Guided Dynamical Systems and Applications to Functional and Partial Differential Equations. Orr Moshe Shalit

Transcription

1 arxiv:math/ v2 [math.ds] 22 May 2006 Guided Dynamical Systems and Applications to Functional and Partial Differential Equations Orr Moshe Shalit

2 Guided Dynamical Systems and Applications to Functional and Partial Differential Equations Research Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Mathematics Orr Moshe Shalit Submitted to the Senate of the Technion - Israel Institute of Technology Sivan, 5765 Haifa June 2005

3 This Research Thesis Was Done Under The Supervision of Professor Boris Paneah in the Department of Mathematics I would like to thank Prof. Paneah for inspiring me and for all that he taught me I would like to thank my friend Daniel Reem for proofreading the manuscript and for making the figures The financial support granted by the Technion during my studies is greatly acknowledged This work is dedicated to the people who supported me during my studies Braha and Ilan Fabian, my in laws Malka and Meir Shalit, my parents

4 Contents 1 Discrete guided dynamical systems with several generators Dynamical systems with several generators Guided dynamical systems Isomorphism of guided dynamical systems Some results in functional equations The Maximum principle for functional equations The maximum principle An application to Cauchy type functional equations Unique solvability The initial value problem for a P-configuration Definition of a P-configuration Generalized P-configuration Some preliminary results in functional analysis and preparations The initial value problem Overdeterminedness of functional equations Overdeterminedness of Cauchy s functional equation A uniqueness/overdeterminedness theorem Boundary value problems for hyperbolic PDE s Formulation of the problem and main result Explicit conditions for solvability Late introduction Chapter Chapter Chapter Chapter Bibliography 55

5 List of Figures 2.1 Generalized P-configuration The solutions of the Cauchy equation are determined on Γ The solution of the functional equation 3.4 is determined on Γ A domain D of the type we consider A domain D of the type we don t consider The guided dynamical system defined on Γ Illustration of proposition Illustration of proposition 4.2.5, solvable case Illustration of proposition 4.2.5, the non-solvable case

6 Abstract In this thesis I present the concept of a guided dynamical system, and then I exploit this idea to solve various problems in functional equations and partial differential equations. The results presented here are in a sense a sequel to a series of papers by B. Paneah published in the years The last chapter in this work is an introduction containing an overview of this work and a comparison between known and new results. In the first chapter I shall first explain what a guided dynamical system is, introducing all notations and definitions to be used in the later chapters. In the second chapter I will use guided dynamical systems to study functional equations which have the form f(x) a i (x)f(δ i (x)) = h(x), x X where the functions a i, δ i and h are given, and f is an unknown, continuous real-valued or vector-valued function defined on (typically) a compact space X. For this type of equations I present here original results regarding uniqueness and solvability, the methods used are extensions of those introduced by Paneah. In the third chapter we make a detor from our main route to treat the more esoteric problem of over-determinedness, for which I also present some new methods and results. In chapter 4 I will use the results of chapters 2 and 3 to give a necessary and sufficient condition for the unique-solvability of the second partly characteristic boundary value problem: (m x + n y ) x y u = 0 in D u = g on D. In this chapter I will use Paneah s reduction of the above problem to a Cauchy type functional equation to give the necessary and sufficient condition in terms of the dynamical properties of a guided dynamical system in the boundary of the problem. For specific families of domains, this necessary and sufficient condition is then translated to explicit conditions for the wellposedness of this hyperbolic boundary problem. 1

7 List of Notations (X, δ) dynamical system generated in X by the maps δ = (δ 1,..., δ N ) (X, δ, Λ) guided dynamical system with guiding sets Λ = (Λ 1,...,Λ N ) Φ δ the semi-group of maps generated by δ id X the identity map on X OS(x) the orbit set of a point x Λ-OS(x) the guided orbit set of a point x l p (R n ) R n equipped with the norm lp lp the norm (x 1,..., x n ) lp = ( n x i p ) 1/p Df the differential (Jacobian matrix) of a map f C(X) the space of all continuous functions on a topological space X C k (M) the space of k times continuously differentiable functions on M L (X,Y) the space of bounded linear operators from X to Y L (X) the space of bounded linear operators from X into itself ImA the image of a linear operator A KerA the kernel {x Ax = 0} of a linear operator A inda the index of a linear operator A I the identity operator on some function space D the boundary of a domain D D the closure of a set D x = differentiation with respect to the variable x x = ( x, y ) gradient operator in the space R 2 T p (Γ) the tangent space of the curve Γ at the point p C0 (D) the space of all infinitely differentiable functions with compact support in D 2

8 Chapter 1 Discrete guided dynamical systems with several generators In this chapter we shall present terminology and notation from the theory of dynamical systems essential for the formulation and derivation of the results presented in later chapters. The notation and terminology we use is not completely consistent with the standard in this field. In particular, pay attention that we use the term orbit-set for what is usually called orbit, and the term orbit will be reserved for a more intuitive concept. 1.1 Dynamical systems with several generators A dynamical system is a pair (X, δ), where X is a metric space with a metric d (usually compact) and δ = (δ 1,...,δ N ) is a set of continuous maps δ i : X X. The maps in δ generate (by composition) a semigroup of maps Φ δ in the following manner: Φ 0 δ = {id X} and Φ m δ = {σ : X X σ 1,...,σ m δ.σ = σ 1 σ m } Φ δ = Φ m δ. m=0 Given any x 1 X, an orbit emanating from x 1 is a sequence O = (x 1, x 2,...,x n ) where for every j = 2,...,n there is some i {1,..., N} such that x j = δ i (x j 1 ) (1.1) 3

9 We consider both finite and infinite orbits. Given any x X, the orbit-set of x is the set OS(x) = {σ(x) σ Φ δ } Equivalently, the orbit-set of a point x may be defined as the set of all y for which there exists an orbit O = (x,...,y) Definitions The following are basic notions relating to a dynamical system (X, δ). (X, δ) is called minimal if for all x X it is true that OS(x) = X. A point x 0 X is called an attractor if there is a neighborhood U of x 0 such that for any x U there is an orbit emanating from x and converging to x 0 1. x 0 is called a global-attractor if for any x X there is an orbit emanating from x and converging to x 0. A point x 0 X is called a weak attractor if x 0 x X OS(x). Remark The term weak attractor is not standard terminology in dynamical systems. Nevertheless, this notion will prove to be of key importance in the sequel, so the author took the right to give this notion a name. Note that every global attractor is a weak attractor. The reader with some experience in the general theory of dynamical systems will note that a weak attractor is nothing but a point lying in the intersection of the ω - limit sets of all the points in X. Example Let X = [ 1, 1] [ 1, 1], put p 1 = (1, 1), p 2 = (1, 1), p 3 = ( 1, 1), p 4 = ( 1, 1). Define for i = 1, 2, 3, 4 the maps δ i : X X by δ i (x) = 1 2 (x + p i) It follows from proposition below that (X, δ) is a minimal dynamical system. For any i = 1, 2, 3, 4, p i is an attractor - actually, a global attractor - and these are the only attractors. On the other hand, as is the case in any minimal system, any point in X is a weak attractor. 1 Some authors define an attractor as a set having this property. Since we shall make no use of attractive sets which contain more than one point, we prefer to regard an attractor as a point. 4

