Compression–Expansion Fixed Point Theorems for Decomposable Maps and Applications to Discontinuous (cid:2) -Laplacian problems

In this paper, we prove new compression–expansion type ﬁxed point theorems in cones for the so-called decomposable maps, that is, compositions of two upper semicontinuous multivalued maps. As an application, we obtain existence and localization of positive solutions for a differential equation with φ -Laplacian and discontinuous nonlinearity subject to multi-point boundary conditions. As far as we are aware, the existence results are new even in the classical case of continuous nonlinearities


Introduction
We are concerned with the existence of positive solutions to the following multi-point boundary value problem involving the φ-Laplacian In recent years, a lot of attention have been paid to the existence of solutions to boundary value problems with φ-Laplacian (see, for instance, [5,9,16,18,19,27] and the references therein). Here, we study in a unified way the classical homeomorphism φ : R → R, the singular homeomorphism φ : (−a, a) → R and the bounded one φ : R → (−b, b).
As usual in the related literature when f is discontinuous (see [18,19]), we consider the following regularized problem (1.2) where F is an usc multivalued map with closed convex values. In order to prove the existence of positive solutions to (1.2), by means of a fixed point approach, it appears the fixed point problem where is a nonlinear continuous map and is an usc multivalued map with closed convex values. Since the values of the composition • can be non-convex, the classical generalization of Krasnosel'skiȋ's fixed point theorem in cones for usc multivalued maps [13] is not applicable here.
To overcome this difficulty we prove a new version of the compression-expansion fixed point theorem in cones for decomposable maps, that is, compositions of two usc multivalued maps which cover the fixed point problem above. Our approach is based on a suitable computation of the fixed point index for this class of maps, which was developed in [27]. Some previous results on fixed point theory for decomposable maps can be found in the pioneering paper [24] and in [11].
Once the existence of positive solutions for the regularized problem (1.2) is established, we wonder whether such solutions are also Carathéodory type solutions for (1.1). This question is not new and was studied, for instance, in case of systems of first-order equations in [10], second-order BVPs in [6] or reaction-diffusion equations in [20]. Here a transversality condition on the discontinuities of f is imposed in order to prove that all the solutions of (1.2) are in fact solutions of (1.1). Roughly speaking, the function f may be discontinuous over the graphs of a countable number of the so-called admissible curves.
To our best knowledge, the main existence result for problem (1.1) is new even in the classical case of a continuous nonlinearity. It is based on a new Harnack type inequality for supersolutions of (1. 1), what enables us to include the boundary value problem (1.1) between the class of problems which can be studied by means of the general technique developed in [14].
Finally, we shall discuss the existence of positive solutions for problem (1.1) under asymptotic conditions on f at 0 and/or infinity, which were inspired by those in [4]. If φ is singular, we prove that (1.1) has at least one positive solution provided that f is superlinear at 0 with respect to φ, that is, In the case of a classical φ, if (1.3) holds and, moreover, f is sublinear at infinity with respect to φ, i.e., then the same conclusion is obtained. Note that, unlike [9] or [15], no monotonicity assumptions on f are required.

Compression-Expansion Fixed Point Theorem for Decomposable Maps
In the recent paper [27], the authors define a fixed point index theory for the composition of two multivalued maps. We recall here its definition and its main properties. Let X and Y be Banach spaces and K X ⊂ X , K Y ⊂ Y be closed convex sets. We are interested into the class of decomposable maps, i.e., multivalued maps T : → 2 K X , where ⊂ K X is open in K X , which can be represented as a composition T = • of two multivalued maps and with the following properties: : K Y → 2 K X is usc with compact convex values.

Definition 2.1
Let and be two multivalued maps satisfying conditions (i), (ii) and such that x / ∈ ( • )(x) for all x ∈ ∂ K X . The fixed point index for the pair of maps ( , ) over with respect to K := K X × K Y is defined as Note that the fixed point index over × K Y with respect to K for the map , i K ( , × K Y ), is well-defined according to the Fitzpatrick-Petryshyn degree theory for multivalued maps [13].
In the sequel we need the following definition. A closed convex subset K of a Banach space X is a cone if it satisfies that K ∩ (−K ) = {0} and λx ∈ K for every x ∈ K and for all λ ≥ 0. Now we prove some sufficient conditions for guaranteeing that the fixed point index for decomposable maps defined above is 0 or 1 in certain open subsets of a cone. Proposition 2.1 Let X and Y be Banach spaces and K X ⊂ X, K Y ⊂ Y cones in X and Y , respectively. Let be a relatively open subset of K X and let T = • : −→ 2 K X be a multivalued operator such that and fulfill conditions (i) and (ii).

