AN EXTENSION OF THE TOPOLOGICAL DEGREE IN HILBERT SPACE

We define classes of mappings of monotone type with respect to a given direct sum decomposition of the underlying Hilbert space H . The new classes are extensions of classes of mappings of monotone type familiar in the study of partial differential equations, for example, the class (S+) and the class of pseudomonotone mappings. We then construct an extension of the Leray-Schauder degree for mappings involving the above classes. As shown by (semi-abstract) examples, this extension of the degree should be useful in the study of semilinear equations, when the linear part has an infinite-dimensional kernel.


Introduction
We will introduce classes of mappings of monotone type with respect to a given projection P : H → E, where E is a closed linear subspace of a real Hilbert space H.We give a systematic classification and show how the new classes are related to each other, how they are related to the corresponding traditional classes and how they stand perturbations.The main result of this note is the construction of a topological degree for mappings of the form conditions; we will not pursue for concrete applications, but will give indications on the kind of context in which the degree theory could be effectively used.The paper is organized as follows.In Section 2, we introduce classes of mappings of monotone type with respect to any bounded linear operator T : H → H.We will mainly deal with the special choice T = P, an orthogonal projection onto a closed linear subspace E. We will give the main properties of the new classes of mappings of monotone type: (S + ) T , T-pseudomonotone, T-quasimonotone and (M) T .We show their relations to each other and to the traditional classes as well.Most of results are direct consequences of the definitions, but are included here for the sake of completeness.In Section 3, we will construct a topological degree for a class of mappings related to the composition E ⊕ E ⊥ of H.The degree is a unique extension of the Leray-Schauder degree in Hilbert space.The construction completes the extension of degree theory given in [3], (see also [4]).Section 4 is devoted to the basic properties and to the uniqueness of the degree.In Section 5, we consider, on a general level, the use of the topological degree to obtain existence results.In Section 6, we will consider systems of semilinear equations, which motivate our definition of degree.It allows indeed to limit the monotonicity hypothesis to some components only of the system.We close this note by a short discussion of semilinear equations with non-symmetric linear part.It turns out that in certain cases we can apply the degree theory, provided the nonlinearity is of class (S + ) T with respect to a suitable operator T.