10 It is useful to have at hand sufficient conditions for the minimality of a dynamical system. The following proposition gives one which will be useful later on. Proposition Let (X, d) be a compact, metric space, and let δ = (δ 1, δ 2,...,δ N ) be a finite family of functions X X satisfying δ 1 (X) δ 2 (X)... δ N (X) = X. (1.2) If δ has the property that for all i = 1,..., N and all x, y X x y d(δ i (x), δ i (y)) < d(x, y) (1.3) then the dynamical system (X, δ) is minimal. Proof. Let us prove a lemma first. Lemma For any ǫ > 0 there exists a constant 0 c ǫ < 1 such that for all i = 1,...,N x, y X. d(x, y) ǫ d(δ i (x), δ i (y)) c ǫ d(x, y) Proof. Let there be given an ǫ > 0 and let Y = X X with the product topology. For every x X let B ǫ (x) denote the open ball around x with radius ǫ. We define a compact subset S Y as follows: [ ( S := Y \ Bǫ/2 (x) B ǫ/2 (x) )]. x X For every i = 1, 2,..., N define a function g i : S R by: g i (x 1, x 2 ) = d(δ i(x 1 ), δ i (x 2 )) d(x 1, x 2 ) for all (x 1, x 2 ) S. For every i, g i is continuous, and so g i attains a maximum c ǫ,i. By (1.3), c ǫ,i < 1, for all i. Set c ǫ to be the maximum of these constants. Now let x, y be two points in X s.t. d(x, y) ǫ. Then we must have (x, y) S so for every i g i (x, y) c ǫ and the lemma follows. Let us complete the proof of the proposition. Fix x 0 X. To prove the proposition we must show that for any y in X and ǫ > 0 there is a z OS(x 0 ) s.t. d(z, y) ǫ. Fix some y X and ǫ > 0. Take some n 5

11 satisfying c n ǫ diam(x) < ǫ, where c ǫ is the constant from the lemma. The lemma tells us that for all σ Φ n δ and all x 1, x 2 X and thus for all σ Φ n δ : d(σ(x 1 ), σ(x 2 )) ǫ But note that by virtue of (1.2), f(x) = X diam(σ(x)) ǫ (1.4) f Φ n δ so that there is an f Φ n δ s.t. y f(x). Now by (1.4) it follows that for all x it is true that d(f(x), y) ǫ so we can choose z = f(x 0 ) and the proof is complete. Definitions Let (X, δ) be a dynamical system. A set Y X is called δ-invariant if δ i (y) Y for all i = 1,...,N and y Y. If Y is a closed, δ-invariant subset of X then δ naturally induces on Y a dynamical system (Y, δ), where δ = (δ 1 Y,...,δ N Y ) (Y, δ) is called a subsystem of (X, δ). Because there is no chance of ambiguity, we shall denote this dynamical system simply by (Y, δ). 1.2 Guided dynamical systems Usually, in the study of dynamical systems, one is interested in the behavior of points under the action of Φ δ, that is, how a point moves under iterations of maps in Φ δ. Such movement may be described by the class of all orbits of point. But in certain applications of dynamical systems it is most profitable to ignore certain, illegal, orbits and to concentrate on a subclass of the orbits. These ideas were introduced by Paneah in [13], [14] and [17], and will be developed below. Definition A guided dynamical system is a dynamical system (X, (δ 1,...,δ N )) together with a system Λ = (Λ 1,...,Λ N ) of N closed subsets of X. 6

12 The sets Λ i are called guiding sets. It will be always assumed that N Λ i =. We shall also denote at some times the set N Λ i by Λ. This will never cause any confusion. Definition An orbit is called a Λ-proper orbit, or, for short, a Λ- orbit, if in (1.1) δ i δ k if x j 1 Λ k. When studying a guided dynamical system we restrict our attention to Λ-proper orbits. One can think of a guided dynamical system as a dynamical system with several generators in which there are points that one can leave using only a subset of δ. A different point of view is to consider δ i as a function with a domain of definition X \ Λ i. For true motivation for this concept the reader must wait until chapters 2 and 4. Remark When dealing with a dynamical system with only two generators, Λ 1 is the set of points which we must leave using δ 2, and vice versa. So one may equivalently define T 1 = Λ 2 and T 2 = Λ 1 to be the guiding sets, that is, to associate the guiding the set with the map which we must use on it. Actually, this is the original notation used by Paneah. Meta-Definition Let (X, δ, Λ) be a guided dynamical system, and let be some concept relating to the dynamical system (X, δ) which may be defined by means of the orbits in (X, δ). Then Λ- is the concept relating to the guided dynamical system (X, δ, Λ) which is defined precisely as with the difference that the phrase orbit is replaced by the phrase Λ-proper orbit. For example, the Λ-orbit set of a point x, denoted Λ-OS(x), is the set of points y for which there exists a Λ-proper orbit O O = (x,...,y) We may similarly define a Λ-attractor, a Λ-minimal dynamical system, etc. Example Let X = S 1, the unit circle in the complex plane, and let δ 1 (z) = e i2πθ 1 z δ 2 (z) = e i2πθ 2 z Define Λ 1 = {1, 1} and Λ 2 = {i, i}. By the well known theorem of Kronecker and Weyl ([21]), the dynamical system (X, δ), (when viewed as an unguided dynamical system), is minimal if and only if at least one of θ 1, θ 2 is irrational. Does this remain true when (X, δ, Λ) is viewed as a guided dynamical system? Let us show that the answer to this is almost yes. To be precise, we shall show that (X, δ, Λ) is Λ-minimal if and only if at least 7

