A Note on Coincidence Degree Theory

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 𝐿𝑥=𝑁𝑥(1.1) 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⊂ℝ^𝑛, 𝑓∈𝐶(Ω), and  𝑝∈ℝ^𝑛⧵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 Π_𝑄 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 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 𝑋 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 𝑋−→^𝑃Dom𝐿−→^𝐿𝑍−→^𝑄𝑍(2.1) is exact, that is, Im𝑃=ker𝐿 and Im𝐿=ker𝑄. Let us define the restriction of 𝐿 to Dom𝐿∩ker𝑃 as 𝐿_𝑃∶Dom𝐿∩ker𝑃→Im𝐿.

Now, let us give the following lemma about 𝐿_𝑃.

Lemma 2.1. 𝐿_𝑃 is an algebraic isomorphism.

Proof. Firstly, let us show that 𝐿_𝑃 is one-to-one mapping. For this let us take 𝑥∈ker𝐿_𝑃⊂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 ker𝐿_𝑃={0}. This means that 𝐿_𝑃 is one-to-one.
Now let us show that 𝐿_𝑃 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 𝐿_𝑃𝑒=𝑧. So the result follows.

Now, let us define 𝐾_𝑃∶=𝐿^𝑃−1. It is clear that 𝐾_𝑃∶Im𝐿⊂𝑍→Dom𝐿∩ker𝑃 is one-to-one, onto, and 𝑃𝐾_𝑃=0.

Lemma 2.2. (1) On Im𝐿, we have 𝐿𝐾_𝑃=𝐼. (2) On Dom𝐿, we have 𝐾_𝑃𝐿=𝐼−𝑃.

Proof. (1) Take 𝑥∈Im𝐿. Therefore, 𝐿𝐾_𝑃𝑥=𝐿(𝐾_𝑃(𝑥))=𝐿_𝑃(𝐾_𝑃(𝑥))=𝐼𝑥.
(2) Since Im𝑃=ker𝐿, then we have 𝐿𝑃=0, so we obtain 𝐾_𝑃𝐿=𝐾_𝑃𝐿(𝐼−𝑃). So that in order to prove (2), we need to show the equality 𝐾_𝑃𝐿(𝐼−𝑃)=𝐾_𝑃𝐿_𝑃(𝐼−𝑃). If we can have Im(𝐼−𝑃)⊆Dom(𝐿_𝑃)=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.
Since 𝑃(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 𝐾_𝑃𝐿(𝐼−𝑃)=𝐾_𝑃𝐿_𝑃(𝐼−𝑃)=𝐼−𝑃 follows.

Now, let us define the canonic surjection Π as Π𝑧∶↦𝑍𝑧→+CImok𝐿er.𝐿(2.2) 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 operator Λ∶Coker𝐿→ker𝐿, then 𝐿𝑥=𝑦,𝑦∈𝑍(2.3) will be equivalent to (𝐼−𝑃)𝑥=ΛΠ+𝐾_𝑃,𝑄𝑦.(2.4) Here, the operator 𝐾_𝑃,𝑄∶𝑍→𝑋 is defined as 𝐾_𝑃,𝑄=𝐾_𝑃(𝐼−𝑄).

Proof. Since Im𝐿=ker𝑄=kerΠ, then for 𝑦∈Im𝐿 we have 𝑄𝑦=0 and ΛΠ𝑦=ΛIm𝐿=Λ0=0. From here, it is seen that ⟺(𝐼−𝐿𝑃𝑥)=𝑥𝑦=⟺𝐾_𝑃𝐿𝑥=𝑦−𝑄𝑦⟺𝐾_𝑃𝐿𝑥=𝐾_𝑃(𝑦−𝑄y)(𝐼−𝑄)𝑦⟺(𝐼−𝑃)𝑥=ΛΠ+𝐾_𝑃(𝐼−𝑄)𝑦.(2.5)

Now, let us consider another projection operator couple (𝑃^,𝑄^) that will make the chain 𝑋−→^𝑃′Dom𝐿−→^𝐿𝑍−→^𝑄′𝑍(2.6) exact, and let us search the relation of this operator couple with (𝑃,𝑄).