On the mappings of monotone type
Throughout, H will denote a real separable Hilbert space, with inner product (ii) demicontinuous, if u j → u (norm convergence) implies F(u j ) F(u) (weak convergence); (iii) compact, if it is continuous and the image of any bounded set is relatively compact; (iv) Leray-Schauder type, if it is of the form I − C, where C is compact; Let T : H → H be a bounded linear operator.We will define classes of mappings of monotone type with respect to T. A mapping F : with u j u and v j → v such that limsup F(u j ), T(u j − u) ≤ 0, it follows that F(u j ) F(u).
In this note we will mainly deal with the special case where T = P, an orthogonal projection onto a closed subspace E ⊂ H. Define Q = I − P : H → E ⊥ .We consider more closely bounded demicontinuous mappings of monotone type with respect to P. For instance, according to the previous definitions, F : H → H is of class (S + ) P , if for any sequence (u n ) such that it follows that u j → u.
The structural properties of the classes (S + ) P , (PM) P , (QM) P and (M) P , respectively, as well as their mutual relations follow easily from the definitions.The first result is a direct consequence of the definitions.
The following comments may clarify the situation for a reader familiar with the general theory of mappings of monotone type: It is easy to see that, for bounded demicontinuous mappings, (M) P = (M) P .
Our main emphasis lies on the classes (S + ) P , (PM) P and (QM) P , which have useful properties as will be seen from the following lemmas.Most proofs are omitted.Lemma 2.2.Let all the mappings considered be bounded and demicontinuous.Then (1) (S + ) P ⊂ (PM) P ⊂ (QM) P .
(2) (S + ) ⊂ (S + ) P , (PM) ⊂ (PM) P , (QM) ⊂ (QM) P . ( Proof.The first two items follow directly from the definitions.If dimE ⊥ < ∞, then Q = I − P : H → E ⊥ is compact and condition u n u is equivalent to Pu n Pu.Hence it is clear that the class (S + ) P coincide with (S + ) with similar conclusion for other classes.On the other hand, the reverse follows from the following simple observation.Assume dimE ⊥ = ∞ and consider the mapping In view of applications, it is useful to notice that any compact mapping is quasimonotone and thus also P-quasimonotone.An important property of the classes (S + ) P , (PM) P and (QM) P is that they have conical structure in the following sense.
The conical structure of the classes (S + ) P , (PM) P and (QM) P is one of the features that make the classes useful and "nice" to deal with.For the class (M) P the sum of two maps is not necessarily in the same class as we will see.
Another important fact is that the class (S + ) P is stable under quasimonotone perturbations and moreover, the class (QM) P is, in a sense, a maximal set of such perturbations.Indeed, we have the following lemma.
Lemma 2.4.Let T : H → H be a given bounded demicontinuous mapping.Then T ∈ (QM) P if and only if F + T ∈ (S + ) P for all F ∈ (S + ) P .
Proof.If T ∈ (QM) P and F ∈ (S + ) P , it is not hard to see that F + T ∈ (S + ) P .On the other hand, assume F + T ∈ (S + ) P for all F ∈ (S + ) P .Clearly P ∈ (S + ) P and thus T + P ∈ (S + ) P for all > 0. A contradiction argument shows that necessarily T ∈ (QM) P .
In view of the degree theory, the conical structure of classes will play a crucial role.Indeed, it is needed to ensure that the family of admissible homotopies will be extensive enough.
The main default of class (M) P is that the sum of two mappings F 1 ,F 2 ∈ (M) P does not necessarily remain in the same class.Similarly, if F 1 ∈ (M) P and F 2 ∈ (S + ) P , then it is possible that F 1 + F 2 / ∈ (S + ) P .Thus the class (S + ) P does not in general stand perturbation satisfying the condition (M) P .These facts can be seen from the next example.
Example 2.5.Let H be a real separable infinite-dimensional Hilbert, E ⊂ H a closed linear subspace such that dim E ⊥ = dimE = ∞.Let P : H → E be the orthogonal projection and denote The map F 1 = −P is weakly continuous and hence F 1 ∈ (M) P .The map F 2 is bounded and continuous.We will show that F 2 ∈ (S + ) P .Assume that Pu n Pu, Qu n → Qu and limsup F 2 (u n ),P(u n − u) ≤ 0. Then limsup Pu n ,P(u n − u) ≤ 0 implying Pu n → Pu and consequently F 2 ∈ (S + ) P and also F 2 ∈ (M) P .However, the sum F 1 + F 2 / ∈ (M) P and thus F 1 + F 2 / ∈ (S + ) P .To see this, let {e n } be an orthonormal basis of E and u n = e 1 + e n .Then, u n e 1 := u and (2.5) ))e 1 := w and limsup ))e 1 and, consequently,

