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Periodic Points of Positive Linear Operators and Perron-Frobenius Operators

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Periodic Points of Positive Linear Operators and Perron-Frobenius Operators Integr. equ. oper. theory 39 (2001) 41-97 0378-620X/01/010041-57 $1.50+0.20/0 I Integral Equations Birlda~iuserVerlag, Basel, 2001 and OperatorTheory PERIODIC POINTS OF POSITIVE LINEAR OPERATORS AND PERRON-FROBENIUS OPERATORS ROGER NUSSBAUM* Let C(S) denote the Banach space of continuous, real-valued maps f : S -+ IR and let A denote a positive linear map of C(S) into itself. We give necessary conditions that the operator A have a strictly positive periodic point of minimal period m. Under mild compactness conditions on the operator A, we prove that these necessary conditions are also sufficient to guarantee existence of a strictly positive periodic point of minimal period m. We study a class of Perron-Frobenius operators defined by c~ (Ax) (t) = ~ bi(t)z(wi(t)), i=I and we show how to verify the necessary compactness conditions to apply our theorems concerning existence of positive periodic points. 1 Introduction Recently, in [10], the following question was raised: Do there exist a nonnegative, continuous function k : [0, 1] x [0, 1] --+ IR and strictly positive, continuous functions f0 : [0, 1] -~ IR and fl : [0, 1] -+ IR with f0 r fl such that for all s E [0, 1] /0 k(s, t)fo(t)dt = f~(s) and /0 k(s, t)fl(t)dt = f0(s)? A simple classical argument shows that this question does not have a positive answer if k(s, t) > 0 for all s and t. In this paper we shall be interested in generalizations of the above question. Let S denote a compact, Hausdorff space, C(S) the Banach space of continuous, real-valued functions z: S --+ IR with lixll = sup{lx(s)l : s E S} and P(S) := P, the set of nonnegative functions in C(S). In this notation, Po(S), the interior of P(S), is the set of functions x such that x(s) > 0 for all s E S. The cone P induces a partial ordering by x _< y if and only if y - x E P. If A : C(S) --+ C(S) is a linear operator, we shall say that A is "positive" if *Partially supported by NSF DMS 97-06891 42 Nussbaum A(P(S)) c P(S). It is well known and easily proved that if A : C(S) ~ C(S) is a positive linear operator, then necessarily A is a bounded linear operator. As usual, if f E C(S), m is a positive integer, and A : C(S) --+ C(S) is a bounded, linear operator, we shall say that "f is a periodic point of A of minimal period m" if Am(f) = f and AJ(f) r f for 0 < j < m. We shall consistently use the above notation. We can now state the main questions of interest here. Question 1. Assume that m is a given positive integer, S is a compact, Hausdorff space and A : C(S) -+ C(S) is a positive, linear operator. Does there exist fo E Po(S) such that f0 is a periodic point of A of minimal period m? Question 2. Let m, S and A be as in Question 1. Does there exist fo c P(S) such that f0 is a periodic point of A of minimal period m? Despite their similarity, there are important differences between these two questions. We shall find necessary conditions that Question I have a positive answer; and we shall prove that, under mild compactness conditions on A, these necessary conditions are also sufficient. We shall prove that a slight variant of the necessary conditions for Question 1 also leads, under mild compactness conditions on A, to sufficient conditions that Question 2 have a positive answer. If bi : S --+ JR, i _> 1, are given nonnegative, continuous functions such that ~i~1 bi(s) := b(s) is a finite, continuous function, and if w~ : S --+ S, i _> 1, are given continuous functions, then one can define a positive linear operator A : C(S) -4 C(S) by (Az)(s) = fi bi(s)x(wi(s)). i=1 Such operators are sometimes called "Perron-Frobenius operators" and represent a major interest of this paper. Establishing appropriate compactness conditions on A so that we can apply our theorems turns out to be a delicate problem, and existing results in the literature seem inadequate for our purposes. These difficulties are already apparent in a simple example studied in Corollary 6.2 below, where S = [0, 1], k > 1 is a real number and A : C(S) --+ C(S) is given