From Lemma 2.2, since 𝐿𝐾_𝑃=𝐼, and 𝐿𝐾_𝑃^′=𝐼 then we have 𝐿(𝐾_𝑃−𝐾_𝑃^′)=0. So for any 𝑧∈Im𝐿, we have (𝐾_𝑃−𝐾_𝑃^′)𝑧∈ker𝐿. Therefore, we can write 𝐾_𝑃−𝐾_𝑃^′∶Im𝐿→ker𝐿=Im𝑃=Im𝑃^. Since the projection operator 𝑃 behaves on Im𝑃 as an identity operator, we have 𝐾_𝑃−𝐾_𝑃^′=𝑃(𝐾_𝑃−𝐾_𝑃^′). As a result, the equality 𝐾_𝑃−𝐾_𝑃^′=𝑃𝐾_𝑃−𝐾_𝑃^′=𝑃^𝐾_𝑃−𝐾_𝑃^′(2.7) follows.

Lemma 2.4. The following relations hold.

(i) 𝑃𝐾_𝑃^′+𝑃^𝐾_𝑃=0,   (ii) 𝐾_𝑃^′=(𝐼−𝑃^)𝐾_𝑃.

Proof. (i) Using 𝑃𝐾_𝑃=0, 𝑃^𝐾_𝑃^′=0, and (2.7), the result 𝑃𝐾_𝑝−−𝐾𝑃𝐾_𝑃^′_𝑃′=𝑃^𝐾_𝑃−𝐾_𝑃^′=𝑃^𝐾_𝑃(2.8) follows.
(ii) Again using (2.7) and (i), we obtain ⇒𝐾𝐾_𝑃𝑃−𝐾_𝑃^′=𝑃^𝐾_𝑃−𝐾_𝑃^′−𝐾_𝑃^′=𝑃^𝐾_𝑃−𝑃^𝐾_𝑃^′⇒⇒𝐾_𝑃𝐾=_𝑃𝑃^𝐾_𝑃+−𝑃𝐾_𝑃^′=𝐼−𝑃𝐾_𝑃^′.(2.9) In a similar manner, the equality 𝐾_𝑃^′=(𝐼−𝑃^𝐾_𝑃) 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 𝑋 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 𝐿∶Dom𝐿⊂𝑋→𝑍,𝑁∶Ω⊂𝑋→𝑍(3.1) 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 operator 𝑁∶Ω⊂𝑋→𝑍 is continuous and Π𝑁(Ω) is bounded,(iv)the operator 𝐾_𝑃,𝑄𝑁∶Ω→𝑍 is compact on Ω.

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

Definition 3.2. The operator 𝑁∶Ω→𝑍 which satisfies the conditions (iii) and (iv) will be called 𝐿-compact operator.

It is clear that if we take 𝑋=𝑍 and 𝐿=𝐼 the operator Π reduced to zero operator and the operator 𝐾_𝑃,𝑄 turns to an identity operator then 𝐿-compactness of 𝑁 on Ω reduced to usual compactness for operators.

Theorem 3.3. Let 𝑍 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 𝑋−→^𝑃Dom𝐿−→^𝐿𝑍−→^𝑄𝑍(3.2) will be exact.

Proof. Assume that ker𝐿 is finite dimensional, and the set {𝑦₁,𝑦₂,…,𝑦_𝑛} is a basis for ker𝐿. Define the vector subspaces as 𝑋_𝑘=span{𝑦₁,𝑦₂,…,𝑦_𝑘−1,𝑦_k+1,…,𝑦_𝑛}. Since 𝑋_𝑘 is finite dimensional so is a closed subspace of 𝑋 (see [35, Theorem  2.4-3] and 𝑦_𝑘∉𝑋_𝑘. Let 𝐵 be a basis of 𝑋 such that {𝑦₁,𝑦₂,…,𝑦_𝑛}⊆𝐵. Now, let us define the linear operators which satisfy the conditions 𝐹𝐹_𝑘∶𝑋→ℝ,_𝑘(𝑦)=1,if𝑦=𝑦^𝑘0,if𝑦∈𝐵−𝑦_𝑘.(3.3) Therefore, the operator 𝑃 defined by 𝑃𝑥=^𝑛_𝑘=1𝐹_𝑘(𝑥)𝑦_𝑘(3.4) is a continuous projection operator (see [36, Remark  2.1.19]).
Now, let us prove the existence of continuous projection operators 𝑄 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 𝑄(𝑧)=𝑄𝑧_ker𝑄+𝑧_𝑍=𝑧^𝑍(3.5) 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 surjection Π∶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]). Assume 𝑋 and 𝑌 are normed spaces and the operator 𝑇∶𝑋→𝑌 is linear. Therefore,(a)If 𝑇 is bounded and dim(Im𝑇)<∞ then 𝑇 is compact.(b)If dim𝑋<∞ then 𝑇 is continuous and compact.

