Schauder’s fixed-point theorem: new applications and a new version for discontinuous operators
© López Pouso; licensee Springer. 2012
Received: 7 June 2012
Accepted: 31 July 2012
Published: 21 August 2012
Schauder’s fixed-point theorem, which applies for continuous operators, is used in this paper, perhaps unexpectedly, to prove existence of solutions to discontinuous problems. Moreover, we introduce a new version of Schauder’s theorem for not necessarily continuous operators which implies existence of solutions for wider classes of problems. Leaning on an abstract fixed-point theorem, our approach is not limited to one-dimensional homogeneous Dirichlet problems, the only type of examples worked out in this paper for coherence and simplicity but yet novelty.
MSC:47H10, 34A36, 34B15.
This paper contains a probably unexpected application of Schauder’s fixed-point theorem to a class of discontinuous problems, and a generalization of it that we have never seen before and proves useful in even more general contexts.
with a -bounded nonlinearity f. The importance of our abstract result is that it allows f to be discontinuous with respect to the dependent variable and does not lean on monotonicity at all. This is a significant contribution to the available literature on existence of solutions to (1.1) with discontinuous f’s which, roughly speaking, consists in rewriting for some function g which is continuous with respect to its second argument and monotone nonincreasing with respect to the third one. Essential references for this approach are [6, 12], and some more recent related results can be looked up in [5, 7, 13, 14].
Removing assumptions from the basic theory on (1.1) can only be useful in applications. Motions of particles in a force field, stationary distributions of temperatures, and many other phenomena can be modeled by means of equations of the form . In real life, external forces often assume only a discrete set of more than one value, so they are often discontinuous (and not necessarily monotone).
For completeness and later references, let us recall Schauder’s theorem , Theorem 2.3.7].
Schauder’s Fixed-Point Theorem Let K be a nonempty, convex, and compact subset of a normed space.
Any continuous operator has at least one fixed point.
The following result can be proven by means of Schauder’s theorem, as we shall see in Section 2.
where is the number of divisors of the integer part of .
Notice that the right-hand side of the differential equation in (1.2) has discontinuities with respect to the unknown in every neighborhood of the boundary condition . This makes it surprising at first sight that Schauder’s theorem can be applied.
The rest of this paper is organized as follows. In Section 2, we show how to apply Schauder’s theorem to derive a pretty easy proof of existence of solutions for a class of discontinuous second-order scalar problems containing (1.2) as a particular case. While our existence result in Section 2 is quite general and has an easy proof, it is somewhat restricted in the type of discontinuities that it admits. In Section 3, we present a generalization of Schauder’s theorem for not necessarily continuous operators which allows working with more general types of discontinuities, as we illustrate in Section 4. Our fixed-point type approach is not limited to second-order differential equations or to homogeneous Dirichlet conditions, which we have considered only for the sake of simplicity.
2 A new application of Schauder’s theorem
under the so-called Carathéodory’s conditions, namely,
(C1) For every , the mapping is measurable;
(C2) For a.a. , the mapping is continuous.
Further conditions are needed in order to apply Schauder’s theorem, and the next one is conveniently simple so as not to hide the main contributions in this paper (which have to do with weak forms of (C2)):
(C3) There exists such that for a.a. and all we have .
The following result is standard.
Proposition 2.1 If f satisfies (C 1), (C 2), and (C 3) then problem (2.1) has at least one solution .
Remark 2.1 We shall identify the set with that of all real-valued functions having an absolutely continuous derivative on I.
where G is the Green’s functiona corresponding to problem (2.1).
The operator T is well defined, maps K into itself, and satisfies the conditions in Schauder’s theorem by virtue of (C1), (C2), and (C3).
Remark 2.2 In fact, proving Proposition 2.1 can be made even easier by working in the Banach space instead of ; see, for instance, . However, working in will be more adequate in next sections, and we have chosen the proof outlined in the previous paragraph because our generalizations will start exactly the same way.
Can we relax condition (C2) in Proposition 2.1 and still get existence of solutions by means of essentially the same proof? We are going to show that the answer is positive, and, moreover, that is the way we are going to generalize (C2) is really meaningful.
