# Fixed Point Theorems and Applications

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1 Fixed Point Theorems and Applications VITTORINO PATA Dipartimento di Matematica F. Brioschi Politecnico di Milano

2 Contents Preface 2 Notation 3 1. FIXED POINT THEOREMS 5 The Banach contraction principle Sequences of maps and fixed points Fixed points of non-expansive maps The Riesz mean ergodic theorem The Brouwer fixed point theorem The Schauder-Tychonoff fixed point theorem The Markov-Kakutani theorem The Kakutani-Ky Fan theorem Notes on Chapter SOME APPLICATIONS OF FIXED POINT THEOREMS 33 The implicit function theorem Ordinary differential equations in Banach spaces Semilinear equations of evolution An abstract elliptic problem The invariant subspace problem Measure preserving maps on compact Hausdorff spaces Invariant means on semigroups Haar measures Game theory Notes on Chapter Bibliography 60 1

3 Preface Fixed point theory is a fascinating subject, with an enormous number of applications in various fields of mathematics. Maybe due to this transversal character, I have always experienced some difficulties to find a book (unless expressly devoted to fixed points) treating the argument in a unitary fashion. In most cases, I noticed that fixed points pop up when they are needed. On the contrary, I believe that they should deserve a relevant place in any general textbook, and particularly, in a functional analysis textbook. This is mainly the reason that made me decide to write down these notes. I tried to collect most of the significant results of the field, and then to present various related applications. This work consists of two chapters which, although rather self-contained, are conceived to be part of a future functional analysis book that I hope to complete in the years to come. Thus some general background is needed to approach the next pages. The reader is in fact supposed to be familiar with measure theory, Banach and Hilbert spaces, locally convex topological vector spaces and, in general, with linear functional analysis. Even if one cannot possibly find here all the information and the details contained in a dedicated treatise, I hope that these notes could provide a quite satisfactory overview of the topic, at least from the point of view of analysis. Vittorino Pata 2

4 Notation Given a normed space X, x X and r > 0 we denote B X (x, r) = { y X : y x < r } B X (x, r) = { y X : y x r } B X (x, r) = { y X : y x = r }. Whenever misunderstandings might occur, we write x X to stress that the norm is taken in X. For a subset Y X, we denote by Y the closure of Y, by Y C the complement of Y, by span(y ) the linear space generated by Y, and by co(y ) the convex hull of Y, that is, the set of all finite convex combinations of elements of Y. We will often use the notion of uniformly convex Banach space. Recall that a Banach space X is uniformly convex if given any two sequences x n, y n X with it follows that x n 1, y n 1, lim n x n + y n = 2 lim x n y n = 0. n In particular, we will exploit the property (coming directly from the definition of uniform convexity) that minimizing sequences in closed convex subsets are convergent. Namely, if C X is (nonvoid) closed and convex and x n C is such that lim x n = inf y n y C then there exists a unique x C such that x = inf y and lim x n = x. y C n Clearly, being Hilbert spaces uniformly convex, all the results involving uniformly convex Banach spaces can be read in terms of Hilbert spaces. A weaker notion is strict convexity : a Banach space X is strictly convex if for all x, y X with x y the relation x = y 1 implies x + y < 2. 3

5 4 It is immediate to check from the definitions that uniform convexity implies strict convexity. Other spaces widely used here are locally convex spaces. A locally convex space X is a vector space endowed with a family P of separating seminorms. Hence for every element x X there is a seminorm p P such that p(x) = 0. Therefore P gives X the structure of (Hausdorff) topological vector space in which there is a local base whose members are covex. A local base B for the topology is given by finite intersections of sets of the form { x X : p(x) < ε } for some p P and some ε > 0. Note that, given U B, there holds U + U := { x + y : x U, y U } = 2U := { 2x : x U }. Good references for these arguments are, e.g., the books [1, 4, 14]. Concerning measure theory, we address the reader to [11, 13].

