A Birkhoﬀ–Kellogg type theorem for discontinuous operators with applications

. By means of ﬁxed point index theory for multivalued maps, we provide an analogue of the classical Birkhoﬀ–Kellogg Theorem in the context of discontinuous operators acting on aﬃne wedges in Banach spaces. Our theory is fairly general and can be applied, for example, to eigenvalues and parameter problems for ordinary diﬀerential equations with discontinuities. We illustrate in details this fact for a class of second order boundary value problem with deviated arguments and discontinuous terms. In a speciﬁc example, we explicitly compute the terms that occur in our theory.


Introduction
The celebrated invariant-direction Theorem due to Birkhoff and Kellogg [2] is an abstract existence result that, roughly speaking, gives conditions for the existence of a "nonlinear" eigenvalue and eigenvector for compact maps in normed linear spaces.Among its various extensions, one is set in cones and is due to Krasnosel'ski ȋ and Ladyženskiȋ [20].These classical functional analytic tools find applications e.g. to eigenvalue problems for ODEs and PDEs (see for example the book [1], the recent papers [18,19] and references therein); typically, the methodology in this context is to reformulate the given boundary value problem as a fixed point problem in a suitable Banach space.
Recently the first two authors developed a Birkhoff-Kellogg type theorem in the framework of affine cones (cf.[4], see also [3,6,9]).The motivation for this new type of results is that the setting of affine cones seems to be helpful when dealing with equations with delay effects.A key ingredient in [4] is the continuity of the involved operator.On the other hand, there has been recently a rising attention towards discontinuous differential equations, that occur when modelling real world phenomena.Here we mention the classical books by Filippov [13], Carl and Heikkilä [7], and Heikkilä and Lakshmikantham [17] and the more recent book by Figueroa, Pouso and RL [12].
In the present paper we provide a discontinuous version of the Birkhoff-Kellogg type result in the setting of affine wedges in Banach spaces, see Theorem 2.13 below.The proof of Theorem 2.13 is based on the fixed point index theory for discontinuous operators developed in [12].We stress that a crucial point in the construction of the index for discontinuous operators is its equivalence with the corresponding one of a suitable multivalued map, for which it is already defined, see [14].Note that this newly constructed topological tool for discontinuous operators inherits the key properties of the classical one.This construction is sketched in Section 2 for completeness.
In Section 3 we illustrate the applicability of our results to boundary value problems, see Theorem 3.4.
In more details, we consider the following second order parameter-dependent differential equation with deviated argument with initial condition and the final homogeneous boundary condition where λ ≥ 0 is a parameter, r ≥ 0, σ : ) may be discontinuous with respect to the second argument in an appropriate sense.We employ a concept of admissible discontinuity curve as in [12].We conclude the paper by illustrating the applicability of our theory by means of a toy model with delay, see Example 3.9.
As far as we know, our results extend and complement the previous literature.This is highlighted in more details in Remarks 2.12, 2.15 and 3.10.
2 Birkhoff-Kellogg type results via fixed point index theory

On fixed point index theory for discontinuous operators
Let K be a nonempty closed and convex subset of a real Banach space (X, • ), U ⊂ K a relatively open subset and T : U ⊂ K −→ K a mapping, not necessarily continuous.

Definition 2.1
The closed-convex envelope of an operator T : U ⊂ K −→ K is the multivalued mapping where B ε (x) denotes the closed ball centered at x and radius ε, and co means closed convex hull.
Example 2.2 1.Consider the real function T : R → R defined as T (x) = x, if x ≤ 0, and 2. The closed-convex envelope of any continuous map T is equal to T .Now we recall some useful properties of closed-convex envelopes (cc-envelopes for short) and the definition of the fixed point index that we will employ throughout this paper.The reader is referred to [11,12] for details.
Proposition 2.3 Let T be the cc-envelope of an operator T : U −→ K. Then the following properties hold: 1.If T : U −→ 2 X is an upper semicontinuous (usc) operator which assumes closed and convex values and T x ∈ Tx for all x ∈ U , then Tx ⊂ Tx for all x ∈ U ; 2. If T maps bounded sets into relatively compact sets, then T assumes compact values and it is usc; 3. If T U is relatively compact, then T U is relatively compact.
The fixed point index for a not necessarily continuous operator T was introduced in [10] by using the degree theory developed in [11] and a retraction trick, just as in the classical case.Both topological degree and fixed point index theories are based on the available results for the multivalued cc-envelope T.
Definition 2.4 Let T : U ⊂ K −→ K be an operator such that T U is relatively compact, T has no fixed points on ∂ U and where T is the cc-envelope of T .
We define the fixed point index of T in K over U as where r is a continuous retraction of X onto K and deg is the degree introduced in [11].
Remark 2.5 Note that condition (2.5) means that the set of fixed points of T (i.e., the set of points x such that x ∈ Tx) is contained in the set of fixed points of T .This is a weaker condition than the continuity of T ; indeed, if T is continuous, then Tx = {T x} for all x ∈ U and thus (2.5) is trivially satisfied.
We now recall a useful proposition from [10] that relates the fixed point index of the discontinuous operator T with that of its associated multivalued mapping T.
Proposition 2.6 [10, Proposition 2.12] Let T be a mapping that satisfies the conditions of Definition 2.4.
Then, the fixed point index of T is such that where the right-hand index is the fixed point index defined for multivalued mappings, see [14].
As a straightforward consequence of the fixed point index theory for usc multivalued mappings, the following properties can be derived (see [12]).

