ω¯→Z which satisfies the conditions (iii) and (iv) will be called L-compact operator.It is clear that if we takeX=Z and L=I the operator Π reduced to zero operator and the operator KP,Q turns to an identity operator then L-compactness of N on Ω¯ reduced to usual compactness for operators.Theorem 3.3.
LetZ be a Banach space. If the operator L is a Fredholm operator of index zero then there exist continuous projections P:X→X and Q:Z→Z such that the chain
(3.2)X→PDomL→LZ→QZ
will be exact.Proof.
Assume thatkerL is finite dimensional, and the set {y1,y2,…,yn} is a basis for kerL. Define the vector subspaces as Xk=span{y1,y2,…,yk-1,yk+1,…,yn}. Since Xk is finite dimensional so is a closed subspace of X (see [35, Theorem  2.4-3] and yk∉Xk. Let B be a basis of X such that {y1,y2,…,yn}⊆B. Now, let us define the linear operators which satisfy the conditions
(3.3)Fk:X→ℝ,Fk(y)={1,ify=yk0,ify∈B-{yk}.
Therefore, the operator P defined by
(3.4)Px=∑k=1n‍Fk(x)yk
is a continuous projection operator (see [36, Remark  2.1.19]).
Now, let us prove the existence of continuous projection operatorsQ on Z with kerQ=ImL. We know that there exists a subspace Z~ such that Z=ImL⊕Z~ (see [37, Proposition  I]). The projection operator Q defined on Z with the rule
(3.5)Q(z)=Q(zkerQ+zZ~)=zZ~
satisfies the relations kerQ=ImL and ImQ=Z~. Since dimCokerL is finite dimensional so is ImQ, therefore it is closed in Z. Since Z is a Banach space, kerQ and ImQ are closed subsets of Z, therefore the projection operator Q is continuous (see [38, Theorem  6.12.6]).Moreover, the canonical surjectionΠ:Z→CokerL is continuous with the quotient topology on CokerL. Now, let us state two theorems that will be used in the proof of following proposition.Theorem 3.4 (see [35]).
AssumeX and Y are normed spaces and the operator T:X→Y is linear. Therefore,(a)
IfT is bounded and dim(ImT)<∞ then T is compact.(b)
IfdimX<∞ then T is continuous and compact.Theorem 3.5 (see [35], Lemma  8.3-2).
LetX be normed space, T:X→X be a linear compact operator, and S:X→X a linear bounded (continuous) operator. So the operators TS and ST are also compact.The following proposition states that the condition (iv) does not depend on the choice of the projection operatorsP and Q.Proposition 3.6.
Assume that the conditions (i), (ii), (iii) are all satisfied. If the condition (iv) is satisfied for the projection operator couple(P,Q) that makes the chain exact then for any projection operator couple (P′,Q′) that makes the chain exact is satisfied.Proof.
Let us denote the restriction ofΠ to ImQ with ΠQ, and let us show that the linear operator ΠQ:ImQ→Z/ImL is one-to-one and onto. Since ker(ΠQ)∈ImL=kerQ and kerQ∩ImQ={0}, then we have ker(ΠQ)={0}. Therefore ΠQ is one-to-one. To show surjection, let us take an arbitrary element z¯∈Z/ImL. So there exists z∈Z such that Πz=z¯ holds. Since the space Z can be written as Z=ImQ⊕kerQ=ImQ⊕ImL there exist unique elements zImQ∈ImQ and zImL∈ImL such that the relation z=zImQ+zImL is satisfied. Since we have
(3.6)z¯=Π(zImQ+zImL)=Π(zImQ)+Π(zImL)=Π(zImQ),
then the surjectivity of ΠQ follows.
Since we havedim(ImQ)≤dim(ImΠQ)≤dim(CokerL)=n<∞, then ImQ is a finite dimensional linear subspace of Z. Similarly, ImQ′ is also a finite dimensional subspace of Z. Therefore, since we have dim(Im(Q-Q′))≤dim(ImQ)+dim(ImQ′), then Im(Q-Q′) is also a finite dimensional subspace of Z.
