On coincidence index for multivalued perturbations of nonlinear Fredholm maps and some applications

We define a nonoriented coincidence index for a compact, fundamentally restrictible, and condensing multivalued perturbations of a map
which is nonlinear Fredholm of nonnegative index on the set of coincidence points. As an application, we consider an optimal controllability problem for a system governed by a second-order integro-differential equation.


Introduction
One of the most efficient methods for the study of boundary and periodic problems for nonlinear differential equations and inclusions, consists in the operator treatment of these problems in suitable functional spaces.
However, for a number of problems of this sort, the maps constructed in functional spaces do not possess "nice" properties on the whole domain, but only on some open neighborhood of the solutions set.As an example, we may note a Monge-Ampere problem arising in geometry of surfaces (see [10]).Moreover, the application of topological methods to the investigation of this kind of problems often requires the embedding of a given equation or inclusion into a corresponding parametric family.In such a situation, either solutions sets or their neighborhoods can vary in dependence of parameters.
In the present paper, we want to study an inclusion of the form where f : E is a multivalued map (multimap), and E, E are real Banach spaces.
In other words, we have to deal with the coincidence points of the triplet ( f ,G,Y).We assume that the set Q of the coincidence points of ( f ,G,Y) is compact, and that f is differentiable on some neighborhood of Q and is nonlinear Fredholm on Q.
We suppose also that the multimap G is closed with convex, compact values and that the triplet ( f ,G,Y) satisfies some conditions of compact restrictibility on some neighborhood of Q; in particular, G may be compact or f -condensing with respect to a certain measure of noncompactness.For the case when G is a single-valued, completely continuous map, we refer the reader to [2].For a pair of this type, a nonoriented index of solutions was defined and studied in [10].For a single-valued and f -compactly restrictible map G, the nonoriented index of solutions was defined in [11].
In the case when G is a completely continuous multimap of acyclic type, f is a nonlinear Fredholm map of zero index, and the domain Y possesses the property of orientability of every Fredholm structure on it, the oriented coincidence index was constructed in [1].Note that the situation considered in the present paper is essentially different since the orientability of Fredholm structures on arbitrary neighborhoods of the coincidence points set cannot be guaranteed in advance.We point out that for the case when f is a linear Fredholm operator, some topological characteristics of the couple ( f ,G) were studied in [3,6,8].
In the present paper we first define a relative coincidence index with respect to a convex, closed set K for triplets ( f ,g, Ū) K and ( f ,G, Ū) K where g and G are, respectively, single-valued and multivalued compact perturbations of a nonlinear Fredholm map f , and U is an open neighborhood of Q.
Using these results as a base, we define a nonoriented coincidence index Ind( f ,G,U) for a triplet which is fundamentally restrictible on certain neighborhood of Q.We pay special attention to the important particular cases when ( f ,G,U) form a condensing or locally condensing triplet.We also describe the main properties of the constructed topological characteristic.As an application we consider an optimal controllability problem for a system governed by a second-order integro-differential equation.

Preliminaries
In the sequel E, E denote real Banach spaces.Everywhere by Y we denote an open set: (2.1) For simplicity we will denote maps and their restrictions by the same symbols.
As a base of our construction, we use the coincidence index of s-admissible pair Ind( f ,g, Ū) defined as an element of the Rohlin-Thom ring of bordisms which has the following main properties (see [9,10]).
Proposition 2.5 (the coincidence point property).If Ind( f ,g, Ū) = 0, the zero element of the ring of bordisms, then Coin( f ,g) = ∅.Proposition 2.6 (the homotopy invariance property). where Proposition 2.7 (additive dependence on the domain property).Let U 0 and U 1 be disjoint open subsets of an open set U ⊆ E, and let ( f ,g, Ū) be an s-admissible triplet such that We recall some notions for multivalued maps (cf.[5]).Denote by Kv(E ) the collection of all compact, convex subsets of E .
Let S ⊆ Ȳ be a closed subset.