Birkhoff-Kellogg theorem and discontinuous operators
The following notions will be used along the text.A closed convex subset K of a Banach space (X, • ) is a wedge if µ x ∈ K for every x ∈ K and for all µ ≥ 0. Furthermore, if a wedge K satisfies that K ∩ (−K) = {0}, then it is said to be a cone.A cone K induces the partial order in X given by u v Let K be a wedge of a Banach space (X, • ).For a given y ∈ X, the translate of the wedge K is defined as follows Given an open bounded subset D ⊂ X with 0 ∈ D, we will denote D Ky := (y + D) ∩ K y , which is a relatively open subset of K y .By D Ky and ∂ D Ky we will mean, respectively, the closure and the boundary of D Ky relative to K y .
For the convenience of the reader, we recall here the classical Birkhoff-Kellogg Theorem [2] and a variant of it set in cones.The latter result is due to Krasnosel'ski ȋ and Ladyženskiȋ [20] (see also [16,Theorem 2.3.6]).Theorem 2.9 (Krasnosel'skiȋ-Ladyženskiȋ) Let X be a real Banach space, U ⊂ X be an open bounded set with 0 ∈ U , K ⊂ X be a cone, T : K ∩ U −→ K be compact and suppose that Then there exist In the context of affine cones a Birkhoff-Kellogg type result was recently proved in [4, Theorem 2].It reads as follows.
Theorem 2.10 Let (X, • ) be a real Banach space, K ⊂ X be a cone and D ⊂ X be an open bounded set with y ∈ D Ky .Assume that T : D Ky −→ K is a compact map and consider the operator Assume that there exists λ > 0 such that i Ky (T (y, λ) , D Ky ) = 0. Then there exist x * ∈ ∂ D Ky and λ * ∈ (0, λ) Now we present a discontinuous version of this Birkhoff-Kellogg type result in affine wedges.
Theorem 2.11 Let D ⊂ X be an open bounded set with 0 ∈ D, y ∈ X be fixed and K be a wedge.Assume that T : D Ky −→ K is a mapping such that T D Ky is relatively compact and consider the operator Moreover, assume that there exists λ > 0 such that i Ky (T (y, λ) , D Ky ) = 1 and for each λ ∈ (0, λ], where T (y,λ) denotes the cc-envelope of T (y,λ) .
Proof.If T (y,λ) has a fixed point on ∂ D Ky for some λ ∈ (0, λ) we are done.Otherwise, suppose that to the context of discontinuous operators, but also an improvement in the continuous case, since the conditions on the index are weakened and the result is extended to the setting of wedges.
We now prove a result in the setting of normal cones which can be of a more direct applicability due to the use of the norm, as in the classical Birkhoff-Kellogg Theorem.
Theorem 2.13 Let K ⊂ X be a normal cone with normal constant c > 0 in a Banach space X, D ⊂ X be an open bounded set with 0 ∈ D and y ∈ X be fixed.Assume that T : D Ky −→ K is a mapping such that T D Ky is relatively compact and If there exists a positive number such that the operator T (y,λ) satisfies condition (2.6) for each λ ∈ (0, λ], then there exist x * ∈ ∂ D Ky and Proof.We shall show that i Ky (T (y, λ) , D Ky ) = 0 and so the conclusion is obtained as a consequence of Theorem 2.11.
Indeed, suppose that there exist x 1 ∈ ∂ D Ky , v ∈ Tx 1 and β 0 ≥ 0 such that Observe that x 1 − y ∈ ∂ D and so x , a contradiction with the choice of λ.
On the other hand, since D Ky and T D Ky are bounded, there exists β > 0 such that x / ∈ y + λ Tx + β x 0 for all x ∈ D Ky .

⊓ ⊔
The following corollary can be seen as an analogue of the classical result of Krasnosel'ski ȋ and Ladyženskiȋ.
Corollary 2.14 Let K ⊂ X be a normal cone in a Banach space X and D ⊂ X be an open bounded set with y ∈ D Ky .Assume that T : D Ky −→ K is a mapping such that T D Ky is relatively compact and, for each λ > 0, the operator T (y,λ) satisfies condition (2.6).If then there exist x * ∈ ∂ D Ky and λ * > 0 such that x * = y + λ * T (x * ).
Remark 2.15 Note that, in the non-affine case, Corollary 2.14 extends Theorem 2.9 within the setting of discontinuous operators in normal cones.We stress that, in the non-affine case, Corollary 2.14 can also be deduced as a consequence of the multivalued generalization of the Birkhoff-Kellogg theorem given in [15].