Theorem 3.5 (see [35], Lemma  8.3-2). Let 𝑋 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 operators 𝑃 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 of Π to Im𝑄 with Π_𝑄, and let us show that the linear operator Π_𝑄∶Im𝑄→𝑍/𝐼𝑚𝐿 is one-to-one and onto. Since ker(Π_𝑄)∈Im𝐿=ker𝑄 and ker𝑄∩Im𝑄={0}, then we have ker(Π_𝑄)={0}. Therefore Π_𝑄 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 𝑧_Im𝑄∈Im𝑄 and 𝑧_Im𝐿∈Im𝐿 such that the relation 𝑧=𝑧_Im𝑄+𝑧_Im𝐿 is satisfied. Since we have 𝑧=Π𝑧_Im𝑄+𝑧_Im𝐿=Π𝑧_Im𝑄+Π𝑧_Im𝐿=Π𝑧_Im𝑄,(3.6) then the surjectivity of Π_𝑄 follows.
Since we have dim(Im𝑄)≤dim(ImΠ_𝑄)≤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 arbitrary 𝛼∈𝑍 the relation Π^𝑄−1Π(𝛼)=Q(𝛼) holds. Since we can write 𝛼=𝑄𝛼+(𝐼−𝑄)𝛼), then Π^𝑄−1Π(𝛼)==ΠΠ^𝑄−1_𝑄−1Π((𝑄𝛼+(𝐼−𝑄)𝛼))=Π^𝑄−1Π(𝑄𝛼)+Π^𝑄−1Π((𝐼−𝑄)𝛼)Π(𝑄𝛼)=Π^𝑄−1Π_𝑄(𝑄𝛼)=𝑄𝛼(3.7) is obtained.
Let 𝐾_𝑃 denote the restriction of the operator 𝐾_𝑃 to the finite dimensional space Im(𝑄−𝑄^). Using the results obtained until here in this proof and using the equality 𝐾_𝑃^′=(𝐼−𝑃^)𝐾_𝑃, 𝐾_𝑃^′_,𝑄^′𝑁==𝐾𝐼−_𝑃^′𝑃𝐼_−𝑄^𝑁=𝐼−𝑃^𝐾_𝑃𝐼−𝑄^𝑁=𝐼−𝑃_𝐾_𝑃𝐼−𝑄+𝑄+𝑄^𝑁=𝐼−𝑃_𝐾_𝑃(𝐼−𝑄)𝑁+𝐼−𝑃^𝐾_𝑃𝑄−𝑄^𝑁=𝐼−𝑃_𝐾_𝑃,𝑄𝑁+𝐼−𝑃^𝐾_𝑃𝑄−𝑄^𝑁=𝐼−𝑃_𝐾_𝑃,𝑄𝑁+𝐼−𝑃^𝐾_𝑃𝑄−𝑄^𝑁=𝐼−𝑃_𝐾_𝑃,𝑄𝑁+𝐼−𝑃^𝐾_𝑃Π^𝑄−1Π−Π^𝑄−1_′Π𝑁𝐾_𝑃,𝑄𝑁+𝐼−𝑃^𝐾_𝑃Π^𝑄−1−Π^𝑄−1_′Π𝑁(3.8) is achieved. Now, let us explain the operator 𝐾_𝑃^′_,𝑄^′𝑁 is compact. Since the operator 𝐾_𝑃,𝑄𝑁 is compact and 𝐼−𝑃^ is continuous, then the operator (𝐼−𝑃^)𝐾_𝑃,𝑄𝑁 is compact. Since dim(Coker𝐿)=𝑛<∞, then the operator Π^𝑄−1∶Coker𝐿→Im𝑄 is compact. From the same reason, the operator Π^𝑄−1_′ is also compact. Since the operators 𝑁, Π, and 𝐼−𝑃^ are all continuous, the compactness of 𝐾_𝑃^′_,𝑄^′𝑁 follows.