Relative coincidence index
3.1.Single-valued perturbation.Let K ⊂ E be a closed, convex set.h1), (h3) of Definition 2.4 hold together with the following condition: (h2 Ks ) g is a continuous compact map.
Our aim is to define a relative coincidence index Ind( f ,g, Ū) K .To this aim we consider first the trivial case We set, by definition the zero element of the Rohlin-Thom ring of bordisms.Now let f −1 (K) = ∅.Let ĝ : Ū → K be an arbitrary extension of g such that ĝ( Ū) is a relatively compact subset of K. Then Ind f ,g, Ū K := Ind f , ĝ, Ū , ( the coincidence index of the s-admissible triplet ( f , ĝ, Ū).
Proof.(a) To verify that ( f , ĝ, Ū) is an s-admissible triplet it is sufficient to be sure that hence x ∈ f −1 (K) and ĝ(x) = g(x), so x ∈ Coin( f ,g) and x ∈ ∂U.
(b) The index does not depend on the choice of the extension ĝ.In fact, let ĝ0 , ĝ1 : Ū → K be two extensions of g.
Consider the map g Therefore x ∈ ∂U.By the homotopy property (Proposition 2.6) We now describe the main properties of the defined characteristic.
In the sequel we will need the following two properties of the relative coincidence index.
Proposition 3.6 (additive dependence on the domain property).
This property follows immediately from Proposition 2.7.
Proposition 3.7 (the map restriction property).

Multivalued perturbation.
Let K, Y , and f : Ȳ → E be as in the previous section, G : Definition 3.8.The maps f , G, and the set Y form a Km-admissible triplet ( f ,G, Ȳ ) K if f satisfies condition (h1) of Definition 2.4 and the following assumptions hold: To define the relative coincidence index Ind( f ,G, Ū) K again we consider first the case f −1 (K) = ∅.In this situation, as before, we set by definition Ind f ,G, Ū = 0. (3.18) To consider the case f −1 (K) = ∅, we introduce the following notions.
Valeri Obukhovskii et al. 301 To prove the existence of a single-valued homotopic approximation and to study its properties, consider any Km-admissible triplet ( f ,G, Ȳ ) K . Let It is easy to see that, by construction, Proof.Suppose the contrary.Then, there exist sequences such that where From (h2 Km ) it follows that we can assume, without loss of generality, that z n → z 0 ∈ K.
By construction, for every n we have where coG( f −1 (K)) is a compact set and, since f is proper, we can assume, without loss of generality, that (x n ,λ n ) → (x 0 ,λ 0 ) ∈ f −1 (K) ∩ ∂Y .From the closedness of G we obtain that z 0 ∈ G(x 0 ,λ 0 ).We can also assume that µ n converges to µ 0 .As f is a continuous map, we get that f Further, by definition of ε-approximation we have hence we can assume, without loss of generality, that g εn (x n ,λ n ) → y 0 ∈ K.
Proof.From Proposition 3.11, we see that we can take the triplet ( f ,g ε , Ū) K , where g ε is an ε-approximation of G and ε > 0 is small enough, as a single-valued homotopic approximation.
We can now justify the following definition.
This notion is well defined.In fact, we can prove the following statement.
Valeri Obukhovskii et al. 303 Proposition 3.14.Let ( f 0 ,g 0 , Ū0 ) K and ( f 1 ,g 1 , Ū1 ) K be two single-valued homotopic approximations of the Km-admissible triplet ( f ,G, Ū) K .Then, where the homotopy is in the class of Ks-admissible triplets.Hence, Proof.From the definition it follows that there exists a Km-admissible triplet From Proposition 3.11, there exists a USC multimap Φ : Furthermore, we can consider the obvious homotopy and, at last, the single-valued map h 1 : and we can apply Proposition 3.5.
As a direct consequence of the definition, we obtain the following property on the homotopy invariance of the relative coincidence index.
We can now formulate the following coincidence point principle.
Proof.In fact, suppose the contrary, repeating the same arguments used in the proof of Proposition 3.11, we can find a single-valued homotopy approximation ( f ,g, Ū) K such that Coin( f ,g) = ∅ and hence, by Proposition 3.3, we have that Ind( f ,g, Ū) = 0.
The use of single-valued approximations in the definition of the index allows the following analogs of Propositions 3.6 and 3.7.
Proposition 3.17.Let ( f ,G, Ū) K be Km-admissible, and It is easy to verify that this notion has the following properties (cf.[5]).
(c) Let {T α } be a system of fundamental sets of ( f ,G,S).The set T = ∩ α T α is also fundamental.f ,G,Y) is said to be τ-admissible if the following conditions are satisfied: Our goal is the definition of a coincidence index, Ind( f ,G,U) for a τ-admissible triplet ( f ,G,U).
First of all we consider the case when the set Q is empty.In this case we set by definition the index Ind( f ,G,U) as the zero element of the Rolin-Thom ring of nonoriented bordisms.
Suppose now that Q = ∅.We can assume, without loss of generality, that the restriction . Selecting a finite subcover {V (x 1 ),...,V(x m )} from the cover {V (x)} x∈Q of Q, we can substitute V with the smaller neighborhood V = ∪ m i=1 V (x i ).Furthermore, since every Φ n C 1 -map is locally proper (see [7]) and Q is compact, we can also assume, without loss of generality, that the restriction f | V is proper.Now, if T is any supporting fundamental set of the triplet ( f ,g, V ), we see that ( f ,G, V ) T is a Tm-admissible triplet in the sense of Definition 3.8.We can now give the following definition of coincidence index.Definition 4.5.Let ( f ,G,U) be τ-admissible with Q = ∅.Then, where T is any supporting fundamental set of ( f ,G, V ).
Lemma 4.6.Definition 4.5 is consistent, that is, the coincidence index does not depend on the choice of the supporting fundamental set T and the neighborhood V with the above mentioned properties.
Proof.(a) Let T 0 and T 1 be two supporting fundamental sets of ( f ,G, V ).Then, the intersection Consider the retraction ρ : E → T and the USC multimap Ĝ : V → Kv(E ), defined as Ĝ(x) = co(ρ • G)(x).It is easy to verify that ( f , Ĝ, V ) T0 forms a T 0 madmissible triplet.Moreover, hence f (x 0 ) ∈ T, Ĝ(x 0 )=G(x 0 ), and f (x 0 ) ∈ G(x 0 ) giving a contradiction.Then, the property (h3 Km ) of Definition 3.8 holds, and from the homotopy property of the relative index (see Proposition 3.15) we have Applying the map restriction property (Proposition 3.18), we see that and therefore The equality follows in the same way.(b) Let V 0 , V 1 be two open neighborhoods of Q with the necessary properties.We can assume without loss of generality that is the consequence of the additive dependence on the domain and coincidence point properties (Propositions 3.16 and 3.17).
The next two properties of the characteristics play a key role in the applications.They follow from Propositions 3.15 and 3.16.