Proposition 2.2
Let X and Y be Banach spaces and K X ⊂ X, K Y ⊂ Y cones in X and Y , respectively. Let be a relatively open and bounded subset of K X and let T = • : −→ 2 K X be a multivalued operator such that and fulfill conditions (i) and (ii).
As a consequence of the previous theory, we generalize to this context the Krasnosel'skiȋ's compression-expansion fixed point theorem in cones, see [1,17]. Theorem 2.2 Let X and Y be Banach spaces and K X ⊂ X, K Y ⊂ Y cones in X and Y , respectively. Let 1 and 2 be two relatively open and bounded subsets of K X , with 0 ∈ 1 ⊂ 1 ⊂ 2 , and let T = • : 2 −→ 2 K X be a multivalued operator such that (i) : 2 → 2 K Y is usc with closed convex values and relatively compact range; Assume that T satisfies either of the following two conditions:

Then T has at least a fixed point in
Otherwise, we have that λx / ∈ T x for all x ∈ ∂ K X 1 and λ ≥ 1, which based on Proposition 2.1 implies ind K ( , , 1 ) = 1, and that x / ∈ T x+μw for all x ∈ ∂ K X 2 and μ ≥ 0, and thus by Proposition 2.2 we obtain that ind K ( , , 2 ) = 0.
Therefore, as a consequence of the additivity property, ind K ( , , 2 \ 1 ) = −1 and so the conclusion is deduced from the existence property of the fixed point index. The reasoning is similar in case that (2) holds.
We note that 1 and 2 are arbitrarily open bounded subsets of a cone instead of just intersections of open balls and cones, which enlarges the applicability of the previous result, cf. [11].

Remark 2.1
In particular, if X = Y , K X = K Y and = I , where I is the identity map in K X , then Theorem 2.2 is the well-known compression-expansion fixed point theorem in cones for usc maps, see [13].

Remark 2.2
In Theorem 2.2 we present an extension of the well-known Krasnosel'skiȋ's fixed point theorem in cones, but clearly we can adapt much more fixed point theorems (all those whose proof is just based on fixed point index properties) to the context of decomposable maps. This is the case, for instance, of Leggett-Williams' fixed point theorem [22] and its generalizations [2,3].

Positive Solutions to a Multi-point BVP Involving the -Laplacian
We consider the following boundary value problem with φ-Laplacian subject to multipoint boundary conditions By a Carathéodory solution of (3.1) we mean a function u In the sequel, the space of continuous functions C(I ) will be endowed with the usual supremum norm · ∞ and the cone of nonnegative continuous functions will be denoted by P.
Let us assume that the function f : I × [0, ∞) → [0, ∞) satisfies the following basic conditions: Now we present the integral formulation of problem (3.1). By integration of the differential equation, Then, integrating from t to 1, we obtain In particular, evaluating at t = η i , and so, taking into account the condition Hence, we define the operator N : whose fixed points correspond with solutions to (3.1).
Since f is discontinuous in both variables, the operator N is not continuous and thus we transform (3.1) into a multi-point boundary value problem with a nonlinear differential inclusion Equivalently, F can be rewritten as Note that the solutions of (3.2) are usually called Krasovskii solutions of (3.1). To solve (3.2), it suffices to consider the operator inclusion and N F denotes the Nemytskii operator We shall need the following result concerning the upper semicontinuity of the Nemytskii operator (see [8,25], for details).

Lemma 3.1 Assume that the function f
Then the Nemytskii operator N F : P → 2 L 1 (I ) is usc.
In order to apply the compression-expansion fixed point theorem in cones, we need the following Harnack type inequality. Its proof is adapted from the ideas in [9,15]. (−a, a) for every t ∈ I , φ(u ) ∈ W 1,1 (I ) and (φ(u )) ≤ 0 on I , one has the following inequality:
Let us fix an arbitrary i ∈ {1, 2, . . . , n}. By its concavity, the graph of u restricted to [0, η i ] lies under the line containing the points (1, u(1)) and (η i , u(η i )). Thus, In the sequel, let us denote α := α n and η := η n . We shall prove the existence of positive solutions to (3.1) in the following subcone of P: In order to apply Theorem 2.2, we let X = Y = C(I ), K X = K and In the following result, we show that the operator T = • is a multivalued decomposable map, what justifies the application of the theory developed in Sect. 2. (H 1) and (H 2).