Now, let us show that for an arbitraryα∈Z the relation ΠQ-1Π(α)=Q(α) holds. Since we can write α=Qα+(I-Q)α), then
(3.7)ΠQ-1Π(α)=ΠQ-1Π((Qα+(I-Q)α))=ΠQ-1Π(Qα)+ΠQ-1Π((I-Q)α)=ΠQ-1Π(Qα)=ΠQ-1ΠQ(Qα)=Qα
is obtained.
LetK~P denote the restriction of the operator KP to the finite dimensional space Im(Q-Q′). Using the results obtained until here in this proof and using the equality KP′=(I-P′)KP,
(3.8)KP′,Q′N=KP′(I-Q′)N=(I-P′)KP(I-Q′)N=(I-P′)KP(I-Q+Q+Q′)N=(I-P′)KP(I-Q)N+(I-P′)KP(Q-Q′)N=(I-P′)KP,QN+(I-P′)KP(Q-Q′)N=(I-P′)KP,QN+(I-P′)K~P(Q-Q′)N=(I-P′)KP,QN+(I-P′)K~P(ΠQ-1Π-ΠQ′-1Π)N=(I-P′)KP,QN+(I-P′)K~P(ΠQ-1-ΠQ′-1)ΠN
is achieved. Now, let us explain the operator KP′,Q′N is compact. Since the operator KP,QN is compact and I-P′ is continuous, then the operator (I-P′)KP,QN is compact. Since dim(CokerL)=n<∞, then the operator ΠQ-1:CokerL→ImQ is compact. From the same reason, the operator ΠQ′-1 is also compact. Since the operators N, Π, and I-P′ are all continuous, the compactness of KP′,Q′N follows.Proposition 3.7.
The elementx∈DomL∩Ω¯ is a solution of the operator equation (1.1) if and only if it satisfies
(3.9)(I-P)x=(ΛΠ+KP,Q)Nx.
In other words, the set of solutions of (1.1) is equal to the set of fixed points of the operator M:Ω¯→X defined by
(3.10)M=P+(ΛΠ+KP,Q)N.
Here, Λ:CokerL→kerL is any isomorphism.Proof.
Clear from Proposition2.3.Remark 3.8.
Note that sinceImP=kerL⊂DomL, ImΛ=kerL⊂DomL, and ImKP,Q⊂DomL∩kerP⊂DomL, then by definition M:Ω¯→DomL. That is, any fixed points of M, if they exist, should be in the set Ω¯∩DomL. Therefore, if (1.1) has a solution in Ω¯, then the solution should be in the set Ω¯∩DomL.Proposition 3.9.
Assume that the conditions (i)–(iv) hold. Then, the operatorM is compact on Ω¯.Proof.
The projection operatorP is bounded and ImP=kerL then ImP is a finite dimensional therefore, from Theorem 3.4 (a), P is compact. By assumption (iv), KP,QN is compact. Beside these the operator Λ:CokerL→kerL is linear isomorphism and dim(CokerL)=n<∞, therefore Λ is compact. Since ΛΠ is continuous then ΛΠN is compact. As a result, we obtained the compactness of the operator M on a set Ω¯.Let∂Ω denote the boundary of a set Ω.(v) If0∉(L-N)(DomL∩∂Ω), then the Leray-Schauder degree d(I-M,Ω,0) is well defined [34], since this condition by Proposition 3.7 gives us 0∉(I-M)(∂Ω).Now, let us search how much the degree depends upon the choice of the operatorsP, Q, and Λ. To show this, we will need the following definition and results.LetΛL will be the set of all linear isomorphism from CokerL to kerL.Definition 3.10.
If there exists a continuousΛ_,
(3.11)Λ_:CokerL×[0,1]→kerLΛ_(·,0)=Λ,Λ_(·,1)=Λ′
such that for any λ∈[0,1] the operator Λ_(·,Λ)∈ΛL then the operator Λ,Λ′∈ΛL is called homotopic in ΛL.Being homotopic is an equivalence relation in the setΛL. Therefore, this equivalence relation divides the set ΛL into homotopy classes.Proposition 3.11.
The operatorsΛ and Λ′ are homotopic in ΛL if and only if det(ΛΛ′)>0.Proof.