6 1. FIXED POINT THEOREMS Fixed point theorems concern maps f of a set X into itself that, under certain conditions, admit a fixed point, that is, a point x X such that f(x) = x. The knowledge of the existence of fixed points has relevant applications in many branches of analysis and topology. Let us show for instance the following simple but indicative example. 1.1 Example Suppose we are given a system of n equations in n unknowns of the form g j (x 1,..., x n ) = 0, j = 1,..., n where the g j are continuous real-valued functions of the real variables x j. Let h j (x 1,..., x n ) = g j (x 1,..., x n ) + x j, and for any point x = (x 1,..., x n ) define h(x) = (h 1 (x),..., h n (x)). Assume now that h has a fixed point x R n. Then it is easily seen that x is a solution to the system of equations. Various application of fixed point theorems will be given in the next chapter. The Banach contraction principle 1.2 Definition Let X be a metric space equipped with a distance d. A map f : X X is said to be Lipschitz continuous if there is λ 0 such that d(f(x 1 ), f(x 2 )) λd(x 1, x 2 ), x 1, x 2 X. The smallest λ for which the above inequality holds is the Lipschitz constant of f. If λ 1 f is said to be non-expansive, if λ < 1 f is said to be a contraction. 1.3 Theorem [Banach] Let f be a contraction on a complete metric space X. Then f has a unique fixed point x X. proof Notice first that if x 1, x 2 X are fixed points of f, then d(x 1, x 2 ) = d(f(x 1 ), f(x 2 )) λd(x 1, x 2 ) which imply x 1 = x 2. Choose now any x 0 X, and define the iterate sequence x n+1 = f(x n ). By induction on n, d(x n+1, x n ) λ n d(f(x 0 ), x 0 ). 5

7 6 1. fixed point theorems If n N and m 1, d(x n+m, x n ) d(x n+m, x n+m 1 ) + + d(x n+1, x n ) (λ n+m + + λ n )d(f(x 0 ), x 0 ) (1) λn 1 λ d(f(x 0), x 0 ). Hence x n is a Cauchy sequence, and admits a limit x X, for X is complete. Since f is continuous, we have f( x) = lim n f(x n ) = lim n x n+1 = x. Remark Notice that letting m in (1) we find the relation d(x n, x) λn 1 λ d(f(x 0), x 0 ) which provides a control on the convergence rate of x n to the fixed point x. The completeness of X plays here a crucial role. Indeed, contractions on incomplete metric spaces may fail to have fixed points. Example Let X = (0, 1] with the usual distance. Define f : X X as f(x) = x/ Corollary Let X be a complete metric space and Y be a topological space. Let f : X Y X be a continuous function. Assume that f is a contraction on X uniformly in Y, that is, d(f(x 1, y), f(x 2, y)) λd(x 1, x 2 ), x 1, x 2 X, y Y for some λ < 1. Then, for every fixed y Y, the map x f(x, y) has a unique fixed point ϕ(y). Moreover, the function y ϕ(y) is continuous from Y to X. Notice that if f : X Y X is continuous on Y and is a contraction on X uniformly in Y, then f is in fact continuous on X Y. proof In light of Theorem 1.3, we only have to prove the continuity of ϕ. For y, y 0 Y, we have d(ϕ(y), ϕ(y 0 )) = d(f(ϕ(y), y), f(ϕ(y 0 ), y 0 )) which implies d(f(ϕ(y), y), f(ϕ(y 0 ), y)) + d(f(ϕ(y 0 ), y), f(ϕ(y 0 ), y 0 )) λd(ϕ(y), ϕ(y 0 )) + d(f(ϕ(y 0 ), y), f(ϕ(y 0 ), y 0 )) d(ϕ(y), ϕ(y 0 )) 1 1 λ d(f(ϕ(y 0), y), f(ϕ(y 0 ), y 0 )). Since the above right-hand side goes to zero as y y 0, we have the desired continuity.