Construction of the degree
Let H be a real separable Hilbert space, E a closed subspace of H and P : H → E the corresponding orthogonal projection.As before, we denote We consider a family of mappings where G is an open bounded set in H, C is compact and N is a bounded demicontinuous mapping of class (S + ) P .Let Since any Leray-Schauder type map is of class (S + ) ⊂ (S + ) P , we have , that is, Ᏺ contains the Leray-Schauder type maps.We will construct the degree theory for the class Ᏺ, which will be a unique extension of the classical Leray-Schauder degree in Hilbert space.Note that in case E is finite dimensional, the projection P is compact, and we can write which is of the Leray-Schauder type.Hence, Ᏺ reduces to the class of Leray-Schauder type maps whenever dim E < ∞.If dimE ⊥ < ∞, then Q is compact and any F ∈ Ᏺ can be written in the form which is of class (S + ) by Lemmas 2.2 and 2.4 (recall that compact maps are quasimonotone).Hence Ᏺ consists of (S + )-type maps, only, whenever dim E ⊥ < ∞.Conversely, if F : G → H is a bounded, demicontinuous map of class (S + ) P and dimE ⊥ < ∞, then (S + ) P = (S + ), Q is compact and The degree for mappings of class (S + ) can be found in [7] in a more general setting.
Hereafter, we are mainly interested in the case where both E and E ⊥ are infinite dimensional.The construction of the new degree can be done by using the Galerkin approximation with respect to the space E (cf.[3]).Note that the Q-component, which is of the Leray-Schauder type, needs no approximation.However, we will give here another, shorter, construction which is based on linear compact mappings.The method is a simplified version of the so called "elliptic super-regularization" used in a more general context elsewhere (see [4]).
Following that approach, let ψ : E → E be some fixed compact linear selfadjoint injection.To any F = Q(I − C) + PN ∈ Ᏺ G , we associate a family of mappings defined by Clearly F λ : G → H is a Leray-Schauder type map.We denote by d LS (F λ ,G,0) the corresponding Leray-Schauder degree (which is well-defined only if 0 / ∈ F λ (∂G)).We start with the following basic result.
Proof.We will argue by contradiction.Assume that the first assertion is not valid.Then we can find sequences and thus Qu n → Qu = Qz and which implies u n → u ∈ ∂G, since N ∈ (S + ) P .Consequently, we have The second assertion follows from the homotopy invariance property of d LS .Indeed, let λ 2 > λ 1 > λ 0 .Then F λ ,λ 1 ≤ λ ≤ λ 2 , defines a Leray-Schauder type homotopy such that F λ (u) = 0 for all λ 1 ≤ λ ≤ λ 2 and u ∈ ∂G.Hence and since λ 1 > λ 0 and λ 2 > λ 0 were arbitrary, the second assertion is proved.
It is relevant to define a new integer-valued function d by setting J. Berkovits and C. Fabry 587 We will show in our next section that the integer-valued function d satisfies the conditions of the classical topological degree.

Properties and uniqueness of the degree
Throughout the next steps we assume that F ∈ Ᏺ G , where G ⊂ H is an open bounded set and h / ∈ F(∂G).We will verify four conditions (a) to (d) -existence of solutions, additivity of domains, normalization, and invariance under homotopies.In fact, it is easy to see that (a) is a consequence of (b).The main argument used to establish the properties of the degree will be the following: for any closed set The proof of (4.1) is an obvious variant of the first part of the proof of Lemma 3.1; it suffices to replace the boundary ∂G by the closed set A.
(a) If d(F,G,h) = 0, then there exists a solution for the equation (c) (Normalizing map) The normalizing map is the identity where G ⊂ H is an open bounded set, we have d(I,G,h) = +1.The above properties of the degree d are easily deduced from the corresponding properties of the Leray-Schauder degree, using (4.1).We give more details for the property of invariance under a homotopy.(d) (Invariance under homotopy) We say that a mapping N : [0,1] × G → H is a bounded homotopy of class (S + ) P , if it is bounded, demicontinuous, and the conditions where N is a bounded homotopy of class (S + ) P .The family of admissible homotopies is We will prove (4.7).Without loss of generality we can assume that h t ≡ 0 by absorbing h t into F(t,x).We omit the proof of the following lemma, which is a straightforward generalization of the proof of Lemma 3.1.
The uniqueness of the degree d will be a consequence of the uniqueness of the Leray-Schauder degree.
assuming 0 / ∈ F(∂G).By definition there exists λ 0 > 0 such that where F λ = I − QC + λψ 2 PN and ψ : E → E is a (arbitrary but fixed) linear self-adjoint compact injection.We know that the Leray-Schauder degree is unique and any Leray-Schauder type map I − C is of class (S + ) P .It is easy to see that defining In particular, The proof will be completed by showing that there exists λ 1 ≥ λ 0 such that This is done by the same arguments as in Lemma 3.1

