AN ORIENTED COINCIDENCE INDEX FOR NONLINEAR FREDHOLM INCLUSIONS WITH NONCONVEX-VALUED PERTURBATIONS

We suggest the construction of an oriented coincidence index for nonlinear Fredholm operators of zero index and approximable multivalued maps of compact and condensing type. We describe the main properties of this characteristic, including applications to coincidence points. An example arising in the study of a mixed system, consisting of a first-order implicit differential equation and a differential inclusion, is given.


Introduction
The necessity of studying coincidence points of Fredholm operators and nonlinear (compact and condensing) maps of various classes arises in the investigation of many problems in the theory of partial differential equations and optimal control theory (see, e.g., [3,4,7,13,17,18,[20][21][22]).The use of topological characteristics of coincidence degree type is a very effective tool for solving such type of problems.For inclusions with linear Fredholm operators, a number of such topological invariants was studied in the works [7,8,13,18,19].In the present paper, we suggest the general construction of an oriented coincidence index for nonlinear Fredholm operators of zero index and approximable multivalued maps of compact and condensing type.A nonoriented analogue of such index was described earlier in the authors work [17].
The paper is organized in the following way.In Section 2, we give some preliminaries.In Section 3, we present the construction of the oriented coincidence index, first for a finite-dimensional case, and later, on that base, we develop the construction in the case of a compact triplet.In Section 4, using the technique of fundamental sets, we give the most general construction of the oriented index for a condensing triplet and describe its main properties, including its application to the existence of coincidence points.In Section 5, we consider an example of a condensing triplet arising in the study of a mixed system, consisting of a first-order implicit differential equation and a differential inclusion.
2 An oriented coincidence index

Preliminaries
By the symbols E, E , we will denote real Banach spaces.Everywhere, by Y we will denote an open bounded set U ⊂ E (case (i) ) or U * ⊂ E × [0,1] (case (ii)).We recall some notions (see, e.g., [3]). ) if for every y ∈ Y the Frechet derivative f (y) is a linear Fredholm map of index k, that is, dimKer f (y) < ∞, dimCoker f (y) < ∞, and dim Ker f (y) − dimCoker f (y) = k. (2.1) We recall now the notion of oriented Fredholm structure on Y .An atlas {(Y i ,Ψ i )} on Y is said to be Fredholm if, for each pair of intersecting charts (Y i ,Ψ i ) and (Y j ,Ψ j ) and every y ∈ Y i ∩ Y j , it is where E is the corresponding model space, and CG( E) denotes the collection of all linear invertible operators in E of the form i + k, where i is the identity map and k is a compact linear operator.The set CG( E) is divided into two connected components.The component containing the identity map will be denoted by CG + ( E).
Two Fredholm atlases are said to be equivalent if their union is still a Fredholm atlas.The class of equivalent atlases is called a Fredholm structure.
A Fredholm structure on U is associated to a at each point y ∈ U, where LC(E ) denotes the collection of all linear operators in E of the form: identity plus a compact map.Let us note that each (2.4) Two oriented Fredholm atlases are called orientally equivalent if their union is an oriented Fredholm atlas on Y .The equivalence class with respect to this relation is said to be the oriented Fredholm structure on Y .
We will need also the following result (see [3]).
Valeri Obukhovskii et al. 3 where M n+k is an n + k dimensional manifold.Moreover, the restriction We describe now some notions of the theory of multivalued maps that will be used in the sequel (details can be found, e.g., in [1,2,9,12]).
Let (X,d X ),(Z,d Z ) be metric spaces.Given a subset A and ε > 0, we denote by O ε (A) the ε-neighborhood of A. Let K(Z) denote the collection of all nonempty compact subsets of Z.Given a multivalued map (multimap) Σ : X K(Z), a continuous map, σ ε : X → Z, ε > 0, is said to be an ε-approximation of Σ if for every x ∈ X, there exists It is clear that the notion can be equivalently expressed saying that for all x ∈ X, or that where Γ σε ,Γ Σ denote the graphs of σ ε and Σ, respectively, while the metric in X × Z is defined in a natural way as The fact that σ ε is an ε-approximation of the multimap Σ will be denoted by σ ε ∈ a(Σ,ε).
We can summarize some properties of ε-approximations in the following statement (see [9]).
To present the class of multimaps which will be considered, we recall some notions.
Definition 2.6 (see, e.g., [1,9,10,15]).A nonempty compact subset A of a metric space Z is said to be aspheric (or UV ∞ , or ∞-proximally connected) if for every ε > 0, there exists δ, 0 < δ < ε, such that for each n = 0,1,2,..., every continuous map g : S n → O δ (A) can be extended to a continuous map g : Definition 2.7 (see [11]).A nonempty compact space A is said to be an R δ -set if it can be represented as the intersection of a decreasing sequence of compact, contractible spaces.
Valeri Obukhovskii et al. 5 Proposition 2.9 (see [9]).Let Z be an ANR-space.In each of the following cases, an u.s.c.multimap Σ : (d) an AR-space.In particular, every continuous map σ : X → Z is a J-multimap.
The next statement describes the approximation properties of J-multimaps.