8 the banach contraction principle 7 Remark If in addition Y is a metric space and f is Lipschitz continuous in Y, uniformly with respect to X, with Lipschitz constant L 0, then the function y ϕ(y) is Lipschitz continuous with Lipschitz constant less than or equal to L/(1 λ). Theorem 1.3 gives a sufficient condition for f in order to have a unique fixed point. Example Consider the map { 1/2 + 2x x [0, 1/4] g(x) = 1/2 x (1/4, 1] mapping [0, 1] onto itself. Then g is not even continuous, but it has a unique fixed point (x = 1/2). The next corollary takes into account the above situation, and provides existence and uniqueness of a fixed point under more general conditions. Definition For f : X X and n N, we denote by f n the n th -iterate of f, namely, f f n-times (f 0 is the identity map). 1.5 Corollary Let X be a complete metric space and let f : X X. If f n is a contraction, for some n 1, then f has a unique fixed point x X. proof Let x be the unique fixed point of f n, given by Theorem 1.3. Then f n (f( x)) = f(f n ( x)) = f( x), which implies f( x) = x. Since a fixed point of f is clearly a fixed point of f n, we have uniqueness as well. Notice that in the example g 2 (x) 1/ Further extensions of the contraction principle There is in the literature a great number of generalizations of Theorem 1.3 (see, e.g., [6]). Here we point out some results. Theorem [Boyd-Wong] Let X be a complete metric space, and let f : X X. Assume there exists a right-continuous function ϕ : [0, ) [0, ) such that ϕ(r) < r if r > 0, and d(f(x 1 ), f(x 2 )) ϕ(d(x 1, x 2 )), x 1, x 2 X. Then f has a unique fixed point x X. Moreover, for any x 0 X the sequence f n (x 0 ) converges to x. Clearly, Theorem 1.3 is a particular case of this result, for ϕ(r) = λr. proof If x 1, x 2 X are fixed points of f, then d(x 1, x 2 ) = d(f(x 1 ), f(x 2 )) ϕ(d(x 1, x 2 ))

9 8 1. fixed point theorems so x 1 = x 2. To prove the existence, fix any x 0 X, and define the iterate sequence x n+1 = f(x n ). We show that x n is a Cauchy sequence, and the desired conclusion follows arguing like in the proof of Theorem 1.3. For n 1, define the positive sequence a n = d(x n, x n 1 ). It is clear that a n+1 ϕ(a n ) a n ; therefore a n converges monotonically to some a 0. From the right-continuity of ϕ, we get a ϕ(a), which entails a = 0. If x n is not a Cauchy sequence, there is ε > 0 and integers m k > n k k for every k 1 such that d k := d(x mk, x nk ) ε, k 1. In addition, upon choosing the smallest possible m k, we may assume that d(x mk 1, x nk ) < ε for k big enough (here we use the fact that a n 0). Therefore, for k big enough, ε d k d(x mk, x mk 1 ) + d(x mk 1, x nk ) < a mk + ε implying that d k ε from above as k. Moreover, d k d k+1 + a mk+1 + a nk+1 ϕ(d k ) + a mk+1 + a nk+1 and taking the limit as k we obtain the relation ε ϕ(ε), which has to be false since ε > 0. Theorem [Caristi] Let X be a complete metric space, and let f : X X. Assume there exists a lower semicontinuous function ψ : X [0, ) such that d(x, f(x)) ψ(x) ψ(f(x)), x X. Then f has (at least) a fixed point in X. Again, Theorem 1.3 is a particular case, obtained for ψ(x) = d(x, f(x))/(1 λ). Notice that f need not be continuous. proof We introduce a partial ordering on X, setting x y if and only if d(x, y) ψ(x) ψ(y). Let = X 0 X be totally ordered, and consider a sequence x n X 0 such that ψ(x n ) is decreasing to α := inf{ψ(x) : x X 0 }. If n N and m 1, d(x n+m, x n ) m 1 i=0 m 1 d(x n+i+1, x n+i ) ψ(x n+i ) ψ(x n+i+1 ) i=0 = ψ(x n ) ψ(x n+m ).

