A Schauder-type theorem for discontinuous operators with applications to second-order BVPs

We prove a new fixed point theorem of Schauder-type which applies to discontinuous operators in non-compact domains. In order to do so, we present a modification of a recent Schauder-type theorem due to Pouso. We apply our result to second-order boundary value problems with discontinuous nonlinearities. We include an example to illustrate our theory.


Introduction
In the recent and interesting paper [9], Pouso proved a novel version of Schauder's theorem for discontinuous operators in compact sets. Pouso used this tool to prove new results on the existence of solutions of a widely studied second order ordinary differential equation (ODE) subject to Dirichlet boundary conditions (BCs), namely In this manuscript, we further develop the ideas of Pouso. Firstly we prove that a Schaudertype theorem for discontinuous operators can be formulated for arbitrary nonempty, closed and convex (not necessarily bounded) subsets of a Banach space. Secondly we apply our new result to prove the existence of solutions of a large class of discontinuous second order ODE subject to separated BCs, complementing the results of [9] and improving them also in the special case of Dirichlet BCs.

Schauder's fixed point theorem for discontinuous operators
For the sake of completeness, we begin this Section by recalling the classical Schauder's fixed point theorem.
Theorem 2.1. [10, Theorem 2.A] Let K be a nonempty, closed, bounded, convex subset of a Banach space X and suppose that T : K −→ K is a compact operator (that is, T is continuous and maps bounded sets into precompact ones). Then T has a fixed point.
A well-known consequence of Theorem 2.1 is the following.
Corollary 2.2. [10, Corollary 2.13] Let K be a nonempty, compact and convex subset of a Banach space X and T : K −→ K a continuous operator. Then T has a fixed point.
The main result in [9] is an improvement of Corollary 2.2, where the continuity of the operator T is replaced by a weaker assumption. We briefly describe the main idea: given a compact subset K of a Banach space X and an operator T : K −→ K that can be discontinuous, it is possible to construct a multivalued mapping T by 'convexifying' T as follows: where B ε (u) denotes the closed ball centered in u and radius ε, and co denotes the closed convex hull. The operator T in (2.1) is an upper semi-continuous mapping with convex and compact values (see [2], [6]), and therefore Kakutani's fixed point theorem guarantees that T has a fixed point in K. If we impose and extra assumption that, roughly speaking, states that a fixed point of T must be a fixed point of T , then we obtain the desired result.
The following characterisation sheds light on the definition of the multivalued operator T.
It is formulated for compact subsets, but it works for arbitrary nonempty subsets of a Banach space (see also [9, Proposition 3.2]). Proposition 2.3. Let K be a compact subset of a Banach space X and T : K −→ K. Then the following statements are equivalent: (1) y ∈ Tu, where T is as in (2.1); (2) for every ε > 0 and every ρ > 0 there exists a finite family of vectors u i ∈ B ε (u) ∩ K and coefficients λ i ∈ [0, 1] (i = 1, . . . , m) such that λ i = 1 and The variant of Schauder's theorem in compact subsets given by Pouso is the following.
(iii) T (u) is nonempty, closed and convex for all u ∈ M.
Then T has a fixed point.

Now we introduce the main result in this Section, which is an extension of Theorem 2.5
to the case of discontinuous operators. Since T(M) is relatively compact, we obtain by application of Theorem 2.6 that T has a fixed point. Finally, the condition (ii) implies that the obtained fixed point of T is a fixed

Second-order BVPs with separated BCs
In this Section we apply the previous abstract result on fixed points for discontinuous operators in order to look for W 2,1 -solutions for the following singular second-order ODE with separated BCs: where α, β, γ, δ ≥ 0 and Γ = γβ + αγ + αδ > 0.  [3] in order to look for extremal solutions for a functional problem with derivative dependence in the nonlinearity. As a main novelty of the present work, we allow the nonlinearity f to have a countable number of discontinuities with respect to its spatial variable and we require no monotonicity conditions. Moreover, the linear part can be singular.
To apply our new fixed point theorem to the BVP (3.1), we recall that a function u ∈ which is given by (see for example [7]) It is known [7] that k is non-negative. Furthermore note that k is continuous (and therefore  (H 3 ) the following estimate holds,  Proof. We have shown in Lemma 3.1 that T (B R ) ⊂ B R , therefore the set T (B R ) is totally bounded in X. Now, to see that T (B R ) is equicontinuous we only have to notice that for a.e. t ∈ I and every u ∈ B R it is which implies that Then T (B R ) is relatively compact in X.

⊓ ⊔
In a similar way as in Definition 4.1 of [9], we introduce the admissible discontinuities for our nonlinearities.
, is an admissible discontinuity curve for the differential equation u ′′ (t) + g(t)f (t, u(t)) = 0 if one of the following conditions holds: (ii) there exist ψ ∈ L 1 ([a, b], [0, +∞)) and ε > 0 such that If (i) holds then we say that γ is viable for the differential equation; if (ii) holds we say that γ is inviable.
The previous definition says, roughly speaking, that a time-dependent discontinuity curve γ is admissible if one of the following holds: either γ solves the differential equation on its domain or, if it does not, the solutions are pushed 'far away' from γ.
The following is the main result in this Section. Then we have that f (t, ·) is continuous for a.e. t ∈ I, and therefore if u k → u in B R we obtain f (t, x k (t)) → f (t, x(t)) for a.e. t ∈ I. This, joint with (H 2 ) and (H 3 ), imply that T u k converges uniformly to T u in X. Then, T is continuous at u and therefore we obtain Case 2 : there exists n ∈ N such that γ n is inviable and m({t ∈ I n : u(t) = γ n (t)}) > 0.
Therefore, assume that γ n satisfies (3.6) (the other case is similar), let ψ ∈ L 1 (I) and ε > 0 given by (3.6) and set Then we repeat the proof done in [9, Theorem 4.4] by taking there M(t) = g(t) H R (t) and it follows that u / ∈ Tu.
Case 3 : m({t ∈ I n : u(t) = γ n (t)}) > 0 only for some of those n ∈ N such that γ n is viable.
Again, it suffices to follow the referred proof by replacing f (t, x(t)) by g(t)f (t, u(t)) to obtain that, in this case, u ∈ Tu implies u = T u. Finally, we illustrate our results in an example.
Example 3.7. For n ∈ N we denote by φ(n) the function such that φ(1) = 2 and for n ≥ 2 φ(n) counts the number of divisors of n. Thus defined, φ(n) ≥ 2 for all n ∈ N, φ is not bounded and, as there are infinite prime numbers, lim inf n→∞ φ(n) = 2. Now we define the function (3.7) (t, u) ∈ (0, 1] × R −→f (t, u) = φ λ (n(t, u)), λ ∈ (0, 1), 7 where n(t, u) := We are concerned with the following ODE We claim that this problem has at least one solution. In order to show this, note that we can rewrite the ODE (3.8) in the form u ′′ (t) + g(t)f (t, u(t)) = 0, where g(t) = 1 √ t and f = −f ,f as in (3.7). We now show that the functions g and f satisfy conditions (H 1 )−(H 5 ).
First, it is clear that g ∈ L 1 (I) and so (H 1 ) holds. On the other hand, as φ(n) ≤ max{2, n} for all n ∈ N, we obtain that for each n ∈ N we have u ∈ [−n, n] ⇒ |f (t, u)| ≤ max{2, n} λ .
We can conclude that the differential equation (3.8) coupled with separated BCs has at least one solution in B R provided that M 1 +M 2 ≤ R 1−λ . Note that the solution is non-trivial since the zero function does not satisfy the ODE.