Oriented coincidence index for compact triplets
We will start from the following notion.
, and the Fredholm structure on Y generated by f is oriented; Let us mention that from hypotheses (h1), (h2), it follows that the coincidence points set

The case of a finite
We can assume, without loss of generality, that E m ⊂ E n .Then clearly Q ⊂ M. Let us mention also that the orientation on Y induces the orientation on M.
A compact triplet ( f ,G,Y ) C such that G is finite-dimensional will be denoted by ( f ,G,Y ) Cm and will be called finite-dimensional.
Consider an open bounded set N satisfying the following conditions: (ii) N is a compact ANR-space.Let us note that as N we can take a union of a finite collection of balls with centers in Q.
Let us take κ > 0 so that O κ ⊂ N. Then the statement follows from Propositions 2.10(i) and 2.4(i).Now, let the neighborhood O κ be chosen so that Σ is approximable on O κ .From Proposition 2.5, we know that provided that σ ε ∈ a(Σ |Oκ ,ε) and ε > 0 is sufficiently small.So, we can consider the following map of pairs of spaces: Now we are in position to give the following notion.
Definition 3.3.The oriented coincidence index of a finite-dimensional triplet ( f ,G = (ϕ • Σ),U) Cm is defined by the equality where κ > 0 and ε > 0 are taken small enough and the right-hand part of equality (3.3) denotes the Brouwer topological degree.Now we will demonstrate that the given definition is consistent, that is, the coincidence index does not depend on the choice of an ε-approximation σ ε and the neighborhood O κ .Lemma 3.4.Let σ ε and σ ε ∈ a(Σ |Oκ .,ε) be two approximations.Then provided that ε > 0 is sufficiently small.
Proof.Let us take any neighborhood and N is an ANR-space.Then, by Propositions 2.4(i) and 2.5, we know that we can take ε > 0 small enough to provide that σ ε|N and σ ε|N are δ 0 -approximations of Since Σ |N is approximable, we can assume that ε > 0 is chosen so small that there exists a map γ : N × [0,1] → Z with the following properties: , transforms the pair (N ,∂N ) into the pair (E n , E n \0) for each λ ∈ [0,1], and by the homotopy property of the Brouwer degree we have deg( Further from (3.5) and the additive property of the Brouwer degree, we have proving equality (3.4).
where ε > 0 is sufficiently small, follows easily from Propositions 2.4(i), 2.5, and the additive property of the Brouwer degree.At last, let us mention also the independence of the construction on the choice of the transversal subspace E n .In fact, if we take two subspaces E n0 and E n1 , we may assume, without loss of generality, that E n0 ⊂ E n1 .As earlier, we assume that G(U) ⊂ E m ⊂ E n0 ⊂ E n1 .Then, from the construction, we obtain two manifolds M n0 , M n1 , M n0 ⊂ M n1 and two neighborhoods κ for κ > 0 sufficiently small.Now, take ε > 0 small enough to provide that the degrees deg( follows from the map restriction property of Brouwer degree.Now, let us mention the main properties of the defined characteristic.Directly from Definition 3.3 and Proposition 2.5, we deduce the following statement. To formulate the topological invariance property of the coincidence index, we will give the following