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Roger Nussbaum

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Papers by Roger Nussbaum

Fixed point theorems and Denjoy-Wolff theorems for Hilbert's projective metric in infinite dimensions

Topological Methods in Nonlinear Analysis, Jun 1, 2007

Let K be a closed, normal cone with nonempty interior int(K) in a Banach space X. Let Σ = {x ∈ in... more Let K be a closed, normal cone with nonempty interior int(K) in a Banach space X. Let Σ = {x ∈ int(K) : q(x) = 1} where q: int(K) → (0, ∞) is continuous and homogeneous of degree 1 and it is usually assumed that Σ is bounded in norm. In this framework there is a complete metric d, Hilbert's projective metric, defined on Σ and a complete metric d, Thompson's metric, defined on int(K). We study primarily maps f : Σ → Σ which are nonexpansive with respect to d, but also maps g: int(K) → int(K) which are nonexpansive with respect to d. We prove under essentially minimal compactness assumptions, fixed point theorems for f and g. We generalize to infinite dimensions results of A. F. Beardon (see also A. Karlsson and G. Noskov) concerning the behaviour of Hilbert's projective metric near ∂Σ := Σ \ Σ. If x ∈ Σ, f : Σ → Σ is nonexpansive with respect to Hilbert's projective metric, f has no fixed points on Σ and f satisfies certain mild compactness assumptions, we prove that ω(x; f ), the omega limit set of x under f in the norm topology, is contained in ∂Σ; and there exists η ∈ ∂Σ, η independent of x, such that (1 -t)y + tη ∈ ∂K for 0 ≤ t ≤ 1 and all y ∈ ω(x; f ). This generalizes results of Beardon and of Karlsson and Noskov. We give some evidence for the conjecture that co(ω(x; f )), the convex hull of ω(x; f ), is contained in ∂K.

Tensor Products, Positive Linear Operators, and Delay-Differential Equations

Journal of Dynamics and Differential Equations, Aug 21, 2013

We develop the theory of compound functional differential equations, which are tensor and exterio... more We develop the theory of compound functional differential equations, which are tensor and exterior products of linear functional differential equations. Of particular interest is the equatioṅ x(t) = −α(t)x(t) − β(t)x(t − 1) with a single delay, where the delay coefficient is of one sign, say δβ(t) ≥ 0 with δ ∈ {−1, 1}. Positivity properties are studied, with the result that if (−1) k = δ then the k-fold exterior product of the above system generates a linear process which is positive with respect to a certain cone in the phase space. Additionally, if the coefficients α(t) and β(t) are periodic of the same period, and β(t) satisfies a uniform sign condition, then there is an infinite set of Floquet multipliers which are complete with respect to an associated lap number. Finally, the concept of u 0-positivity of the exterior product is investigated when β(t) satisfies a uniform sign condition.