Assume thatΛ and Λ′ are homotopic in ΛL. From the condition (ii) we know that we have dimkerL=dimCokerL=n. Let Λ_ be the operator defined in Definition 3.10, [a1,a2,…,an] and [b1,b2,…,bn] be bases of the spaces CokerL and kerL, respectively. If for any λ∈[0,1], Δ(λ) denotes the determinant of the matrix corresponding to Λ_(·,λ) with respect to these bases, then, for any λ∈[0,1], Δ(λ)≠0 since Λ_(·,λ) is an isomorphism for any λ∈[0,1]. Beside this, since Λ_ is continuous, then Δ is also continuous with respect to λ. Using continuity and the fact that for any λ∈[0,1], Δ(λ)≠0 we have the number Δ(λ) is always positive or negative, that is it has always same sign. In particular Δ(0) and Δ(1) have the same signs, therefore we have(3.12)det(Λ′Λ-1)=det(Λ′)det(Λ-1)=det(Λ′)det(Λ)=Δ(1)Δ(0)>0.
Conversely assume that det(Λ′Λ-1)>0. With respect to bases of CokerL and kerL, let Λ~ and Λ'~ denote the matrix representations of the operators Λ and Λ′, respectively. By assumption det(Λ~) and det(Λ′~) have the same sign. Therefore, they belong to same connected component of the topological group GL(n,r). Since GL(n,r) is locally arcwise connected then the corresponding component is also path connected. Therefore, there exists a continuous operator
(3.13)Λ_~:CokerL×[0,1]→kerLΛ_~(·,0)=Λ~,Λ_~(·,1)=Λ′~.
Therefore, for any λ∈[0,1], if we take Λ_(·,λ) as a family of isomorphisms corresponding to continuous matrices defined from CokerL to kerL, then the proof will be completed.Corollary 3.12.
Λ
L is separated into two homotopy classes.Therefore, the set of all isomorphismsΛ:CokerL→kerL 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: letΛ:CokerL→kerL be any isomorphism from the set ΛL. The sign of determinant of the matrix corresponding to Λ depends upon not only the basis chosen for CokerL and kerL but also the order of the elements in these basis. If the operators Λ and Λ′ are homotopic with respect to chosen bases for CokerL and kerL, then they are homotopic with respect to  any basis chosen for these spaces.Now, let us fix an orientation onCokerL and kerL, and let [a1,a2,…,an] be a basis for CokerL for the chosen orientation.Definition 3.13.
Let the operatorΛ:CokerL→kerL be given. If [Λa1,Λa2,…,Λan] has the same orientation with basis chosen in kerL, then the operator Λ is said to be an orientation preserving transformation. Otherwise, it is said to be an orientation reversing transformation.Proposition 3.14.
IfCokerL and kerL are oriented, then the operators Λ and Λ′ are homotopic in ΛL if and only if they are both orientation preserving or both orientation reversing transformations.Proof.
Assume that[a1,a2,…,an] and [b1,b2,…,bn] are, respectively, bases of CokerL and kerL with respect to chosen the orientation. The basis [Λa1,Λa2,…,Λan] on kerL has the same orientation with [b1,b2,…,bn] if and only if the determinant of the matrix S=(sij) defined by
(3.14)Λaj=∑j=1n‍sijbj
will be positive. Namely, let M1 be the transition matrix from the basis [a1,a2,…,an] to the basis [b1,b2,…,bn] and M2 be the transition matrix from the basis [a1,a2,…,an] to the basis [Λa1,Λa2,…,Λan],
(3.15)[a1,a2,…,an]→M1[b1,b2,…,bn],[a1,a2,…,an]→M2[Λa1,Λa2,…,Λan],[Λa1,Λa2,…,Λan]→S[b1,b2,…,bn],
then we have M1=SM2 and det(M1)=det(S)det(M2). [Λa1,Λa2,…,Λan] has the same orientation with [b1,b2,…,bn] if and only if the determinants of the matrixes M1 and M2 have the same sign. This is only possible in the case the determinant of S is positive. Therefore, since the determinant of the matrixes M1 and M2 have the same sign, using the relation det(M1)=det(S)det(M2), we obtain that det(S)>0.