13 one of θ 1, θ 2, say θ 1, is irrational and the other one, say θ 2, is not an integer multiple of 1 2. The only if part is clear. Now assume, without loss of generality, that θ 1 / Q. We also assume that θ 2 Q, as the proof in the case θ 2 / Q is similar. Let z 1 be a point on the circle. We have to prove that Λ OS(z 1 ) = S 1. Consider the maximal Λ-proper orbit of the type O = (z 1, δ 1 (z 1 ), δ 1 (δ 1 ((z 1 )),...) Denote this maximal orbit by Õ. Note that Õ is at least one point long. There are only two possibilities: 1. Õ is infinite (this happens when Õ never intersects Λ 1). In this case, by the Kronecker-Weyl theorem, Õ is dense in S1, so Λ-OS(z 1 ) is, too. 2. Õ is finite. This means that for some m N, δ m 1 (z 1 ) Λ 1 2, but also δ m 1 (z 1) Λ-OS(z 1 ). Now, θ 2 is not an integer multiple of 1 2, so δ 2 (δ m 1 (z 1 )) / Λ 1, and (δ 2 (δ m 1 (z 1)), δ 1 (δ 2 (δ m 1 (z 1))), δ 1 (δ 1 (δ 2 (δ m 1 (z 1)))),...) is now an infinite orbit that doesn t intersect Λ 1, therefore it is dense in S 1. Because δ m 1 (z 1 ) Λ OS(z 1 ) this implies that the orbit set of z 1 is dense in S 1. Examining the above proof one sees that even if θ 2 is equal to 1, the points 2 z = 1 and z = 1 are Λ-weak attractors if θ 1 is irrational. The next concept we shall introduce turns out to be crucial for stating necessary and sufficient conditions for unique solvability of functional equations and boundary value problems, so we shall be explicit when defining it. Definition A set Y X is called (Λ, δ)-invariant if y Y. i.y / Λ i δ i (y) Y In words, any Λ-orbit that begins in Y also ends there. It is a well known fact in the theory of dynamical systems that any compact dynamical system 3 (X, δ) has a closed subsystem (A, δ) that is minimal (see [7]). It is interesting to note that with some care this result carries over to guided dynamical systems as well. 2 By δ m 1 we mean the mth iterate of δ 1. 3 By compact dynamical system we mean a dynamical system (X, δ) where X is compact. 8

14 Lemma Let (X, δ) be a dynamical system, and let Y be a (Λ, δ)- invariant subset of X. Then Y is also (Λ, δ)-invariant. Proof. Let y Y, and assume that I {1,..., N} is the set of indices i for which y / Λ i. we have to show that i I.δ i (y) Y Fix i I. There is a sequence (y n ) n=1 of points in Y such that y n y. Since Λ i is closed, for sufficiently large n, y n / Λ i. Since Y is (Λ, δ)-invariant, for these n we have δ i (y n ) Y. By continuity of δ i, δ i (y n ) δ i (y), so δ i (y) Y, as required. Now since i was an arbitrary element of I, the proof is complete. Theorem Every compact guided dynamical system (X, δ, Λ) has a closed, Λ-minimal, (Λ, δ)-invariant subsystem. Proof. Let (X, δ, Λ) be a compact guided dynamical system. Denote by M the collection consisting of all closed, non-empty, (Λ, δ)-invariant subsets of X. M is not empty, because X M. We shall use Zorn s lemma to prove that M has a minimal 4 element. Assume that {A α } α is a chain in M. A lower bound for this chain is given by B α A α Indeed, let us prove that B M. Obviously, B is closed. Also, B, because if it is empty then, X being compact, there must A α1,...,a αm such that M A αk = k=1 But the above intersection is decreasing and thus equals one of the A α s, contradicting the assumption that for all α, A α. Finally, B is (Λ, δ)- invariant. Indeed, let b B, and assume that I {1,...,N} is the set of indices i such that b / Λ i. For all α and all i I, b A α and b / Λ i. By the (Λ, δ)-invariance of A α we have that δ i (b) A α. This is true for all α, so δ i (b) α A α = B. Since this is true for all i I, B is (Λ, δ)-invariant, and thus is in M. 4 Here, of course, we are using the word minimal in the usual sense, that is, minimal with respect to inclusion. 9

15 Now Zorn s lemma guaranties the existence of a closed, non-empty, (Λ, δ)- invariant A X. It is left to show that A is Λ-minimal. Take any x A. Λ-OS(x) is definitely (Λ, δ)-invariant. By the previous lemma, so is Λ OS(x). As A is invariant, Λ OS(x) A. By the minimality of A, Λ OS(x) = A and, since x was arbitrary in this discussion, this means that (A, δ, (Λ 1 A,...,Λ N A)) is a minimal dynamical system. Proposition A guided dynamical system (X, δ, Λ) is Λ-minimal if and only if it has no Λ-subsystem other than itself. Proof. Taking into account and the fact that a subsystem is nothing but a closed, non-empty, invariant subset, the assertion is clear. 1.3 Isomorphism of guided dynamical systems For every abstract mathematical structure it is always useful to define the maps between two instances of the same type of structure that preserve the essential features of that structure. In the standard theory of dynamical systems, there are the important concepts of a factor and an isomorphism of dynamical system. More details are to be found in [7]. We shall restrict our attention only to isomorphism of two (guided) dynamical systems, as this term will be very useful later on. Definition Two dynamical systems (X, (δ 1,...,δ N )) and (Y, (γ 1,...,γ N )) are said to be isomorphic if there exists a homeomorphism ϕ : X Y satisfying ϕ δ i ϕ 1 = γ i for i = 1,..., N ϕ is called an isomorphism, of the dynamical systems (X, δ) and (Y, γ). Loosely speaking, isomorphic dynamical systems exhibit the same dynamical behavior. For instance, x X is an attractor if and only if ϕ(x) is an attractor in (Y, γ), and (X, δ) is minimal if and only if (Y, γ) is minimal, and so on. Definition Two guided dynamical systems (X, δ, (Λ 1,...,Λ N )) and (Y, γ, (Ω 1,...,Ω N )) are said to be isomorphic if (X, δ) and (Y, γ) are isomorphic as dynamical systems and ϕ from definition maps each Λ i onto Ω i. For completeness of this exposition, let us prove two results regarding isomorphic guided dynamical systems. 10

