Concave iteration semigroups of linear continuous set-valued functions

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Andrzej Smajdor ◽  
Wilhelmina Smajdor
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Nonempty Interior ◽  
Iteration Semigroup

AbstractLet {F t: t ≥ 0} be a concave iteration semigroup of linear continuous set-valued functions defined on a convex cone K with nonempty interior in a Banach space X with values in cc(K). If we assume that the Hukuhara differences F 0(x) − F t (x) exist for x ∈ K and t > 0, then D t F t (x) = (−1)F t ((−1)G(x)) for x ∈ K and t ≥ 0, where D t F t (x) denotes the derivative of F t (x) with respect to t and $$G(x) = \mathop {\lim }\limits_{s \to 0} {{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} \mathord{\left/ {\vphantom {{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} {\left( { - s} \right)}}} \right. \kern-\nulldelimiterspace} {\left( { - s} \right)}}$$ for x ∈ K.

Commutativity of set-valued cosine families

2014 ◽  
Vol 12 (12) ◽  
Author(s):  
Andrzej Smajdor ◽  
Wilhelmina Smajdor
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Real Banach Space ◽  
Nonempty Interior ◽  
Convex Compact ◽  
Closed Convex Cone ◽  
Cosine Family ◽  
Cosine Families ◽  
Nonempty Convex ◽  
The Family

AbstractLet K be a closed convex cone with nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. If {F t: t ≥ 0} is a regular cosine family of continuous additive set-valued functions F t: K → cc(K) such that x ∈ F t(x) for t ≥ 0 and x ∈ K, then $F_t \circ F_s (x) = F_s \circ F_t (x)fors,t \geqslant 0andx \in K$.

scholarly journals Best Simultaneous approximation of quasi-continuous functions by monotone functions

1991 ◽  
Vol 50 (3) ◽  
pp. 391-408 ◽  
Author(s):  
Salem M. A. Sahab
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Simultaneous Approximation ◽  
Continuous Functions ◽  
Unit Interval ◽  
Closed Convex Cone ◽  
Monotone Functions ◽  
Measurable Functions ◽  
Best Simultaneous Approximation ◽  
Nondecreasing Functions

AbstractLet Q denote the Banach space (under the sup norm) of quasi-continuous functions on the unit interval [0, 1]. Let ℳ denote the closed convex cone comprised of monotone nondecreasing functions on [0, 1]. For f and g in Q and 1 < p < ∞, let hp denote the best Lp-simultaneous approximant of f and g by elements of ℳ. It is shown that hp converges uniformly as p → ∞ to a best L∞-simultaneous approximant of f and g by elements of ℳ. However, this convergence is not true in general for any pair of bounded Lebesgue measurable functions. If f and g are continuous, then each hp is continuous; so is limp→∞ hp = h∞.

Some families of sublinear correspondences

2019 ◽  
Vol 25 (1) ◽  
pp. 91-95
Author(s):  
Parvaneh Najmadi ◽  
Masoumeh Aghajani
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Real Banach Space ◽  
Closed Convex Cone ◽  
Nonempty Convex

Abstract Let K be a closed convex cone in a real Banach space, {H\colon K\to\operatorname{cc}(K)} a continuous sublinear correspondence with nonempty, convex and compact values in K, and let {f\colon\mathbb{R}\to\mathbb{R}} be defined by {f(t)=\sum_{n=0}^{\infty}a_{n}t^{n}} , where {t\in\mathbb{R}} , {a_{n}\geq 0} , {n\in\mathbb{N}} . We show that the correspondence {F^{t}(x)\mathrel{\mathop{:}}=\sum_{n=0}^{\infty}a_{n}t^{n}H^{n}(x),(x\in K)} is continuous and sublinear for every {t\geq 0} and {F^{t}\circ F^{s}(x)\subseteq\sum_{n=0}^{\infty}c_{n}H^{n}(x)} , {x\in K} , where {c_{n}=\sum_{k=0}^{n}a_{k}a_{n-k}t^{k}s^{n-k}} , {t,s\geq 0} .

