Applications of variable discounting dynamic programming to iterated function systems and related problems

Cioletti, L.; Oliveira, Elismar R.

doi:10.1088/1361-6544/aaf087

Cited by 8 publications

(12 citation statements)

References 32 publications

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“…Remark 13.1 Let us observe here that X and A are nonempty sets. In the literature, discount maps δ and variable δ-discounted Bellman equations are studied in the case when X is a complete metric space, A is a metric space, B(X) is the set of all continuous bounded real-valued maps on X, (B(X), P) is a complete metric space, u and f are continuous, Ψ (x) is a compact set for each x ∈ X, and the dynamic system (B(X), B), B : B(X) → B(X), is a continuous generalized Matkowski contraction (see, e.g., [14,26,34]).…”

Section: Discount Maps δ and Convergence Existence And Uniqueness Rementioning

confidence: 99%

“…Concerning the Bellman functional equations (see [6][7][8]), variable δ-discounted Bellman equations (see, e.g., [14,26,34]) and Volterra integral equations (see [50]), most of the results contained in several works and books require such assumptions which (by using various techniques or by utilizing various known fixed point theorems) imply that the appropriate Bellman and Volterra operators are continuous (on suitable Banach spaces or complete metric spaces or sequentially complete locally convex vector spaces).…”

Section: Introductionmentioning

confidence: 99%

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Set-valued Leader type contractions, periodic point and endpoint theorems, quasi-triangular spaces, Bellman and Volterra equations

Włodarczyk

2020

Fixed Point Theory Appl

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Set-valued contractions of Leader type in quasi-triangular spaces are constructed, conditions guaranteeing the existence of nonempty sets of periodic points, fixed points and endpoints of such contractions are established, convergence of dynamic processes of these contractions are studied, uniqueness properties are derived, and single-valued cases are considered. Investigated dynamic systems are not necessarily continuous and spaces are not necessarily sequentially complete or Hausdorff. Obtained results suggest, in particular, strategies to new studies of functional Bellman equations and variable discounted Bellman equations in metric spaces and integral Volterra equations in locally convex spaces. Results in this direction are also presented in this paper. More precisely, without continuity of Bellman and Volterra appropriate operators, the sets of solutions of these equations (which are periodic points of these operators) are studied and new and general convergence, existence and uniqueness theorems concerning such equations are proved. MSC: 47H04; 47H10; 54A05; 65J15; 65Q20; 45D05 (A) We say that a family N C;A = {N α : α ∈ A} of maps N α : X → [0, ∞), α ∈ A, is a locally convex quasi-triangular family on X if(B) A locally convex quasi-triangular space (X, N C;A ) is a set X together with the locally convex quasi-triangular family N C;A = {N α : α ∈ A} on X. (C) Let (X, N C;A ) be a locally convex quasi-triangular space. We say that N C;A is separating onRemark 2.2 We see that each locally convex quasi-triangular space is a symmetric quasitriangular space. Indeed, if X is a vector space over K and N C;is a symmetric and quasi-triangular family and (X, P C;A ) is a symmetric quasi-triangular space.Remark 2.3 Let (X, P C;A ) be a quasi-triangular space. In general, the distances P α , α ∈ A, do not vanish on the diagonal, are asymmetric, and do not satisfy the triangle in-We will use (x m : m ∈ N) ⊂ X as a sequence and as a set as the situation demands. Asymmetry of P α , α ∈ A, justify the use of term "left" and term "right". When the symmetry holds, then term "left" and term "right" are identical. Definition 2.3 Let (X, P C;A ) be a quasi-triangular space.(A) We say that (Let (X, P C;A ) be a quasi-triangular space. The set-valued dynamic system on (X, P C;A ) is defined as a pair (X, T), where T : X → 2 X ; here, 2 X denotes the family of all nonempty subsets of X. The single-valued dynamic system on (X, P C;A ) is defined as a pair (X, T), where T is a single-valued map T : X → X, i.e., ∀ x∈X {T(x) ∈ X}.For q ∈ N and for set-valued and single-valued dynamic systems (X, T), we defineLet (X, T) be a set-valued dynamic system on (X, P C;A ). For each w 0 ∈ X, we denote by O X,T (w 0 ) the set of all dynamic processes or trajectories starting at w 0 or motions of the system (X, T) (see [1][2][3]56]), i.e.,By Fix X (T), Per X (T) and End X (T) we denote the sets of all fixed points, periodic points and endpoints (stationary points) of (X, T), respectively, i.e., Fix X (T) = {w ∈ X : w ∈ T(w)}, Per X (T) = {w ∈ X...