Let us assume thatS′ is a matrix related to a basis [Λ′a1,Λ′a2,…,Λ′an]. In this case if the matrix G=(gij) is the matrix represent the operator Λ′Λ-1 with respect to basis [b1,b2,…,bn], then we have
(3.16)∑j=1n‍sijbj=Λ′ai=Λ′(ΛΛ-1)ai=(Λ′Λ-1)Λai,(Λ′Λ-1)∑k=1n‍skibk=∑k=1n‍ski(Λ′Λ-1)(bk),∑k=1n‍ski∑j=1n‍gjkbj.
Therefore, (S′)T=STGT and S′=GS are obtained. Since det(S)>0 and det(S′)>0, then det(G)>0, that is det(Λ′Λ-1)>0. This means that Λ′ and Λ-1 have the same orientation.
Conversely, if the operatorsΛ′ and Λ-1 have the same orientation, then det(Λ′Λ-1)>0. Therefore, from the Proposition 3.11, Λ′ and Λ-1 are homotopic.Lemma 3.15.
LetY be a vector space and S,S′:Y→Y be two projection operators with ImS=ImS′≠0. Therefore the operator S′′ defined by S′′=aS+bS', a,b∈ℝ, is a projection operator with the property ImS′′=ImS if and only if a+b=1.Proof.
Leta and b are real numbers and assume that the operator S′′ defined S′′=aS+bS' is a projection operator with its image is equal to ImS=ImS'. Since for any x∈ImS we have Sx=x and for any y∈Y, S'y∈ImS'=ImS then, for any y∈Y we have SS'y=S(S'y)=S'y. Therefore, we get the relation SS'=S'. In a similar manner, the equality S'S=S can be shown. So
(3.17)aS+bS′=S′′=(S′′)2=(aS+bS′)(aS+bS′)=a2S2+abSS'+abS'S+b2(S′)2=a2S+abSS'+abS'S+b2S'=a2S+abS'+abS+b2S'=a(aS+bS′)+b(aS+bS′)=(a+b)(aS+bS′)
is obtained. From here, we get the result (a+b-1)(aS+bS')=0, that is (a+b-1)S′′=0. The assumption ImS′′≠0 forces the fact that a+b=1.
Conversely, ifa+b=1, then
(3.18)S′′=(S′′)2=(aS+bS′)(aS+bS′)=a2S2+abSS'+abS'S+b2(S′)2=a2S+abSS'+abS'S+b2S'=a2S+abS'+abS+b2S'=(a+b)(aS+bS′)=aS+bS'=S′′
is obtained. Therefore, S′′ is a projection operator. Since ImS=ImS′ is a vector space and S′′=aS+bS′, then we have ImS′′⊆ImS. Now, let us take an arbitrary element x∈ImS=ImS′′≠{0}. Therefore,
(3.19)S′′x=aSx+bS'x=ax+bx=(a+b)x=x
and from here we obtain x∈ImS′′ and ImS⊆ImS′′. So the result ImS=ImS′′ follows.Lemma 3.16.
IfP and P′ are projection operators onto kerL, a+b=1 and P′′=aP+bP′, then KP′′=aKP+bKP′.Proof.
In the casekerL={0}, the proof is clear. Assume that ImP=ImP′=kerL≠{0}. Since a+b=1 by Lemma 3.15, P′′ is a projection operator and ImP′′=ImP. Since KP′=(I-P′)KP and PKP=0, then the relation
(3.20)KP′′=(I-P′′)KP=(I-aP-bP′)KP=KP-aPKP-bP′KP=1·KP-bP′KP=(a+b)KP-bP′KP=aKP+bKP-bP′KP=aKP+b(I-P′)KP=aKP+bKP′
is obtained.Lemma 3.17.
LetZ be a vector space, S,S′:Z→Z two projection operators with kerS=kerS′, then for a,b∈ℝ, a+b=1 the operator S′′ defined by S′′=aS+bS′ is a projection operator with kerS′′=kerS.Proof.
