In the text on Brouwer’s Fixed Point Theorem, I had confidently stated that Schauder’s Theorem follows from it with less effort and that one may easily conclude things like Peano’s Theorem from there. As a matter of fact, things lie considerably deeper than I had naively thought. If sound proofs are to be given, there is technical work to be done in many instances. The ideas are not tough themselves, but the sheer number of steps to be taken and the methodological machinery cannot be neglected. Let’s see.
We shall follow the lines of Heuser’s books (both this one and this one), as we did before, to collect the ingredients to give a proof of Schauder’s Fixed Point Theorem. It involves a statement about convex sets, on which we will focus first, followed by an excursion on approximation in normed vector spaces. We shall also need the theorem named after Arzelà and Ascoli, being a basic glimpse into the ways of thinking of functional analysis. All of this will allow us to prove Schauder’s Theorem in a rather strong flavour. For the conclusion, we split our path: we give both Heuser’s treatment of Peano’s Theorem in the spirit of functional analysis and Walter’s more elementary approach (which, however, also makes use of the Theorem of Arzelà-Ascoli).
Remember that a set is called convex, if for any and for any , we have . This formalizes the intuition that the line connecting and be contained in as well.
Lemma (on convex sets): Let be a normed space and let . Let
the convex hull. Then,
and is compact.
Proof: Let us first prove the representation of the convex hull. For the ““-direction, we will show that the set on the right-hand side of is convex. Let and , with . Let , then
where . Hence, is part of the set on the right-hand side of .
We now turn to the ““-direction. Let . We show that if . That means any point that as a representation as in the right-hand side of must be in . This is clear for . For , we take to see
Note that . By induction,
Finally, we shall prove compactness. Let , with a representation . The sequences are bounded by and hence have convergent subsequences. Choosing subsequences times (for each ), we find a subsequence that converges to some for each . Besides,
We will now prove a result that extends Brouwer’s Fixed Point Theorem to a more general setting. This is the one that I had skimmed earlier, believing it consisted only of standard arguments; in principle, this is true. But let’s have a closer look at it and how these standard arguments work together.
Theorem (on fixed points in real convex sets): Let be convex, compact, and let be continuous. Then has a fixed point.
Proof: 0th step. As is compact, it is bounded and thus there is some such that .
1st step. Let us construct the best approximation of some within ; that means we look for a with .
Taking , there is a sequence with .
We wish to prove that is a Cauchy sequence. From the basic properties of any scalar product, we find
In our case, this shows
Since is convex, . Therefore,
Therefore, is a Cauchy sequence, having a limit , say. As is closed, . In total, we have seen (noting that the absolute value is continuous)
is the best approximation to within .
2nd step. The best approximation is unique.
If there were two of them, and , say, then . If we consider the sequence that alternates between and , we’d find for all , and hence is a Cauchy sequence by what we found in step 1. Therefore must be convergent, which implies .
3rd step. The mapping that takes to its best approximation, is continuous.
Let with . Let . For sufficiently large , we have
These inequalities give us
Hence, is the best approximation to . As this is unique, we have shown that is sequentially continuous.
4th step. The quest for the fixed point.
The mapping is continuous. By Brouwer’s Fixed Point Theorem, has a fixed point: there is some with . As only takes images in , we must have . By construction, for points in , the mapping does not do anything: hence
Corollary (on fixed points in convex sets of normed spaces): Let a normed vector space, let convex, compact, and let continuous. Then has a fixed point.
Proof: Let us choose a base for from the . We take w.l.o.g. for a certain . Then any has a unique representation as , and the maping
is a bijection. As all norms on are equivalent, convergence issues are not affected by this bijection. Hence, the theorem and its proof work out in the setting of this corollary, too. q.e.d.
Note that this Corollary may deal with an infinite-dimensional space, however we make use of a finite-dimensional subspace only. This will become relevant in Schauder’s Theorem as well.
Theorem (Arzelà 1895, Ascoli 1884): Let compact, let be a family of continuous real-valued functions on , which satisfies two properties:
- it is pointwise bounded: for any , there is some with , for all .
- it is equicontinuous: for any there is some such that for any with we have , for all .
Then, is relatively compact, that means every sequence in has a uniformly convergent subsequence.
Note, that we do not demand the limit of the convergent subsequence to be contained in ; that would mean compact, instead of relatively compact.
Proof: 1st step. We get hold of a countable dense subset of .
For our immediate uses of the theorem, it should suffice to choose , since we will take to be intervals and there will not be any need for more exotic applications. However, to show up something a little more general, have a look at the sets . This is a covering of and finitely many of them will suffice to cover , for instance . The set is countable. By construction, for any and any , we can find some point that has . Therefore, is dense in .
2nd step. We construct a certain subsequence to a given sequence .