by (Ax)(t) - tx((1 - t) k) + (1 - t)x(1 - tk). This paper is rather long, so a guide to the principal results may be in order. In Section 2 we present the following theorem, which gives necessary conditions for Question 1 to have a positive solution. Theorem 1.1 Let S be a compact, Hausdorff space, A : C(S) --+ C(S) a positive, linear operator and m a positive integer. Assume that there exists fo ~ Po(S) such that fo is a periodic point of A of minimal period m. Then the following conditions hold: (1) There exists 0 e Po(S) with A(O) = O. (2) There exist closed, nonempty proper subsets Ej C S, 0 < j < rn, with E,~ = Eo, such that (a) N~-_-olEj = 0 and (b) whenever f E C(S) and flEi = 0 for some 0 < j < m- 1, it follows that AflEj+I = O. Nussbaum 43 In Corollary 2.1 below we prove that, if m is a prime, the sets E~ in Theorem 1.1 can be chosen pairwise disjoint for 0 < i < m. For general m we can always arrange that, for 0 < i < k < m, either Ei = E~ or Ei N Ek = O. In Section 3 we start from the necessary condition (2) in Theorem 1.1 and ask what further conditions are necessary to insure a positive answer to Question 1 or Question 2. We observe (see Theorem 3.5 below) that Questions 1 and 2 for general m can be reduced to the case that m is a prime, so we restrict attention to the case that m is a prime. The following theorem is a special case of Theorem 3.1 in Section 3. Theorem 1.2 Let S be a compact, Hausdorff space, A: C(S) -+ C(S) a positive linear oper- ator and m a prime number. Assume that there exist closed, nonempty sets Ey, 0 << j < m, as in condition (2) of Theorem 1.1. Suppose, also, that there exists 0 P(S) such that A(O) = 0 and O(so) > 0 for some so E := Ur~s Finally, assume that at least one of the following compactness conditions on A is satisfied: (a) For every M > 0 and for every f C(S) such that -MO <_ f < MO, the norm closure of {(Am)J(f)l j > 1} is compact in the norm topology. (b) p(A) < 1, where p(A) denotes the essential spectral radius of A. Then it follows that there exist positive reals a and b and fo P(S) such that aO <_ fo <<_ bO and fo is a periodic point of A of minimal period m. In Theorem 3.1A below we also show that a version of Theorem 1.2 holds for a class of nonlinear maps A : P(S) -+ P(S), although we do not pursue this point in this paper. As a very special case of Theorems i.i and 1.2 (see Remark 3.2), we obtain a complete answer to our original question about integral operators. Corollary 1.1 Let m be a prime number, S a compact, Hausdorff space and # a regular, Borel measure on S such that it(G) > 0 for every nonempty, open subset G of S. Let k : S • S --~ 1R be a continuous, nonnegative function and define A : C(S) ~ C(S) by (Ax)(s) = / k(s, t)x(t)p(dt). Then A has a periodic point fo C Po(S) of minimal period m if and only if the following conditions are satisfied: (1) There exists 0 ~ Fo(S) with A(O) = a. (2) There exist compact, nonempty, proper subsets ]~ of S, 0 < i < m, with E,~ = E0, such that (a) nm-iEi=0 = 0 and (b) k(s,t) = O for (s,t) U?=IE~ • ~_~, where L~ denotes the complement of Ej. Much of the rest of this paper is devoted to finding useful hypotheses which imply the compactness condition (a) of Theorem 1.2. It is straightforward to prove that condition (b) of Theorem 1.2 implies condition (a) of Theorem 1.2. However, it will generally be the 44 Nussbaum case that for Perron-Frobenius operators A of interest, p(A), the essential spectral radius of A, equals r(A), the spectral radius of A; and condition (b) of Theorem 1.2 cannot be satisfied. The reader's attention is particularly directed to Theorem 3.3, which gives a framework applicable to Perron-Frobenius operators and gives conditions under which condition (a) of Theorem 1.2 is satisfied. In general, under the conditions we consider, the Krein-Rutman theorem and its generalizations only insure the existence of an eigenvector 0 E P(S) of A with eigenvalue r(A), although it may well happen that 0 E Po(S). Theorem 3.4 of Section 3 gives hypotheses which insure that 0 satisfies the hypotheses of Theorem 1.2. One can also ask whether A*, the Banch space adjoint of A : C(S) ~ C(S), has periodic point #* C P(S)* of minimal period m.
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