10 the banach contraction principle 9 Hence x n is a Cauchy sequence, and admits a limit x X, for X is complete. Since ψ can only jump downwards (being lower semicontinuous), we also have ψ(x ) = α. If x X 0 and d(x, x ) > 0, then it must be x x n for large n. Indeed, lim n ψ(x n ) = ψ(x ) ψ(x). We conclude that x is an upper bound for X 0, and by the Zorn lemma there exists a maximal element x. On the other hand, x f( x), thus the maximality of x forces the equality x = f( x). If we assume the continuity of f, we obtain a slightly stronger result, even relaxing the continuity hypothesis on ψ. Theorem Let X be a complete metric space, and let f : X X be a continuous map. Assume there exists a function ψ : X [0, ) such that d(x, f(x)) ψ(x) ψ(f(x)), x X. Then f has a fixed point in X. Moreover, for any x 0 X the sequence f n (x 0 ) converges to a fixed point of f. proof Choose x 0 X. Due the above condition, the sequence ψ(f n (x 0 )) is decreasing, and thus convergent. Reasoning as in the proof of the Caristi theorem, we get that f n (x 0 ) admits a limit x X, for X is complete. The continuity of f then entails f( x) = lim n f(f n (x 0 )) = x. We conclude with the following extension of Theorem 1.3, that we state without proof. Theorem [Čirič] Let X be a complete metric space, and let f : X X be such that d(f(x 1 ), f(x 2 )) { } λ max d(x 1, x 2 ), d(x 1, f(x 1 )), d(x 2, f(x 2 )), d(x 1, f(x 2 )), d(x 2, f(x 1 )) for some λ < 1 and every x 1, x 2 X. Then f has a unique fixed point x X. Moreover, d(f n (x 0 ), x) = O(λ n ) for any x 0 X. Also in this case f need not be continuous. However, it is easy to check that it is continuous at the fixed point. The function g of the former example fulfills the hypotheses of the theorem with λ = 2/ Weak contractions We now dwell on the case of maps on a metric space which are contractive without being contractions. Definition Let X be a metric space with a distance d. A map f : X X is a weak contraction if d(f(x 1 ), f(x 2 )) < d(x 1, x 2 ), x 1 x 2 X.

11 10 1. fixed point theorems Being a weak contraction is not in general a sufficient condition for f in order to have a fixed point, as it is shown in the following simple example. Example Consider the complete metric space X = [1, + ), and let f : X X be defined as f(x) = x + 1/x. It is easy to see that f is a weak contraction with no fixed points. Nonetheless, the condition turns out to be sufficient when X is compact. Theorem Let f be a weak contraction on a compact metric space X. Then f has a unique fixed point x X. Moreover, for any x 0 X the sequence f n (x 0 ) converges to x. proof The uniqueness argument goes exactly as in the proof of Theorem 1.3. From the compactness of X, the continuous function x d(x, f(x)) attains its minimum at some x X. If x f( x), we get d( x, f( x)) = min d(x, f(x)) d(f( x), f(f( x))) < d( x, f( x)) x X which is impossible. Thus x is the unique fixed point of f (and so of f n for all n 2). Let now x 0 x be given, and define d n = d(f n (x 0 ), x). Observe that d n+1 = d(f n+1 (x 0 ), f( x)) < d(f n (x 0 ), x) = d n. Hence d n is strictly decreasing, and admits a limit r 0. Let now f n k (x0 ) be a subsequence of f n (x 0 ) converging to some z X. Then r = d(z, x) = lim k d nk = lim k d nk +1 = lim k d(f(f n k (x 0 )), x) = d(f(z), x). But if z x, then d(f(z), x) = d(f(z), f( x)) < d(z, x). Therefore any convergent subsequence of f n (x 0 ) has limit x, which, along with the compactness of X, implies that f n (x 0 ) converges to x. Obviously, we can relax the compactness of X by requiring that f(x) be compact (just applying the theorem on the restriction of f on f(x)). Arguing like in Corollary 1.5, it is also immediate to prove the following Corollary Let X be a compact metric space and let f : X X. If f n is a weak contraction, for some n 1, then f has a unique fixed point x X. 1.8 A converse to the contraction principle Assume we are given a set X and a map f : X X. We are interested to find a metric d on X such that (X, d) is a complete metric space and f is a contraction on X. Clearly, in light of Theorem 1.3, a necessary condition is that each iterate f n has a unique fixed point. Surprisingly enough, the condition turns out to be sufficient as well.