16 Lemma Let (X, δ, Λ) and (Y, γ, Ω) be two guided dynamical systems. If ϕ : X Y is an isomorphism of guided dynamical systems then the orbit is Λ-proper if and only if is Ω-proper. O = (x 1, x 2,...,x n ) Õ = (ϕ(x 1 ), ϕ(x 2 ),...,ϕ(x n )) Proof. Note that (x 1, x 2,...,x n ) is Λ-proper if and only if (x 1, x 2 ), (x 2, x 3 ),..., (x n 1, x n ) are all Λ-proper. So we may assume that O = (x 1, x 2 ). Also, by the symmetry of the relation isomorphic, it suffices to show that Λ- properness of O implies Ω-properness of Õ. Assume then that O = (x 1, x 2 ) is Λ-proper. We must have x 2 = δ i (x 1 ), for some i {1,..., N}, and x 1 / Λ i. Because ϕ is an isomorphism of dynamical systems ϕ(x 2 ) = ϕ(δ i (x 1 )) = γ i (ϕ(x 1 )) and this shows that Õ = (ϕ(x 1), ϕ(x 2 )) is an orbit. To see that it is Ω-proper, we just note that as ϕ is a 1-1 function that maps guiding sets onto guiding sets and x 1 / Λ i, ϕ(x 1 ) cannot be in Ω i. Theorem Let (X, δ, Λ) and (Y, γ, Ω) be two isomorphic guided dynamical systems. 1. x 0 is a Λ-weak attractor in (X, δ, Λ) if and only if ϕ(x 0 ) is an Ω-weak attractor in (Y, γ, Ω). 2. (X, δ, Λ) is Λ-minimal if and only if (Y, γ, Ω) is Ω-minimal. Proof. Assume that x 0 is a Λ-weak attractor in (X, δ, Λ). Let y Y. We must show that ϕ(x 0 ) Ω OS(y). Choose any ǫ > 0 and define x = ϕ 1 (y). By the continuity of ϕ, there is a µ > 0 such that d X (z, x 0 ) < µ implies d Y (ϕ(z), ϕ(x 0 )) < ǫ 5 for all z X. There is a Λ-proper orbit in X O = (x, x 1,..., x n ) such that d X (x n, x 0 ) < µ. But then by the lemma Õ = (ϕ(x) = y, ϕ(x 1 ),...,ϕ(x n )) is Ω-proper and d Y (ϕ(x n ), ϕ(x 0 )) < ǫ. Since this argument is valid for any ǫ > 0, we have that ϕ(x 0 ) Ω OS(y). We have established the only if half of the first part of the theorem. The if part follows by interchanging the roles of (X, δ, Λ) and (Y, γ, Ω). Finally, the second part of the theorem clearly follows from the first. 5 Here d X and d Y denote the metrics on X and Y, respectively. 11

17 Chapter 2 Some results in functional equations In the following sections we will show how the notions and results of chapter 1 are applied in the field of functional equations. We shall not attempt to explain what a functional equation is, the history of functional equations, and so forth. Such information may be found in the fundamental works of two of the leading specialist in this field in the 20th century: Janos Aczél ([1], [3]) and Marek Kuczma ([8], [10]). 2.1 The Maximum principle for functional equations Maximum principles appeared in analysis long ago. In the theory of functions of a complex variable, the maximum modulus principle for analytical functions helps to establish further results - e.g. Schwartz s lemma. In partial differential equations they serve as a tool for proving uniqueness theorems, approximating solutions, etc. A maximum principle in the field of functional equations appeared for the first time only a few years ago. In 2003 Paneah showed in [14] and [13] that under certain assumptions, if a function F satisfies F(t) a 1 (t)f(δ 1 (t)) a 2 (t)f(δ 2 (t)) = 0, t [ 1, 1] then F attains its maximum and minimum values on the boundary of [ 1, 1]. This theorem proved useful for applications in integral geometry, partial differential equations and, of course, in functional equations ([16] and [17]). The purpose of this section is to extend Paneah s maximum principle as far as we can in order to prove a uniqueness theorem for a conditional cauchy equation in R n. Throughout this section (X, δ, Λ) will be a guided dynamical 12

18 system The maximum principle To begin with, let us recall the notion of a semi-continuous function. Definition Let X be a metric space and x 0 X. A real valued function f : X R is said to be upper semi-continuous at x 0 if lim sup f(x) f(x 0 ). x x 0 f is a said to be upper semi-continuous if it is upper semi-continuous at any point x X. A real valued function f is called lower semi-continuous (at a point x 0 ) if x f(x) is upper semi-continuous (at the point x 0 ). Lemma Let f : X R be an upper (lower) semi-continuous function that satisfies the following functional equation: f(x) a i (x) f(δ i (x)) = 0, x X (2.1) where a i : X R satisfy : i. x.a i (x) 0 (2.2) i. x / Λ i.a i (x) > 0 (2.3) x. a i (x) = 1. (2.4) Then if f attains its maximum (minimum) at some point y 0 X, then it attains its maximum (minimum) at any point x Λ OS(y 0 ). Proof. Put M = f(y 0 ) = maxf, and let I {1,...,N} be a subset of indices i such that y 0 / Λ i. Then there are numbers ǫ 1,...,ǫ N 0 such that f(δ i (y 0 )) = M ǫ i, i = 1,..., N. Combining these relations with (2.1) and using (2.2), (2.3), (2.4) results in a i (y 0 ) ǫ i = 0. i I Thus ǫ i = 0 and so f(δ i (y 0 )) = M for all i I. Now by induction, for any point x Λ OS(y 0 ) we have f(x) = M. If x Λ OS(y 0 ) then there is a 13

19 sub-sequence x n x from Λ OS(y 0 ), and since f is upper semi-continuous, we have M = lim n M = lim n f(x n ) f(x) so f(x) = M, which was to be proved. Corollary Let (X, δ, Λ) be a compact, Λ-minimal dynamical system. Assume that f : X R is an upper semi-continuous function that satisfies (2.1) where the coefficients a i satisfy (2.2), (2.3), (2.4). Then f is constant. Proof. Being an upper semi-continuous function on a compact space, f attains a maximum M = max X f at some point y 0 X. The Λ-minimality of (X, δ, Λ) gives us Λ OS(y 0 ) = X. Using the lemma we assert that f M. We now proceed to prove a lemma which will be useful when proving the main result of this section. Lemma Assume that (X, δ, Λ) is a compact guided dynamical system having a Λ-weak attractor x 0 X. Assume that f : X R is a continuous solution of equation (2.1) where all the coefficients a i satisfy relations (2.2), (2.3), (2.4). Then the function f is constant. Proof. As the function f is continuous, there are points y 0, y 1 X for which f(y 0 ) = min X f and f(y 1 ) = max X f. Being a Λ-weak attractor, the point x 0 belongs to both sets Λ OS(y 0 ) and Λ OS(y 1 ). Being continuous, the function f is simultaneously upper and lower semicontinuous, and hence, by lemma 2.1.2, it takes its maximum and minimal values at x 0. It follows that f f(x 0 ) = const, and this completes the proof of the lemma. Example Let S 1 be the unit circle in the complex plane. Consider the functional equation f(z) = sin 2 (arg z)f(e iτ1 z) + cos 2 (arg z)f(e iτ2 z), z S 1 (2.5) where τ 1, τ 2 R are fixed constants. We claim that equation (2.5) has a nonconstant continuous solution if and only if both numbers τ 1 /2π and τ 2 /2π are rational. Indeed, if τ 1 /2π, τ 2 /2π Q then we may write τ 1 = 2πk 1 /n τ 2 = 2πk 2 /n 14