First of all, let us show thatS′′ is a projection operator. Since
(3.21)S:Z→ZS′:Z→ZZ=kerS⊕ImS,Z=kerS′⊕ImS′,
then for any z∈Z there exist unique elements z0∈kerS, z1∈ImS, z0′∈kerS′, and z1′∈ImS′ such that z=z0+z1 and z=z0′+z1′ hold. Therefore,
(3.22)(S′′)2(z)=a2S(z)+abSS′(z)+abS′S(z)+b2S′(z)=a2S(z0′+z1′)+abSS′(z0′+z1′)+abS′S(z0+z1)+b2S′(z0+z1)=a2S(z1′)+abSS′(z1′)+abS′S(z1)+b2S′(z1)=a2S(z1′)+abS(z1′)+abS′(z1)+b2S′(z1)=(a+b)(aS(z1′)+bS′(z1))=aS(z1′)+bS′(z1)=aS(z0′)+aS(z1′)+bS′(z0)+bS′(z1)=aS(z)+bS′(z)=S′′(z)
is obtained.
Now, let us show thatkerS′′=kerS=kerS′. For this, take an arbitrary element x∈kerS=kerS′. Therefore,
(3.23)S′′(x)=aS(x)+bS(x)=a.0+b.0=0.
This means that kerS=kerS′⊆kerS′′. Now, take x∈kerS′′. Since Z=kerS⊕ImS, then there exist unique elements e∈kerS and f∈ImS such that x=e+f holds. Therefore, we obtain
(3.24)0=S′′(x)=aS(x)+bS′(x)=aS(e+f)+bS′(e+f)=aS(f)+bS′(f)=af+bS′(f).
That is af+bS′(f)=0. If b=0, since a+b=1 then a=1 and then f=0. So that x∈kerS=kerS'. If b≠0, then S'(f)=-(a/b)f. Then,
(3.25)S′(f)=S2(f)=S′(S′(f))=S′(-abf)=-abS′(f)
is obtained. In this case, we get S′(f)=-(a/b)S′(f). Since a+b=1, this gives us S′(f)=0. From here, we get that kerS′′⊆kerS=kerS′ is obtained. So in any case we showed that kerS′′=kerS=kerS′.Proposition 3.18.
If the assumptions (i)−(v) hold, then Leray-Schauder degreed[I-M,Ω,0] depends on only L, N, Ω and homotopy class of Λ in ΛL.Proof.
Let the operatorsP, P′, Q, Q′ be the projection operators with the properties ImP=ImP′=kerL, kerQ=kerQ′=ImL and Λ, Λ′ two isomorphisms from CokerL to kerL in the same homotopy class. From Lemma 3.15 and Lemma 3.16, it is clear that for each λ∈[0,1] the operators
(3.26)P(λ)=(1-λ)P+λP′,Q(λ)=(1-λ)Q+λQ′
are the projection operators with the property of for each λ∈[0,1], ImP(λ)=kerL and kerQ(λ)=ImL. Beside this, from Lemma 3.16, we have KP(λ)=(1-λ)KP+λKP′. Let the operator Λ_:CokerL×[0,1]→kerL be the operator given in Definition 3.1. Using Proposition 3.7, we see that for each λ∈[0,1] the fixed points of the operator
(3.27)M_(·,λ):Ω¯→XM_(·,λ)=P(λ)+Λ_ΠN(·,λ)+KP(λ),Q(λ)N
coincide with the solutions of the operator equation (1.1). From the condition (v), we have 0∉(L-N)(DomL∩∂Ω) and
(3.28)x∉M_(x,λ),∀x∈∂Ω,∀λ∈[0,1].
Clearly, we have
(3.29)M_(·,0)=M=P+(ΛΠN+KP,QN),M_(·,1)=M'=P'+(Λ′ΠN+KP′,Q′N).
Now let us show that M_ is compact on Ω¯×[0,1]. From the open form
(3.30)M_(·,λ)=(1-λ)Px+λP′x+Λ_ΠN(·,λ)+((1-λ)KP+λKP′)(I-(1-λ)Q-λQ′)N,
it is clear that M_ is continuous. So, in order to show that the set M_(Ω¯×[0,1]) is relatively compact the only delicate point is the last term. Using the fact that KP′=(I-P′)KP, we obtain the last term as
(3.31)(I-λP′)KP(I-Q)N+λ(I-λP′)KP(Q-Q′)N.