Applications
Consider the second order parameter-dependent differential equation with initial conditions of the form and the final homogeneous boundary condition where λ is a positive parameter, r ≥ 0, and σ : ) may be discontinuous with respect to the second argument in a sense which will be specified later.
In order to study the problem (3.7)-(3.9),we shall use a superposition principle as in [5].To do so, first consider the Dirichlet BVP whose unique solution is given by where G is the corresponding Green's function.It is well-known that and, moreover (see [21]), with Φ(s) := s(1 − s).Associated to the Green's function, we consider the kernel k : On the other hand, note that the function ŷ(t) = 1 − t solves the Dirichlet BVP so we define the function which will be the vertex of our affine cone.
In order to apply the theory of the previous section, we will work in the Banach space of continuous functions X = C([−r, 1]), endowed with the usual sup-norm, • [−r,1] , and the cone Observe that K is a normal cone with normal constant c = 1 and that u [−r,1] = u [0,1] for all u ∈ K.
Now, for the vertex y defined in (3.11), we consider the translate of the cone K given by and, for each ρ > 0, we denote by K y,ρ the relatively open bounded set We will look for solutions of the following perturbed Hammerstein integral equation Before doing so, we need to define the type of regions where f is allowed to be discontinuous.The concept of admissible discontinuity curve used here has been widely employed in [12].
Definition 3.2 A λ-admissible discontinuity curve for the second-order parameter dependent differential Indeed, one may check that condition (3.13) holds with Let us now state and prove the main result of this Section.
Proof.Let us divide the proof in several steps: Step 1.The operator T , defined in (3.12), maps the set K y,ρ into the cone K and, moreover, T K y,ρ is relatively compact.
First, let u ∈ K y,ρ be arbitrarily fixed and let us show that T u ∈ K.By definition, The continuity of the kernel k, jointly with hypothesis (H 1 ), (H 2 ) and the constant sign of f and k, imply . Moreover, since k(t, s) = 0 for all t ≤ 0, we have that T u(t) = 0 for all as a consequence of the properties of the Green's function G stated above.In conclusion, T u ∈ K.
On the other hand, the compactness of the set T K y,ρ follows from assumption (H 2 ) and the continuity of the kernel k, combined with a careful use of the Arzelà-Ascoli theorem (see [22]).
Let us prove that T is continuous at u, which implies that T(u) = {T (u)} and thus condition (3.15) holds for such u.Indeed, in this case we have that for a.a.
which implies, due to Lebesgue's dominated convergence theorem, that In this case, one can show that u / ∈ y + λ T(u), which implies that condition (3.15) holds for such u.The proof is based on condition (H 4 ) and the fact that the function γ n is a λ-admissible discontinuity curve for the problem.It can be replicated following the reasoning in the proof of Proposition 4.7, Case 2, in [10].
Proof.It follows from Theorem 3.4 together with Remark 3.3.

⊓ ⊔
Consider the special case of (3.7) where the nonlinearity can be seen as a discontinuous perturbation of a Carathéodory function, that is, where g : For each λ ∈ (0, λ] with λ > ρ/ δ, each function γ n is a λ-admissible discontinuity curve and each function Γ j satisfies that ) satisfies either of the following conditions: there exists Then there exist λ ρ ∈ (0, λ) and u ρ ∈ ∂ K y,ρ that satisfy the integral equation (3.12), that is, they solve the problem (3.7)-(3.9).
Proof.It follows in line of the proof of Theorem 3.4 as a consequence of Theorem 2.13.Observe that it suffices to rewrite Step 2. Let us prove that for each λ ∈ (0, λ], the operator y + λ T satisfies that {u} ∩ {y + λ T(u)} ⊂ {y + λ T (u)} for every u ∈ K y,ρ , where λ is fixed by assumption (D).
Fix arbitrary λ ∈ (0, λ] and u ∈ K y,ρ .Now, consider three different cases: ) and so T is continuous at u.
Case 2. m ({t ∈ I j : u(σ(t)) = Γ j (t)}) > 0 for some j ∈ N. Let us prove that u / ∈ y + λ Tu, which can be justified as in the proof of [10,Proposition 4.7], but we include the reasoning here again for completeness.

Theorem 2 . 8 (
Birkhoff-Kellogg)  Let U be a bounded open neighborhood of 0 in an infinite-dimensional normed linear space X, and T : ∂ U −→ X a compact map satisfying T x ≥ α > 0 for all x ∈ ∂ U .Then there exist x 0 ∈ ∂ U and λ 0 > 0 such that x 0 = λ 0 T x 0 .