On a Multivalued Differential Problem

2003 ◽  
Vol 13 (07) ◽  
pp. 1877-1882 ◽  
Author(s):  
Andrzej Smajdor

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.

2014 ◽  
Vol 12 (12) ◽  
Author(s):  
Andrzej Smajdor ◽  
Wilhelmina Smajdor

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


2019 ◽  
Vol 25 (1) ◽  
pp. 91-95
Author(s):  
Parvaneh Najmadi ◽  
Masoumeh Aghajani

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} .


2005 ◽  
Vol 2005 (17) ◽  
pp. 2749-2756 ◽  
Author(s):  
D. B. Rokhlin

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.


Author(s):  
Salem M. A. Sahab

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∞.


1967 ◽  
Vol 7 (2) ◽  
pp. 129-134 ◽  
Author(s):  
Sadayuki Yamamuro

Let E be a real Banach space. The set of all continuous linear mappings of E into E is a Banach algebra under the usual algebraic operations and the operator bound as norm. We denote this Banach algebra by ℒ, if E is a separate Hilbert space.


2020 ◽  
Vol 1664 (1) ◽  
pp. 012038
Author(s):  
Saied A. Jhonny ◽  
Buthainah A. A. Ahmed

Abstract In this paper, we ⊥ B J C ϵ -orthogonality and explore ⊥ B J C ϵ -symmetricity such as a ⊥ B J C ϵ -left-symmetric ( ⊥ B J C ϵ -right-symmetric) of a vector x in a real Banach space (𝕏, ‖·‖𝕩) and study the relation between a ⊥ B J C ϵ -right-symmetric ( ⊥ B J C ϵ -left-symmetric) in ℐ(x). New results and proofs are include the notion of norm attainment set of a continuous linear functionals on a reflexive and strictly convex Banach space and using these results to characterize a smoothness of a vector in a unit sphere.


2018 ◽  
Vol 27 (1) ◽  
pp. 63-70
Author(s):  
Adesanmi Alao Mogbademu ◽  

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.


1988 ◽  
Vol 38 (3) ◽  
pp. 401-411 ◽  
Author(s):  
Salem M.A. Sahab

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.


1968 ◽  
Vol 9 (2) ◽  
pp. 123-127 ◽  
Author(s):  
I. Tweddle

In [2], R. C. James proved that a weakly closed subset X of a real Banach space is weakly compact if and only if each continuous linear form attains its supremum on X. He also extended the result to the locally convex case, and, in [5], J. D. Pryce gave a simplified proof of the general result that is recorded below for reference in the sequel.


2020 ◽  
Vol 53 (2) ◽  
pp. 192-205
Author(s):  
T. Vasylyshyn ◽  
A. Zagorodnyuk

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 $L_^{(\mathbb{R})}\infty[0,1]$.


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