A Schaudertype theorem for discontinuous operators with applications to secondorder BVPs
 Rubén Figueroa^{1}Email author and
 Gennaro Infante^{2}
https://doi.org/10.1186/s136630160547y
© Figueroa and Infante 2016
Received: 4 December 2015
Accepted: 22 April 2016
Published: 30 April 2016
Abstract
We prove a new fixed point theorem of Schauder type, which applies to discontinuous operators in noncompact domains. In order to do so, we present a modification of a recent Schaudertype theorem of Pouso. We apply our result to secondorder boundary value problems with discontinuous nonlinearities. We include an example to illustrate our theory.
Keywords
MSC
1 Introduction
In this manuscript, we further develop the ideas of Pouso. First, 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. Second, we apply our new result to prove the existence of solutions of a large class of discontinuous secondorder ODEs subject to separated BCs, complementing the results of [1] and improving them also in the special case of Dirichlet BCs.
2 Schauder’s fixed point theorem for discontinuous operators
For completeness, we begin this section by recalling the classical Schauder fixed point theorem.
Theorem 2.1
([2], Theorem 2.A)
Let K be a nonempty, closed, bounded, convex subset of a Banach space X and suppose that \(T:K \longrightarrow K\) is a compact operator (that is, T is continuous and maps bounded sets into precompact ones). Then T has a fixed point.
A wellknown consequence of Theorem 2.1 is the following.
Corollary 2.2
([2], Corollary 2.13)
Let K be a nonempty, compact and convex subset of a Banach space X, and \(T:K \longrightarrow K\) a continuous operator. Then T has a fixed point.
The following characterization sheds light on the definition of the multivalued operator \(\mathbb {T}\). It is formulated for compact subsets, but it works for arbitrary nonempty subsets of a Banach space (see also [1], Proposition 3.2).
Proposition 2.3
 (1)
\(y \in \mathbb {T}u\), where \(\mathbb {T}\) is as in (2.1);
 (2)for every \(\varepsilon>0\) and every \(\rho>0\), there exists a finite family of vectors \(u_{i} \in B_{\varepsilon}(u) \cap K\) and coefficients \(\lambda_{i} \in[0,1]\) (\(i=1, \ldots,m\)) such that \(\sum \lambda_{i} =1\) and$$\Biggl\Vert y  \sum_{i=1}^{m} \lambda_{i} Tu_{i} \Biggr\Vert < \rho. $$
The variant of Schauder’s theorem in compact subsets given by Pouso is the following.
Theorem 2.4
([1], Theorem 3.1)
Theorem 2.4 is very interesting and powerful; however, when we want to look for solutions for a certain boundary value problem (BVP), the fact of working in a compact domain could be quite restrictive. In order to overcome this difficulty, we first recall that Theorem 2.1 admits the following extension to unbounded domains.
Theorem 2.5
([5], Theorem 4.4.10)
Let M be nonempty, closed and convex subset of a Banach space X, and \(T:M \longrightarrow M\) a continuous operator. If \(T(M)\) is precompact, then T has a fixed point.
Secondly, we recall the following result due to Bohnenblust and Karlin.
Theorem 2.6
([2], Corollary 9.8)
 (i)
\(T:M\to2^{M}\) is upper semicontinuous;
 (ii)
\(T(M)\) is relatively compact in X;
 (iii)
\(T(u)\) is nonempty, closed, and convex for all \(u \in M\).
Now we introduce the main result in this section, which is an extension of Theorem 2.5 to the case of discontinuous operators.
Theorem 2.7
 (i)
\(T(M)\) is relatively compact in X;
 (ii)
\(\{u\} \cap \mathbb {T}u \subset\{Tu\}\) for all \(u \in M\), where \(\mathbb {T}\) is as in (2.1).
Then T has a fixed point in M.
Proof
Since \(\mathbb {T}(M)\) is relatively compact, we obtain by application of Theorem 2.6 that \(\mathbb {T}\) has a fixed point. Finally, condition (ii) implies that the obtained fixed point of \(\mathbb {T}\) is a fixed point of T. □
Remark 2.8
Notice that if T is continuous then \(\mathbb {T}u =\{Tu\}\) for all u, and so previous results regarding operator \(\mathbb {T}\) actually generalize known results about singlevalued operators.
3 Secondorder BVPs with separated BCs
This kind of secondorder BVPs have received a lot of attention in the literature. For example, in the monograph [7] the method of lower and upper solutions is used to look for \(\mathcal {C}^{2}\)solutions in the case of continuous nonlinearities and \(W^{2,1}\)solutions in the case of Carathéodory ones. This method is also applied in [8] to a continuous φLaplacian problem with separated BCs. On the other hand, a monotone method is applied in [9] 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 function g can be singular.
Lemma 3.1
 (H_{1}):

