fatou property
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2021 ◽  
Vol 110 (1-2) ◽  
pp. 196-209
Author(s):  
G. A. Karagulyan ◽  
I. N. Katkovskaya ◽  
V. G. Krotov

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Awad A. Bakery ◽  
O. M. Kalthum S. K. Mohamed

In the recent past, some researchers studied some fixed point results on the modular variable exponent sequence space ℓ r . ψ , where ψ v = ∑ a = 0 ∞ 1 / r a v a r a and r a ≥ 1 . They depended on their proof that the modular ψ has the Fatou property. But we have explained that this result is incorrect. Hence, in this paper, the concept of the premodular, which generalizes the modular, on the Nakano sequence space such as its variable exponent in 1 , ∞ and the operator ideal constructed by this sequence space and s -numbers is introduced. We construct the existence of a fixed point of Kannan contraction mapping and Kannan nonexpansive mapping acting on this space. It is interesting that several numerical experiments are presented to illustrate our results. Additionally, some successful applications to the existence of solutions of summable equations are introduced. The novelty lies in the fact that our main results have improved some well-known theorems before, which concerned the variable exponent in the aforementioned space.


2021 ◽  
Author(s):  
Shengzhong Chen

The problem of optimal capital and risk allocation among economic agents, has played a predominant role in the respective academic and industrial research areas for decades. Typically as risk occurs in face of randomness the risks which are to be measured are identified with real-valued random variables on some probability space (Ω, F, P). Consider a model space X , and n economic agents with initial endowments X1, · · · , Xn ∈ X who assess the riskiness of their positions by means of law-invariant convex risk measures ρi : X → (−∞,∞]. In order to minimize total and individual risk, the agents redistribute the aggregate endowment X = X1 + · · · + Xn among themselves. An optimal capital and risk allocation Y1, · · · , Yn satisfies Y1 + · · · + Yn = X and ρ1(Y1) + · · · + ρ(Yn) = inf nXn i=1 ρi(Xi) : Xi ∈ X , i = 1, . . . , n, and Xn i=1 Xi = X o , (0.1) where n i=1ρi(X) = inf nPn i=1 ρi(Xi) : Xi ∈ X , i = 1, . . . , n, and Pn i=1 Xi = X o is the inf-convolution of ρ1, ..., ρn. In 2008, Filipovi´c and Svindland proved that if X is an L p (P) for some 1 ≤ p ≤ ∞ and ρi satisfy a suitable continuity condition (i.e. Fatou property), then Problem (0.1) always admits a solution. To reflect the fact of randomness of risk, we should consider the model space X chosen for risk evaluations to be as general as possible. The main contribution of this thesis is Theorem 4.10 has been published in [9]. It extends Filipovi´c and Svindland’s result from L p spaces to general rearrangement invariant (r.i.) spaces.


2021 ◽  
Author(s):  
Shengzhong Chen

The problem of optimal capital and risk allocation among economic agents, has played a predominant role in the respective academic and industrial research areas for decades. Typically as risk occurs in face of randomness the risks which are to be measured are identified with real-valued random variables on some probability space (Ω, F, P). Consider a model space X , and n economic agents with initial endowments X1, · · · , Xn ∈ X who assess the riskiness of their positions by means of law-invariant convex risk measures ρi : X → (−∞,∞]. In order to minimize total and individual risk, the agents redistribute the aggregate endowment X = X1 + · · · + Xn among themselves. An optimal capital and risk allocation Y1, · · · , Yn satisfies Y1 + · · · + Yn = X and ρ1(Y1) + · · · + ρ(Yn) = inf nXn i=1 ρi(Xi) : Xi ∈ X , i = 1, . . . , n, and Xn i=1 Xi = X o , (0.1) where n i=1ρi(X) = inf nPn i=1 ρi(Xi) : Xi ∈ X , i = 1, . . . , n, and Pn i=1 Xi = X o is the inf-convolution of ρ1, ..., ρn. In 2008, Filipovi´c and Svindland proved that if X is an L p (P) for some 1 ≤ p ≤ ∞ and ρi satisfy a suitable continuity condition (i.e. Fatou property), then Problem (0.1) always admits a solution. To reflect the fact of randomness of risk, we should consider the model space X chosen for risk evaluations to be as general as possible. The main contribution of this thesis is Theorem 4.10 has been published in [9]. It extends Filipovi´c and Svindland’s result from L p spaces to general rearrangement invariant (r.i.) spaces.


