Some fixed point theorems for multifunctions with applications in game theory

Būi Cong Cuōng

Displaying similar documents to “Some fixed point theorems for multifunctions with applications in game theory”

CM-Selectors for pairs of oppositely semicontinuous multivalued maps with $_{p}$ -decomposable values

Hôǹg Thái Nguyêñ, Maciej Juniewicz, Jolanta Ziemińska (2001)

Studia Mathematica

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We present a new continuous selection theorem, which unifies in some sense two well known selection theorems; namely we prove that if F is an H-upper semicontinuous multivalued map on a separable metric space X, G is a lower semicontinuous multivalued map on X, both F and G take nonconvex $L_{p} (T, E)$ -decomposable closed values, the measure space T with a σ-finite measure μ is nonatomic, 1 ≤ p < ∞, $L_{p} (T, E)$ is the Bochner-Lebesgue space of functions defined on T with values in a Banach space E, F(x)...

Parametrization of Riemann-measurable selections for multifunctions of two variables with application to differential inclusions

Giovanni Anello, Paolo Cubiotti (2004)

Annales Polonici Mathematici

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We consider a multifunction $F : T \times X \to 2^{E}$ , where T, X and E are separable metric spaces, with E complete. Assuming that F is jointly measurable in the product and a.e. lower semicontinuous in the second variable, we establish the existence of a selection for F which is measurable with respect to the first variable and a.e. continuous with respect to the second one. Our result is in the spirit of [11], where multifunctions of only one variable are considered.

Lindelöf indestructibility, topological games and selection principles

Marion Scheepers, Franklin D. Tall (2010)

Fundamenta Mathematicae

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Arhangel’skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most $2^{ℵ ₀}$ . Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are $G_{δ}$ has been more elusive. In this paper we continue the agenda started by the second author, [Topology Appl. 63 (1995)], of considering the cardinality problem for spaces satisfying stronger versions of the Lindelöf property. Infinite games and selection principles, especially the Rothberger...

On the topological dimension of the solutions sets for some classes of operator and differential inclusions

Ralf Bader, Boris D. Gel&amp;#039;man, Mikhail Kamenskii, Valeri Obukhovskii (2002)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In the present paper, we give the lower estimation for the topological dimension of the fixed points set of a condensing continuous multimap in a Banach space. The abstract result is applied to the fixed point set of the multioperator of the form $= S_{F}$ where $_{F}$ is the superposition multioperator generated by the Carathéodory type multifunction F and S is the shift of a linear injective operator. We present sufficient conditions under which this set has the infinite topological dimension....

Selection principles and upper semicontinuous functions

Masami Sakai (2009)

Colloquium Mathematicae

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In connection with a conjecture of Scheepers, Bukovský introduced properties wQN* and SSP* and asked whether wQN* implies SSP*. We prove it in this paper. We also give characterizations of properties S₁(Γ,Ω) and $S_{f i n} (Γ, Ω)$ in terms of upper semicontinuous functions

The ω-problem

Stanisław Kowalczyk

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Let (X,) be any T₁ topological space. Given a function F: X → ℝ and x ∈ X, we define the oscillation of F at x to be $ω (F, x) = i n f_{U} s u p_{x ₁, x ₂ \in U} | F (x ₁) - F (x ₂) |$ , where the infimum is taken over all neighborhoods U of x. It is well known that ω(F,·): X → [0,∞] is upper semicontinuous and vanishes at all isolated points of X. Suppose an upper semicontinuous function f: X → [0,∞] vanishing at isolated points of X is given. If there exists a function F: X → ℝ such that ω(F,·)=f, then we call F an ω-primitive for f. By the ’ω-problem’...

Non-Hausdorff Ascoli theory

Pedro Morales

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CONTENTSIntroduction......................................................................................................................................... 5Chapter I. THEORY IN CLASSICAL POEM FOB FUNCTIONS 1. Equicontinuity in quasi-uniform context................................................................... 6 2. Quasi-uniform convergence on compacta............................................................. 8 3. k-spaces and $k_{3}$ ,-spaces......................................................................................

Semicontinuous integrands as jointly measurable maps

Oriol Carbonell-Nicolau (2014)

Commentationes Mathematicae Universitatis Carolinae

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Suppose that $(X, 𝒜)$ is a measurable space and $Y$ is a metrizable, Souslin space. Let $𝒜^{u}$ denote the universal completion of $𝒜$ . For $x \in X$ , let $\underset{̲}{f} (x, \cdot)$ be the lower semicontinuous hull of $f (x, \cdot)$ . If $f : X \times Y \to \bar{ℝ}$ is $(𝒜^{u} \otimes ℬ (Y), ℬ (\bar{ℝ}))$ -measurable, then $\underset{̲}{f}$ is $(𝒜^{u} \otimes ℬ (Y), ℬ (\bar{ℝ}))$ -measurable.