Inequivalent measures of noncompactness and the radius of the essential spectrum

Proceedings of the American Mathematical Society, Mar 1, 2011

The Kuratowski measure of noncompactness α on an infinite dimensional Banach space (X, •) assigns... more The Kuratowski measure of noncompactness α on an infinite dimensional Banach space (X, •) assigns to each bounded set S in X a nonnegative real number α(S) by the formula α(S) = inf{δ > 0 | S = n i=1 S i for some S i with diam(S i) ≤ δ, for 1 ≤ i ≤ n < ∞}. In general a map β which assigns to each bounded set S in X a nonnegative real number and which shares most of the properties of α is called a homogeneous measure of noncompactness or homogeneous MNC. Two homogeneous MNC's β and γ on X are called equivalent if there exist positive constants b and c with bβ(S) ≤ γ(S) ≤ cβ(S) for all bounded sets S ⊂ X. There are many results which prove the equivalence of various homogeneous MNC's. Working with X = p (N) where 1 ≤ p ≤ ∞, we give the first examples of homogeneous MNC's which are not equivalent. Further, if X is any complex, infinite dimensional Banach space and L : X → X is a bounded linear map, one can define ρ(L) = sup{|λ| | λ ∈ ess(L)}, where ess(L) denotes the essential spectrum of L. One can also define β(L) = inf{λ > 0 | β(LS) ≤ λβ(S) for every S ∈ B(X)}. The formula ρ(L) = lim m→∞ β(L m) 1/m is known to be true if β is equivalent to α, the Kuratowski MNC; however, as we show, it is in general false for MNC's which are not equivalent to α. On the other hand, if B denotes the unit ball in X and β is any homogeneous MNC, we prove that ρ(L) = lim sup m→∞ β(L m B) 1/m = inf{λ > 0 | lim m→∞ λ −m β(L m B) = 0}. Our motivation for this study comes from questions concerning eigenvectors of linear and nonlinear cone-preserving maps. If (X, d) is a complete metric space and S is a bounded subset of X, then K. Kuratowski [10] has defined α(S), the Kuratowski measure of noncompactness of S, by α(S):=inf{δ > 0 | S = n i=1 S i for some S i with diam(S i) ≤ δ, for 1 ≤ i ≤ n<∞}.

Some remarks on differential equations of quadratic type

Journal of Dynamics and Differential Equations, Jul 1, 1991

In this paper we study differential equations of the formx′(t) + x(t)=f(x(t)), x(0)=x0ε C HereC i... more In this paper we study differential equations of the formx′(t) + x(t)=f(x(t)), x(0)=x0ε C HereC is a closed, bounded convex subset of a Banach spaceX,f(C) ⊂C, and it is often assumed thatf(x) is a quadratic map. We study the differential equation by using the general theory of nonexpansive maps and nonexpansive, non-linear semigroups, and we obtain sharp results in a number of cases of interest. We give a formula for the Lipschitz constant off: C → C, and we derive a precise explicit formula for the Lipschitz constant whenf is quadratic,C is the unit simplex inRn, and thel1 norm is used. We give a new proof of a theorem about nonexpansive semigroups; and we show that if the Lipschitz constant off: C→C is less than or equal to one, then limt→∞∥f(x(t))−x(t)∥=0 and, if {x(t):t ⩾ 0} is precompact, then limt→∞x(t) exists. Iff¦C=L¦C, whereL is a bounded linear operator, we apply the nonlinear theory to prove that (under mild further conditions on C) limt→∞ ∥f(x(t))−x(t)∥=0 and that limt→∞x(t) exists if {x(t):t⩾ 0} is precompact. However, forn ⩾ 3 we give examples of quadratic mapsf of the unit simplex ofRn into itself such that limt→∞x(t) fails to exist for mostx0ε C andx(t) may be periodic. Our theorems answer several questions recently raised by J. Herod in connection with so-called model Boltzmann equations.

Intricate Structure of the Analyticity Set for Solutions of a Class of Integral Equations

Journal of Dynamics and Differential Equations, 2019

We consider a class of compact positive operators L : X → X given by (Lx)(t) = t η(t) x(s) ds, ac... more We consider a class of compact positive operators L : X → X given by (Lx)(t) = t η(t) x(s) ds, acting on the space X of continuous 2π-periodic functions x. Here η is continuous with η(t) ≤ t and η(t+2π) = η(t)+2π for all t ∈ R. We obtain necessary and sufficient conditions for the spectral radius of L to be positive, in which case a nonnegative eigensolution to the problem κx = Lx exists for some κ > 0 (equal to the spectral radius of L) by the Krein-Rutman Theorem. If additionally η is analytic, we study the set A ⊆ R of points t at which x is analytic; in general A is a proper subset of R, although x is C ∞ everywhere. Among other results, we obtain conditions under which the complement N = R \ A of A is a generalized Cantor set, namely, a nonempty closed set with empty interior and no isolated points. The proofs of this and of other such results depend strongly on the dynamical properties of the map t → η(t).