Theorem 3.1 Assume that f satisfies conditions
Then the operators is usc with closed convex values and maps bounded sets into relatively compact sets; and is a single-valued continuous operator.
Take an arbitrary function v ∈ K Y and let u := v, then u ∈ P. Moreover, φ(u ) = −v, and thus φ(u ) is nonincreasing.
In addition, as a linear operator from L 1 (I ) to C (I ) is compact and, by Lemma 3.1, N F is usc from the topology of C(I ) to that of L 1 (I ). Hence, is usc and maps bounded sets into relatively compact sets. Also, has closed and convex values, see [26,Theorem 2].
Finally, the continuity and the compactness of the operator are standard consequences of Lebesgue's dominated convergence theorem and Ascoli-Arzela's theorem. Now we present some sufficient conditions about f for guaranteeing the conecompression or cone-expansion conditions for the operator T = • on the boundary of two nested neighborhoods of the origin.
Let us introduce some notations. For r > 0 and ε ∈ (0, r ), denote We also make use of the following relatively open set of the cone K , which is similar to that introduced in [21]. Notice that B r ⊂ V r ⊂ B r /c , where B ρ := {u ∈ K : u ∞ < ρ}.

Lemma 3.3 Assume that f satisfies conditions (H 1) and (H 2).
If there exist r > 0 and ε > 0 such that

4)
then we have that λu / ∈ T u for all u ∈ K , u ∞ = r , and all λ > 1.
Proof Let us show that v ∞ ≤ r for all v ∈ T u and u ∈ K with u ∞ = r , which implies that λu / ∈ T u for all λ > 1 and u ∈ K with u ∞ = r .
Assume to the contrary that there exist v ∈ T u and u ∈ K with u ∞ = r such that r < v ∞ . Observe that if w ∈ N F (u) and u ∞ = r , by the definition of the regularized multivalued map F, we have Finally, inequality (3.4) yields the contradiction r < v ∞ ≤ r .

Remark 3.1 Of course, condition (3.4) holds for r large enough if φ is singular. Therefore, in this case, problem (3.2) is always solvable under the basic assumptions (H 1)
and (H 2). Moreover, it is worth mentioning that if φ is singular and f is continuous, then problem (3.1) has a nonnegative (maybe trivial) solution without additional hypotheses, as a consequence of Proposition 2.1. This is similar to what happens for the Dirichlet problem, see [5].

5)
then u / ∈ T u + μw for all u ∈ ∂ K V r and all μ > 0 with w ≡ 1.
Proof Suppose that there exist u ∈ ∂ K V r and μ > 0 such that u ∈ T u + μw. Since by definition T = • • N F , then there exists v ∈ N F (u) such that Notice that if v ∈ N F (u) and u ∈ ∂ K V r , by the definition of the regularized multivalued map F, we have Therefore, since 0 < η i ≤ η < 1 for every i ∈ {1, . . . , n − 1}, we have for t ∈ [0, η], Hence, condition (3.5) implies that u(t) ≥ r + μ for t ∈ [0, η]. Taking the infimum in [0, η], we get the contradiction r ≥ r + μ.
Now we are in a position to prove the existence of positive solutions to the inclusion problem (3.2) based on the previous lemmas and the compression-expansion fixed point theorem in Sect. 2. (H 1) and (H 2). Moreover, assume that there exist r 1 , r 2 > 0 and ε ∈ (0, r 2 ) such that

Theorem 3.2 Assume that f satisfies conditions
(3.6) • If r 1 < r 2 , then problem (3.2) has one positive solution u such that r 1 ≤ u ∞ ≤ r 2 /c. • If r 2 /c < r 1 , then problem (3.2) has one positive solution u such that r 2 ≤ u ∞ ≤ r 1 .
Proof By Lemmas 3.3 and 3.4 we obtain, thanks to condition (3.6), that λu / ∈ T u for all u ∈ ∂ K B r 1 and all λ > 1, and u / ∈ T u + μ for all u ∈ ∂ K V r 2 and all μ > 0.
First, if r 1 < r 2 , it follows that 0 ∈ B r 1 ⊂ B r 1 ⊂ V r 2 and thus Theorem 2.2 ensures that the operator T has at least one fixed point in V r 2 \B r 1 . Now, the fact that V r 2 ⊂ B r 2 /c gives the conclusion.
On the other hand, in case that r 2 /c < r 1 , we have that 0 ∈ V r 2 ⊂ V r 2 ⊂ B r 1 and as a consequence of Theorem 2.2 we obtain that T has at least one fixed point in B r 1 \V r 2 ⊂ B r 1 \B r 2 .

Remark 3.2
We use the open set V r in Lemma 3.4 to prove the index zero result in Proposition 2.2. As explained in [17], the requirement about the growth condition of f is less stringent than if the set B r /c is employed instead.