definition.Definition 3.6.Two finite-dimensional triplets ( f 0 ,G 0 = (ϕ 0 • Σ 0 ),U 0 ) Cm and ( f 1 ,G 1 = (ϕ 1 • Σ 1 ),U 1 ) Cm are said to be homotopic, if there exists a finite-dimensional triplet ( f * ,G * ,U * ) Cm , where where , is a continuous map, and Theorem 3.7 (the homotopy invariance property).If then where κ > 0 is sufficiently small.Take σ * ε ∈ a(Σ * |O * κ ,ε) for ε > 0 sufficiently small.Applying Propositions 2.4 and 2.5, we can verify that the map ) is a δapproximation of G * |O * κ for δ > 0 arbitrary small and, moreover, and where δ > 0 is arbitrary small.Denoting σ * ε|Oκi = σ i , i = 0,1, we have (see [22]), proving the theorem.
Remark 3.8.If the Fredholm map f is constant under the homotopy, that is, U * has the form where (see [21,22]).Hence Ind f ,G 0 ,U Cm = Ind f ,G 1 ,U Cm . (3.17) From Definition 3.3 and the additive property of the Brouwer degree, we obtain the following property of the oriented coincidence index.
Valeri Obukhovskii et al. 9 Theorem 3.9 (additive dependence on the domain property).Let U 0 and U 1 be disjoint open subsets of an open bounded set U ⊂ E and let ( f ,G,U) Cm be a finite-dimensional triplet such that (3.18) Then 3.2.The case of a compact triplet.Now, we want to define the oriented coincidence degree for the general case of a compact triplet ( From the properness property of f and the compactness of G, one can easily deduce the following statement. (3.20) Then, for every closed subset Y 1 ⊂ Y , the set Λ(Y 1 ) is closed.
From the above assertion, it follows that, given a compact triplet ( f ,G,U) C , there exists δ > 0 such that where B δ (0) ⊂ E is a δ-neighborhood of the origin.
Let us take a continuous map i δ : G(U) → E m , where E m ⊂ E is a finite-dimensional subspace, with the property that for each v ∈ G(U).As i δ , we can choose the Schauder projection (see, e.g., [14]).Now, if G has the representation 21) and (3.22), it follows that f , G m and U form a finite-dimensional triplet ( f ,G m ,U) Cm .
We can now define the oriented coincidence index for a compact triplet in the following way.
Definition 3.11.The oriented coincidence index for a compact triplet ( f ,G = (ϕ • Σ),U) C is defined by the equality where G m = i δ • ϕ • Σ and the map i δ satisfies condition (3.22).
To prove the consistency of the given definition, it is sufficient to mention that, given two different maps i 0 δ , i 1 δ : G(U) → E m satisfying property (3.22), we have the homotopy of the corresponding finite-dimensional triplets where (It is clear that the finite-dimensional space E m can be taken the same for both maps i 0 δ , i 1 δ .)In fact, the homotopy is realized by the multimap G * : U × [0,1] → K(E m ), defined as Applying Proposition 3.10 and Theorem 3.5, we can deduce the following coincidence point property.
The definition of homotopy for compact triplets ( f ,G 0 ,U) C ∼ ( f ,G 1 ,U) C has the same form as in Definition 3.6 with the only difference that the connected triplet ( f * ,G * , U * ) is assumed to be compact.
Again, if f and U are constant, we have the equality An analog of the additive dependence on the domain property (see Theorem 3.9) for compact triplets also holds.