Eigenvectors and eigenvalues of nonlinear cone maps

Nonlinear Perron–Frobenius Theory

Birkhoff's version of Hilbert's metric and its application in analysis

Birkhoff's version of Hilbert's metric is a distance between pairs of rays in a closed cone, and ... more Birkhoff's version of Hilbert's metric is a distance between pairs of rays in a closed cone, and is closely related to Hilbert's classical cross-ratio metric. The version we discuss here was popularized by Bushell and can be traced back to the work of Garrett Birkhoff and Hans Samelson. It has found numerous applications in mathematical analysis, especially in the analysis of linear, and nonlinear, mappings on cones. Some of these applications are discussed in this chapter. Birkhoff's version of Hilbert's metric provides a different perspective on Hilbert geometries and naturally leads to infinite-dimensional generalizations. We illustrate this by showing some of its uses in the geometric analysis of Hilbert geometries.

Handbook of Hilbert Geometry

We survey the Hilbert geometry of convex polytopes. In particular we present two important charac... more

Dynamics of subhomogeneous maps

Nonlinear Perron–Frobenius Theory

Topological Methods for Ordinary Differential Equations

Lecture Notes in Mathematics, 1993

Denjoy-Wolff theorems for Hilbert's and Thompson's metric spaces

We study the dynamics of fixed point free mappings on the interior of a normal, closed cone in a ... more We study the dynamics of fixed point free mappings on the interior of a normal, closed cone in a Banach space that are nonexpansive with respect to Hilbert's metric or Thompson's metric. We establish several Denjoy-Wolff type theorems that confirm conjectures by Karlsson and Nussbaum for an important class of nonexpansive mappings. We also extend and put into a broader perspective results by Gaubert and Vigeral concerning the linear escape rate of such nonexpansive mappings.

Analytic Solutions of Delay-Differential Equations

Journal of Dynamics and Differential Equations, 2021

Nussbaum proved that certain bounded solutions of autonomous delay-differential equations with an... more Nussbaum proved that certain bounded solutions of autonomous delay-differential equations with analytic nonlinearities are themselves analytic. On the other hand, the two authors of this paper more recently showed that bounded solutions of certain delay-differential equations, again with analytic nonlinearities, can be C ∞ smooth, yet not be analytic for certain ranges of the independent variable t. In this paper we extend the 1973 results to obtain analytic solutions of a broader class of delay-differential equations, including a wide variety of nonautonomous equations. Nevertheless, there are still equations with analytic nonlinearities possessing global bounded C ∞ solutions for which analyticity is unknown. This is the case, for example, for the equation where g is analytic and where y = 0 is a hyperbolic equilibrium when ε = 0.

Hausdorff dimension of invariant sets and positive linear operators

OF THE DISSERTATION Hausdorff Dimension of Invariant Sets and Positive Linear Operators by Amit P... more OF THE DISSERTATION Hausdorff Dimension of Invariant Sets and Positive Linear Operators by Amit Priyadarshi Dissertation Director: Roger D. Nussbaum In this thesis we obtain theorems which give the Hausdorff dimension of the invariant set for a family of contraction mappings on a complete, perfect metric space. We obtain results for the graph-directed systems similar to that of Mauldin and Williams [17] but for more general contraction mappings which are “infinitesimal similitudes” rather than just “similitudes”. Also the underlying spaces are not assumed to be finite dimensional. For finite graph-directed systems our results are valid on any bounded, complete, perfect metric space. We use the theory of positive linear operators and generalizations of the Krein-Rutman theorem to characterize the Hausdorff dimension as the unique value of σ > 0 for which r(Lσ) = 1, where Lσ, σ > 0, is a naturally associated family of positive linear operators and r(Lσ) denotes the spectral radi...