12 sequences of maps and fixed points 11 Theorem [Bessaga] Let X be an arbitrary set, and let f : X X be a map such that f n has a unique fixed point x X for every n 1. Then for every ε (0, 1), there is a metric d = d ε on X that makes X a complete metric space, and f is a contraction on X with Lipschitz constant equal to ε. proof Choose ε (0, 1). Let Z be the subset of X consisting of all elements z such that f n (z) = x for some n N. We define the following equivalence relation on X \ Z: we say that x y if and only if f n (x) = f m (y) for some n, m N. Notice that if f n (x) = f m (y) and f n (x) = f m (y) then f n+m (x) = f m+n (x). But since x Z, this yields n + m = m + n, that is, n m = n m. At this point, by means of the axiom of choice, we select an element from each equivalence class. We now proceed defining the distance of x from a generic x X by setting d( x, x) = 0, d(x, x) = ε n if x Z with x x, where n = min{m N : f m (x) = x}, and d(x, x) = ε n m if x Z, where n, m N are such that f n (ˆx) = f m (x), ˆx being the selected representative of the equivalence class [x]. The definition is unambiguous, due to the above discussion. Finally, for any x, y X, we set { d(x, x) + d(y, x) if x y d(x, y) = 0 if x = y. It is straightforward to verify that d is a metric. To see that d is complete, observe that the only Cauchy sequences which do not converge to x are ultimately constant. We are left to show that f is a contraction with Lipschitz constant equal to ε. Let x X, x x. If x Z we have If x Z we have d(f(x), f( x)) = d(f(x), x) ε n = εε (n+1) = εd(x, x). d(f(x), f( x)) = d(f(x), x) = ε n m = εε n (m+1) = εd(x, x) since x f(x). The thesis follows directly from the definition of the distance. Sequences of maps and fixed points Let (X, d) be a complete metric space. We consider the problem of convergence of fixed points for a sequence of maps f n : X X. Corollary 1.5 will be implicitly used in the statements of the next two theorems. 1.9 Theorem Assume that each f n has at least a fixed point x n = f n (x n ). Let f : X X be a uniformly continuous map such that f m is a contraction for some m 1. If f n converges uniformly to f, then x n converges to x = f( x). proof We first assume that f is a contraction (i.e., m = 1). Let λ < 1 be the Lipschitz constant of f. Given ε > 0, choose n 0 = n 0 (ε) such that d(f n (x), f(x)) ε(1 λ), n n 0, x X.

13 12 1. fixed point theorems Then, for n n 0, d(x n, x) = d(f n (x n ), f( x)) d(f n (x n ), f(x n )) + d(f(x n ), f( x)) ε(1 λ) + λd(x n, x). Therefore d(x n, x) ε, which proves the convergence. To prove the general case it is enough to observe that if d(f m (x), f m (y)) λ m d(x, y) for some λ < 1, we can define a new metric d 0 on X equivalent to d by setting d 0 (x, y) = m 1 k=0 1 λ k d(f k (x), f k (y)). Moreover, since f is uniformly continuous, f n converges uniformly to f also with respect to d 0. Finally, f is a contraction with respect to d 0. Indeed, d 0 (f(x), f(y)) = m 1 k=0 1 λ k d(f k+1 (x), f k+1 (y)) m 1 1 = λ λ d(f k (x), f k (y)) + 1 k λ d(f m (x), f m (y)) m+1 k=1 m 1 λ k=0 1 λ k d(f k (x), f k (y)) = λd 0 (x, y). So the problem is reduced to the previous case m = 1. The next result refers to a special class of complete metric spaces Theorem Let X be locally compact. Assume that for each n N there is m n 1 such that fn mn is a contraction. Let f : X X be a map such that f m is a contraction for some m 1. If f n converges pointwise to f, and f n is an equicontinuous family, then x n = f n (x n ) converges to x = f( x). proof Let ε > 0 be sufficiently small such that K( x, ε) := { x X : d(x, x) ε } X is compact. As a byproduct of the Ascoli theorem, f n converges to f uniformly on K( x, ε), since it is equicontinuous and pointwise convergent. Choose n 0 = n 0 (ε) such that d(f m n (x), f m (x)) ε(1 λ), n n 0, x K( x, ε)

