diff --git "a/0dAyT4oBgHgl3EQfbfdc/content/tmp_files/load_file.txt" "b/0dAyT4oBgHgl3EQfbfdc/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/0dAyT4oBgHgl3EQfbfdc/content/tmp_files/load_file.txt" @@ -0,0 +1,694 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf,len=693 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='00263v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='CA] 31 Dec 2022 ON BOCHNER’S ALMOST-PERIODICITY CRITERION PHILIPPE CIEUTAT Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We give an extension of Bochner’s criterion for the almost periodic func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' By using our main result, we extend two results of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Haraux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The first is a generalization of Bochner’s criterion which is useful for periodic dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The second is a characterization of periodic functions in term of Bochner’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' 2020 Mathematic Subject Classification: 35B10, 35B40, 42A75, 47H20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Keywords: Bochner almost periodicity, periodic function, almost periodic function, asymptotically almost periodic function, nonlinear semigroup, periodic dynamical sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Introduction The almost periodic functions in the sense of Bohr have been characterized by Bochner by means of a compactness criterion in the space of the bounded and continuous functions [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The Bochner’s criterion plays an essential role in the theory and in applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We give a new almost-periodicity criterion for functions with values in a given complete metric space which is useful to study the almost periodicity of solutions of dynamical systems governed by a family of operators with a positive parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' This criterion is an extension of Bochner’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then Haraux gave a generalization of Bochner’s criterion [9, Theorem 1], called a simple almost-periodicity criterion which is useful for periodic dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' From our result, we deduce an extension of this criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We also obtain an extension of an other result of Haraux which characterizes the periodic functions in terms of the Bochner’s criterion [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' In the same spirit, we treat the asymptotically almost periodic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We give a description of this article, the precise definitions will be given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Throughout this section (X, d) is a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' An almost periodic function u : R → X in the sense of Bohr is characterized by the Bochner’s criterion which is the following: u is bounded and continuous, and from any real sequence of real numbers (τn)n, there exists a subsequence (τφ(n))n such that the sequence of functions (u(t + τφ(n)))n is uniformly convergent on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' In Section 3, we give two extensions of Bochner’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' First u : R → X is an almost periodic if and if only if in the Bochner’s criterion, we impose that the terms of the sequence of real numbers (τn)n are all positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Second u : R → X is an almost periodic if and if only if in the Bochner’s criterion, the convergence of the subsequence of functions (u(t + τφ(n)))n is uniform only on [0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' These improvements are useful to study the almost periodicity of solutions of an evolution equation governed by a family of operators with a positive parameter, in particular for a C0-semigroup of linear operators or more generally, for an autonomous dynamical system (nonlinear semigroup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Universit´e Paris-Saclay, UVSQ, CNRS, Laboratoire de math´ematiques de Versailles, 78000, Versailles, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' E-mail address: philippe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='cieutat@uvsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='fr 1 2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Cieutat From our extension of Bochner’s criterion, we give new proofs which are direct and simpler on known results on the almost periodicity of solutions of autonomous dynamic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Haraux gave a generalization of Bochner’s criterion the called a simple almost-periodicity criterion [9, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' This criterion makes it possible to choose in the Bochner’s cri- terion, the sequence of real numbers (τn)n in a set of the type ωZ which is very useful for periodic dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' From our extension of Bochner’s criterion, in Section 4, we deduce an improvement of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' An asymptotically almost periodic function u : R+ → X is a perturbation of almost periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' A such function is characterized by a property of the type of the Bochner’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' In the same spirit, we extend this char- acterization of asymptotically almost periodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then we apply these results to study the almost periodicity of solutions of periodic dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Bochner’s criterion can also be expressed in terms of the relative compactness of the set {u(· + τ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ R} in a suitable set of continuous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' A periodic function is a special case of almost periodic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' A direct consequence of [8, Proposition 2] given by Haraux characterizes a periodic function in terms of the Bochner’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' This characterization is the following: u : R → X is continuous is periodic if and if only if the set {u(· + τ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ R} is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' In Section 5, By using our improvement of Bochner’s criterion, we give an extension of the Haraux’s characterization of periodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We will also give a result on asymptotically periodic functions of the type of Haraux result described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then we apply these results to study the periodicity of solutions of autonomous dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Notation Let us now give some notations, definitions and properties which will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Throughout this section (X, d) is a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' R, Z and N stand re- spectively for the real numbers, the integers and the natural integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We