Functional Differential Equations

Bifurcation Theory and Periodic Solutions of Some Autonomous Functional Differential Equations

Dynamical Systems, 1976

Publisher Summary This chapter focuses on the bifurcation theory and periodic solutions of some a... more Publisher Summary This chapter focuses on the bifurcation theory and periodic solutions of some autonomous functional differential equations. A number of authors have studied nonlinear, autonomous functional differential bifurcation theorem that implies the known existence results for periodic solutions and also new information concerning the structure of the set of periodic solutions, for example, how the period of a periodic solution varies with parameters in the equation. The chapter also reviews study a van der Pol equation with time lag. I If C is a topological space, x 0 ∈ C , W an open neighborhood of x 0 , and f: W — { x 0 } → C a continuous map, then x 0 is an ejective point of ƒ if there exists an open neighborhood U 0 of x 0 such that for every x ɛ U 0 — {x 0 }, there is a positive integer m ( x ) = m such that F m ( x ) is defined and F m ( x ) ∉ U 0 . If C is a topological space, y 0 ɛ C, W an open neighborhood of y 0 , and ƒ : W → C a continuous map, then y 0 is called an attractive point for ƒ if there exists an open neighborhood U 0 of y 0 and for any open neighborhood V of y 0 there exists an integer m ( V ) = m such that f j ( U 0 ) ⊂ W for 0 ≤ j ≤ m and f j ( U 0 ) ⊂ V for j ≥ m .

Periodic solutions of autonomous functional differential equations

Bulletin of the American Mathematical Society, 1973

The purpose of this note is to indicate some applications of a new fixed point theorem to the que... more The purpose of this note is to indicate some applications of a new fixed point theorem to the question of periodic solutions of nonlinear autonomous functional differential equations. The techniques developed give the standard periodicity examples in the literature and some new results, notably for the neutral case, which do not seem accessible by previous methods.

Periodic points of nonexpansive maps

Idempotency, 1998

Extension of order-preserving maps on a cone

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2003

We examine the problem of extending, in a natural way, order-preserving maps that are defined on ... more We examine the problem of extending, in a natural way, order-preserving maps that are defined on the interior of a closed cone K1 (taking values in another closed cone K2) to the whole of K1.We give conditions, in considerable generality (for cones in both finite- and infinite-dimensional spaces), under which a natural extension exists and is continuous. We also give weaker conditions under which the extension is upper semi-continuous.Maps f defined on the interior of the non-negative cone K in RN, which are both homogeneous of degree 1 and order preserving, are non-expanding in the Thompson metric, and hence continuous. As a corollary of our main results, we deduce that all such maps have a homogeneous order-preserving continuous extension to the whole cone. It follows that such an extension must have at least one eigenvector in K – {0}. In the case where the cycle time χ(f) of the original map does not exist, such eigenvectors must lie in ∂K – {0}.We conclude with some discussions...

Generalizations of the Perron-Frobenius theorem for nonlinear maps

Memoirs of the American Mathematical Society, 1999

Let K n = fx 2 R n j x i 0; 1 i ng and suppose that f : K n ! K n is nonexpansive with respect to... more Let K n = fx 2 R n j x i 0; 1 i ng and suppose that f : K n ! K n is nonexpansive with respect to the l 1 -norm, kxk 1 = P n i=1 x i , and f (0) = 0. It is known (see 1]) that for every x 2 K n there exists a periodic point = x 2 K n (so f p ( ) = for some minimal positive integer p = p ) and f k (x) approaches ff j ( ) j 0 j < pg as k tends to in nity. In a previous paper 13] the set P 2 (n) of positive integers p for which there exists a map f as above and a periodic point 2 K n of minimal period p was related to the idea of \admissible arrays" and a set Q(n) determined by certain arithmetical and combinatorial constraints. In a sequel to this paper 14] it is proved that P 2 (n) = Q(n) for all n, but the computation of Q(n) is highly nontrivial. Here we derive a variety of theorems about admissible arrays and use these theorems to compute Q(n) explicitly for n 50 and prove that P (n) = P 2 (n) = Q(n) for n 50, where P (n) is a naturally occurring set de ned below.