Proposition 3.7. The element 𝑥∈Dom𝐿∩Ω is a solution of the operator equation (1.1) if and only if it satisfies (𝐼−𝑃)𝑥=ΛΠ+𝐾_𝑃,𝑄𝑁𝑥.(3.9) 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 𝑀=𝑃+ΛΠ+𝐾_𝑃,𝑄𝑁.(3.10) Here, Λ∶Coker𝐿→ker𝐿 is any isomorphism.

Proof. Clear from Proposition 2.3.

Remark 3.8. Note that since Im𝑃=ker𝐿⊂Dom𝐿, ImΛ=ker𝐿⊂Dom𝐿, and Im𝐾_𝑃,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 operator 𝑀 is compact on Ω.

Proof. The projection operator 𝑃 is bounded and Im𝑃=ker𝐿 then Im𝑃 is a finite dimensional therefore, from Theorem 3.4 (a), 𝑃 is compact. By assumption (iv), 𝐾_𝑃,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) If 0∉(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 operators 𝑃, 𝑄, and Λ. To show this, we will need the following definition and results.

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

Definition 3.10. If there exists a continuous Λ, ΛΛ∶Coker𝐿×[0,1]→ker𝐿(⋅,0)=Λ,Λ(⋅,1)=Λ^(3.11) such that for any 𝜆∈[0,1] the operator Λ(⋅,Λ)∈Λ_𝐿 then the operator Λ,Λ^∈Λ_𝐿 is called homotopic in Λ_𝐿.

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

Proposition 3.11. The operators Λ and Λ^ are homotopic in Λ_𝐿 if and only if det(ΛΛ^)>0.

Proof. Assume that Λ and Λ^ are homotopic in Λ_𝐿. From the condition (ii) we know that we have dimker𝐿=dimCoker𝐿=𝑛. Let Λ be the operator defined in Definition 3.10, [𝑎₁,𝑎₂,…,𝑎_𝑛] and [𝑏₁,𝑏₂,…,𝑏_𝑛] 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 havedetΛ^Λ⁻¹=detΛ^detΛ⁻¹=detΛ^det(Λ)=ΔΔ((01))>0.(3.12) 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 ΛΛ∶Coker𝐿×[0,1]→ker𝐿(⋅,0)=Λ,Λ(⋅,1)=Λ^.(3.13) 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 isomorphisms Λ∶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: let Λ∶Coker𝐿→ker𝐿 be any isomorphism from the set Λ_𝐿. 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 on Coker𝐿 and ker𝐿, and let [𝑎₁,𝑎₂,…,𝑎_𝑛] be a basis for Coker𝐿 for the chosen orientation.

Definition 3.13. Let the operator Λ∶Coker𝐿→ker𝐿 be given. If [Λ𝑎₁,Λ𝑎₂,…,Λ𝑎_𝑛] 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. If Coker𝐿 and ker𝐿 are oriented, then the operators Λ and Λ^ are homotopic in Λ_𝐿 if and only if they are both orientation preserving or both orientation reversing transformations.