Oriented coincidence index for condensing triplets
In this section, we extend the notion of the oriented coincidence index to the case of condensing triplets.At first we recall some notions (see, e.g., [12]).Denote by P(E ) the collection of all nonempty subsets of a Banach space E .Let (Ꮽ,≥) be a partially ordered set.

Definition 4.1. A map β
Valeri Obukhovskii et al. 11 with the natural ordering, and β(D) < +∞ for every bounded set D ∈ P(E ).
Among the known examples of MNC satisfying all the above properties we can consider the Hausdorff MNC Let again for every Ω ⊆ Y such that G(Ω) is not relatively compact.
The entire space E and the set coG(Y ) are natural examples of fundamental sets for ( f ,G,U) β .
It is easy to verify the following properties of a fundamental set.(c) Let {T α } be a system of fundamental sets of ( f ,G,U) β .The set T = ∩ α T α is also fundamental.Proposition 4.5.Each β-condensing triplet ( f ,G,U) β , where β is a monotone, nonsingular MNC, admits a nonempty, compact fundamental set T.
Proof.Consider the collection {T α } of all fundamental sets of ( f ,G,U) β containing an arbitrary point a ∈ E .This collection is nonempty since it contains E .Then, taking T = ∩ α T α = ∅, we obviously have and hence so G( f −1 (T)) is relatively compact and T is compact.
Everywhere from now on, we assume that the MNC β is monotone and nonsingular.
Hence, f , G, and where ( f , G,U) C is a compact approximation of ( f ,G,U) β .
To prove the consistency of the above definition, consider two nonempty, compact fundamental sets T 0 and T 1 of the triplet ( f ,G = ϕ • Σ,U) β with retractions ρ 0 : E → T 0 and ρ 1 : Otherwise, we can assume, without loss of generality, that T 0 ⊆ T 1 .In this case, consider the map ϕ : Indeed, the only fact that we need to verify is that where f (x,λ) ≡ f (x) is the natural extension.
Remark 4.9.Let us mention that in case of invariable f and U: the condition of β-condensivity for a triplet ( f ,G * , U × [0,1]) β may be weakened: for the existence of a nonempty, compact fundamental set T, it is sufficient to demand that 19) for every Ω ⊆ U such that G * (Ω × [0,1]) is not relatively compact.
In fact, it is enough to notice that in this case f −1 * (T) = f −1 (T) × [0,1] and to follow the line of reasoning of Proposition 4.5.
Taking into consideration the corresponding property of compact triplets, we can precise the above property of homotopy invariance.
As an example of application of Theorems 4.8 and 4.10, consider the following coincidence point result.Proof.From the condition it follows that there exists a ball Ꮾ ⊂ E centered at the origin whose boundary ∂Ꮾ does not contain solutions of (4.21).Let ϕ • Σ be a representation of then f ,G * , and implying that G(Ω), and hence G * (Ω × [0,1]), is relatively compact.
In conclusion of this section, let us formulate the additive dependence on the domain property for β-condensing triplets. then,

Example
Consider a mixed problem of the following form: where It should be noted that problem (5.1)-(5.3)may be treated as the law of evolution of a system x(t), whose dynamics is described by the implicit differential equation (5.1) and the control y(t) is the subject of the feedback relation (5.2).Our aim is to show that, under appropriate conditions, the problem of solving problem (5.1)-(5.3)can be reduced to the study of a condensing triplet of the above-mentioned form (see Section 4).
Consider the following condition: (A) for each (t,u,v)∈[0,a]×R n ×R n , there exist continuous partial derivatives A u (t,u, v), A v (t,u,v), and moreover, det is a Fredholm map of index zero, whose restriction to each closed bounded set Consider the continuous map σ : (5. 13) and the multimap Σ : From Propositions 5.2 and 2.9, it follows that Σ is a J-multimap, and hence the composition consists of all functions of the form B(t,x(t),x (t), y(t) ), where y ∈ Π(x).
Define now the CJ-multimap G : The solvability of problem (5.1)-( 5.3) is equivalent to the existence of a coincidence point ) is an open bounded set, then to show that ( f ,G,U) form a condensing triplet with respect to the Kuratowski MNC, it is sufficient to prove the following statement.
(5. 19) The proved statement implies that the coincidence index theory, developed in the previous sections, can be applied to the study of the solvability of problem (5.1)-(5.3).Moreover, it is easy to see that the coincidence point set Coin( f ,G) of a condensing triplet ( f ,G,U) β is a compact set.In case when problem (5.1)-(5.3) is a model for a control system, this approach can be used also to obtain the existence of optimal solutions.As an example, we can consider the following statement.Proposition 5.4.Under the above conditions, suppose that the map A is odd: A(t,−u,−v) = A(t,u,v) for all t ∈ [0,a]; u,v ∈ R n and the set of functions x ∈ C 1 ([0,a];R n ) satisfying the family of relations A t,x(t),x (t) = λB t,x(t),x (t), y(t) , λ ∈ [0,1], y (t) ∈ C t,x(t), y(t) , (5. 21)

. 2 )
and the Kuratowski MNC α(D) = inf d > 0 : D has a finite partition with sets of diameter less than d .(4.3)

Proposition 4 . 4 .
(a) The set Coin( f ,G) is included in f −1 (T) for each fundamental set T of ( f ,G,U) β .(b)Let T be a fundamental set of ( f ,G,U) β , and P ⊂ T, then the set T = co(G( f −1 (T)) ∪ P) is also fundamental.