Positive solutions of nonlinear elliptic boundary value problems

Journal of Mathematical Analysis and Applications, 1975

Fixed point theorems and Denjoy-Wolff theorems for Hilbert's projective metric in infinite dimensions

Topological Methods in Nonlinear Analysis, Jun 1, 2007

Let K be a closed, normal cone with nonempty interior int(K) in a Banach space X. Let Σ = {x ∈ in... more Let K be a closed, normal cone with nonempty interior int(K) in a Banach space X. Let Σ = {x ∈ int(K) : q(x) = 1} where q: int(K) → (0, ∞) is continuous and homogeneous of degree 1 and it is usually assumed that Σ is bounded in norm. In this framework there is a complete metric d, Hilbert's projective metric, defined on Σ and a complete metric d, Thompson's metric, defined on int(K). We study primarily maps f : Σ → Σ which are nonexpansive with respect to d, but also maps g: int(K) → int(K) which are nonexpansive with respect to d. We prove under essentially minimal compactness assumptions, fixed point theorems for f and g. We generalize to infinite dimensions results of A. F. Beardon (see also A. Karlsson and G. Noskov) concerning the behaviour of Hilbert's projective metric near ∂Σ := Σ \ Σ. If x ∈ Σ, f : Σ → Σ is nonexpansive with respect to Hilbert's projective metric, f has no fixed points on Σ and f satisfies certain mild compactness assumptions, we prove that ω(x; f ), the omega limit set of x under f in the norm topology, is contained in ∂Σ; and there exists η ∈ ∂Σ, η independent of x, such that (1 -t)y + tη ∈ ∂K for 0 ≤ t ≤ 1 and all y ∈ ω(x; f ). This generalizes results of Beardon and of Karlsson and Noskov. We give some evidence for the conjecture that co(ω(x; f )), the convex hull of ω(x; f ), is contained in ∂K.

Tensor Products, Positive Linear Operators, and Delay-Differential Equations

Journal of Dynamics and Differential Equations, Aug 21, 2013

We develop the theory of compound functional differential equations, which are tensor and exterio... more We develop the theory of compound functional differential equations, which are tensor and exterior products of linear functional differential equations. Of particular interest is the equatioṅ x(t) = −α(t)x(t) − β(t)x(t − 1) with a single delay, where the delay coefficient is of one sign, say δβ(t) ≥ 0 with δ ∈ {−1, 1}. Positivity properties are studied, with the result that if (−1) k = δ then the k-fold exterior product of the above system generates a linear process which is positive with respect to a certain cone in the phase space. Additionally, if the coefficients α(t) and β(t) are periodic of the same period, and β(t) satisfies a uniform sign condition, then there is an infinite set of Floquet multipliers which are complete with respect to an associated lap number. Finally, the concept of u 0-positivity of the exterior product is investigated when β(t) satisfies a uniform sign condition.