This step is at the heart of the Arzelà-Ascoli-Theorem, with a diagonal argument to make it work. Let us enumerate the set from the step 1 as .
As is pointwise bounded, the sequence is bounded as well. By Bolzano-Weierstrass, it has a convergent subsequence that we will call .
If we evaluate this new sequence in , we arrive at , which is bounded as well. Again, we find a convergent subsequence that is now called . As this is a subsequence of , it converges in as well.
We continue this scheme and we find an array of sequences like this
where each row is a subsequence of the row above. Row is convergent in the point by Bolzano-Weierstrass and convergent in the points by construction.
Now, consider the sequence . It will converge in any point of .
3rd step. Our subsequence of the 2nd step converges uniformly on . We will use equicontinuity to expand the convergence from to the whole of .
As is equicontinuous, we will find for any some with , for all , as long as . Since is compact, there are some points with . And as is dense in , we can find some for any .
Let , then
We have already seen that is convergent on , and hence (convergent sequences are Cauchy-sequences)
Now, let , no longer restricted. Then, there is some such that , and
Thus, for sufficiently large . This sequence is a Cauchy sequence and hence convergent. q.e.d.
This was our last stepping stone towards Schauder’s Theorem. Let’s see what we can do.
Theorem (Schauder, 1930): Let be a normed vector space, convex and closed, let continuous, relatively compact. Then has a fixed point.
Proof: 1st step. As is relatively compact, its closure is compact. We construct a finite approximating subset of .
Let . There are some finitely many points with . In particular, for any , there is some with . Let us consider the function for
It is obviously continuous, and as is covered by these ,
This allows to be well-defined, and by construction . Hence, the function
is continuous (the Lemma on convex sets tells us that this actually maps into the convex hull). Now, let . We find
and therefore, for any ,
This shows that uniformly approximates the identity on . Note that depends on the choice of .
2nd step. Reference to the Theorem on fixed points in convex sets and approximation of the fixed point.
We set , which is a continuous mapping
We can restrict it to and then re-name it . By the Lemma on convex sets, is compact, it is finite-dimensional, and by the Corollary on fixed point sets in normed spaces, has a fixed point :
Note that depends on and hence on .
3rd step. Construction of the fixed point.
For any , by step 2, we find some with
As is relatively compact, the sequence has a convergent subsequence: there is some with . As is closed, we get . Now,
which means that and get arbitrarily close: . Since is continuous, we arrive at
It is apparent that Schauder’s Theorem already has very general conditions that are tough to weaken further. Obviously the Theorem gets false if is not continuous. If were not closed, we’d get the counter-example of , , which doesn’t have any fixed points. If were not convex, we’d get the counter-example of , . It is hard to give a counter-example if is not relatively compact – in fact I would be interested to hear of any such counter-example or of the generalization of Schauder’s Theorem to such cases. Which is the most general such fixed point theorem?
Now, we are able to harvest the ideas of all this work and apply it to differential equations. Usually, in courses on ordinary differential equations, the famous Picard-Lindelöf-Theorem is proved, which states that for well-behaved functions (meaning that they satisfy a Lipschitz-condition), the initial-value problem
has a unique solution. This is a powerful theorem which simplifies the entire theory of differential equations. However, a little more holds true: it suffices that is continuous to guarantee a solution. However, uniqueness is lost in general. While in many applications one can assume continuity of without remorse (especially in physics), a Lipschitz-condition is much harder to justify. This is not to diminish the usefulness of Picard and Lindelöf, as any model has assumptions to be justified – the Lipschitz-condition is just one of them (if one even bothers to demand for a proper justification of existence and uniqueness – sometimes this would seem obvious from the start).
Let us have a look at what Peano told us:
Theorem (Peano, 1886/1890): Let be continuous, where
let , .
Then, the initial value problem , , has a solution on the interval .
Concerning the interval on which we claim the solution to exist, have a look at how such a solution might behave: as we vary , the solution may “leave” either to the vertical bounds (to left/right) or to the horizontonal bounds (up/down). A solution can at most have a slope of , and thus, if it leaves on the horizontal bounds, this will happen at as the earliest point. If it doesn’t leave there, it will exist until . Of course, it might exist even further, but we have only demanded to be defined till there. A little more formally, the mean value theorem tells us
This guarantees that the solution is well-defined on , because is defined there.
Proof: 0th step. To simplify notation, let us set
1st step. We twist the problem to another equivalent shape, making it more accessible to our tools.
First, let be a solution to the initial value problem on a sub-interval . Then, for any ,
On the other hand, if we start from this equation and suppose that it holds for any , must be differentiable with and .
We have seen that a function solves the initial value problem on if and only if it satisfies the equation on .
2nd step. We try to give a representation of the problem as a fixed-point-problem.