20 with k 1, k 2, n Z.Then for an arbitrary continuous ( 2π ) periodic function n g : R R the function f(z) = g(argz) is a continuous solution of (2.5). On the other hand, if, for example, τ 1 /2π / Q, then as we have shown in example 1.2.5, the guided dynamical system on the circle generated by the functions δ 1 (z) = e iτ1 z and δ 2 (z) = e iτ2 z and guided by Λ 1 = {z sin 2 (arg z) = 0} = {1, 1} and Λ 2 = {z cos 2 (arg z) = 0} = {i, i} has the point z = 1 is a Λ-weak attractor. Thus by lemma equation (2.5) has no non-constant continuous solutions, the requirements on the coefficients being clearly fulfilled. Theorem Assume that (X, δ, Λ) is compact guided dynamical system that has a Λ-weak attractor x 0 X. Assume that a function F : X R n is a continuous solution of the equation F(x) A i (x) F(δ i (x)) = 0, x X. (2.6) The coefficients A i : X R n n are assumed to be lower triangular matrices with non-negative entries on the diagonal for all x X and satisfy: Then F is constant. i. x / Λ i det(a i (x)) > 0 (2.7) x. A i (x) = I. (2.8) Proof. We write F(x) = (f 1 (x),...,f n (x)), with f k : X R continuous for every k. We also write A k,m i for the entry in the kth row and mth column in the matrix A i (x). Equation (2.6) may now be written as a system of n functional equations: f k (x) k m=1 A k,m 1 f m (δ 1 (x))... for k = 1,..., n. The first equation is k m=1 A k,m N f m(δ N (x)) = 0 (2.9) f 1 (x) A 1,1 1 f 1(δ 1 (x))... A 1,1 N f 1(δ N (x)) = 0 (2.10) and, using (2.7) and (2.8) the lemma tells us that f 1 c 1. 15

21 Assume that f 1 c 1,...,f k c k. Let us show that f k+1 c k+1. Indeed, for k + 1 we may rewrite (2.9) as f k+1 (x) A k+1,k+1 i f k+1 (δ i (x)) k m=1 A k+1,m i c m = 0. (2.11) But (2.8) means that N Ak+1,m i c m = 0 for m < k + 1 so (2.11) reduces to (2.1) and again the lemma ensures that f k+1 c k+1. This completes the proof of the theorem. Assume now that for every x, {A 1 (x),...,a N (x)} is a commuting family of matrices with only (real) positive eigenvalues. A basic result from linear algebra says that for every x there is an invertible matrix P x R n n such that T i (x) = Px 1A i(x)p x is lower triangular for all i 1. In some (very rare, unfortunately) cases, P x can be chosen to be constant throughout X, that is, P x = P. Assume that this is the case. If F satisfies (2.6) we may equivalently write that equation as: PP 1 F(x) PT i (x)p 1 F(δ i (x)) = 0, x X or P[P 1 F(x) T i (x)p 1 F(δ i (x))] = 0, x X. (2.12) We define a new function G(x) = P 1 F(x). Because P is invertible, (2.12) can be re-written as G(x) T i (x) G(δ i (x)) = 0, x X and this is exactly the situation of theorem We record this result as Corollary Let the assumptions of theorem hold with the single change that now A 1 (x),..., A N (x) form a commuting family of matrices with only (real) positive eigenvalues for which there exists a constant triangulating matrix (that is good for all x). Then F is constant. In the above discussion we assumed the existence of a matrix P that triangulates A i (x) for all x X, i = 1,...,N. When can we be sure that such a matrix exists? Trivially, when A i (x) are already triangular. Also, if A i (x) = ϕ i (x)b i, where B i is a constant matrix for all i, and ϕ i is some real valued function, we can find a constant matrix P that does the job. However, the latter class of matrix-functions will never arise non-trivially in our applications. 1 [5] 16

22 2.1.2 An application to Cauchy type functional equations We now use the results of the previous section to find the C 1 2 solutions to certain functional equations of the type f(x) f(a 1 (x)) f(a 2 (x)) = 0, x K. (2.13) Following Paneah we shall call equations of the above type Cauchy type functional equations. Theorem Let K be a compact, connected subset of R n. Let a 1, a 2 : K K be C 1 maps that generate a dynamical system in K with a weak attractor and satisfy x K.a 1 (x) + a 2 (x) = x Assume that the differentials A 1 (x) and A 2 (x) of a 1 (x) and a 2 (x) have only (real) positive eigenvalues. Assume also that there exists an invertible matrix P such that for any point x X there exists two lower triangular matrices T 1 (x) and T 2 (x) such that T i (x) = P 1 A i (x)p, i = 1, 2. If f C 1 (K, R) is a solution of (2.13), then there exists a vector c R n such that f(x) = c x. Proof. Assume that f is a solution of (2.13). Denote by f(x), A 1 (x) and A 2 (x) the differentials of f a 1 and a 2, respectively, at the point x. Put g(x) = ( f(x)) T and B i (x) = (A i (x)) T. Then differentiating (2.13) we obtain g(x) B 1 (x) g(a 1 (x)) B 2 (x) g(a 2 (x)) = 0. Note that B 1 = I n n B 2, so that these matrices commute, and they also have exactly the same eigenvalues as A 1 and A 2. By corollary 2.1.7, g must be constant. So f(x) = c x + b (2.14) for some c R n and b R. Direct substitution in (2.13) shows that (2.14) is a solution if and only if b = 0. 2 Given a compact subset K of R n, we denote by C 1 (K) the space of all functions on K that have continuously differentiable extensions to every neighborhood U of K. 17

23 Example Let K = {(x, y) R 2 : x + y 1}, (K is the unit ball in l 1 (R 2 )), and let ( 1 a 1 (x, y) = 2 x sin y, 1 ) 3 y ( 1 a 2 (x, y) = 2 x 1 4 sin y, 2 ) 3 y Denote by l1 the norm in l 1 (R 2 ). A straightforward computation shows that for i = 1, 2 a i (x, y) l (x, y) l 1 and this shows that the a i s are maps in K with an attractor 0. The differentials of these maps are given by Da 1 (x, y) = ( cosy ) ( 1 1 Da 1 (x, y) = cosy ) Having all of the conditions of theorem (2.1.8), we assert that the Cauchy type functional equation ( 1 f(x, y) f 2 x sin y, 1 ) ( 1 3 y f 2 x 1 4 sin y, 2 ) 3 y = 0 has only as C 1 solutions. f(x) = c x. Example Let K = {(x 1, x 2 ) T R 2 : 1 2 x2 1 + x 2 2 1}, α = π 3, ( ) cos(α) sin(α) L α = sin(α) cos(α) and R α = L T α. Let θ denote the angle between the positive x axis and the line that connects the point (x 1, x 2 ) T to the origin. Let C 1 (r, θ), C 2 (r, θ) be smooth functions that are periodic with period π in the second variable. 3 Then f(x 1, x 2 ) = C 1 (x x 2 2, θ)x 1 + C 2 (x x 2 2, θ)x 2 is a solution to f(x) f(l α x) f(r α x) = 0, x K. 18