denote by R+ := {t ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' t ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let E be a topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We denote by C(E, X) the space of all continuous functions from E into X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' When J = R or J = R+, we de- note by BC(J, X) the space of all bounded and continuous functions from J into X equipped with the sup-distance, denoted by d∞(u, v) := sup t∈R d(u(t), v(t)) when J = R and d∞,+(u, v) := sup t≥0 d(u(t), v(t))) when J = R+ for u, v ∈ BC(J, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The metric spaces (BC(R, X), d∞) and (BC(R+, X), d∞,+)) are complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We now give some definitions and properties on almost periodic, asymptotically almost periodic functions with values in a given complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' A subset D of R (respectively of R+) is said to be relatively dense if there exists ℓ > 0 such that D ∩ [α, α + ℓ] ̸= ∅ for all α ∈ R (respectively α ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' A continuous function u : R → X is said to be almost periodic (in the sense of Bohr) if for each ε > 0, the set of ε-almost periods: P(u, ε) = � τ ∈ R ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' sup t∈R d(u(t + τ), u(t)) ≤ ε � is relatively dense in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' An almost periodic function u has its range u(R) relatively compact, that is its closure denoted by cl (u(R)) is a compact set of (X, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We denote the space of all such functions by AP(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' It is a closed metric subspace of (BC(R, X), d∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' An almost periodic function u is uniformly recurrent, that is there exists a sequence of real On Bochner’s almost-periodicity criterion 3 numbers (τn)n such that lim n→+∞ sup t∈R d(u(t + τn), u(t)) = 0 and lim n→+∞ τn = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' To see that consider the Bohr’s definition of u ∈ AP(R, X), then the set of 1 n-almost periods satisfies P(u, 1 n)∩[n, +∞) ̸= ∅, for each integer n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' A useful characterization of almost periodic functions was given by Bochner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The Bochner’s criterion which may be found in [12, Bochner’s theorem, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' 4] in the context of metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Before to cite this criterion, we need to introduce the translation mapping of a function of BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For τ ∈ R and u ∈ BC(R, X), we define the translation mapping Tτu ∈ BC(R, X) by Tτu(t) = u(t + τ) for t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1 (Bochner’s criterion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For u ∈ BC(R, X), the following statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) u ∈ AP(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) The set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ R} is relatively compact in (BC(R, X), d∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Haraux gave a generalization of Bochner’ criterion the called a simple almost-periodicity criterion [9, Theorem 1] which is useful for periodic dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2 (Haraux’s criterion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let D be a relatively dense subset of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The following statements are equivalent for u ∈ BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) u ∈ AP(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) The set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ D} is relatively compact in (BC(R, X), d∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Periodic functions, which are a special case of almost periodic functions, are also char- acterized in terms of Bochner’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' This criterion is a direct consequence of a result of Haraux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' [8, Consequence of Proposition 2] The following statements are equivalent for u ∈ BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) u is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) The set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ R} is a compact set of (BC(R, X), d∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For some preliminary results on almost periodic functions with values in a given com- plete metric space, we refer to the book of Levitan-Zhikov [12] and in the special case of Banach spaces to the book of Amerio-Prouse [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The notion of asymptotic almost periodicity was first introduced by Fr´echet [6] in 1941 in the case where X = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' A continuous function u : R+ → X is said to be asymp- totically almost periodic if there exists v ∈ AP(R, X) such that lim t→∞ d(u(t), v(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' An asymptotic almost periodic function u has its range u(R+) relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We denote the space of all such functions by AAP(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' It is a closed metric subspace of (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' An asymptotic almost periodicity function u : R+ → X is char- acterized by u ∈ APP(R+, X) if and only if u ∈ C(R+, X) and for each ε > 0, there exists M ≥ 0 such that the � τ ≥ 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' sup t≥M d(u(t + τ), u(t)) ≤ ε � is relatively dense in R+ [15, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' In the context of metric spaces, Ruess and Summers give a charac- terization of asymptotically almost periodic functions in the spirit of Bochner’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' 4 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Cieutat To prove this characterization, Ruess and Summers use results from the paper [16] by the same authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For τ ≥ 0 and u ∈ BC(R+, X), we define the translation mapping T + τ u ∈ BC(R+, X) by T + τ u(t) = u(t + τ) for t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' [15, a part of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2 & 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3] Let (X, d) be a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For u ∈ BC(R+, X), the following statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) u ∈ AAP(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) The set {T + τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is relatively compact in (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For some preliminary results on asymptotically almost periodic functions, we refer to the book of Yoshizawa [17] in the case where X is a finite dimensional space, to the book of Zaidman [18] where X is a Babach space and to Ruess and Summers [14, 15, 16] in the general case: X is a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' An improvement of Bochner’s criterion An almost periodic function is characterized by the