Inequivalent measures of noncompactness and the radius of the essential spectrum

Proceedings of the American Mathematical Society, Mar 1, 2011

The Kuratowski measure of noncompactness α on an infinite dimensional Banach space (X, •) assigns... more The Kuratowski measure of noncompactness α on an infinite dimensional Banach space (X, •) assigns to each bounded set S in X a nonnegative real number α(S) by the formula α(S) = inf{δ > 0 | S = n i=1 S i for some S i with diam(S i) ≤ δ, for 1 ≤ i ≤ n < ∞}. In general a map β which assigns to each bounded set S in X a nonnegative real number and which shares most of the properties of α is called a homogeneous measure of noncompactness or homogeneous MNC. Two homogeneous MNC's β and γ on X are called equivalent if there exist positive constants b and c with bβ(S) ≤ γ(S) ≤ cβ(S) for all bounded sets S ⊂ X. There are many results which prove the equivalence of various homogeneous MNC's. Working with X = p (N) where 1 ≤ p ≤ ∞, we give the first examples of homogeneous MNC's which are not equivalent. Further, if X is any complex, infinite dimensional Banach space and L : X → X is a bounded linear map, one can define ρ(L) = sup{|λ| | λ ∈ ess(L)}, where ess(L) denotes the essential spectrum of L. One can also define β(L) = inf{λ > 0 | β(LS) ≤ λβ(S) for every S ∈ B(X)}. The formula ρ(L) = lim m→∞ β(L m) 1/m is known to be true if β is equivalent to α, the Kuratowski MNC; however, as we show, it is in general false for MNC's which are not equivalent to α. On the other hand, if B denotes the unit ball in X and β is any homogeneous MNC, we prove that ρ(L) = lim sup m→∞ β(L m B) 1/m = inf{λ > 0 | lim m→∞ λ −m β(L m B) = 0}. Our motivation for this study comes from questions concerning eigenvectors of linear and nonlinear cone-preserving maps. If (X, d) is a complete metric space and S is a bounded subset of X, then K. Kuratowski [10] has defined α(S), the Kuratowski measure of noncompactness of S, by α(S):=inf{δ > 0 | S = n i=1 S i for some S i with diam(S i) ≤ δ, for 1 ≤ i ≤ n<∞}.

Some remarks on differential equations of quadratic type

Journal of Dynamics and Differential Equations, Jul 1, 1991

In this paper we study differential equations of the formx′(t) + x(t)=f(x(t)), x(0)=x0ε C HereC i... more In this paper we study differential equations of the formx′(t) + x(t)=f(x(t)), x(0)=x0ε C HereC is a closed, bounded convex subset of a Banach spaceX,f(C) ⊂C, and it is often assumed thatf(x) is a quadratic map. We study the differential equation by using the general theory of nonexpansive maps and nonexpansive, non-linear semigroups, and we obtain sharp results in a number of cases of interest. We give a formula for the Lipschitz constant off: C → C, and we derive a precise explicit formula for the Lipschitz constant whenf is quadratic,C is the unit simplex inRn, and thel1 norm is used. We give a new proof of a theorem about nonexpansive semigroups; and we show that if the Lipschitz constant off: C→C is less than or equal to one, then limt→∞∥f(x(t))−x(t)∥=0 and, if {x(t):t ⩾ 0} is precompact, then limt→∞x(t) exists. Iff¦C=L¦C, whereL is a bounded linear operator, we apply the nonlinear theory to prove that (under mild further conditions on C) limt→∞ ∥f(x(t))−x(t)∥=0 and that limt→∞x(t) exists if {x(t):t⩾ 0} is precompact. However, forn ⩾ 3 we give examples of quadratic mapsf of the unit simplex ofRn into itself such that limt→∞x(t) fails to exist for mostx0ε C andx(t) may be periodic. Our theorems answer several questions recently raised by J. Herod in connection with so-called model Boltzmann equations.

Intricate Structure of the Analyticity Set for Solutions of a Class of Integral Equations

Journal of Dynamics and Differential Equations, 2019

We consider a class of compact positive operators L : X → X given by (Lx)(t) = t η(t) x(s) ds, ac... more We consider a class of compact positive operators L : X → X given by (Lx)(t) = t η(t) x(s) ds, acting on the space X of continuous 2π-periodic functions x. Here η is continuous with η(t) ≤ t and η(t+2π) = η(t)+2π for all t ∈ R. We obtain necessary and sufficient conditions for the spectral radius of L to be positive, in which case a nonnegative eigensolution to the problem κx = Lx exists for some κ > 0 (equal to the spectral radius of L) by the Krein-Rutman Theorem. If additionally η is analytic, we study the set A ⊆ R of points t at which x is analytic; in general A is a proper subset of R, although x is C ∞ everywhere. Among other results, we obtain conditions under which the complement N = R \ A of A is a generalized Cantor set, namely, a nonempty closed set with empty interior and no isolated points. The proofs of this and of other such results depend strongly on the dynamical properties of the map t → η(t).