Minimax theorems without changeless proportion

Liang-Ju Chu, Chi-Nan Tsai (2003)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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The so-called minimax theorem means that if X and Y are two sets, and f and g are two real-valued functions defined on X×Y, then under some conditions the following inequality holds: $i n f_{y \in Y} s u p_{x \in X} f (x, y) \leq s u p_{x \in X} i n f_{y \in Y} g (x, y)$ . We will extend the two functions version of minimax theorems without the usual condition: f ≤ g. We replace it by a milder condition: $s u p_{x \in X} f (x, y) \leq s u p_{x \in X} g (x, y)$ , ∀y ∈ Y. However, we require some restrictions; such as, the functions f and g are jointly upward, and their upper sets are connected. On the other hand, by using some...

Semicontinuity and continuous selections for the multivalued superposition operator without assuming growth-type conditions

Hông Thái Nguyêñ (2004)

Studia Mathematica

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Let Ω be a measure space, and E, F be separable Banach spaces. Given a multifunction $f : Ω \times E \to 2^{F}$ , denote by $N_{f} (x)$ the set of all measurable selections of the multifunction $f (\cdot, x (\cdot)) : Ω \to 2^{F}$ , s ↦ f(s,x(s)), for a function x: Ω → E. First, we obtain new theorems on H-upper/H-lower/lower semicontinuity (without assuming any conditions on the growth of the generating multifunction f(s,u) with respect to u) for the multivalued (Nemytskiĭ) superposition operator $N_{f}$ mapping some open domain G ⊂ X into $2^{Y}$ , where X and Y are...

On convex and *-concave multifunctions

Bożena Piątek (2005)

Annales Polonici Mathematici

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A continuous multifunction F:[a,b] → clb(Y) is *-concave if and only if the inclusion $1 / (t - s) \int_{s}^{t} F (x) d x \subset (F (s) \frac{*}{+} F (t)) / 2$ holds for every s,t ∈ [a,b], s < t.

Selections and representations of multifunctions in paracompact spaces

Alberto Bressan, Giovanni Colombo (1992)

Studia Mathematica

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Let (X,T) be a paracompact space, Y a complete metric space, $F : X \to 2^{Y}$ a lower semicontinuous multifunction with nonempty closed values. We prove that if $T^{+}$ is a (stronger than T) topology on X satisfying a compatibility property, then F admits a $T^{+}$ -continuous selection. If Y is separable, then there exists a sequence $(f_{n})$ of $T^{+}$ -continuous selections such that $F (x) = \bar{f_{n} (x); n \geq 1}$ for all x ∈ X. Given a Banach space E, the above result is then used to construct directionally continuous selections on arbitrary subsets...

On complex interpolation and spectral continuity

Karen Saxe (1998)

Studia Mathematica

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Let ${[X_{0}, X_{1}]}_{t}$ , 0 ≤ t ≤ 1, be Banach spaces obtained via complex interpolation. With suitable hypotheses, linear operators T that act boundedly on both $X_{0}$ and $X_{1}$ will act boundedly on each ${[X_{0}, X_{1}]}_{t}$ . Let $T_{t}$ denote such an operator when considered on ${[X_{0}, X_{1}]}_{t}$ , and $σ (T_{t})$ denote its spectrum. We are motivated by the question of whether or not the map $t \to σ (T_{t})$ is continuous on (0,1); this question remains open. In this paper, we study continuity of two related maps: $t \to {(σ (T_{t}))}^{\land}$ (polynomially convex hull) and $t \to \partial_{e} (σ (T_{t}))$ (boundary of the polynomially...

Correction to : “Precompact contraction of metric uniformities and the continuity of $F (t, x)$ ”

C. J. Himmelberg (1974)

Rendiconti del Seminario Matematico della Università di Padova

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$α$ -continuous multivalued maps

J. Eweret (1989)

Matematički Vesnik

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Non-autonomous implicit integral equations with discontinuous right-hand side

Giovanni Anello, Paolo Cubiotti (2004)

Commentationes Mathematicae Universitatis Carolinae

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We deal with the implicit integral equation $h (u (t)) = f (t, \int_{I} g (t, z) u (z) d z) for a.a. t \in I,$ where $I : = [0, 1]$ and where $f : I \times [0, λ] \to ℝ$ , $g : I \times I \to [0, + \infty [$ and $h :] 0, + \infty [\to ℝ$ . We prove an existence theorem for solutions $u \in L^{s} (I)$ where the contituity of $f$ with respect to the second variable is not assumed.