Bochner’s criterion, recalled in Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Our main result is an extension of Bochner’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then we deduce new proofs which are direct and simpler on known results on the solutions of autonomous dynamic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Before to state our extension of Bochner’s criterion, we need to introduce the restriction operator R : BC(R, X) → BC(R+, X) defined by R(u)(t) := u(t) for t ≥ 0 and u ∈ BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let (X, d) be a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For u ∈ BC(R, X) the following statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) u ∈ AP(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) The set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is relatively compact in (BC(R, X), d∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' iii) The set {R(Tτu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ R} is relatively compact in (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' In our results, the compactness and the relative compactness of a set often intervene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' To prove them, we will often use the following result whose proof is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Recall that a set A of a metric space (E, d) is relatively compact if its closure denoted by cl (A) is a compact set of (E, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let E be a set, (G1, d1) and (G2, d2) be two metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let u : E → G1 and v : E → G2 be two functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Assume there exists M > 0 such that ∀x1, x2 ∈ E, d1(u(x1), u(x2)) ≤ Md2(v(x1), v(x2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) If the metric space (G1, d1) is complete and v(E) is relatively compact in (G2, d2), then u(E) is relatively compact in (G1, d1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) If v(E) is a compact set of (G2, d2), then u(E) is a compact set of (G1, d1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) =⇒ iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' It is obvious by using the Bochner’s criterion and the continuity of the restriction operator R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' iii) =⇒ ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The set u(R) = {R(Tτu)(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ R} is relatively compact in X as the range of {R(Tτu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ R} by the continuous evaluation map at 0 from BC(R+, X) into X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' By assumption, H := cl ({R(Ttu) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' t ∈ R}) is a compact set of (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' On Bochner’s almost-periodicity criterion 5 For all τ ≥ 0, we define φτ : H → X by φτ(h) = h(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The functions φτ are 1-Lipschitz continuous and for each t ∈ R, the set {φτ(R(Ttu)) = u(τ + t) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is included in the relatively compact set u(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' By density of {R(Ttu) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' t ∈ R} in H and the continuity of φτ, it follows that {φτ(h) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is relatively compact in X for each h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' According to Arzel`a-Ascoli’s theorem [11, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' 57], the set {φτ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is relatively compact in C(H, X) equipped with the sup-norm denoted by dC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' From the density of {R(Ttu) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' t ∈ R} in H and the continuity of φτ, we deduce that for τ1 and τ2 ≥ 0, sup h∈H d(φτ1(h), φτ2(h)) = sup t∈R d (φτ1(R(Ttu)), φτ2(R(Ttu))) = sup t∈R d (u(τ1 + t), u(τ2 + t)) = sup t∈R d (Tτ1u(t), Tτ2u(t)), then dC(φτ1, φτ2) = d∞ (Tτ1u, Tτ2u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2, it follows that {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is relatively compact in the complete metric space (BC(R, X), d∞) since {φτ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is also one in (C(H, X), dC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) =⇒ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For τ1, τ2 ≥ 0, d∞(Tτ1u, Tτ2u) := sup t∈R d(u(τ1 + t), u(τ2 + t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Replacing t by t − τ1 − τ2 in the upper bound, we get d∞(Tτ1u, Tτ2u) = d∞(T−τ1u, T−τ2u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then the set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≤ 0} = {T−τu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is relatively compact in BC(R, X) since {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is also one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Therefore the set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ R} is relatively compact in BC(R, X) as the union of two relatively compact sets in BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' According to Bochner’s criterion, u ∈ AP(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' □ The connection between the almost periodicity of a solution of a dynamical system and its stability is well known (see the monograph by Nemytskii & Stepanov [13, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' This weakened form of Bochner’s criterion: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1 makes it possible to obtain direct and simpler proofs on these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let us start by recalling some definitions on dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' A dynamical system or nonlinear semigroup on a complete metric space (X, d) is a one parameter family (S(t))t≥0 of maps from X into itself such that i) S(t) ∈ C(X, X) for all t ≥ 0, ii) S(0)x = x for all x ∈ X, iii) S(t + s) = S(t) ◦ S(s) for all s, t ≥ 0 and iv) the mapping S(·)x ∈ C([0, +∞), X) for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For each x ∈ X, the positive trajectory of x is the map S(·)x : R+ → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' A function u : R → X is called a complete trajectory if we have u(t + τ) = S(τ)u(t), for all t ∈ R and τ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We will need a notion of Lagrange-type stability to ensure that a solution with a relatively compact range is almost periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Recall that (S(t))t≥0 is equicontinuous on a compact set K of X, if forall ε > 0, there exists δ > 0, such that ∀x1, x2 ∈ K, d(x1, x2) ≤ δ =⇒ sup t≥0 d(x1, x2) ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1, we give a new proof which is direct and simpler of the following result which can be found in [10, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' 51] or partly in [12, Markov’s theorem, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let (S(t))t≥0 be a dynamical system on a complete metric space (X, d) and u be a complete trajectory such that u(R) is relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then u is almost periodic if and only if (S(t))t≥0 is