On the continuation method
The standard application of the topological degree theory in order to obtain existence results is through the use of the homotopy invariance property.Hence a typical, and in fact the most useful, form of the continuation method based on affine homotopy can be stated as follows.The underlying idea is to replace the equation F(u) = h which is "hard to solve" by some equation T(u) = 0 which is "easy to solve".Note that in practice F and T are somehow connected, the easy problem T(u) = 0 being a simplification (e.g.linearization) of the hard one.
To illuminate the general argumentation, we consider the semilinear equation where L : D(L) ⊂ H → H is a linear, densely defined, closed operator with ImL = (KerL) ⊥ and N : H → H a nonlinearity.The inverse K : ImL → Im L of the restriction of L to Im L ∩ D(L) is assumed to be compact.It is easy to see that (5.2) can be written equivalently as where P : H → Ker L and Q = I − P : H → Im L are orthogonal projections.Above we have used the fact that KQ − P is the right inverse of L − P. In case N ∈ (S + ) P the mapping F = Q(I − KQN) + PN ∈ Ᏺ and the degree theoretic approach is possible.As an example, we present the following generalization of a continuation theorem of Mawhin [9,10].(5.5) Arguing as in Lemma 3.1, we can show that there exists λ 0 > 0 such that, for (5.7) The next lemma is an immediate generalization of a classical result about linear operators.Let L be as above.Assume that B : H → H is linear bounded operator such that B ∈ (S + ) P and L − B is injective.Then T = Q(I − KQB) + PB ∈ Ᏺ is linear and injective.The next lemma may be helpful for the verification of one of the conditions about the linear map B in Lemma 5. implying Pu n → Pu, which completes the first part of the proof.On the other hand, if B ∈ (S + ) P , then clearly PB |KerL ∈ (S + ).

Semilinear systems of equations
In this section we will provide abstract examples, which will show that the degree theory constructed in this paper has practical value in view of applications.The aim is to consider the nature of mappings which are of class (S + ) P , but not necessarily of class (S + ).For the general treatment of semilinear problems we refer to [6,8].Let V be a real separable Hilbert space and denote H = V n with n ≥ 2. For k = 1,2,..., n, let L k : D(L k ) ⊂ V → V be a linear, densely defined, closed operator with ImL k = (KerL k ) ⊥ .The inverse K k : ImL k → Im L k of the restriction of each L k to Im L k ∩ D(L k ) is assumed to be a compact linear operator.We define the diagonal operator L : D(L) ⊂ V n → V n by setting For further applications to problems involving non-symmetric linear part, we refer to [2].

Lemma 4 . 2 .
Let F,T ∈ Ᏺ G , where G is an open bounded set.Then the affine homotopy

Theorem 4 . 3 .
There exists one and only one degree function for the class Ᏺ satisfying properties (a), (b) and (d), which is invariant under the homotopy class Ᏼ. Proof.The existence of the degree is already established.Let d be given by (3.10).Assume that d 1 is another degree function satisfying conditions (a)-(d) above.Let G ⊂ X be any open bounded set and F dLS (I − C,D, y) = d 1 (I − C,D, y) (4.12) for any y ∈ H\(I − C)(∂D), where D ⊂ H is an open bounded set, gives a degree function for the Leray-Schauder type mappings in H.By the uniqueness of the Leray-Schauder degree dLS = d LS and thus d LS (I − C,D, y) = d 1 (I − C,D, y).(4.13)

Lemma 5 . 3 .
Let B : H → H be a linear bounded operator such that B ∈ (S + ) P and L − B is injective.Then for any open bounded set G, d(T,G, y) = 0 for any y ∈ T(G).(5.8) Proof.For y ∈ T(G) we have y = Tv for some unique v ∈ H.To prove (5.8) we notice first, by the injectivity of T and the additivity of degree, that d(T,G, y) = d T,B(0,R), y for any R > v .(5.9)It is clear that Tu = ty for all u = R and 0 ≤ t ≤ 1. Hence d(T,B(0,R),ty) is constant in t ∈ [0,1] and consequently d T,B(0,R), y = d T,B(0,R),0 .(5.10) Since B is linear, the mapping T is linear and thus odd.The same is true for the approximation T λ = I − QKQB + λψ 2 PB used in the construction of the degree.By the standard property of the Leray-Schauder degree we then obtain d T,B(0,R),0 = lim λ→∞ d LS T λ ,B(0,R),0 , an odd integer, (5.11) completing the proof.

3 .
Lemma 5.4.Let B : H → H be a linear bounded operator.Then B ∈ (S + ) P if and only if PB |KerL : KerL → Ker L is of class (S + ). .Assume that PB |KerL : KerL → Ker L is of class (S + ).Let u n u, Qu n → Qu and limsup Bu n ,P u n − u ≤ 0.