Eigenvectors and eigenvalues of nonlinear cone maps

Nonlinear Perron–Frobenius Theory

Birkhoff's version of Hilbert's metric and its application in analysis

Birkhoff's version of Hilbert's metric is a distance between pairs of rays in a closed cone, and ... more Birkhoff's version of Hilbert's metric is a distance between pairs of rays in a closed cone, and is closely related to Hilbert's classical cross-ratio metric. The version we discuss here was popularized by Bushell and can be traced back to the work of Garrett Birkhoff and Hans Samelson. It has found numerous applications in mathematical analysis, especially in the analysis of linear, and nonlinear, mappings on cones. Some of these applications are discussed in this chapter. Birkhoff's version of Hilbert's metric provides a different perspective on Hilbert geometries and naturally leads to infinite-dimensional generalizations. We illustrate this by showing some of its uses in the geometric analysis of Hilbert geometries.

Handbook of Hilbert Geometry

We survey the Hilbert geometry of convex polytopes. In particular we present two important charac... more

Dynamics of subhomogeneous maps

Nonlinear Perron–Frobenius Theory

Topological Methods for Ordinary Differential Equations

Lecture Notes in Mathematics, 1993

Denjoy-Wolff theorems for Hilbert's and Thompson's metric spaces

We study the dynamics of fixed point free mappings on the interior of a normal, closed cone in a ... more We study the dynamics of fixed point free mappings on the interior of a normal, closed cone in a Banach space that are nonexpansive with respect to Hilbert's metric or Thompson's metric. We establish several Denjoy-Wolff type theorems that confirm conjectures by Karlsson and Nussbaum for an important class of nonexpansive mappings. We also extend and put into a broader perspective results by Gaubert and Vigeral concerning the linear escape rate of such nonexpansive mappings.

Analytic Solutions of Delay-Differential Equations

Journal of Dynamics and Differential Equations, 2021

Nussbaum proved that certain bounded solutions of autonomous delay-differential equations with an... more Nussbaum proved that certain bounded solutions of autonomous delay-differential equations with analytic nonlinearities are themselves analytic. On the other hand, the two authors of this paper more recently showed that bounded solutions of certain delay-differential equations, again with analytic nonlinearities, can be C ∞ smooth, yet not be analytic for certain ranges of the independent variable t. In this paper we extend the 1973 results to obtain analytic solutions of a broader class of delay-differential equations, including a wide variety of nonautonomous equations. Nevertheless, there are still equations with analytic nonlinearities possessing global bounded C ∞ solutions for which analyticity is unknown. This is the case, for example, for the equation where g is analytic and where y = 0 is a hyperbolic equilibrium when ε = 0.

Hausdorff dimension of invariant sets and positive linear operators

OF THE DISSERTATION Hausdorff Dimension of Invariant Sets and Positive Linear Operators by Amit P... more OF THE DISSERTATION Hausdorff Dimension of Invariant Sets and Positive Linear Operators by Amit Priyadarshi Dissertation Director: Roger D. Nussbaum In this thesis we obtain theorems which give the Hausdorff dimension of the invariant set for a family of contraction mappings on a complete, perfect metric space. We obtain results for the graph-directed systems similar to that of Mauldin and Williams [17] but for more general contraction mappings which are “infinitesimal similitudes” rather than just “similitudes”. Also the underlying spaces are not assumed to be finite dimensional. For finite graph-directed systems our results are valid on any bounded, complete, perfect metric space. We use the theory of positive linear operators and generalizations of the Krein-Rutman theorem to characterize the Hausdorff dimension as the unique value of σ > 0 for which r(Lσ) = 1, where Lσ, σ > 0, is a naturally associated family of positive linear operators and r(Lσ) denotes the spectral radi...