equicontinuous on cl (u(R)) the closure of u(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let us denote the compact set K := cl (u(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' It follows by density of u(R) in K and the continuity of S(t), that {S(t)x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' t ≥ 0} ⊂ K for each x ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' According 6 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Cieutat to Arzel`a-Ascoli’s theorem, (S(t))t≥0 is equicontinuous on K if and only if (S(t))t≥0 is relatively compact in C(K, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1, we have u ∈ AP(R, X) if and only if {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is relatively compact in BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then it remains to prove that (S(t))t≥0 is relatively compact in C(K, X) equipped with the sup-norm if and only if {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is relatively compact in (BC(R, X), d∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' This results from the following equalities, for τ1 and τ2 ≥ 0, sup t∈R d (Tτ1u(t), Tτ2u(t)) = sup t∈R d (S(τ1)u(t), S(τ2)u(t)) = sup x∈K d (S(τ1)x, S(τ2)x) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' a) The condition of equicontinuity required by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3 is satisfied by a bounded dynamical system : d (S(t)x1, S(t)x2) ≤ Md (x1, x2) for some M ≥ 1 and in particular for a C0 semigroup of contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' In this case, the almost periodicity of a complete trajectory u having a relatively compact range results from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We can also obtain this result with the implication iii) =⇒ i) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1 and the inequality sup t≥0 d(R(Tτ1u)(t), R(Tτ2u)(t)) = sup t≥0 d (S(t)u(τ1), S(t)u(τ2) ≤ Md(u(τ1), u(τ2)) for τ1, τ2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' b) For a bounded C0-semigroup (S(t))t≥0, the main result of Zaidman [19] asserts that a positive trajectory u with relatively compact range satisfies a condition called the generalized normality property in Bochner’s sense, without concluding that u is almost periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' This condition is nothing but hypothesis iii) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1, so u is almost periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4, we give a new proof which is direct and simpler of the following result which can be found in [15, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' 149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let (S(t))t≥0 be a dynamical system on a complete metric space (X, d) and u be a positive trajectory such that u(R+) is relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then u is asymptotically almost periodic if and only if (S(t))t≥0 is equicontinuous on cl (u(R+)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The proof is analogous to that of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3, using Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4 instead of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1 and by replacing R by R+ and AP(R, X) by AAP(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' An improvement of Haraux’s criterion Haraux gave a generalization of Bochner’scriterion [9, Theorem 1], the called a simple almost-periodicity criterion which is useful for periodic dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' From our main result, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1, we deduce an extension of the Haraux’s criterion, recalled in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' In the same spirit, we extend the well-known characterization of asymptotically almost periodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' To end this section, we give an exemple of application on a periodic dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We give an extension of the Haraux’s criterion (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Recall that we denote by R the restriction operator R : BC(R, X) → BC(R+, X) defined by R(u)(t) := u(t) for t ≥ 0 and u ∈ BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let (X, d) be a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For u ∈ BC(R, X) the following statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) u ∈ AP(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) The set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ D} is relatively compact in (BC(R, X), d∞) where D be a relatively dense subset of R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' On Bochner’s almost-periodicity criterion 7 iii) The set {R(Tτu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ D} is relatively compact in (BC(R+, X), d∞,+) where D be a relatively dense subset of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Our main result, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1 is obviously a particular case of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' But to present our results, it was easier to start with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' To prove Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1, we use Haraux’s criterion and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) =⇒ iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' It is a consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' iii) =⇒ ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' To establish this implication, using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1, it suffices to show that assertion iii) implies that {R(Tτu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ R} is relatively compact in BC(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The proof of this last implication is a slight adaptation of those of those of the Haraux’s criterion given in [9, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' A similar proof will be detailed in the following result as there will be technical issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' To demonstrate that {R(Tτu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ R} is relatively compact, it suffices in the proof of ii) =⇒ i) of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3 to take ℓ = 0 and replace {T + τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ D} by {R(Tτu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) =⇒ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For τ1, τ2 ≥ 0, d∞(Tτ1u, Tτ2u) = sup t∈R d(u(τ1 + t), u(τ2 + t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Replacing t by t − τ1 − τ2 in the upper bound, we get d∞(Tτ1u, Tτ2u) = d∞(T−τ1u, T−τ2u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then the set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ −D} = {T−τu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ D} is relatively compact in BC(R, X) since {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ D} is also one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Therefore the set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ D ∪ (−D)} is relatively compact in BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Moreover D ∪(−D) is a relatively dense subset of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' According to Haraux’s criterion, we