Proof. Assume that [𝑎₁,𝑎₂,…,𝑎_𝑛] and [𝑏₁,𝑏₂,…,𝑏_𝑛] are, respectively, bases of Coker𝐿 and ker𝐿 with respect to chosen the orientation. The basis [Λ𝑎₁,Λ𝑎₂,…,Λ𝑎_𝑛] on ker𝐿 has the same orientation with [𝑏₁,𝑏₂,…,𝑏_𝑛] if and only if the determinant of the matrix 𝑆=(𝑠_𝑖𝑗) defined by Λ𝑎_𝑗=^𝑛_𝑗=1𝑠_𝑖𝑗𝑏_𝑗(3.14) will be positive. Namely, let 𝑀₁ be the transition matrix from the basis [𝑎₁,𝑎₂,…,𝑎_𝑛] to the basis [𝑏₁,𝑏₂,…,𝑏_𝑛] and 𝑀₂ be the transition matrix from the basis [𝑎₁,𝑎₂,…,𝑎_𝑛] to the basis [Λ𝑎₁,Λ𝑎₂,…,Λ𝑎_𝑛], 𝑎₁𝑎₁,𝑎₂,…,𝑎_𝑛−−→^𝑀1𝑏₁,𝑏₂,…,𝑏_𝑛,Λ,𝑎𝑎₂₁,…,𝑎_𝑛−−→^𝑀2Λ𝑎₁,Λ𝑎₂,…,Λ𝑎_𝑛,,Λ𝑎₂,…,Λ𝑎_𝑛−→^𝑆𝑏₁,𝑏₂,…,𝑏_𝑛,(3.15) then we have 𝑀₁=𝑆𝑀₂ and det(𝑀₁)=det(𝑆)det(𝑀₂). [Λ𝑎₁,Λ𝑎₂,…,Λ𝑎_𝑛] has the same orientation with [𝑏₁,𝑏₂,…,𝑏_𝑛] 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 that 𝑆^ is a matrix related to a basis [Λ^𝑎₁,Λ^𝑎₂,…,Λ^𝑎_𝑛]. In this case if the matrix 𝐺=(𝑔_𝑖𝑗) is the matrix represent the operator Λ^Λ⁻¹ with respect to basis [𝑏₁,𝑏₂,…,𝑏_𝑛], then we have ^𝑛_𝑗=1𝑠Λ_𝑖𝑗𝑏_𝑗=Λ^𝑎_𝑖=Λ^ΛΛ⁻¹𝑎_𝑖=Λ^Λ⁻¹Λ𝑎_𝑖,Λ⁻¹^𝑛_𝑘=1𝑠_𝑘𝑖𝑏_𝑘=^𝑛_𝑘=1𝑠_𝑘𝑖Λ^Λ⁻¹𝑏_𝑘,_𝑛_𝑘=1𝑠_𝑛𝑘𝑖_𝑗=1𝑔_𝑗𝑘𝑏_𝑗.(3.16) Therefore, (𝑆^)^𝑇=𝑆^𝑇𝐺^𝑇 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 operators Λ^ 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. Let 𝑌 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. Let 𝑎 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 𝑎𝑆+𝑏𝑆^=𝑆^==𝑎𝑆^₂²=𝑎𝑆+𝑏𝑆^𝑎𝑆+𝑏𝑆^=𝑎₂𝑆²+𝑎𝑏𝑆𝑆′+𝑎𝑏𝑆′𝑆+𝑏²𝑆^²=𝑎²𝑆+𝑎𝑏𝑆𝑆′+𝑎𝑏𝑆′𝑆+𝑏²𝑆′=(𝑎𝑆++𝑏)𝑎𝑏𝑎𝑆𝑆′++𝑏𝑎𝑆𝑏𝑆_+𝑏²𝑆′=𝑎𝑎𝑆+𝑏𝑆^+𝑏𝑎𝑆+𝑏𝑆^(3.17) 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, if 𝑎+𝑏=1, then 𝑆^==𝑎𝑆^₂²=𝑎𝑆+𝑏𝑆^𝑎𝑆+𝑏𝑆^=𝑎₂𝑆²+𝑎𝑏𝑆𝑆′+𝑎𝑏𝑆′𝑆+𝑏²𝑆^²=𝑎²𝑆+𝑎𝑏𝑆𝑆′+𝑎𝑏𝑆′𝑆+𝑏²𝑆′𝑆+𝑎𝑏𝑆′+𝑎𝑏𝑆+𝑏²𝑆′=(𝑎+𝑏)𝑎𝑆+𝑏𝑆^=𝑎𝑆+𝑏𝑆′=𝑆^(3.18) 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, 𝑆^𝑥=𝑎𝑆𝑥+𝑏𝑆′𝑥=𝑎𝑥+𝑏𝑥=(𝑎+𝑏)𝑥=𝑥(3.19) and from here we obtain 𝑥∈Im𝑆^ and Im𝑆⊆Im𝑆^. So the result Im𝑆=Im𝑆^ follows.

Lemma 3.16. If 𝑃 and 𝑃^ are projection operators onto ker𝐿, 𝑎+𝑏=1 and 𝑃^=𝑎𝑃+𝑏𝑃^, then 𝐾_𝑃^′′=𝑎𝐾_𝑃+𝑏𝐾_𝑃^′.