Coincidence index for condensing triplets.
We consider now some important examples of τ-admissible triplets.
The first one is rather simple.Suppose that ( f ,G,Y) satisfies assumptions (H 1 ), (H 2 ) of Definition 4.4 and the following one: (H 3 ) there exists an open neighborhood It is clear that in such a situation we can consider E as a supporting fundamental set of ( f ,G,Y), hence ( f ,G,Y) is τ-admissible.
To deal with more consistent examples, we recall some notions (cf.[5]).Denote by P(E ) the collection of all nonempty subsets of E .Definition 4.9.A function β : for every Ω ∈ P(E ), and β(Ω) < +∞ for each bounded set

Among the known examples of MNC satisfying all the above properties:
The Hausdorff MNC   f ,G,M) is said to be β-condensing if, for every Ω ⊆ M such that G(Ω) is not relatively compact, we have (4.15) We now introduce the following important class of β-condensing triplets.
for each Ω ⊆ M.
We can now give new sufficient conditions under which ( f ,G,Y) is τ-admissible.
Theorem 4.12.Let ( f ,G,Y) satisfy conditions (H 1 ), (H 2 ) of Definition 4.4 and the following: Proof.Let {T α } be the collection of all fundamental sets of ( f ,G,V).Consider the set T = ∩ α T α .From Proposition 4.2(b), (c) it follows that T is the fundamental set satisfying
The condensivity condition may take only a local form.
The notion of locally (k,β)-condensing triplet is defined analogously.
We can now formulate the following statement.Proof.We prove that condition (H 3 ) implies condition (H 3 ).Choose a finite subcover Further, the condition of local condensivity implies that and, from the monotonicity of β we have , and ( f ,G,U) satisfies conditions (H 1 ), (H 2 ) and either (H 3 ), (H 3 ), or (H 3 ), then ( f ,G,U) is τ-admissible and the coincidence index Ind( f ,G,U) is well defined and satisfies all the properties described in Section 4.1.
We now select the property of topological invariance which we will use in applications.