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
(3.32)d(I-M(·,0),Ω,0)=d(I-M(·,1),Ω,0),
that is,
(3.33)d(I-M,Ω,0)=d(I-M′,Ω,0).Now, let us indicate how the degreed(I-M,Ω,0) depends on homotopy class of Λ. For this, let us prove the following lemma.Lemma 3.19.
IfG:kerL→kerL is any automorphism and
(3.34)M′=P+(GΛΠ+KP,Q)N
then the relation
(3.35)I-M′=(I-P+GP)(I-M)
is satisfied.Proof.
SinceP2=P, PKP,Q=0, and PΛ=Λ, then
(3.36)(I-P+GP)(I-M)=I-M-P+PM+GP-GPM=I-P-ΛΠN-KP,QN-P+P2+PΛΠN+PKP,QN+GP-GPM=I-P-ΛΠN-KP,QN+PΛΠN+GP-GPM=I-P-ΛΠN-KP,QN+PΛΠN+GP-GP(P+(ΛΠN+KP,QN))=I-P-ΛΠN-KP,QN+PΛΠN+GP-GP2-GPΛΠN-GPKP,QN=I-P-ΛΠN-KP,QN+PΛΠN-GPΛΠN=I-(P+(GPΛΠ+KP,Q)N)=I-M′.Proposition 3.20.
IfΛ,Λ′∈ΛL and
(3.37)M'=P+(Λ′Π+KP,Q)N
then we have
(3.38)d(I-M′,Ω,0)=sgn(det(Λ′Λ-1))d(I-M,Ω,0).Proof.
In Lemma3.19, if we take G=Λ′Λ-1, then
(3.39)I-M'=(I-P+Λ′Λ-1P)(I-M)
is obtained. Now let us show that the operator A=I-P+Λ′Λ-1P is an automorphism on X. For this take, x∈kerL, then we have
(3.40)x-Px+Λ′Λ-1Px=0.
If we apply the operator P to both sides, we get Λ′Λ-1Px=0. Since Λ′Λ-1 is one-to-one, this result gives Px=0. If we substitute this result in (3.40) we obtain that x=0, and therefore A is one-to-one. For surjectivity, take y∈X. Therefore, there exists unique elements k∈kerP, j∈ImP such that y=k+j. Now, we are looking for x∈X, k′∈kerP, j′∈ImP such that x=k′+j′ and Ax=y. So
(3.41)Ax=y(I-P+(Λ′Λ-1)P)x=y(I-P+(Λ′Λ-1)P)(k′+j′)=k+jk′+Λ′Λ-1j′=k+j.
Using uniqueness in direct sum and the fact that Λ′Λ-1 is an automorphism on kerL=ImP, we get k′=k and Λ′Λ-1j′=j. Therefore, taking x=k′+Λ′Λ-1j′ ontoness of the operator A is proved. Therefore, A is an automorphism on X. So using the identity I-M′=(I-P+Λ′Λ-1P)(I-M) and Leray Product Theorem, we have
(3.42)d(I-M′,Ω,0)=d(A,Bϵ(0),0)·d(I-M,Ω,0).
Therefore the result
(3.43)d(I-M′,Ω,0)=d(I-P+Λ′Λ-1P,Bϵ(0),0)·d(I-M,Ω,0)=d((I-P+Λ′Λ-1P)kerL,Bϵ(0)∩kerL,0)·d(I-M,Ω,0)=d(Λ′Λ-1,Bϵ(0)∩kerL,0)·d(I-M,Ω,0)=sgn(det(Λ′Λ-1))·d(I-M,Ω,0)
is achieved.Corollary 3.21.
Under the assumptions of Proposition3.18, Leray-Schauder degree |d(I-M,Ω,0)| only depends upon L, N, and Ω.Now, if the orientation on the spaceskerL and CokerL is fixed, then we can give the following beautiful and fruitful definition.Definition 3.22.