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Section: Discount Maps δ and Convergence Existence And Uniqueness Rementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Set-valued Leader type contractions, periodic point and endpoint theorems, quasi-triangular spaces, Bellman and Volterra equations

Włodarczyk

2020

Fixed Point Theory Appl

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Section: Dynamic Programming and The Boundary Of λmentioning

confidence: 99%

“…The notation and the main results in dynamic programming presented here are from [JMN14] and the results on discounted limits are from [CO19]. We consider a decision-making process S = {X, A, ψ, f, u, δ} given by: Assuming such hypothesis we can show that for each fixed 0 < λ < 1 there exists an unique Lipschitz continuous function v + λ which satisfies the Bellman equation…”

Section: Dynamic Programming and The Boundary Of λmentioning

confidence: 99%

On the connection between a skew product IFS and the ergodic optimization for a finite family of potentials

Oliveira

2019

Dynamical Systems

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We study a skew product IFS on the cylinder defined by Baker-like maps associated to a finite family of potential functions and the doubling map. We show that there exist a compact invariant set with attractive behavior and a random SRB measure whose support is in that set. We also study the IFS ergodic optimization problem for that finite family of potential functions and characterize the maximizing measures and the critical value through a discounting limit. This shows the connection between this maximization problem and the superior boundary of the compact invariant set, which is described as a graph of the solution of the Bellman equation.

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“…In this section we show how one can construct equilibrium states using the spectral analysis of the transfer operator. The results present here are based on Theorem 3.27 of [CO17] which is a result about Dynamic Programming. In order to make its statement and this section as self-contained as possible we provided here the needed background.…”

Section: Applications and Constructive Approach To Equilibrium Statesmentioning

confidence: 99%

Thermodynamic Formalism for Iterated Function Systems with Weights

Cioletti,

Oliveira

2017

Preprint

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This paper introduces an intrinsic theory of Thermodynamic Formalism for Iterated Functions Systems with general positive continuous weights (IFSw). We study the spectral properties of the Transfer and Markov operators and one of our first results is the proof of the existence of at least one eigenprobability for the Markov operator associated to a positive eigenvalue. Sufficient conditions are provided for this eingenvalue to be the spectral radius of the transfer operator and we also prove in this general setting that positive eigenfunctions of the transfer operator are always associated to its spectral radius.We introduce variational formulations for the topological entropy of holonomic measures and the topological pressure of IFSw's with weights given by a potential. A definition of equilibrium state is then natural and we prove its existence for any continuous potential. We show, in this setting, a uniqueness result for the equilibrium state requiring only the Gâteaux differentiability of the pressure functional. We also recover the classical formula relating the powers of the transfer operator and the topological pressure and establish its uniform convergence. In the last section we present some examples and show that the results obtained can be viewed as a generalization of several classical results in Thermodynamic Formalism for ordinary dynamical systems.

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Applications of variable discounting dynamic programming to iterated function systems and related problems

Cited by 8 publications

References 32 publications

Set-valued Leader type contractions, periodic point and endpoint theorems, quasi-triangular spaces, Bellman and Volterra equations

Set-valued Leader type contractions, periodic point and endpoint theorems, quasi-triangular spaces, Bellman and Volterra equations

On the connection between a skew product IFS and the ergodic optimization for a finite family of potentials

Thermodynamic Formalism for Iterated Function Systems with Weights

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