Before going into detail, let us recall that (C1) and (C2) imply
(H1) Any composition is measurable whenever .
Next, we present a nontrivial generalization of Proposition 2.1 which has a remarkably simple proof. For the convenience of readers, we recall the following technical result: if two absolutely continuous functions agree on a given measurable set, then their derivatives coincide almost everywhere in that set; see, for instance, , Exercise 5(i), p.332].
Theorem 2.2 Assume that satisfies (H 1), (C 3), and
where M is as in (C 3).
Then problem (2.1) has at least one solution .
Proof Let and be as in (2.2). Operator T is well-defined and maps K into itself by (H1) and (C3).
To prove T has at least one fixed point by means of Schauder’s theorem, it suffices to show that T is continuous. To do it, let in K.
a contradiction with (C2)∗.
We now deduce that in thanks to standard properties of the Green’s function and a straightforward application of Lebesgue’s dominated convergence theorem. □
As an example, we show that Proposition 1.1 is a particular case to Theorem 2.2.
and for and we define . This definition ensures that is continuous on for each , and the corresponding problem (2.1) can only have strictly convex solutions, which would then be solutions of (1.2).
It suffices to show that f satisfies every condition in Theorem 2.2.
To show that f satisfies (H1) and (C2)∗, we use the sequence of all discontinuity points of ϕ, and we define .
where is measurable for each . Therefore, is measurable whenever γ is measurable and nonnegative, and then the composition is measurable for any continuous function . Hence, (H1) is satisfied.
and is continuous on . □
3 Schauder’s theorem for discontinuous operators
This section is devoted to introducing and proving a new fixed-point result of Schauder’s type for not necessarily continuous operators. Despite its important implications (one of which we illustrate in Section 4) it is nothing but a straightforward corollary of Kakutani’s fixed-point theorem for multivalued upper semicontinuous operators; see , Theorem 9.2.2] or , Theorem 3, p.232].
Theorem 3.1 Let K be a nonempty, convex, and compact subset of a normed space X.
where stands for the closed ball in X with center x and radius , and denotes the closed convex hull.
whose values are nonempty, convex, and compact subsets of K.
It is just routine to check that is upper semicontinuous, i.e., if in K, for all , and , then we have .
Kakutani’s fixed-point theorem guarantees that has at least one fixed point, i.e., at least one such that . Now condition (3.1) trivially implies that x is a fixed point of T. □
Remark 3.1 One of the referees correctly pointed out that condition (3.1) can be rephrased simply as follows: either x is a fixed point of T, or , where is defined as in (3.2). In applications of Theorem 3.1, we should then prove that every is a fixed point of T.
However, in order to highlight the roles of the different types of admissible discontinuity curves (which we shall define in our next section), we are going to use a different, not so simple, reformulation of (3.1).
Notice that the definition of ensures that when T is continuous at x, so (3.1) is also equivalent to the following condition: for each either T is continuous at x, or , or . We shall consider separately these three situations in our application of Theorem 3.1 in the proof of Theorem 4.4.
Finally, note also that many known fixed-point theorems could be extended exactly the same way we generalized Schauder’s to Theorem 3.1.
considering the corresponding side limits for .
It is difficult to have a view on how is in higher dimensions. Let us content ourselves with the following analytical characterization. The proof is trivial.
Proposition 3.2 In the conditions of Theorem 3.1, let be fixed.
as defined in (3.2);
- 2.For every and every there exists a finite family of vectors and coefficients () such that and
4 Application to Dirichlet problems
In this section, we illustrate the applicability of Theorem 3.1 to deduce the existence of -solutions to the Dirichlet problem (2.1) with a function which may be discontinuous with respect to both arguments.
Basically, we allow f to be discontinuous over countably many graphs of functions in the conditions of the following definition. The reader is referred to [3, 8, 15] for similar ideas for first-order problems.
We say that the admissible discontinuity curve γ is inviable for the differential equation if it satisfies (4.1) or (4.2).
Remark 4.1 It should be already apparent that this paper owes many ideas to set-valued analysis and viability theory. It is therefore fair (and reasonable) to acknowledge it by using the adjectives viable or inviable for our admissible discontinuity curves.
Roughly, inviable curves push solutions away from them, and viable curves allow solutions slide over them.