14 fixed points of non-expansive maps 13 where λ < 1 is the Lipschitz constant of f m. Then, for n n 0 and x K( x, ε) we have d(f m n (x), x) = d(f m n (x), f m ( x)) d(f m n (x), f m (x)) + d(f m (x), f m ( x)) ε(1 λ) + λd(x, x) ε. Hence fn m (K( x, ε)) K( x, ε) for all n n 0. Since the maps fn mn are contractions, it follows that, for n n 0, the fixed points x n of f n belong to K( x, ε), that is, d(x n, x) ε. Fixed points of non-expansive maps Let X be a Banach space, C X nonvoid, closed, bounded and convex, and let f : C C be a non-expansive map. The problem is whether f admits a fixed point in C. The answer, in general, is false. Example Let X = c 0 with the supremum norm. Setting C = B X (0, 1), the map f : C C defined by f(x) = (1, x 0, x 1,...), for x = (x 0, x 1, x 2,...) C is non-expansive but clearly admits no fixed points in C. Things are quite different in uniformly convex Banach spaces Theorem [Browder-Kirk] Let X be a uniformly convex Banach space and C X be nonvoid, closed, bounded and convex. If f : C C is a non-expansive map, then f has a fixed point in C. We provide the proof in the particular case when X is a Hilbert space (the general case may be found, e.g., in [6], Ch.6.4). proof Let x C be fixed, and consider a sequence r n (0, 1) converging to 1. For each n N, define the map f n : C C as f n (x) = r n f(x) + (1 r n )x. Notice that f n is a contractions on C, hence there is a unique x n C such that f n (x n ) = x n. Since C is weakly compact, x n has a subsequence (still denoted by x n ) weakly convergent to some x C. We shall prove that x is a fixed point of f. Notice first that ( lim f( x) xn 2 x x n 2) = f( x) x 2. n

15 14 1. fixed point theorems Since f is non-expansive we have f( x) x n f( x) f(x n ) + f(x n ) x n x x n + f(x n ) x n = x x n + (1 r n ) f(x n ) x. But r n 1 as n and C is bounded, so we conclude that ( lim sup f( x) xn 2 x x n 2) 0 n which yields the equality f( x) = x. Proposition In the hypotheses of Theorem 1.11, the set F of fixed points of f is closed and convex. proof The first assertion is trivial. Assume then x 0, x 1 F, with x 0 x 1, and denote x t = (1 t)x 0 + tx 1, with t (0, 1). We have f(x t ) x 0 = f(x t ) f(x 0 ) x t x 0 = t x 1 x 0 f(x t ) x 1 = f(x t ) f(x 1 ) x t x 1 = (1 t) x 1 x 0 that imply the equalities f(x t ) x 0 = t x 1 x 0 f(x t ) x 1 = (1 t) x 1 x 0. The proof is completed if we show that f(x t ) = (1 t)x 0 +tx 1. This follows from a general fact about uniform convexity, which is recalled in the next lemma. Lemma Let X be a uniformly convex Banach space, and let α, x, y X be such that α x = t x y, α y = (1 t) x y, for some t [0, 1]. Then α = (1 t)x + ty. proof Without loss of generality, we can assume t 1/2. We have (1 t)(α x) t(α y) = (1 2t)(α x) t(x y) t x y (1 2t) α x = 2t(1 t) x y. Since the reverse inequality holds as well, and (1 t) α x = t α y = t(1 t) x y from the uniform convexity of X (but strict convexity would suffice) we get α (1 t)x ty = (1 t)(α x) + t(α y) = 0 as claimed.