and (H 3 ). Then
where It may be convenient for applications to formulate the condensivity and local condensivity conditions in terms of Fréchet derivative f .We start from the following notion.Let f : for each Ω ⊂ B δ (x).
We recall that a linear operator A is said to be a Φ + -operator if ImA is closed and kerA is finite dimensional.It is clear that every linear Fredholm operator is a Φ + -operator.
Denote by α E , α E the Kuratowski MNC in spaces E and E , respectively.
Lemma 4.17 (see [4]).Let A : E → E be a Φ + -operator.Then the number is finite and different from zero.
where ω is the residual term in the representation , for every ε, 0 < ε < 1 there exists a ball B δ (0) such that Without loss of generality, we can also suppose that Then, from (4.26) we have, for all Ω ⊂ B δ (0), and therefore Since ε is arbitrary, we also have Proof.We prove that ( f (x),G,Y) are (k 1 (x),α E )-bounded at every point In fact, let x ∈ Q, then there exists a bounded open neighborhood V (x) such that, for every Ω ⊂ V (x), we have

.35)
The statement now follows from Corollary 4.19.

5.
1.An optimal controllability problem.We consider a control system governed by a second-order integro-differential equation.For simplicity we restrict ourselves to the one-dimensional model; the generalization to the n-dimensional case is straightforward.Denote by C[0,1] the space of continuous functions x : [0,1] → R endowed with the usual norm x 0 = max t∈[0,1] |x(t)|, and by C k [0,1], k = 1,2 the space of k times continuously differentiable functions with norms respectively.We suppose that the dynamic of the control system is the following: We assume also that a lower semicontinuous cost functional is given.
We want to find a control u * such that the corresponding solution x * of (5.2) and (5.3) satisfies the controllability relation for given c 0 ,c 1 ∈ R, and minimizes j 0 : where Σ ⊂ C 2 [0,1] denotes the set of all solutions of (5.2), (5.3), and (5.5).
We now describe the assumptions on the given control problem.First of all, suppose the following.
(L) The polynomial has no multiple roots, its degree m is an odd number, and a 0 > 0.
Using estimate (5.29), we have (5.35) Supposing, without loss of generality, that ẍ 0 > 1, we obtain for some positive constants a and b .Therefore, since f (x) ∈ λG(x) for some λ ∈ [0,1] we have the following estimate: (5.38) Taking into account condition (L), it becomes clear that the estimate (5.38) holds only if ẍ 0 is a priori bounded and then x 2 is a priori bounded.
Lemma 5.5.The set Q is compact.
Proof.The continuity of f and the closedness of G imply that Q is closed.Taking into account Lemma 5.3, it is sufficient to show that the set f (Q) is compact in Ꮿ.To this aim we demonstrate that the set (5.40) From Lemma 5.4 it follows, without loss of generality, that we can assume that the function ϕ is defined on the set [0,1] × [−N,N] 3 for some N > 0, hence it is uniformly continuous.From the mean value theorem, it follows that (5.42) Using Lemma 5.2 and condition (ϕ 2 ) we can estimate for some i, 0 ≤ i ≤ l.
Taking into account the uniform continuity of ϕ and estimate (5.41), given ε > 0, we can find δ > 0 such that We want to prove now that the multimap G is locally (k V (x) ,α C 2 ,α C )-bounded on Q with k V (x) ≤ C α ( f (x)) for every x ∈ Q.In order to get this result we need first to demonstrate the following statement.

.42) 4 . Coincidence index for noncompact triplets 4 . 1 .
Coincidence index for fundamentally restrictible triplets.Let f : The entire space E and coG(S) are natural examples of fundamental sets of ( f ,G,S).Valeri Obukhovskii et al. 305 Definition 4.3.A triplet ( f ,G,S) is called fundamentally restrictible if there exists a fundamental set T such that the restriction G | f −1 (T) is compact.Such fundamental set is called supporting.Definition 4.4.A triplet (

. 13 )
The Kuratowskii MNC α(Ω)=inf d >0:Ω has a finite partition with sets of diameter less than d .(4.14) Let M ⊆ Y be a bounded set; f : M → E a map; G : M → Kv(E ) a multimap; β a MNC in E .