\(g \in L^{1}(I)\);
 (H_{2}):

there exist \(R >0\) and \(H_{R} \in L^{\infty}(I)\) such that for a.e. \(t \in I\) and all \(u \in[R,R]\) we have \(f(t,u) \le H_{R}(t)\);
 (H_{3}):

the following estimate holds:where$$\H_{R}\_{L^{\infty}} (M_{1}+M_{2}) \le R, $$$$ M_{1}=\sup_{t \in I} \int_{0}^{1} k(t,s) \biglg(s)\bigr\,ds,\qquad M_{2}= \sup _{t \in I} \int_{0}^{1} \biggl\vert \frac{\partial k}{\partial t}(t,s) g(s) \biggr\vert \,ds; $$(3.4)
 (H_{4}):

for each \(u \in\overline{B}_{R}=\{u \in X : \u\ \le R\}\) the composition \(t \in I \longmapsto f(t,u(t))\) is a measurable function.
Remark 3.2
Since k is the Green’s function related to a homogeneous secondorder BVP, \(Tu \in W^{2,1}(I)\) for all u, so, in particular, \((Tu)'\) is absolutely continuous (then \(Tu \in X\)), and \((Tu)''\) exists almost everywhere on I. This will be used later in our argumentations.
Proof of Lemma 3.1
Let \(R>0\) given by condition (H_{2}). First, note that the kernel k has the form (3.3). Therefore, for each \(t \in[0,1]\), \(k(t,\cdot)\) is a continuous function, and for \(s\neq t\), the function \(s \in[0,1] \to\frac {\partial k}{\partial t}(t,s)\) is well defined and integrable. Then, conditions (H_{1}), (H_{2}) and (H_{4}) imply that for \(u \in \overline{B}_{R}\) the function Tu is well defined.
Lemma 3.3
Under the assumptions of Lemma 3.1, \(T(\overline{B}_{R})\) is relatively compact in X.
Proof
In a similar way as in Definition 4.1 of [1], we introduce the admissible discontinuities for our nonlinearities.
Definition 3.4
 (i)
\(\gamma''(t)=g(t) f(t,\gamma(t))\) for a.e. \(t \in[a,b]\);
 (ii)there exist \(\psi\in L^{1}([a,b])\), \(\psi>0\) almost everywhere, and \(\varepsilon>0\) such that$$\begin{aligned}& \begin{aligned}[b] &\mbox{either }{}\gamma''(t) + \psi(t) < g(t)f(t,y)\\ &\quad\mbox{for a.e. }t \in[a,b]\mbox{ and all } y \in\bigl[\gamma (t)\varepsilon, \gamma(t)+\varepsilon\bigr], \end{aligned} \end{aligned}$$(3.5)$$\begin{aligned}& \mbox{or }{}\gamma''(t)  \psi(t) > g(t)f(t,y) \\& \quad\mbox{for a.e. }t \in[a,b]\mbox{ and all }y \in\bigl[\gamma (t)\varepsilon, \gamma(t)+\varepsilon\bigr]. \end{aligned}$$(3.6)
The previous definition says, roughly speaking, that a timedependent 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.
To prove our main result on the existence of solutions for problem (3.1) by using admissible discontinuity curves, we need some auxiliary theoretical results on integrable functions. The reader can see their proofs in [1].
Lemma 3.5
([1], Lemma 4.1)
Let \(a,b \in \mathbb {R}\), \(a< b\), and let \(g, h \in L^{1}(a,b)\), \(g \ge0\) a.e., and \(h>0\) a.e. in \((a,b)\).
Corollary 3.6
([1], Corollary 4.2)
Let \(a,b \in \mathbb {R}\), \(a< b\), and let \(h \in L^{1}(a,b)\) be such that \(h>0\) a.e. in \((a,b)\).
Corollary 3.7
([1], Corollary 4.3)
If there exists \(M \in L^{1}(a,b)\) such that \(f'(t) \le M(t)\) a.e. in \([a,b]\) and also \(f_{n}'(t) \le M(t)\) a.e. in \([a,b]\) (\(n \in \mathbb {N}\)), then \(f'(t)=g(t)\) for a.a. \(t \in A\).
Now we can show the main result in this section.
Theorem 3.8
 (H_{5}):