Proof. In the case ker𝐿={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 𝐾_𝑃^′=(𝐼−𝑃^)𝐾_𝑃 and 𝑃𝐾_𝑃=0, then the relation 𝐾_𝑃^′′=𝐼−𝑃^𝐾_𝑃=𝐼−𝑎𝑃−𝑏𝑃^𝐾_𝑃=𝐾_𝑃−𝑎𝑃𝐾_𝑃−𝑏𝑃^𝐾_𝑃=1⋅𝐾_𝑃−𝑏𝑃^𝐾_𝑃=(𝑎+𝑏)𝐾_𝑃−𝑏𝑃^𝐾_𝑃=𝑎𝐾_𝑃+𝑏𝐾_𝑃−𝑏𝑃^𝐾_𝑃=𝑎𝐾_𝑃+𝑏𝐼−𝑃^𝐾_𝑃=𝑎𝐾_𝑃+𝑏𝐾_𝑃^′(3.20) is obtained.

Lemma 3.17. Let 𝑍 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 that 𝑆^ is a projection operator. Since 𝑍=ker𝑆𝑆∶⊕𝑍Im→𝑆𝑍,𝑍𝑆^=∶k𝑍er→𝑆_𝑍⊕Im𝑆^,(3.21) then for any 𝑧∈𝑍 there exist unique elements 𝑧₀∈ker𝑆, 𝑧₁∈Im𝑆, 𝑧^0∈ker𝑆^, and 𝑧^1∈Im𝑆^ such that 𝑧=𝑧₀+𝑧₁ and 𝑧=𝑧^0+𝑧^1 hold. Therefore, 𝑆^²(𝑧)==𝑎𝑎²₂𝑆(𝑧)+𝑎𝑏𝑆𝑆^(𝑧)+𝑎𝑏𝑆^𝑆(𝑧)+𝑏²𝑆^(𝑧)=𝑎₂𝑆𝑧^0+𝑧^1+𝑎𝑏𝑆𝑆^𝑧^0+𝑧^1+𝑎𝑏𝑆^𝑆𝑧₀+𝑧₁+𝑏²𝑆^𝑧₀+𝑧₁=𝑎₂𝑆𝑧^1+𝑎𝑏𝑆𝑆^𝑧^1+𝑎𝑏𝑆^𝑆𝑧₁+𝑏²𝑆^𝑧₁=𝑎𝑆𝑆𝑧𝑧^1_1+𝑎𝑏𝑆𝑧^1+𝑎𝑏𝑆^𝑧₁+𝑏²𝑆^𝑧₁=(𝑎+𝑏)𝑎𝑆𝑧^1+𝑏𝑆^𝑧₁=𝑎𝑆(𝑧)++𝑏𝑆𝑏𝑆^_𝑧₁=𝑎𝑆𝑧^0+𝑎𝑆𝑧^1+𝑏𝑆^𝑧₀+𝑏𝑆^𝑧₁(𝑧)=𝑆^(𝑧)(3.22) is obtained.
Now, let us show that ker𝑆^=ker𝑆=ker𝑆^. For this, take an arbitrary element 𝑥∈ker𝑆=ker𝑆^. Therefore, 𝑆^(𝑥)=𝑎𝑆(𝑥)+𝑏𝑆(𝑥)=𝑎.0+𝑏.0=0.(3.23) 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 0==𝑆𝑎𝑆^((𝑒𝑥)+=𝑓𝑎)𝑆+(𝑏𝑥𝑆)_+𝑏𝑆^(𝑥)(𝑒+𝑓)=𝑎𝑆(𝑓)+𝑏𝑆^(𝑓)=𝑎𝑓+𝑏𝑆^(𝑓).(3.24) 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, 𝑆^(𝑓)=𝑆²(𝑓)=𝑆^𝑆^(𝑓)=𝑆^−𝑎𝑏𝑓=−𝑎𝑏𝑆^(𝑓)(3.25) 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 degree 𝑑[𝐼−𝑀,Ω,0] depends on only 𝐿, 𝑁, Ω and homotopy class of Λ in Λ_𝐿.