24 In the above example, two conditions from theorem (2.1.8) were violated: the eigenvalues are not positive and the dynamical system generated by L α and R α has no weak attractor. It would be interesting to find a connection between the condition on the eigenvalues of the differentials and the existence of a weak attractor. Perhaps theorem (2.1.8) can be refined in such a way that only conditions on the eigenvalues are given. Theorem was proved for general equations in which appear general maps a i, at the price of being able to deal with compact domains only. But for a restricted family of maps a i we can actually prove the uniqueness of solutions to the Cauchy type functional equation in the entire space R n. Theorem Let A 1 and A 2 be two commuting, positive definite (symmetric) n n matrices and let b 1, b 2 R n. Define for any x R n T i x = A i x + b i, i = 1, 2 All C 1 solutions f : R n R of the Cauchy type functional equation are of the form f(t 1 x + T 2 x) = f(t 1 x) + f(t 2 x), x R n (2.15) for some constant vector c R n. f(x) = c x Proof. Let f C 1 be a solution of Define a new variable then we may rewrite equation 2.15 as y = Sx (A 1 + A 2 )x + b 1 + b 2 f(y) = f(t 1 S 1 y) + f(t 2 S 1 y), y R n (2.16) Note that T 1 S 1 y = A 1 ((A 1 + A 2 ) 1 (y b 1 b 2 )) + b 1, so we introduce a matrix B 1 B 1 = A 1 (A 1 + A 2 ) 1 and a vector d 1 R n d 1 = B 1 ( b 1 b 2 ) + b 1 to obtain the convenient form T 1 S 1 y = B 1 y + d 1. Similarly, T 2 S 1 y = B 2 y + d 2, and we re-write (2.16) as f(y) = f(b 1 y + d 1 ) + f(b 2 y + d 2 ), y R n (2.17) From the definitions it follows that B 1 + B 2 = I, and that all the eigenvalues of B 1, B 2 are strictly between 0 and 1. Being symmetric, the B i s 19

25 are diagonalizable, thus there exists a γ < 1 such that B i y l2 γ y l2 for i = 1, 2. Introduce the notation δ i (y) = B i y + d i and d i = k=0 Bk i d i, i = 1, 2. For any y 1, y 2 R n we have δ i (y 1 ) δ i (y 2 ) l2 γ y 1 y 2 l2. On the other hand δ i ( d i ) = B i ( Bi k d i ) + d i = Bi k d i = d i k=0 k=0 This means that for any point in z R n the orbit (z, δ i (z), δ 2 i (z),...) converges exponentially to d i. Now let N be a positive integer such that N > d 1 d 2 l2 1 γ. For each m N define K m = B( d 1, m) B( d 2, m) where B(x, r) denotes the closed ball centered at x with radius r. We note that the condition on N insures that if x B( d i, m), i = 1, 2, and i j = 1, 2 then δ j (x) d j γ x d ( j γ x d i + d i d ) j m It turns out that K m is compact, connected and δ invariant, so for each m we apply theorem with K = K m to infer that f(y) = c m y for all y K m. But {K m } is an increasing sequence of sets whose union is R n, so there is some c such that c m = c for all m N. This shows what we claimed above. 20

26 2.2 Unique solvability In the preceding section we used a maximum principle to assert that, under some appropriate conditions, the only solutions of the homogeneous equation f(x) a i (x)f(δ i (x)) = 0 (2.18) are constants. This clearly implies that, under the same conditions, if the following non-homogeneous equation f(x) a i (x)f(δ i (x)) = h(x) (2.19) has two solutions f 1 and f 2, then f 1 = f 2 + C for some constant C. In this section we shall also concern ourselves with the solvabilty, as well the uniqueness of solutions, of functional equations of the type (2.19). Theorem Let (X, δ) be a compact dynamical system. For i = 1,...,N, let a i : X R be non-negative, continuous functions such that Define the guiding sets x X. a i (x) 1. (2.20) Λ i = {x X : a i (x) = 0}. Assume that there is in X a Λ-weak attractor x 0, and that x 0 {x X : N a i(x) < 1}. Then for any h C(X) the functional equation (2.19) has a unique solution f C(X). Remark This theorem was essentially proved by Paneah in [13], (Theorem 3). There X was the interval I = [ 1, 1] and the existence of an attractive set in I was a consequence of explicit assumptions on δ. The proof we give is a modification of the proof given in [13]. Proof. Define a linear operator A : C(X) C(X) by Af = a i f δ i It is enough to prove that 3 m N. A m < 1. (2.21) 3 In this proof, will denote both the sup norm on C(X) and the operator norm on L (C(X)), the space of bounded linear operators on C(X). 21

27 Indeed, if this is the case, then the operator f f Af is invertible 4, and this is exactly the content of the theorem. We shall prove 2.21 by a series of lemmas. Lemma Let T : C(X) C(X) be a positive linear operator. Then T = T1. Proof. Let f C(X) be of norm 1. Then 1 f 0 thus T(1 f) 0 or T1 Tf. Similarly, T1 Tf. This clearly implies T1 Tf, and the lemma follows. For every n N, define a continuous function g n on X by g n (x) = (A n 1) (x). Note that A is a positive operator. By the above lemma, it suffices to show that m N. g m < 1. (2.22) Let s take a closer look at the functions g n. Lemma Explicitly, for n 2, g n is given by g n (x) = i 1,...,i n a in (x) a in 1 (δ in (x)) a i1 (δ i2 δ in (x)) (2.23) where the sum is over all multi indices (i 1,...,i n ) {1,..., N} n. Proof. We use induction. g 1 (x) = a i (x) and g 2 (x) = (Ag 1 )(x) = a j (x) j=1 4 See [17] for a concise proof of this fact. a i (δ j (x)) 22

28 and this is (2.23) for n = 2. Now let n > 2. g n (x) = = a in (x)g n 1 (δ in (x)) i n=1 a in (x) i n=1 = and (2.23) is proved. i 1,...,i n 1 a in 1 (δ in (x)) a i1 (δ i2 δ in (x)) i 1,...,i n a in (x) a in 1 (δ in (x)) a i1 (δ i2 δ in (x)) Lemma For all x X, if n < k then g k (x) g n (x). Proof. By the previous lemma, g n (x) = a in (x) a i2 (δ i3 δ in (x)) a i1 (y i2,...,i n ) i 2,...,i n i 1 g n 1 (x) where we have denoted y i2,...,i n = δ i2 δ in (x) and used (2.20) for the inequality. The following lemma will make the conclusion of the theorem quite clear. Lemma For any x X there exists a positive integer m(x) such that g m(x) (x) < 1. (2.24) Proof. Fix x X. Since {x X : N a i(x) < 1} is open, there exists an open neighborhood V of x 0 that is contained in {x X : N a i(x) < 1}. x 0 is a Λ-weak attractor, so there exists a Λ-proper orbit ( x, δjn (x), δ jn 1(δ jn (x)),...,δ j2 δ jn (x) ) emanating from x and terminating in V. This means that δ j2 δ jn (x) V. Now, as we have noted before, g n (x) = a in (x) a i2 (δ i3 δ in (x)) a i1 (δ i2 δ in (x)). i 2,...,i n i 1 We may write the right hand side as a jn (x) a j2 (δ j3 δ jn (x)) i 1 a i1 (δ j2 δ jn (x)) + a in (x) a i2 (δ i3 δ in (x)) a i1 (δ i2 δ in (x)) i 2,...,i n j 2,...,j n 23 i 1