Convergence of inexact orbits of monotone nonexpansive mappings

2018 ◽  
Vol 34 (3) ◽  
pp. 405-410
Author(s):  
SIMEON REICH ◽  
◽  
ALEXANDER J. ZASLAVSKI ◽  
Keyword(s):  
Banach Space ◽  
Asymptotic Behavior ◽  
Convex Cone ◽  
Nonexpansive Mappings ◽  
Ordered Banach Space

We study monotone nonexpansive self-mappings of a closed and convex cone in an ordered Banach space with particular emphasis on the asymptotic behavior of their inexact iterates.

On a Multivalued Differential Problem

2003 ◽  
Vol 13 (07) ◽  
pp. 1877-1882 ◽  
Author(s):  
Andrzej Smajdor
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Real Banach Space ◽  
Convex Compact ◽  
Closed Convex Cone ◽  
Continuous Linear ◽  
Differential Problem ◽  
Linear Set ◽  
Nonempty Convex ◽  
The Family

Let K be a closed convex cone with the nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. Assume that two continuous linear set-valued functions G, Ψ : K → cc(K) are given. The following problem is considered: [Formula: see text] for t ≥ 0 and x ∈ K, where DtΦ(t, x) denotes the Hukuhara derivative of Φ(t, x). with respect to t.

scholarly journals On the monotone simultaneous approximation on [0, 1]

1988 ◽  
Vol 38 (3) ◽  
pp. 401-411 ◽  
Author(s):  
Salem M.A. Sahab
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Simultaneous Approximation ◽  
Closed Interval ◽  
Continuous Functions ◽  
Closed Convex Cone ◽  
Measurable Functions ◽  
Decreasing Functions

Let Ω denote the closed interval [0, 1] and let bA denote the set of all bounded, approximately continuous functions on Ω. Let Q denote the Banach space (sup norm) of quasi-continuous functions on Ω. Let M denote the closed convex cone in Q comprised of non-decreasing functions. Let hp, 1 < p < ∞, denote the best Lp-simultaneaous approximation to the bounded measurable functions f and g by elements of M. It is shown that if f and g are elements of Q, then hp converges unifornily to a best L1-simultaneous approximation of f and g. We also show that if f and g are in bA, then hp is continuous.

scholarly journals Isometric Reflection Vectors and Characterizations of Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-3
Author(s):  
Donghai Ji ◽  
Senlin Wu
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Number Theory ◽  
Recent Result ◽  
Hilbert Spaces ◽  
Unit Sphere ◽  
Substantial Reduction ◽  
Nonempty Interior

A known characterization of Hilbert spaces via isometric reflection vectors is based on the following implication: if the set of isometric reflection vectors in the unit sphereSXof a Banach spaceXhas nonempty interior inSX, thenXis a Hilbert space. Applying a recent result based on well-known theorem of Kronecker from number theory, we improve this by substantial reduction of the set of isometric reflection vectors needed in the hypothesis.

scholarly journals The Kreps-Yan theorem forL∞

2005 ◽  
Vol 2005 (17) ◽  
pp. 2749-2756 ◽  
Author(s):  
D. B. Rokhlin
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Linear Functional ◽  
Closed Convex Cone ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
Usual Approach

We prove the following version of the Kreps-Yan theorem. For any norm-closed convex coneC⊂L∞such thatC∩L+∞={0}andC⊃−L+∞, there exists a strictly positive continuous linear functional, whose restriction onCis nonpositive. The technique of the proof differs from the usual approach, applicable to a weakly Lindelöf Banach space.

scholarly journals Monotone Flows with Dense Periodic Orbits

2019 ◽  
Vol 12 (4) ◽  
pp. 1350-1359
Author(s):  
Morris W. Hirsch
Keyword(s):  
Periodic Orbits ◽  
Convex Cone ◽  
Periodic Points ◽  
Nonempty Interior ◽  
Closed Convex Cone ◽  
Partially Ordered ◽  
Open Set

Let R d be partially ordered by a closed convex cone K ⊂ R d having nonempty interior: y x ⇐⇒ y − x ∈ K. Assume X ⊂ R d is a connected open set, and ϕ a flow on X that is monotone for this order: If y x and t ≥ 0, then ϕ t y ϕ t y. Theorem: If periodic points are dense, ϕ is globally periodic.

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