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 $F_t \circ F_s (x) = F_s \circ F_t (x)fors,t \geqslant 0andx \in K$.

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.

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} .

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∞.

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.

Fixed points of nearly weak uniformly L-Lipschitzian mappings in real Banach spaces

2018 ◽  
Vol 27 (1) ◽  
pp. 63-70
Author(s):  
Adesanmi Alao Mogbademu ◽  
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Real Banach Space ◽  
Unique Fixed Point ◽  
Stability Result ◽  
Iteration Scheme ◽  
Lipschitzian Mapping ◽  
Nonempty Convex Subset ◽  
Lipschitzian Mappings ◽  
Nonempty Convex

Let K be a nonempty convex subset of a real Banach space X. Let T be a nearly weak uniformly L-Lipschitzian mapping. A modified Mann-type iteration scheme is proved to converge strongly to the unique fixed point of T. Our result is a significant improvement and generalization of several known results in this area of research. We give a specific example to support our result. Furthermore, an interesting equivalence of T-stability result between the convergence of modified Mann-type and modified Mann iterations is included.

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 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.

scholarly journals Symmetric polynomials on the Cartesian power of the real Banach space $L_\infty[0,1]$

2020 ◽  
Vol 53 (2) ◽  
pp. 192-205
Author(s):  
T. Vasylyshyn ◽  
A. Zagorodnyuk
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Analytic Functions ◽  
Symmetric Functions ◽  
Real Banach Space ◽  
Symmetric Polynomials ◽  
The Real ◽  
Cartesian Power ◽  
The Family ◽  
Algebraic Basis

We construct an algebraic basis of the algebra of symmetric (invariant under composition of the variable with any measure preserving bijection of $[0,1]$) continuous polynomials on the $n$th Cartesian power of the real Banachspace $L_^{(\mathbb{R})}\infty[0,1]$ of Lebesgue measurable essentially bounded real valued functions on $[0,1].$ Also we describe the spectrum of the Fr\'{e}chet algebra $A_s(L_^{(\mathbb{R})}\infty[0,1])$ of symmetric real-valued functions on the space $L_^{(\mathbb{R})}\infty[0,1]$, which is the completion of the algebra of symmetric continuous real-valued polynomials on  $L_^{(\mathbb{R})}\infty[0,1]$ with respect to the family of norms of uniform convergence of complexifications of polynomials. We show that $A_s(L_^{(\mathbb{R})}\infty[0,1])$ contains not only analytic functions. Results of the paper can be used for investigations of algebras of symmetric functions on the $n$th Cartesian power of the Banach space $L_^{(\mathbb{R})}\infty[0,1]$.

scholarly journals The local spectral radius of a nonnegative orbit of compact linear operators

2016 ◽  
Vol 66 (3) ◽  
Author(s):  
Mihály Pituk
Keyword(s):  
Banach Space ◽  
Spectral Radius ◽  
Additional Assumption ◽  
Real Banach Space ◽  
Linear Operators ◽  
Partial Ordering ◽  
Positive Eigenvector ◽  
Local Spectral Radius

AbstractWe consider orbits of compact linear operators in a real Banach space which are nonnegative with respect to the partial ordering induced by a given cone. The main result shows that under a mild additional assumption the local spectral radius of a nonnegative orbit is an eigenvalue of the operator with a positive eigenvector.

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