16 the riesz mean ergodic theorem 15 The Riesz mean ergodic theorem If T is a non-expansive linear map of a uniformly convex Banach space, then all the fixed points of T are recovered by means of a limit procedure Projections Let X be a linear space. A linear operator P : X X is called a projection in X if P 2 x = P P x = P x for every x X. It is easy to check that P is the identity operator on ran(p ), and the relations ker(p ) = ran(i P ), ran(p ) = ker(i P ) and ker(p ) ran(p ) = {0} hold. Moreover every element x X admits a unique decomposition x = y + z with y ker(p ) and z ran(p ). Proposition If X is a Banach space, then a projection P is continuous if and only if X = ker(p ) ran(p ). The notation X = A B is used to mean that A and B are closed subspaces of X such that A B = {0} and A + B = X. proof If P is continuous, so is I P. Hence ker(p ) and ran(p ) = ker(i P ) are closed. Conversely, let x n x, and P x n y. Since ran(p ) is closed, y ran(p ), and therefore P y = y. But P x n x n ker(p ), and ker(p ) is closed. So we have x y ker(p ), which implies P y = P x. From the closed graph theorem, P is continuous Theorem [F. Riesz] Let X be a uniformly convex Banach space. Let T : X X be a linear operator such that Then for every x X the limit T x x, x X. p x = lim n x + T x + + T n x n + 1 exists. Moreover, the operator P : X X defined by P x = p x is a continuous projection onto the linear space M = {y X : T y = y}. proof Fix x X, and set C = co({x, T x, T 2 x, T 3 x,...}). C is a closed nonvoid convex set, and from the uniform convexity of X there is a unique p x C such that µ = p x = inf { z : z C }. Select ε > 0. Then, for p x C, there are m N and nonnegative constants α 0, α 1,..., α m with m j=0 α j = 1 such that, setting z = m α j T j x j=0

17 16 1. fixed point theorems there holds p x z < ε. In particular, for every n N, z + T z + + T n z n + 1 z µ + ε. Notice that z + T z + + T n z = (α 0 x + + α m T m x) + (α 0 T x + + α m T m+1 x) Thus, assuming n m, we get where Therefore + + (α 0 T n x + + α m T m+n x). z + T z + + T n z = x + T x + + T n x + r r = (α 0 1)x + + (α 0 + α α m 1 1)T m 1 x +(1 α 0 )T 1+n x + + (1 α 0 α 1 α m 1 )T m+n x. x + T x + + T n x n + 1 = z + T z + + T n z n + 1 Since r n + 1 2m x n + 1 r n + 1. upon choosing n enough large such that 2m x < ε(n + 1) we have x + T x + + T n x n + 1 z + T z + + T n z n r n + 1 µ + 2ε. On the other hand, it must be x + T x + + T n x n + 1 µ. Then we conclude that lim x + T x + + T n x n + 1 = µ. n This says that the above is a minimizing sequence in C, and due to the uniform convexity of X, we gain the convergence x + T x + + T n x lim n n + 1 = p x.

18 the brouwer fixed point theorem 17 We are left to show that the operator P x = p x is a continuous projection onto M. Indeed, it is apparent that if x M then p x = x. In general, T p x = lim n T x + T 2 x + + T n+1 x n + 1 = p x + lim n T n+1 x x n + 1 = p x. Finally, P 2 x = P P x = P p x = p x = P x. The continuity is ensured by the relation p x x. When X is a Hilbert space, P is actually an orthogonal projection. This follows from the next proposition. Proposition Let H be a Hilbert space, P = P 2 : H H a bounded linear operator with P 1. Then P is an orthogonal projection. proof Since P is continuous, ran(p ) is closed. Let then E be the orthogonal projection having range ran(p ). Then P = E + P (I E). Let now x ran(p ). For any ε > 0 we have P (P x + εx) P x + εx which implies that P x 2 ε 2 + ε x 2. Hence P x = 0, and the equality P = E holds. The role played by uniform convexity in Theorem 1.13 is essential, as the following example shows. Example Let X = l, and let T L(X) defined by T x = (0, x 0, x 1,...), for x = (x 0, x 1, x 2,...) X. Then T has a unique fixed point, namely, the zero element of X. Nonetheless, if y = (1, 1, 1,...), for every n N we have y + T y + + T n y (1, 2,..., n, n + 1, n + 1,...) n + 1 = = 1. n + 1 The Brouwer fixed point theorem Let D n = {x R n : x 1}. A subset E D n is called a retract of D n if there exists a continuous map r : D n E (called retraction) such that r(x) = x for every x E.