have u ∈ AP(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' □ We extend Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4, the well-known characterization of asymptotically almost pe- riodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For τ ∈ R+ and u ∈ BC(R+, X), we define the translation mapping T + τ u ∈ BC(R+, X) by T + τ u(t) = u(t + τ) for t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let (X, d) be a complete metric space and let D be a relatively dense subset of R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For u ∈ BC(R+, X) the following statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) u ∈ AAP(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) The set {T + τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ D} is relatively compact in (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' To establish implication ii) =⇒ i), by using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4, it suffices to prove that assertion ii) implies that {T + τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is relatively compact in (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The proof of this last implication is an adaptation of those of the Haraux’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' But contrary to the proof of implication iii) =⇒ ii) in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1, there are technical issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' These technical difficulties come from the fact that when D is a relatively dense subset in R+, the sets D and [t − ℓ, t] can be disjoint for some 0 ≤ t ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For this reason we give the complete proof of this implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) =⇒ ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' It is a consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) =⇒ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We will prove that assumption ii) implies {T + τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is relatively compact in BC(R+, X), then we conclude by using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The subset D being relatively dense in R+, there exists ℓ > 0 such that D ∩ [α, α + ℓ] ̸= ∅ for all α ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We prove that u is uniformly continuous on [ℓ, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let us fix ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' By assump- tion the set {T + τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ D} is in particular relatively compact in C([0, 2ℓ], X), then it is uniformly equicontinuous on [0, 2ℓ], that is there exists δ > 0 such that s1, s2 ∈ [0, 2l], |s1 − s2| ≤ δ =⇒ sup τ∈D d(u(s1 + τ), u(s2 + τ)) ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1) 8 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Cieutat Let t1, t2 be two real numbers such that t1, t2 ≥ ℓ and |t1 − t2| ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We can assume without loss of generality that ℓ ≤ t1 ≤ t2 ≤ t1 + ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We have D ∩ [t1 − ℓ, t1] ̸= ∅ since t1 − ℓ ≥ 0, then there exists τ ∈ D such that 0 ≤ t1 − τ ≤ t2 − τ ≤ 2l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Taking account (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1), we deduce that d(u(t1), u(t2)) = d(u((t1 − τ) + τ), u((t2 − τ) + τ)) ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Hence u is uniformly continuous on [ℓ, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We prove that {T + τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ ℓ} is relatively compact in BC(R+, X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let (tn)n be a sequence of real numbers such that tn ≥ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We have D ∩ [tn − ℓ, tn] ̸= ∅ for each n ∈ N, since tn − ℓ ≥ 0, then there exist τn ∈ D and σn ∈ [0, l] such that tn = τn + σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' By compactness of the sequences (σn)n in [0, ℓ] and (T + τnu)n in BC(R+, X), it follows that lim n→+∞ σn = σ and lim n→+∞ sup t≥0 d(u(t + τn), v(t)) = 0 (up to a subsequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' From the following inequality sup t≥0 d(u(tn + t), v(σ + t)) ≤ sup t≥0 d(u(τn + σn + t), u(τn + σ + t) + sup t≥0 d(u(τn + t), v(t)) and the uniform continuity of u, we deduce that lim n→+∞ sup t≥0 d(u(tn +t), v(σ +t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then {T + τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ ℓ} is relatively compact in BC(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We prove that u ∈ AAP(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The function u is uniformly continuous on R+, since u is continuous on [0, ℓ] and uniformly continuous on [ℓ, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then the map ˆu : R+ → BC(R+, X) defined by ˆu(τ) = T + τ u for τ ≥ 0 is continuous, consequently the set {T + τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' 0 ≤ τ ≤ ℓ} is relatively compact in BC(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The set {T + τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is relatively compact in BC(R+, X) as the union of two relatively compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' According to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4, u ∈ AAP(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' □ Using corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3, we give a proof which is direct and simpler of the following result which can be found in [7, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Before we recall some definitions on process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' A process on a complete metric space (X, d) according to Dafermos [4] is a two pa- rameter family U(t, τ) of maps from X into itself defined for (t, τ) ∈ R × R+ and such that i) U(t, 0)x = x for all (t, x) ∈ R × X, ii) U(t, σ + τ) = U(t + σ, τ) ◦ U(t, σ) for all (t, σ, τ) ∈ R×R+×R+ and iii) the mapping U(t, ·)x ∈ C([0, +∞), X) for all (t, x) ∈ R×X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For each x ∈ X, the positive trajectory starting of x is the map U(0, ·)x : R+ → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' A function u : R → X is called a complete trajectory if we have u(t + τ) = U(t, τ)u(t) for all (t, τ) ∈ R × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' A process U is said ω-periodic (ω > 0) if U(t + ω, τ) = U(t, τ) for all (t, τ) ∈ R × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' A process U is said bounded if we have for some M ≥ 1 for all (τ, x1, x2) ∈ R+ ×X ×X d (U(0, τ)x1, U(0, τ)x2) ≤ Md (x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' [7, 9], [10, Th´eor`eme 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' 84] Let U be a ω-periodic process on a complete metric space (X, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' If U is bounded, then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) If u is a complete trajectory of U such that u(−ωN) is relatively compact, then u is almost periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) If u is a positive trajectory of U such that u(ωN) is relatively compact, then u is asymptotically almost periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) The process U is ω-periodic, then we have u(nω) = U(−mω, (n+m)ω)u(−mω) = U(0, (n + m)ω)u(−mω) for all n, m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' From the boundedness assumption on U, we On Bochner’s almost-periodicity criterion 9 deduce that d (u(nω), u(mω)) ≤ Md (u(−mω), u(−nω)), then u(ωN) is relatively compact since u(−ωN) is also one, therefore u(ωZ) is relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' From assumptions on the process U, it follows that for all n, m ∈ Z, sup τ≥0 d (u(τ + nω), u(τ + mω)) ≤ Md (u(nω), u(mω)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2) From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2, {R(Tnωu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' n ∈ Z} is relatively compact in (BC(R+, X), d∞,+) since u(ωZ) is also one in (X, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We conclude with Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1 by setting D = ωZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) For all n, m ∈ N, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2) holds on the positive trajectory u, then from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2, {T + nωu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' n ∈ N} is relatively compact in (BC(R+, X), d∞,+) since u(ωN) is also one in (X, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We conclude with Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3 by setting D = ωN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Bochner’s criterion in the periodic case Periodic functions are a special case of almost periodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Haraux gave a charac- terization of periodic functions in terms of Bochner’s criterion which is recalled in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' This criterion is a direct consequence of [8, Proposition 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Haraux established a general result [8, Th´eor`eme 1] implying as a special case a characterization of periodic functions and the fact that any compact trajectory of a one-parameter continuous group is automatically periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' In this section, we give an extension of this characterization of periodic functions in the spirit of the main result of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We also treat the asymptotically periodic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then we apply these results to study the periodicity of solutions of dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Recall that we denote by R the restriction operator R : BC(R, X) → BC(R+, X) defined by R(u)(t) := u(t) for t ≥ 0 and u ∈ BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let (X, d) be a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For u ∈ BC(R, X) the following statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) The function u is ω-periodic (ω > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) The set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is a compact set of (BC(R, X), d∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' iii) The set {R(Tτu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ R} is a compact set of (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) =⇒ ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' From assumption, it follows that the function τ → Tτu from R into BC(R, X) is continuous and ω-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then the {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} = {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' 0 ≤ τ ≤ ω} is a compact set of (BC(R, X), d∞) as the range of a compact set by a continuous map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) =⇒ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For τ1, τ2 ∈ R, d∞(Tτ1u, Tτ2u) := sup t∈R d(u(τ1 + t), u(τ2 + t)), we get d∞(Tτ1u, Tτ2u) = d∞(T−τ1u, T−τ2u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then the set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≤ 0} = {T−τu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is compact in BC(R, X) since {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is also one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Therefore the set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ R} is a compact set of BC(R, X) as the union of two compact sets in BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' According to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3, u is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) =⇒ iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' It is obvious by using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3 and the continuity of the restriction operator R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' iii) =⇒ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' By using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3, we have to prove that K := {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ R} is a compact set of (BC(R, X), d∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' As consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1 and Bohner’s criterion, the set K is relatively compact in (BC(R, X), d∞) and the function u is almost periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' 10 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Cieutat It remains to prove that K is closed in (BC(R, X), d∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let (τn)n be a sequence of real numbers such that lim n→+∞ d∞(Tτnu, v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let us prove that v = Tτu for some τ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' By continuity of the operator R, we have lim n→+∞ d∞,+(R(Tτnu), R(v)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' By assumption, the set {R(Tτu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ R} is in particular closed in (BC(R+, X), d∞,+), then R(v) = R(Tτu) for some τ ∈ R, that is ∀t ≥ 0, v(t) = Tτu(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1) We have to prove that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1) holds on the whole real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The function Tτu is almost periodic as translation of an almost periodic function and v is also one as uniform limit on R of almost periodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let us denote by φ : R → R the function defined by φ(t) := d(Tτu(t), v(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The function φ is almost periodic [12, Property 4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' 3 & 7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' An almost periodic function is uniformly recurrent, then there exists a sequence of real numbers such that lim n→+∞ τn = +∞ and lim n→+∞ φ(t+ τn) = φ(t) for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1), it follows φ(t) = 0 for all t ≥ 0, so we deduce that φ(t) = lim n→+∞ φ(t + τn) = 0 for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then v(t) = Tτu(t) for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' This ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' □ According Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4, if the set {T + τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is relatively compact in BC(R+, X), then the function u is asymptotically almost periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We now give