there exist admissible discontinuity curves \(\gamma_{n}: I_{n} =[a_{n},b_{n}] \longrightarrow \mathbb {R}\), \(n \in \mathbb {N}\), such that for a.e. \(t \in I\) the function \(f(t,\cdot)\) is continuous in \([R,R] \setminus\bigcup_{n : t \in I_{n}} \{ \gamma_{n}(t)\}\).
Then problem (3.1) has at least one solution in \(\overline{B}_{R}\).
Proof
We consider the multivalued operator \(\mathbb {T}\) associated to T as in (2.1). Therefore, \(\mathbb {T}\) is upper semicontinuous with nonempty, convex and compact values and, as T, maps \(\overline{B}_{R}\) into itself. Moreover, \(\mathbb {T}(\overline{B}_{R})\) is relatively compact in X by Lemma 3.3. Therefore, if we show that \(\{u\} \cap \mathbb {T}u \subset\{Tu\}\), then we obtain by Theorem 2.7 that T has a fixed point in \(\overline{B}_{R}\), which corresponds to a solution of the BVP (3.1). This part of the proof now follows the lines of [1], Theorem 4.4, but we include it for completeness and for highlighting the main differences between the two results. Thus, we fix \(u \in\overline{B}_{R}\) and consider three cases.
Case 1: \(m(\{t \in I_{n} : u(t)=\gamma_{n}(t)\} )=0\) for all \(n \in \mathbb {N}\).
Then we have that \(f(t,\cdot)\) is continuous for a.e. \(t \in I\), and therefore if \(u_{k} \to u\) in \(\overline{B}_{R}\) then we obtain \(f(t,u_{k}(t)) \to f(t,u(t))\) for a.e. \(t \in I\). This, together with (H_{2}) and (H_{3}), implies that \(Tu_{k}\) converges uniformly to Tu in X. Then, T is continuous at u, and therefore we obtain \(\mathbb {T}u= \{ Tu\}\).
Case 2: there exists \(n \in \mathbb {N}\) such that \(\gamma _{n}\) is inviable and \(m(\{t \in I_{n} : u(t)=\gamma_{n}(t)\})>0\). We will show that, in this case, \(u \notin \mathbb {T}u\).
Claim
Similar computations with \(t_{+}\) instead of \(t_{}\) show that if \(v'(\tau _{0}) \le u'(\tau_{0})\), then we also have \(\uv\_{{\mathcal {C}}^{1}} \ge \rho\). The claim is proven.
Case 3: \(m(\{t \in I_{n} : u(t)=\gamma_{n}(t)\} )>0\) only for some of those \(n \in \mathbb {N}\) such that \(\gamma_{n}\) is viable. We will show that, in this case, \(u \in \mathbb {T}u\) implies \(u=Tu\).
Now we assume that \(u \in{\mathbb {T}}u\), and we prove that it also implies that \(u''(t)=g(t) f(t,u(t))\) a.e. in \(I \setminus J\), thus showing that \(u=Tu\).
Then, we have proven that \(\{u\} \cap \mathbb {T}u \subset\{Tu\}\) for all \(u \in\overline{B}_{R}\). By application of Theorem 2.7 we obtain that T has at least one fixed point in \(\overline{B}_{R}\), which corresponds to a solution of the BVP (3.1) in \(\overline{B}_{R}\). □
Remark 3.9
Note that if \(g(t)f(t,0)=0\) for almost all \(t \in[0,1]\), then 0 is a solution of the BVP (3.1). Therefore, when \(g(t)f(t,0)\neq 0\) in a set of positive measure, then Theorem 3.8 provides the existence of a nontrivial solution. In this case, since the kernel k is nonnegative and if, moreover, \(g(t)f(t,u)\geq0\) almost everywhere, then we obtain the existence of a nonnegative solution with a nontrivial norm.
Remark 3.10
The improvement with respect to Theorem 4.4 of [1] relies not only on the fact that we can deal with a more general set of BCs but also on the fact that we do not require global \(L^{1}\) estimates on f, allowing a more general class of nonlinearities. On the other hand, notice that our result can be extended to other type of BCs whenever condition (H_{3}) makes sense for the corresponding Green’s function.
Finally, we illustrate our results by an example.
Example 3.11
We claim that this problem has at least one solution. In order to show this, note that we can rewrite the ODE (3.19) in the form \(u''(t)+g(t)f(t,u(t))=0\), where \(g(t)=\frac{1}{\sqrt{t}}\) and \(f=\tilde{f}\) with f̃ as in (3.18). We now show that the functions g and f satisfy conditions (H_{1})(H_{5}).
First, it is clear that \(g \in L^{1}(I)\), and so (H_{1}) holds. On the other hand, since for all \(n \in \mathbb {N}\) it is \(\phi(n) \le\max\{2,n\}\), we obtain that we have \(u \in[n,n] \Rightarrow f(t,u) \le\max\{2,n\}^{\lambda}\) for each \(n \in \mathbb {N}\). Then, if we take \(R \in \mathbb {N}\), \(R \ge2\), large enough such that \(M_{1}+M_{2} \le R^{1\lambda}\) (with \(M_{1}\), \(M_{2}\) as in (3.4)), then we can guarantee that (H_{2}) and (H_{3}) hold.
We can conclude that the differential equation (3.19), coupled with separated BCs, has at least one solution in \(\overline{B}_{R}\) provided that \(M_{1}+M_{2} \le R^{1\lambda}\). Note that the solution is nontrivial since the zero function does not satisfy the ODE.
In the special case of \(\alpha=\beta=\gamma=\delta=1\) and \(\lambda =1/3\), we obtain (rounded to the third decimal place) \(M_{1}+M_{2}=2{,}336\) and \(R=4\).
Declarations
Acknowledgements
The authors wish to thank Professor Rodrigo López Pouso and both referees of the manuscript for their constructive comments. R Figueroa was partially supported by Xunta de Galicia, Consellería de Cultura, Educación e Ordenación Universitaria, through the project EM2014/032 ‘Ecuacións diferenciais non lineares’. G Infante was partially supported by G.N.A.M.P.A.  INdAM (Italy). This paper was partially written during a visit of R Figueroa to the Dipartimento di Matematica e Informatica, Università della Calabria. R Figueroa wants to thank all the people of this Dipartimento for their kind and warm hospitality.
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Authors’ Affiliations
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