19 18 1. fixed point theorems 1.14 Lemma The set S n 1 = {x R n : x = 1} is not a retract of D n. The lemma can be easily proved by means of algebraic topology tools. Indeed, a retraction r induces a homomorphism r : H n 1 (D n ) H n 1 (S n 1 ), where H n 1 denotes the (n 1)-dimensional homology group (see, e.g., [8]). The natural injection j : S n 1 D n induces in turn a homomorphism j : H n 1 (S n 1 ) H n 1 (D n ), and the composition rj is the identity map on S n 1. Hence (rj) = r j is the identity map on H n 1 (S n 1 ). But since H n 1 (D n ) = 0, j is the null map. On the other hand, H n 1 (S n 1 ) = Z if n 1, and H 0 (S 0 ) = Z Z, leading to a contradiction. The analytic proof reported below is less evident, and makes use of exterior forms. Moreover, it provides a weaker result, namely, it shows that there exist no retraction of class C 2 from D n to S n 1. This will be however enough for our scopes. proof Associate to a C 2 function h : D n D n the exterior form ω h = h 1 dh 2 dh n. The Stokes theorem (cf. [12], Ch.10) entails D h := ω h = S n 1 dω h = D n dh 1 dh n = D n det[j h (x)] dx D n where J h (x) denotes the (n n)-jacobian matrix of h at x. Assume now that there is a retraction r of class C 2 from D n to S n 1. From the above formula, we see that D r is determined only by the values of r on S n 1. But r S n 1 = i S n 1, where i : D n D n is the identity map. Thus D r = D i = vol (D n ). On the other hand r 1, and this implies that the vector J r (x)r(x) is null for every x D n. So 0 is an eigenvalue of J r (x) for every x D n, and therefore det[j r ] 0 which implies D r = 0. Remark Another way to show that det[j r ] 0 is to observe that the determinant is a null lagrangian (for more details see, e.g., [5], Ch.8) 1.15 Theorem [Brouwer] Let f : D n D n be a continuous function. Then f has a fixed point x D n. proof Since we want to rely on the analytic proof, let f : D n D n be of class C 2. If f had no fixed point, then where r(x) = t(x)f(x) + (1 t(x))x t(x) = x 2 x, f(x) ( x 2 x, f(x) ) 2 + (1 x 2 ) x f(x) 2 x f(x) 2 is a retraction of class C 2 from D n to S n 1, against the conclusion of Lemma Graphically, r(x) is the intersection with S n 1 of the line obtained extending the

20 the schauder-tihonov fixed point theorem 19 segment connecting f(x) to x. Hence such an f has a fixed point. Finally, let f : D n D n be continuous. Appealing to the Stone-Weierstrass theorem, we find a sequence f j : D n D n of functions of class C 2 converging uniformly to f on D n. Denote x j the fixed point of f j. Then there is x D n such that, up to a subsequence, x j x. Therefore, f( x) x f( x) f( x j ) + f( x j ) f j ( x j ) + x j x 0 as j, which yields f( x) = x. Remark An alternative approach to prove Theorem 1.15 makes use of the concept of topological degree (see, e.g., [2], Ch.1). The Schauder-Tychonoff fixed point theorem We first extend Theorem 1.15 to a more general situation Lemma Let K be a nonvoid compact convex subset of a finite dimensional real Banach space X. Then every continuous function f : K K has a fixed point x K. proof Since X is homeomorphic to R n for some n N, we assume without loss of generality X = R n. Also, we can assume K D n. For every x D n, let p(x) K be the unique point of minimum norm of the set x K. Notice that p(x) = x for every x K. Moreover, p is continuous on D n. Indeed, given x n, x D n, with x n x, x p(x) x p(x n ) x x n + inf k K x n k x p(x) as n. Thus x p(x n ) is a minimizing sequence as x n x in x K, and this implies the convergence p(x n ) p(x). Define now g(x) = f(p(x)). Then g maps continuously D n onto K. From Theorem 1.15 there is x K such that g( x) = x = f( x). As an immediate application, consider Example 1.1. If there is a compact and convex set K R n such that h(k) K, then h has a fixed point x K. Other quite direct applications are the Frobenius theorem and the fundamental theorem of algebra. Theorem [Frobenius] Let A be a n n matrix with strictly positive entries. Then A has a strictly positive eigenvalue. proof The matrix A can be viewed as a linear transformation from R n to R n. Introduce the compact convex set K = { x R n : n j=1 } x j = 1, x j 0 for j = 1,..., n

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