Let us consider the mapping
This is a functional where we plug in a continuous function and where we get a continuous function back. In particular, and to make it even more painfully obvious,
Therefore, is a solution to the intial value problem, if it is a fixed point of , meaning .
3rd step. We show that maps to itself. We have defined only on , so let and ; then:
The second-to-last inequality follows from , the last one from the definition of .
This shows that .
4th step. is obvious, as the constant function is in .
5th step. is convex. Let and let . Then, for any ,
This proves .
6th step. is a closed set in , where we use the topology of uniform convergence.
Consider the sequence which converges uniformly to some . Remember that is complete, which is why we can do this. Then, for any ,
This shows that .
7th step. Using the topology of uniform convergence, the mapping is continuous.
Let . The function is continuous on the compact set and hence uniformly continuous. Therefore, there is some such that for ,
Now, let with . Then we have just seen that for any
8th step. The set is relatively compact. Note that is a set of continuous functions.
Let and let . Then, every function of is bounded pointwise, since
Besides, is equicontinuous, because of
Arzelà and Ascoli now tell us that any sequence in has a uniformly convergent subsequence.
9th and final step. From Schauder’s Fixed Point Theorem and steps 3 to 8, has a fixed point in . From step 2, the initial value problem has a solution. q.e.d.
There was a lot of technical work that we have only needed to invoke Schauder’s Theorem. Some of this could have been avoided, if we had a more elementary proof of Schauder’s Theorem. Such a proof exists, however, some of our machinery is still needed – the proof cannot honestly be called elementary. In some way, the proof matches our procedure given above, however, not everything is needed in such a fine manner. Let’s have short look at it; this is taken from Walter’s book.
Proof (Peano’s Theorem in a more elementary fashion): We proceed in two parts. First, we shall prove the weaker statement that if is continuous and bounded on the (non-compact) set , then there is a solution to the intial-value problem on . Afterwards, we extend this to our compact set . We won’t deal with extending the solution to the left of as it’s neither important nor difficult. In the previous proof we didn’t need to bother about this.
1st step. Let us define a function on using some parameter by
This is well-defined, since on the sub-interval we have , and thus has been recursively defined; hence is defined as well.
Let us denote
2nd step. is equicontinuous. Let and let . Then we get, as is bounded by some ,
which doesn’t depend on , or (only on their distance). Hence, if ,
3rd step. is pointwise bounded. This is obvious from , which doesn’t depend on or .
4th step. We determine a solution to the intial-value problem.
From steps 2 and 3 and from Arzelà-Ascoli, we know that the sequence has a uniformly convergent subsequence. Let us denote its limit by , which is defined for all . This allows us to get
for any and for sufficiently large . It should be clear what we intend to say with (let’s bring in a little sloppiness here, shall we). Since is continuous in its second component, this proves
Hence, as every participant here converges uniformly,
This shows that
5th step. Extension to the general case: Let be defined on the compact rectangle .
We give a continuation of beyond for all via
Obviously, is continuous and bounded. From Step 1, has a solution on . For , we get
Therefore, the solution of is well-defined if , and thus the solution is guaranteed to exist for . q.e.d.
As a sort of last-minute addendum, I have stumbled upon two articles from the 1970s that shed some more light on the issue of elementary proofs to Peano’s Theorem which completely avoid the technicalties of Schauder’s Theorem and of Arzelà-Ascoli. One is called “There is an Elementary Proof of Peano’s Existence Theorem” by Wolfgang Walter (the author of the book we cited earlier; Amer. Math. Monthly 78 1971, 170-173), the other is “On Elementary Proofs of Peano’s Existence Theorems” by Johann Walter (Amer. Math. Monthly 80, 1973, 282-286). The issue of whether Arzelà-Ascoli can be avoided is solved by both papers positively: they give proofs of Peano’s Theorem which only deal with standard calculus methods. Basically, the employ the Euler polygon method to construct a solution of the intitial value problem. However, again, the proofs are not constructive. Besides, “elementary” is not to be confused with “easy”, Peano’s Theorem is still nothing that lies directly on the surface of things. A brief look at the second of those articles (to the best of my knowledge the identical names of the authors are a coincidence) raises hope that this proof is actually not too hard – it should be understandable with a lot less effort than the proof via Schauder’s Theorem that we gave above in full detail; remember that Schauder itself required many non-standard theorems on its way. The elementary proof will only work for one-dimensional differential equations, but we bothered only with those anyway; it uses monotonicity of its approximating sequence which is only applicable in the real numbers. On the plus side, the proof explicitly constructs a solution via the Euler method.
The papers also shed some light on the history of Peano’s Theorem and the quest for its proof (together with some rather unusual disagreement on whether an earlier proof is valid or not; some interesting lines to read in passing). This should be enough on this matter for now. If the interest holds up (which is, to this extent, rather unlikely), we’ll return to it. But not for now.