29 But i 2,...,i n a in (x) a i2 (δ i3 δ in (x)) = g n 1 (x) and because ( x, δ jn (x), δ jn 1 (δ jn (x)),...,δ j2 δ jn (x) ) is Λ-proper we have that a jn (x) a i2 (δ j3 δ jn (x)) 0. Moreover, δ j2 δ jn (x) V, so i 1 a i1 (δ j2 δ jn (x)) < 1 and thus g n (x) < g n 1 (x) 1. Taking m(x) = n the proof is complete. We are now in a position to finish the proof of the theorem. For every x X there is an m(x) such that g m(x) (x) < 1. Since g m(x) is continuous, there is a neighborhood V x of x where y V x.g m(x) (y) < 1. The neighborhoods {V x } x X form an open covering of the space X, and therefore, by compactness of X, there is a finite sub-covering {V x1,...,v xk }. Denote m j = m(x j ), j = 1,...,k, and put m = max{m 1,...,m k }. Then for any y X there is a j {1,...,k} such that y V xj. So g mj (y) < 1. But by lemma g m (y) g mj (y) < 1 so that the inequality g m (y) < 1 holds for all y X. Consequently g m < 1, and this completes the proof of theorem

30 2.3 The initial value problem for a P-configuration In the previous sections we dealt with rather general dynamical systems and functional equations. Now we will concentrate on a very specific family of dynamical systems and their corresponding Cauchy type functional equations. In fact, we shall prove a necessary and sufficient condition for the existence of a unique solution f C 2 (I) to the problem f(t) f(δ 1 (t)) f(δ 2 (t)) = h(t), t I (2.25) f (c) = µ (2.26) where I = [a, b], c (a, b), µ is some real number, h C 2 satisfies h(a) = h(b), and δ 1, δ 2 form a P-configuration in I. This problem is of great importance for us for two reasons: 1) it is equivalent to a boundary value problem which we treat in chapter 4, and 2) historically the dynamical system in this problem is the origin of the theory of guided dynamical systems. The history of this problem can be found in Paneah s papers [12] - [17], where certain conditions for unique solvability of the problem (2.25)-(2.26) are proved Definition of a P-configuration Let I = [a, b] be a fixed closed interval in R, c (a, b), and let δ 1, δ 2 : I I be two C 2 maps satisfying the following conditions: δ 1 (t) + δ 2 (t) = 1, t I ; (2.27) δ i (t) 0, t I, i = 1, 2 ; (2.28) δ 2 (a) = a, δ 2 (b) = δ 1 (a) = c, δ 1 (b) = b. (2.29) If all these assumptions hold, then the maps δ 1 and δ 2 are said to form a P-configuration in I. We introduce the guiding sets Λ 1 = {t I δ 1(t) = 0} and Λ 2 = {t I δ 2 (t) = 0} Generalized P-configuration At the same cost of proving the necessary and sufficient conditions for unique solvability of (2.25)-(2.26), we may prove the same type of theorem for a class 25

31 a 3 δ (t) a 2 δ (t) a 1 δ (t) a 0 a 0 a 1 a 2 a 3 t Figure 2.1: Generalized P-configuration. of a equations that is a little more general. To this end, we make the following definitions. Let a 0 < a 1 <... < a N be N + 1 points in R. Define I = [a 0, a N ]. Let δ 1,...,δ N be C 2 functions such that δ i maps I onto [a i 1, a i ], for i = 1,...,N. Assume that δ i (t) = 1, t I, (2.30) δ i (t) 0, t I, i = 1,..., N, (2.31) δ i (a 0 ) = a i 1, δ i (a N ) = a i, i = 1,...,N. (2.32) We say that the maps δ 1,..., δ N generate a generalized P-configuration in I. For i = 1,..., N, introduce the guiding sets Λ i = {t I δ i (t) = 0}. See figure 2.1. Our aim now will be to prove a necessary and sufficient condition for the 26

32 existence of a unique solution f to the following problem: f(t) f(δ i (t)) = h(t), t I (2.33) f (c) = µ, (2.34) where the point c I and the number µ are given, and h is an arbitrary C 2 function satisfying h(a 0 ) = h(a N ) Some preliminary results in functional analysis and preparations In this section we shall use without explanation results from functional analysis. Our reference for facts regarding Fredholm operators and Riesz-Schauder theory is [22]. Let us just recall the following two facts: 1. If A is a Fredholm operator and K is compact then A + K is also Fredholm and ind(a + K) = ind(a) If A : V V and B : V V are Fredholm, then BA is also Fredholm and ind(ba) = ind(a) + ind(b) Before proceeding it is worth noting that the idea to use Riesz-Schauder theory in this problem is due to Paneah and was introduced in the papers cited above. However, as is the case in papers many times, this idea was not explained in great detail. Therefore, the rest of this subsection is devoted to making the necessary preparations that will justify the use we shall make later on of Paneah s idea. Fix some point c [a 0, a N ]. We introduce the function spaces 6 and X = { ϕ C 2 (I) N 1 ϕ(a i ) = ϕ (c) = 0 Y = { ψ C 2 (I) ψ(a 0 ) = ψ(a N ) = 0 } W = { ξ C 1 (I) ξ(c) = 0 } Z = { an } ω C 1 (I) ω(t)dt = 0. a 0 5 For a Fredholm operator A we denote by ind(a) the index of A 6 These are Banach spaces when equipped with the usual norm. For example, f X = sup I f + sup I f + sup I f, etc. } 27