Functional Differential Equations

Bifurcation Theory and Periodic Solutions of Some Autonomous Functional Differential Equations

Dynamical Systems, 1976

Publisher Summary This chapter focuses on the bifurcation theory and periodic solutions of some a... more Publisher Summary This chapter focuses on the bifurcation theory and periodic solutions of some autonomous functional differential equations. A number of authors have studied nonlinear, autonomous functional differential bifurcation theorem that implies the known existence results for periodic solutions and also new information concerning the structure of the set of periodic solutions, for example, how the period of a periodic solution varies with parameters in the equation. The chapter also reviews study a van der Pol equation with time lag. I If C is a topological space, x 0 ∈ C , W an open neighborhood of x 0 , and f: W — { x 0 } → C a continuous map, then x 0 is an ejective point of ƒ if there exists an open neighborhood U 0 of x 0 such that for every x ɛ U 0 — {x 0 }, there is a positive integer m ( x ) = m such that F m ( x ) is defined and F m ( x ) ∉ U 0 . If C is a topological space, y 0 ɛ C, W an open neighborhood of y 0 , and ƒ : W → C a continuous map, then y 0 is called an attractive point for ƒ if there exists an open neighborhood U 0 of y 0 and for any open neighborhood V of y 0 there exists an integer m ( V ) = m such that f j ( U 0 ) ⊂ W for 0 ≤ j ≤ m and f j ( U 0 ) ⊂ V for j ≥ m .

Periodic solutions of autonomous functional differential equations

Bulletin of the American Mathematical Society, 1973

The purpose of this note is to indicate some applications of a new fixed point theorem to the que... more The purpose of this note is to indicate some applications of a new fixed point theorem to the question of periodic solutions of nonlinear autonomous functional differential equations. The techniques developed give the standard periodicity examples in the literature and some new results, notably for the neutral case, which do not seem accessible by previous methods.

Periodic points of nonexpansive maps

Idempotency, 1998

Extension of order-preserving maps on a cone

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2003

We examine the problem of extending, in a natural way, order-preserving maps that are defined on ... more We examine the problem of extending, in a natural way, order-preserving maps that are defined on the interior of a closed cone K1 (taking values in another closed cone K2) to the whole of K1.We give conditions, in considerable generality (for cones in both finite- and infinite-dimensional spaces), under which a natural extension exists and is continuous. We also give weaker conditions under which the extension is upper semi-continuous.Maps f defined on the interior of the non-negative cone K in RN, which are both homogeneous of degree 1 and order preserving, are non-expanding in the Thompson metric, and hence continuous. As a corollary of our main results, we deduce that all such maps have a homogeneous order-preserving continuous extension to the whole cone. It follows that such an extension must have at least one eigenvector in K – {0}. In the case where the cycle time χ(f) of the original map does not exist, such eigenvectors must lie in ∂K – {0}.We conclude with some discussions...

Generalizations of the Perron-Frobenius theorem for nonlinear maps

Memoirs of the American Mathematical Society, 1999

Let K n = fx 2 R n j x i 0; 1 i ng and suppose that f : K n ! K n is nonexpansive with respect to... more Let K n = fx 2 R n j x i 0; 1 i ng and suppose that f : K n ! K n is nonexpansive with respect to the l 1 -norm, kxk 1 = P n i=1 x i , and f (0) = 0. It is known (see 1]) that for every x 2 K n there exists a periodic point = x 2 K n (so f p ( ) = for some minimal positive integer p = p ) and f k (x) approaches ff j ( ) j 0 j < pg as k tends to in nity. In a previous paper 13] the set P 2 (n) of positive integers p for which there exists a map f as above and a periodic point 2 K n of minimal period p was related to the idea of \admissible arrays" and a set Q(n) determined by certain arithmetical and combinatorial constraints. In a sequel to this paper 14] it is proved that P 2 (n) = Q(n) for all n, but the computation of Q(n) is highly nontrivial. Here we derive a variety of theorems about admissible arrays and use these theorems to compute Q(n) explicitly for n 50 and prove that P (n) = P 2 (n) = Q(n) for n 50, where P (n) is a naturally occurring set de ned below.

Positive solutions of nonlinear elliptic boundary value problems

Journal of Mathematical Analysis and Applications, 1975