Remark 3.3 Observe that condition (3.6) holds if there exists two constants
, as required in [16].
The existence of positive solutions for the differential inclusion (3.2) is meaningful itself when f is a discontinuous function, see [7,18,19]. However, we are concerned with the existence of positive Carathéodory type solutions for (3.1) and so we need some additional condition about f which implies that all the solutions of (3.2) are in fact Carathéodory solutions of (3.1). It is given by the notion of admissible discontinuity curves presented in the following definition, which can be traced back to [23] (see also [12,25]).
The transversality condition (3.7) is the key ingredient to show that all solutions of (3.2) are Carathéodory solutions of (3.1). The proof of this result can be looked up in [27,Lemma 3.3]. Now, as a straightforward consequence of Theorem 3.2 and Lemma 3.5, we derive the existence of positive solutions for problem (3.1). (H 1), (H 2) and (H 3). Moreover, assume that there exist r 1 , r 2 > 0 and ε ∈ (0, r 2 ) such that
We illustrate the applicability of our result with the following example.

Example 3.1 Consider the problem
where [x] denotes the integer part of x.
Note that the function f satisfies that Hence inequalities (3.6) hold, for instance, with r 1 = 4/5, r 2 = 1/20 and ε small enough. Because of this localization, it suffices to define f in I × [0, 1]. Moreover, the function f is discontinuous over the graphs of a countable number of lines, which are those given by They are admissible discontinuity curves in the sense of Definition 3.1 since (φ(γ n )) ≡ 0 for all n ∈ N and thus inequality (3.8) is trivially satisfied for any δ ∈ (0, 1/4) and any > 0. This implies that f satisfies condition (H 3). Then Theorem 3.3 ensures that problem (3.10) has a positive solution u such that We highlight that, as far as we are aware, the previous existence result is new even in the case of continuous nonlinearities. Note that in this case, since F(t, x) = { f (t, x)}, the positive number ε can be omitted in conditions (3.4) and (3.5).

Remark 3.4
Obviously, from Theorem 3.3, one can obtain multiplicity results for problem (3.1) provided that there exist several pairs of numbers (r 1 , r 2 ) satisfying conditions (3.4) and (3.5), respectively.
Finally, we show that problem (3.1) has at least one positive solution under suitable asymptotic conditions. Next results are inspired by those in [4], but our arguments are slightly different since they rely on Krasnosel'skiȋ's type fixed point theorem in cones.

Proposition 3.1 Assume that f satisfies conditions (H 1), (H 2) and that
Then there exists r > 0 such that u / ∈ T u + μw for all u ∈ ∂ K V r and all μ > 0 with w ≡ 1.
If φ is singular (i.e., a < +∞, b = +∞), then problem (3.1) has at least one positive solution provided that f is superlinear at 0 with respect to φ. A remarkable particular problem is that given by the relativistic operator, namely, Then problem (3.14) has at least one positive solution.
Proof The conclusion follows from Theorem 3.4 with the homeomorphism φ : and that, in this case, condition (3.15) implies (3.11).
Unlike the singular case, if φ is a classical homeomorphism (i.e., a = b = +∞), then some additional condition to the asymptotic behavior of f at 0 is needed to ensure the existence of a positive solution for (3.1). To this end, it is enough to assume that f is sublinear at infinity with respect to φ.

Proposition 3.2 Assume that f satisfies conditions (H 1), (H 2) and that
= 0 uniformly with t ∈ I (3. 16) and φ is a classical homeomorphism such that Then there exists R > 0 such that λu / ∈ T u for all u ∈ K , u ∞ = R, and all λ > 1.
Proof First, note that (3.17), with τ = 2 1 − α i /3, implies that there exist positive numbers ξ and ρ such that Next, by (3.16), for each L > 0 there exists m > 0 such that We can choose L > 0 small enough such that 2 L ξ ≤ 1. Since φ : R → R is an increasing unbounded homeomorphism, we can also take R > 0 large enough (suppose R > ρ) so that m ≤ Lφ(3R/2). Hence, we have that f (t, x) ≤ 2 Lφ 3 2 R for all t ∈ I and all x ∈ 0, 3 2 R .
From the choice of L and inequality (3.18), we obtain that f (t, x) ≤ φ 1 − α i R for all t ∈ I and all x ∈ 0, 3 2 R .
Therefore, condition (3.4) holds with r = R and ε = R/2, so the conclusion is obtained by application of Lemma 3.3. Proof Propositions 3.1 and 3.2 allow us to deduce, by application of Theorem 2.2, that problem (3.2) has at least one positive solution which belongs to B R \V r . Finally, Lemma 3.5 implies that it is also a Carathéodory solution for (3.1).
As a consequence, in the classical p-Laplacian case, the existence of one positive solution is ensured provided that f is superlinear at 0 and sublinear at infinity with respect to φ p (x) = |x| p−2 x, p > 1.
Then problem (3.19) has at least one positive solution.
Funding Jorge Rodríguez-López was partially supported by Xunta de Galicia ED431C 2019/02. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.
Availability of data and material Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.