33 Define B 0 L(X,Y), B 1 L(W,Z) and B 2 L (C(I)) by and (B 2 h)(t) = h(t) (B 0 f)(t) = f(t) (B 1 g)(t) = g(t) f (δ i (t)) δ i (t)g (δ i(t)) δ i2 h(δ i (t)) δ i δi (t) c h(s)ds. An easy check shows that these operators are bounded (with respect to the standard norms of these spaces) and that they map into the right spaces. For example, if f X, then (B 0 f)(a 0 ) = f(a 0 ) f (δ i (a 0 )) = f(a 0 ) f(a 0 ) N 1 f(a i ) = 0 so (B 0 f)(a 0 ) = 0 and (B 0 f)(a N ) = 0 is shown in a similar manner, thus (B 0 f) Y. There are four different invertible bounded linear operators X W, Y Z, W C(I) and Z C(I) representing differentiation. Let us make a convenient abuse of notation by denoting all of these operators by D. Differentiating equation (2.33) once and twice gives and (if f X) ((B 0 f)(t)) = (B 1 f )(t) (2.35) = f (t) δ i (t) f δ i (t) = h (t) ((B 0 f)(t)) = (B 2 f )(t) (2.36) = f (t) δ i2 f (δ i (t)) δ i f (δ i (t)) = f (t) = h (t). δ i2 f (δ i (t)) 28 δ i δi (t) c f (s)ds

34 (We used the fact that f (c) = 0). From this it follows that DB 0 = B 1 D (2.37) and DB 1 = B 2 D. (2.38) Lemma If one of B 0, B 1 or B 2 is injective (surjective), then all of B 0, B 1 and B 2 are injective (surjective). If one of B 0, B 1 or B 2 is Fredholm, then all of B 0, B 1 and B 2 are Fredholm and indb 0 = indb 1 = indb 2. Proof. Assume, for instance, that KerB 0 = {0}. Then KerDB 0 = {0}, whereas KerB 1 D = D 1 (KerB 1 ). From (2.37) we infer that KerB 1 = {0}. The rest of the first statement is proved in a similar manner. Abusing our notation a little more we may write, e.g., B 0 = D 1 B 1 D Taking into account the second fact that we cited above 7 this shows that B 0 is Fredholm if and only if B 1 is, and that their indices agree, since ind(b 0 ) = ind ( D 1) + ind(b 1 ) + ind(d) = 0 + ind(b 1 ) + 0 = ind(b 1 ) The initial value problem Theorem ImB 0 = Y if and only if (I, δ, Λ) is Λ-minimal. When this is the case, B 0 is an isomorphism. Proof. Let us begin by showing necessity. Assume that (I, δ, Λ) is not Λ- minimal. We have to show that ImB 0 Y. By lemma 2.3.1, it is enough to show that B 1 is not surjective. By proposition 1.2.9, there exists a closed, non-empty (Λ, δ)-invariant set A I. Let G be a C 1 function such that an a 0 G(t)dt = 0 and G A 1. Attempting to arrive at a contradiction we assume that F W is a solution to the equation B 1 F = G. (2.39) 7 and also the fact that D is bounded and invertible on the relevant spaces 29

35 Denote M = max t I F(t). Define a linear operator T : C(I) C(I) by TF = δ i F δ i. By (2.30) and (2.31), T 1. As A is (Λ, δ)-invariant, we also have that for all k, (T k G) A 1. Fix some t 0 A, and let I denote the identity operator on C(I). Operating on both sides of (2.39) with the operator I+T +T T n at the point t 0, and noting thatb 1 = I T, we obtain thus for all n we have that (I T n+1 )F(t 0 ) = (I + T + + T n )G(t 0 ) 2M (I T n+1 )F(t 0 ) = (I + T + + T n )G(t 0 ) = n + 1 a contradiction. Following Paneah, the sufficiency will be established by proving that: 1. KerB 1 = {0} 2. B 2 is a Fredholm operator and indb 2 = 0. Recall that lemma translates these facts to the invertibility of B 0. Proof of 1. Let F W satisfy B 1 F = 0. Note that B 1 F = 0 is precisely the functional equation studied in section 2.1. The conditions on the maps in a P-configuration, and the existence of Λ-weak attractor, (which is a trivial consequence of Λ-minimality), all add up to the fact that F and (I, δ, Λ) satisfy the conditions of lemma 2.1.4, and thus F = const. But, being in W, F(c) = 0, thus F = 0. This proves 1. Proof of 2. Define the operators L, K : C(I) C(I) and (LF)(t) = (KF)(t) = δ i2 F(δ i (t)) δ i δi (t) c F(s)ds. With this new notation we can decompose B 2 as B 2 = I L K. Now, δ 1,...,δ N C 2, so the set where at least two of the δ i are positive is nonempty. But this set is exactly { } t I δ i2 (t) < 1 30

36 and by the assumed Λ-minimality this set contains a Λ-weak attractor. We can now employ theorem to conclude that I L is an invertible operator 8. 2 now follows from the fact that K is a compact operator, and from the first fact from functional analysis cited at the beginning of Remark For applications in partial differential equations it is worth noting that the operator B0 1 is bounded if the operator B 0 is invertible. This, of course, follows from Banach s open mapping theorem. Note that in the above proof for sufficiency we used the Λ-minimality only to infer the existence of a Λ-weak attractor in A = {t I δ i2 (t) < 1}. This set A contains {t i. δ i (t) > 0} = I \Λ. Thus the existence of a Λ-weak attractor in I \ Λ is a sufficient condition for the solvability of the equation B 0 f = h. But we have just shown that the solvability of this problem implies that (I, δ, Λ) is Λ-minimal! Thus we arrive at the very unexpected result: Proposition In a P-configuration (I, δ, Λ) the following are equivalent: 1. (I, δ, Λ) is Λ-minimal. 2. There exists a Λ-weak attractor in I \ Λ. Now we return to the problem (2.33) - (2.34). Theorem Let (I, δ, Λ) be a generalized P-configuration that has a Λ- weak attractor in I \ Λ. Then for any h C 2 (I) with h(a 0 ) = h(a N ), and for any µ R, c [a 0, a N ], there exists a unique solution f C 2 (I) of the problem f(t) f(δ i (t)) = h(t), t I (2.40) f (c) = µ (2.41) Remark Substituting t = a 0 and t = a N in (2.40) we see, using the properties of the δ s, that if f is a solution to (2.40) then N 1 f(a i ) = h(a 0 ) = h(a N ). 8 In this work, a function (or operator) is called invertible if it is both injective and surjective. 31

37 Proof. Let h C 2 (I) satisfy h(a 0 ) = h(a N ). Define h(t) = h(t) h(a 0 ). Using the notation introduced in 2.3.3, we have that h Y. By theorem 2.3.2, there exists an f X such that f(t) f(δ i (t)) = h(t), t I. Put f(t) = f(t) h(a 0) + µc N 1 where C satisfies N δ i(t) = t + C. Now f (c) = µ, and + µt (B 0 f)(t) = (B 0 f)(t) B0 ( h(a0 ) + µc N 1 = h(t) + h(a 0 ) + µc µc = h(t). Uniqueness follows from ) + B 0 (µt) 32