If the operatorsL and N satisfy the conditions (i)–(v) then the coincidence degree of L and N in Ω defined by
(3.44)d[(L,N),Ω]=d(I-M,Ω,0).
Here, Λ in M 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 whereX=Z and L=I. In this situation, kerL={0} and ImL=Z=X, so that CokerL=Z/ImL={0¯}. Therefore dimkerL=dimCokerL=0 and then the assumptions (i) and (ii) are clearly satisfied. Since ImP=kerL=0 and kerQ=ImL=Z, then P=0, and Q=0. Thus KP,Q=KP(I-Q)=I(I-0)=I, and Π=0. Therefore, the conditions (iii) and (iv) reduced to the compactness of N on Ω¯. Since L=I and DomL=X, then the condition (v) in this case means that N has no fixed point on ∂Ω. Since P=0, Π=0, and KP,Q=0, then M=P+(ΛΠ+KP,Q)N=N. Therefore,
(4.1)d[(L,N),Ω]=d[(I,N),Ω]=d(I-N,Ω,0).
That is the coincidence degree of L and N in this case is nothing but the Leray-Schauder degree of I-N.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: ifd[(L,N),Ω]≠0, then 0∈(L-N)(domL∩Ω).(2)
Excision property: ifΩ0∈Ω is an open set such that (L-N)-1(0)∈Ω0, then
(4.2)d[(L,N),Ω]=d[(L,N),Ω0].(3)
Additivity property: ifΩ=Ω1∪Ω2 with Ω1 and Ω2 are open, bounded, disjoint subsets of X, then
(4.3)d[(L,N),Ω]=d[(L,N),Ω1]+d[(L,N),Ω2].(4)
Invariance under homotopy property: if the operator(4.4)N~:Ω¯×[0,1]→Z(x,λ)↦N~(x,λ)
is L-compact in Ω¯×[0,1] and such that for each λ∈[0,1], 0∉[L-N~(·,λ)](domL∩∂Ω), then coincidence degree d[(L,N(·,λ)),Ω], is independent of λ in [0,1]. In particular
(4.5)d[(L,N(·,1)),Ω]=d[(L,N(·,0)),Ω].Proof.
(1) Ifd(I-M,Ω,0)=d[(L,N),Ω]≠0, then ∃x∈Ω such that (I-M)x=0. But in fact, we know that x∈DomL∩Ω. Also, by Proposition 3.7, Lx=Nx. That is 0∈(L-N)(domL∩Ω).
(2) Assume thatΩ0∈Ω is an open set such that (L-N)-1(0)∈Ω0, then by Proposition 3.7,  (I-M)-1(0)∈Ω0. Therefore, by the excision property of the Leray-Schauder degree, d(I-M,Ω,0)=d(I-M,Ω0,0). So, by the definition of coincidence degree, we have d[(L,N),Ω]=d[(L,N),Ω0].
(3) IfΩ=Ω1∪Ω2 with Ω1 and Ω2 are open, bounded, disjoint subsets of X, then additive property of Leray-Schauder degree we have d(I-M,Ω,0)=d(I-M,Ω1,0)+d(I-M,Ω2,0). So the result follows from the definition of coincidence degree.
(4) Since the operatorN~(·,λ) is L-compact for each λ∈[0,1] and for each λ∈[0,1],  0∉[L-N~(·,λ)](domL∩∂Ω), then for each λ∈[0,1] the coincidence degree d[(L,N(·,λ)),Ω] is well defined. Since the operator N~ is L-compact on Ω¯×[0,1] then it is a homotopy of compact operators on Ω¯. 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.
IfΩ is symmetric with respect to 0 and contains it and if N(-x)=-N(x) in Ω, then coincidence degree d[(L,N),Ω] is an odd integer.Proof.
We proved that the operatorM is compact on Ω¯. Since a projection operator P is linear then it is odd in Ω and N is odd in Ω then the operator M=P+(ΛΠ+KP,Q)N is odd in Ω. Therefore, the result follows from the validity of Borsuck theorem in the Leray-Schauder degree.



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*Source: 370946-2012-09-05.xml*