2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Awad A. Bakery ◽  
O. M. Kalthum S. K. Mohamed

In this article, we explore the concept of the prequasi norm on Nakano special space of sequences (sss) such that its variable exponent in 0 , 1 . We evaluate the sufficient setting on it with the definite prequasi norm to configuration prequasi Banach and closed (sss). The Fatou property of different prequasi norms on this (sss) has been investigated. Moreover, the existence of a fixed point of Kannan prequasi norm contraction maps on the prequasi Banach (sss) and the prequasi Banach operator ideal constructed by this (sss) and s − numbers have been examined.


2020 ◽  
Vol 13 (5) ◽  
pp. 1162-1175
Author(s):  
B. V. Senthil Kumar ◽  
Hemen Dutta ◽  
S. Sabarinathan

In the present work, we propose a dierent reciprocal second power Functional Equation (FE) which involves the arguments of functions in rational form and determine its stabilities in the setting of modular spaces with and without using Fatou property. We also prove the stabilities in beta-homogenous spaces. As an application, we associate this equation with the electrostatic forces of attraction between unit charges in various cases using Coloumb's law.


2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Massoomeh Rahsepar ◽  
Foivos Xanthos

Abstract Let 𝒳 be a subset of L 1 L^{1} that contains the space of simple random variables ℒ and ρ : X → ( - ∞ , ∞ ] \rho\colon\mathcal{X}\to(-\infty,\infty] a dilatation monotone functional with the Fatou property. In this note, we show that 𝜌 extends uniquely to a σ ⁢ ( L 1 , L ) \sigma(L^{1},\mathcal{L}) lower semicontinuous and dilatation monotone functional ρ ¯ : L 1 → ( - ∞ , ∞ ] \overline{\rho}\colon L^{1}\to(-\infty,\infty] . Moreover, ρ ¯ \overline{\rho} preserves monotonicity, (quasi)convexity and cash-additivity of 𝜌. We also study conditions under which ρ ¯ \overline{\rho} preserves finiteness on a larger domain. Our findings complement extension and continuity results for (quasi)convex law-invariant functionals. As an application of our results, we show that transformed norm risk measures on Orlicz hearts admit a natural extension to L 1 L^{1} that retains robust representations.


2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Choonkil Park ◽  
Abasalt Bodaghi

AbstractIn this article, we study n-variable mappings which are cubic in each variable. We also show that such mappings can be described by an equation, say, multi-cubic functional equation. Furthermore, we study the stability of such functional equations in the modular space $X_{\rho }$Xρ by applying $\Delta _{2}$Δ2-condition and the Fatou property (in some cases) on the modular function ρ. Finally, we show that, under some mild conditions, one of these new multi-cubic functional equations can be hyperstable.


Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 2897-2905
Author(s):  
Omid Zabeti

Suppose G is a locally solid lattice group. It is known that there are non-equivalent classes of bounded homomorphisms on G which have topological structures. In this paper, our attempt is to assign lattice structures on them. More precisely, we use of a version of the remarkable Riesz-Kantorovich formulae and Fatou property for bounded order bounded homomorphisms to allocate the desired structures. Moreover, we show that unbounded convergence on a locally solid lattice group is topological and we investigate some applications of it. Also, some necessary and sufficient conditions for completeness of different types of bounded group homomorphisms between topological rings have been obtained, as well.


Filomat ◽  
2020 ◽  
Vol 34 (3) ◽  
pp. 919-930
Author(s):  
Murali Ramdoss ◽  
Divyakumari Pachaiyappan ◽  
Hemen Dutta

Mixed type is a further step of development in functional equations. In this paper, the authors made an attempt to introduce such equation of the following form with its general solution h(py + z) + h(py-z) + h(y + pz) + h(y-pz) = (p + p2)[h(y + z) + h(y-z)] + 2h(py)- 2(p2 + p-1)h(y) for all y,z ? R, p ? 0,?1. Also, without Fatou property authors investigate its various stabilities related to Ulam problem in modular space by considering with and without ?p-condition.


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