Proof. Let the operators 𝑃, 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 𝑃(𝜆)=(1−𝜆)𝑃+𝜆𝑃^,𝑄(𝜆)=(1−𝜆)𝑄+𝜆𝑄^(3.26) 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 𝐾_𝑃(𝜆)=(1−𝜆)𝐾_𝑃+𝜆𝐾_𝑃^′. 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 𝑀𝑀(⋅,𝜆)∶Ω→𝑋(⋅,𝜆)=𝑃(𝜆)+ΛΠ𝑁(⋅,𝜆)+𝐾_{𝑃(𝜆),𝑄(𝜆)}𝑁(3.27) coincide with the solutions of the operator equation (1.1). From the condition (v), we have 0∉(𝐿−𝑁)(Dom𝐿∩𝜕Ω) and 𝑥∉𝑀(𝑥,𝜆),∀𝑥∈𝜕Ω,∀𝜆∈[0,1].(3.28) Clearly, we have 𝑀𝑀(⋅,0)=𝑀=𝑃+ΛΠ𝑁+𝐾_𝑃,𝑄𝑁,(⋅,1)=𝑀′=𝑃′+Λ^Π𝑁+𝐾_𝑃^′_,𝑄^′𝑁.(3.29) Now let us show that 𝑀 is compact on Ω×[0,1]. From the open form 𝑀(⋅,𝜆)=(1−𝜆)𝑃𝑥+𝜆𝑃^𝑥+ΛΠ𝑁(⋅,𝜆)+(1−𝜆)𝐾_𝑃+𝜆𝐾_𝑃^′𝐼−(1−𝜆)𝑄−𝜆𝑄^𝑁,(3.30) 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 𝐾_𝑃^′=(𝐼−𝑃^)𝐾_𝑃, we obtain the last term as 𝐼−𝜆𝑃^𝐾_𝑃(𝐼−𝑄)𝑁+𝜆𝐼−𝜆𝑃^𝐾_𝑃𝑄−𝑄^𝑁.(3.31) 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 𝑑(𝐼−𝑀(⋅,0),Ω,0)=𝑑(𝐼−𝑀(⋅,1),Ω,0),(3.32) that is, 𝑑(𝐼−𝑀,Ω,0)=𝑑𝐼−𝑀^,Ω,0.(3.33)

Now, let us indicate how the degree 𝑑(𝐼−𝑀,Ω,0) depends on homotopy class of Λ. For this, let us prove the following lemma.

Lemma 3.19. If 𝐺∶ker𝐿→ker𝐿 is any automorphism and 𝑀^=𝑃+𝐺ΛΠ+𝐾_𝑃,𝑄𝑁(3.34) then the relation 𝐼−𝑀^=(𝐼−𝑃+𝐺𝑃)(𝐼−𝑀)(3.35) is satisfied.

Proof. Since 𝑃²=𝑃, 𝑃𝐾_𝑃,𝑄=0, and 𝑃Λ=Λ, then (𝐼−𝑃+𝐺𝑃)(𝐼−𝑀)==𝐼𝐼−−𝑃𝑀−−Λ𝑃Π+𝑁𝑃−𝑀𝐾+_𝑃,𝑄𝐺𝑃−𝐺𝑃𝑀=𝐼−𝑃−ΛΠ𝑁−𝐾_𝑃,𝑄𝑁−𝑃+𝑃²+𝑃ΛΠ𝑁+𝑃𝐾_𝑃,𝑄𝑁+𝐺𝑃−𝐺𝑃𝑀=𝐼−𝑃−ΛΠ𝑁−𝐾_𝑃,𝑄𝑁+𝑃ΛΠ𝑁+𝐺𝑃−𝐺𝑃𝑀=𝐼−𝑃−ΛΠ𝑁−𝐾_𝑃,𝑄𝑁+𝑃ΛΠ𝑁+𝐺𝑃−𝐺𝑃𝑃+ΛΠ𝑁+𝐾_𝑃,𝑄𝑁=𝐼−𝑃−ΛΠ𝑁−𝐾_𝑃,𝑄𝑁+𝑃ΛΠ𝑁+𝐺𝑃−𝐺𝑃²−𝐺𝑃ΛΠ𝑁−𝐺𝑃𝐾_𝑃,𝑄𝑁=𝐼−𝑃+𝐺𝑃ΛΠ+𝐾𝑁_𝑃,𝑄+𝑃ΛΠ𝑁−𝐺𝑃ΛΠ𝑁𝑁=𝐼−𝑀^.(3.36)