an answer to the question what can be said about the function u when {T + τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is a compact set of BC(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For u ∈ BC(R+, X), we say that u is ω-periodic (ω > 0) on [t0, +∞) for some t0 ≥ 0 if u(t + ω) = u(t) for all t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let (X, d) be a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For u ∈ BC(R+, X) the following statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) There exists t0 ≥ 0 such that u is ω-periodic on [t0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) The set {T + τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is a compact set of (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let u be a function which satisfies condition i) of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) Let us denote by v ∈ C(R, X) the ω-periodic function satisfying u(t) = v(t) for t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' A such function v exists and is unique, v is defined by v(t) = u(t − [ t−t0 ω ]ω) where [ t−t0 ω ] denotes the integer part of t−t0 ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) The function u is a special case of asymptotic almost periodic function where the almost periodic function v is periodic and d(u(t), v(t)) = 0 for t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Proof of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) =⇒ ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let us denote by v the function defined in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1 and the periodicity of v, we have {R(Tτv);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ t0} = {R(Tτv);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ R} is a compact set of (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' First, we have T + τ u = R(Tτv) for τ ≥ t0, then {T + τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ t0} is a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Second, the function u is uniformly continuous on R+, then the function from R+ to BC(R+, X) defined by τ → T + τ u is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then the set {T + τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' 0 ≤ τ ≤ t0} is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Therefore the set {T + τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is a compact set of BC(R+, X) as the union of two compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) =⇒ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' As consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4, the function u is asymptotically almost periodic, that is lim t→∞ d(u(t), v(t)) = 0 for some v ∈ AP(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' An almost periodic On Bochner’s almost-periodicity criterion 11 function is uniformly recurrent, then there exists a sequence of real numbers (tn)n such that lim n→+∞ tn = +∞ and lim n→+∞ v(t + tn) = v(t) for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We deduce that ∀t ∈ R, lim n→+∞ u(t + tn) = v(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2) First we prove that v is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For t ∈ R, τ1, τ2 ≥ 0, we have for n enough large d(u(t + tn + τ1), u(t + tn + τ2)) ≤ sup s≥0 d(u(s + τ1), u(s + τ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2), it follows that sup t∈R d(v(t+τ1), v(t+τ2)) ≤ sup s≥0 d(u(s+τ1), u(s+τ2)) for each τ1 and τ2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' According to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2, {Tτv ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is a compact set of (BC(R, X), d∞) since {T + τ u ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is also one in (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' As consequence of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1, the function v is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Second we prove that: ∃t0 ≥ 0 such that ∀t ≥ 0, v(t) = u(t + t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' By compactness of {T + τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0}, there exists a subsequence (T + tφ(n)u)n such that lim n→+∞ d∞,+(T + tφ(n)u, T + t0 u) = 0 for some t0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2) we deduce that R(v) = T + t0 u, that is v(t) = u(t + t0) for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then u(t) = v(t − t0) for each t ≥ t0 where the function v(· − t0) is periodic on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' □ Now we give an example of application on dynamical systems of Corollary of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1 and Corollary of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For the definition of a dynamical system, see above Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let (S(t))t≥0 be a dynamical system on a complete metric space (X, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) If u is a positive trajectory, then u is periodic on [t0, +∞) for some t0 ≥ 0 if and only if u(R+) is a compact set and (S(t))t≥0 is equicontinuous on u(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) If u is a complete trajectory, then u is periodic if and only if u(R) is a compact set and (S(t))t≥0 is equicontinuous on u(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' iii) There exists a complete trajectory which is periodic if and only if there exists a positive trajectory u such that u(R+) is a compact set and (S(t))t≥0 is equicontinuous on u(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Thus under the assumption of equicontinuity, a complete trajectory of a dynamical system with a compact range is necessarily periodic, although there are almost periodic functions with a compact range, which are not periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' An example of such function is given by Haraux in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Proof of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) Remark that if u is a positive trajectory which is periodic on [t0, +∞) for some t0 ≥ 0, then first u(R+) is compact and second u ∈ AAP(R+, X) (see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' As consequence of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='5, the set (S(t))t≥0 is equicontinuous on u(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Reciprocally assume the positive trajectory u is such that (S(t))t≥0 is equicontinu- ous on the compact set u(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' It remains to prove that the positive trajectory u is periodic on [t0, +∞) for some t0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For each x ∈ u(R+), the map S(·)x is continuous and satisfies S(t)x ∈ u(R+) for each t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then the map S(·)x is bounded and continuous , so the map Φ : u(R+) → BC(R+, X) with Φ(x) = S(·)x is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The continuity of Φ results of the equicontinuity of (S(t))t≥0 on u(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then the set Φ(u(R+)) = {Φ(u(τ)) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is a compact of BC(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Moreover Φ(u(τ))(t) = S(t)u(τ) = u(t + τ) for t and τ ≥ 0, 12 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Cieutat then Φ(u(τ)) = T + τ u, so {T + τ u ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ≥ 0} is a compact set of BC(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' According to Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2, the function u is periodic on [t0, +∞) for some t0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) The proof of ii) is similar to that of i) by using Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3 instead of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='5, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='1 instead of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='2 and by replacing the map Φ : u(R+) → BC(R+, X) with Φ(x) = S(·)x by the map Φ : u(R) → BC(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' This permits to prove that the set {Φ(u(τ)) = R(Tτu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' τ ∈ R} is a compact set of (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' iii) If v is a complete trajectory which is periodic, then v(R) is compact and according to ii), (S(t))t≥0 is equicontinuous on v(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' So the restriction u of v on R+ is a posi- tive trajectory such that u(R+) = v(R+) = v(R) since v is periodic, then (S(t))t≥0 is equicontinuous on the compact set u(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Reciprocally, assume that u is a positive tra- jectory such that (S(t))t≥0 is equicontinuous on the compact set u(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' According to i), u is ω-periodic on [t0, +∞) for some t0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let us denote by v the function defined in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For t ≥ s, there exists n0 ∈ N such that s + n0ω ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The function v is ω-periodic and u is a positive trajectory satisfying u(τ) = v(τ) for τ ≥ t0, then v(t) = v(t + n0ω) = u(t + n0ω) = T(t − s)u(s + n0ω) = T(t − s)v(s + n0ω) = T(t − s)v(s) for t ∈ R and n enough large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then v is a periodic complete trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Under i) of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4, one can have t0 > 0, that is the positive trajectory u is not the restriction of a periodic complete trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For example, consider the bounded dynamical system (S(t))t≥0 on L1(0, 1) defined by (S(t)x)(s) = \uf8f1 \uf8f2 \uf8f3 x(s − t) if t < s < 1 0 if 0 < s < t for x ∈ L1(0, 1) and 0 < t < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For t ≥ 1, we set S(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then all positive trajectories have a compact range and the alone complete trajectory is the null function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Thus all positive trajectories are not the restriction of a periodic complete trajectory except the null function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Not all dynamical systems have this pathology, some systems are such that if two positive trajectories have the same value at the same time, then they are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' If we consider such systems, we get more refined results from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' A dynamical system (S(t))t≥0 has the backward uniqueness property if any two positive trajectories having the same value at t = t0 ≥ 0 coincide for any other t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' This property is equivalent to S(t) ∈ C(X, X) is injective for each t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' We say that a positive trajectory u is extendable to a periodic complete trajectory, if there exists a periodic complete trajectory such that its restriction on R+ is u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let (S(t))t≥0 be a dynamical system on a complete metric space (X, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Assume that (S(t))t≥0 has the backward uniqueness property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) If u is a positive trajectory, then u is periodic on R+ if and only if u(R+) is a compact set and (S(t))t≥0 is equicontinuous on u(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' In this case the positive trajectory u is extendable to a periodic complete trajectory v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) If v is a complete trajectory, then v is periodic if and only if v(R+) is a compact set and (S(t))t≥0 is equicontinuous on v(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' On Bochner’s almost-periodicity criterion 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' i) The direct implication results of i) of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For the reciprocal implica- tion we use i) of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then the positive trajectory u is ω-periodic on [t0, +∞) for some t0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let us denote by v ∈ C(R, X) the ω-periodic function satisfying u(t) = v(t) for t ≥ t0 (see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The restriction of v on R+ and u are two positive trajectories having the same value at t = t0 (t0 ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' From the backward uniqueness property, we have u(t) = v(t) for t ≥ 0, then u is periodic on R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' By build, v is periodic and as in the proof of iii) of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4, we deduce that v is a complete trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' ii) The direct implication results of ii) Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4, since v(R+) = v(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' For the reciprocal implication, we consider v a complete trajectory such that v(R+) is a compact set and (S(t))t≥0 is equicontinuous on v(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Then the restriction u of the complete trajectory v on R+ is a positive trajectory such that u(R+) is compact and (S(t))t≥0 is equicontinuous on u(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' According to i), u is ω-periodic on R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Let us denote by w ∈ C(R, X) the ω-periodic function satisfying u(t) = w(t) for t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' As in proof of iii) Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content='4, we deduce that w is a complete trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Fix T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' The two maps ˜v and ˜w : R+ → X defined by ˜v = v(·, −T) and ˜w = w(·, −T) are two positive trajectories having the same value at t = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' From the backward uniqueness property, we have ˜v = ˜w, that is v(t) = w(t) for t ≥ −T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Since T is arbitrary, then v(t) = w(t) for each t ∈ R where w is a periodic complete trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' This proves that v is a periodic complete trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' □ References [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'} +page_content=' Amerio, G.' metadata={'source': 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