If function f were continuous, then inviable admissible discontinuity curves would be just strict lower (or upper) solutions on subintervals of I. Of course, the interest of admissible discontinuity curves is Theorem 4.4 below, which concerns discontinuous f’s.
Viable admissible discontinuity curves are nothing but solutions of the differential equation. It can be reasonably argued that viable discontinuity curves are unlikely to be found in applications. It will, however, remain clear in our final examples that viable curves can be in some cases more useful in applications than inviable ones.
Discontinuity curves in Theorem 2.2 cannot be viable but, curiously, need not be inviable.
Working with admissible discontinuity curves involves some technicalities gathered in the next lemma and its subsequent corollaries. Here, we mimic the ideas in , Lemma 2.3]. In the sequel m stands for the Lebesgue measure in .
Lemma 4.1 Let , , and let , a.e., and a.e. in .
so taking limit when we obtain the first identity in (4.3). The second identity admits a similar proof. □
Corollary 4.2 Let , , and let be such that a.e. in .
and the other identity can be proven in the same way. □
A second consequence of Lemma 4.1 has independent interest (notice that the set A in our next corollary need not be an interval).
If there exists such that a.e. in and also a.e. in (), then for a.a. .
has positive Lebesgue measure.
which implies that does not tend to zero because of (4.6), a contradiction.
and therefore a.e. in A. □
We are now ready for the proof of the main result in this section.
Theorem 4.4 Problem (2.1) has at least one solution in provided that satisfies (H 1), (C 3), and
(H2) There exist admissible discontinuity curves () such that for a.a. the function is continuous on .
Proof We start (exactly as in the proof of Theorem 2.2) considering and as in (2.2). Operator T is well defined and maps K into itself by (H1) and (C3).
The proof will be over once we have checked that condition (3.1) in Theorem 3.1 is satisfied. To do so, we fix an arbitrary function and we consider three different cases (remember Remark 3.1). For simplicity, we use the notation as introduced in (3.2).
Case 1 - for all . Let us prove that then T is continuous at x.
which, along with (C3), yield in .
Case 2 - for some such that is inviable. In this case we can prove that .
First, we fix some notation. Let us assume that for some we have and there exist and , for a.a. , such that (4.2) holds with γ replaced by . (The proof is similar if we assume (4.1) instead of (4.2), so we omit it.)
and we are now in a position to prove that . By virtue of Proposition 3.2, it suffices to prove the following claim:
Claim - Let be given by our assumptions over and let ρ be as in ( 4.13 ). For every finite family and ( ), with , we have .
hence provided that .
Similar computations with instead of show that if then we also have . The claim is proven.
Case 3 - only for some of those such that is viable. Let us prove that in this case the relation implies .
and, therefore, a.e. in .
Now we assume that and we prove that it implies that a.e. in , thus showing that .
Let us denote , and notice that uniformly in I and for all and all .
Hence, for a.a. , and then Corollary 4.3 guarantees that for a.a. . □
Finally, we go back to problem (2.3) for an illustrative example. In this case we can quickly prove the existence of solutions by redefining the nonlinear part over the discontinuity curves so that all of them become viable, and then we show that solutions of the modified problem are solutions of the former one (which is not true in general).
Proposition 4.5 Problem (2.3) has at least one solution.
and for and (this definition makes be continuous for all and for a.a. , and, moreover, any possible solution of (2.1) is nonpositive and, therefore, it is a solution of (2.3)).
and is the sequence of all discontinuity points of ϕ.
Similar arguments to those in the proof of Proposition 1.1 show that satisfies (H1) and (C3) (take for a.a. ).
Plainly, for each , is a viable discontinuity curve for , and therefore Theorem 4.4 ensures that (2.1) with f replaced by has at least one solution .
Since x is convex and , then x can only intersect each once, so x is also a solution to (2.1). □
but we have decided not to include it here because it was just a particular case to , Theorem 2.4] (although easier to prove).
We recommend readers to visit Alberto Cabada’s webpage where a very useful program for computing Green’s functions can be downloaded .
This work was partially supported by FEDER and Ministerio de Educación y Ciencia, Spain, project MTM2010-15314.
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