Proposition 3.20. If Λ,Λ^∈Λ_𝐿 and 𝑀′=𝑃+Λ^Π+𝐾_𝑃,𝑄𝑁(3.37) then we have 𝑑𝐼−𝑀^,Ω,0=sgndetΛ^Λ⁻¹𝑑(𝐼−𝑀,Ω,0).(3.38)

Proof. In Lemma 3.19, if we take 𝐺=Λ^Λ⁻¹, then 𝐼−𝑀′=𝐼−𝑃+Λ^Λ⁻¹𝑃(𝐼−𝑀)(3.39) 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 𝑥−𝑃𝑥+Λ^Λ⁻¹𝑃𝑥=0.(3.40) 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 𝐼−𝑃+𝐴Λ𝑥_=𝑦𝐼−𝑃+Λ_Λ⁻¹𝑃𝑥=𝑦𝑘Λ⁻¹𝑃𝑘^+𝑗^=𝑘+𝑗_+Λ^Λ⁻¹𝑗^=𝑘+𝑗.(3.41) 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 𝑑𝐼−𝑀^,Ω,0=𝑑𝐴,𝐵_𝜖(0),0⋅𝑑(𝐼−𝑀,Ω,0).(3.42) Therefore the result 𝑑𝐼−𝑀^,Ω,0==𝑑𝑑𝐼𝐼−−𝑃𝑃++ΛΛ^Λ⁻¹_𝑃,𝐵_𝜖(0),0⋅𝑑(𝐼−𝑀,Ω,0)=𝑑Λ_Λ⁻¹𝑃_ker𝐿,𝐵_𝜖(0)∩ker𝐿,0⋅𝑑(𝐼−𝑀,Ω,0)=sgnΛdet⁻¹,𝐵Λ_𝜖(0)∩ker𝐿,0⋅𝑑(𝐼−𝑀,Ω,0)Λ⁻¹⋅𝑑(𝐼−𝑀,Ω,0)(3.43) is achieved.

Corollary 3.21. Under the assumptions of Proposition 3.18, Leray-Schauder degree |𝑑(𝐼−𝑀,Ω,0)| only depends upon 𝐿, 𝑁, and Ω.

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

Definition 3.22. If the operators 𝐿 and 𝑁 satisfy the conditions (i)–(v) then the coincidence degree of 𝐿 and 𝑁 in Ω defined by 𝑑[(𝐿,𝑁),Ω]=𝑑(𝐼−𝑀,Ω,0).(3.44) 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 where 𝑋=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 𝐾_𝑃,𝑄=𝐾_𝑃(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 𝐾_𝑃,𝑄=0, then 𝑀=𝑃+(ΛΠ+𝐾_𝑃,𝑄)𝑁=𝑁. Therefore, 𝑑[(𝐿,𝑁),Ω]=𝑑[(𝐼,𝑁),Ω]=𝑑(𝐼−𝑁,Ω,0).(4.1) 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: if 𝑑[(𝐿,𝑁),Ω]≠0, then 0∈(𝐿−𝑁)(dom𝐿∩Ω). (2)Excision property: if Ω₀∈Ω is an open set such that (𝐿−𝑁)⁻¹(0)∈Ω₀, then 𝑑[(𝐿,𝑁),Ω]=𝑑(𝐿,𝑁),Ω₀.(4.2)(3) Additivity property: if Ω=Ω₁∪Ω₂ with Ω₁ and Ω₂ are open, bounded, disjoint subsets of 𝑋, then 𝑑[(𝐿,𝑁),Ω]=𝑑(𝐿,𝑁),Ω₁+𝑑(𝐿,𝑁),Ω₂.(4.3)(4)Invariance under homotopy property: if the operator 𝑁(∶𝑥,Ω𝜆)×↦[0𝑁,1(]𝑥→,𝜆)𝑍(4.4) 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 𝑑[(𝐿,𝑁(⋅,1)),Ω]=𝑑[(𝐿,𝑁(⋅,0)),Ω].(4.5)

Proof. (1) If 𝑑(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 that Ω₀∈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) If Ω=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 operator 𝑁(⋅,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. If Ω 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 operator 𝑀 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+𝐾_𝑃,𝑄)𝑁 is odd in Ω. Therefore, the result follows from the validity of Borsuck theorem in the Leray-Schauder degree.