Some fixed point theorems in generating spaces of quasi-metric family

M.H.M. Rashid

Displaying similar documents to “Some fixed point theorems in generating spaces of quasi-metric family”

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Based on a completely distributive lattice $L$ , degrees of compatible $L$ -subsets and compatible mappings are introduced in an $L$ -approximation space and their characterizations are given by four kinds of cut sets of $L$ -subsets and $L$ -equivalences, respectively. Besides, some characterizations of compatible mappings and compatible degrees of mappings are given by compatible $L$ -subsets and compatible degrees of $L$ -subsets. Finally, the notion of complete $L$ -sublattices is introduced and it is shown...

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Let $ℳ^{d}$ be a d-dimensional normed space with norm ||·|| and let B be the unit ball in $ℳ^{d}$ . Let us fix a Lebesgue measure $V_{B}$ in $ℳ^{d}$ with $V_{B} (B) = 1$ . This measure will play the role of the volume in $ℳ^{d}$ . We consider an arbitrary simplex T in $ℳ^{d}$ with prescribed edge lengths. For the case d = 2, sharp upper and lower bounds of $V_{B} (T)$ are determined. For d ≥ 3 it is noticed that the tight lower bound of $V_{B} (T)$ is zero.

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In 1984 J. Clunie and T. Sheil-Small proved ([2, Corollary 5.8]) that for any complex-valued and sense-preserving injective harmonic mapping $F$ in the unit disk $𝔻$ , if $F (𝔻)$ is a convex domain, then the inequality $| G (z_{2}) - G (z_{1}) | < | H (z_{2}) - H (z_{1}) |$ holds for all distinct points $z_{1}, z_{2} \in 𝔻$ . Here $H$ and $G$ are holomorphic mappings in $𝔻$ determined by $F = H + \bar{G}$ , up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain $Ω$ in $ℂ$ and improve it provided $F$ is additionally a quasiconformal mapping...

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We describe some recent developments concerning the Jacobian Conjecture (JC). First we describe Drużkowski’s result in [6] which asserts that it suffices to study the JC for Drużkowski mappings of the form $x + {(A x)}^{* 3}$ with A² = 0. Then we describe the authors’ result of [2] which asserts that it suffices to study the JC for so-called gradient mappings, i.e. mappings of the form x - ∇f, with $f \in k^{[n]}$ homogeneous of degree 4. Using this result we explain Zhao’s reformulation of the JC which asserts the...

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Fixed point approximation under Mann iteration beyond Ishikawa

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Consider the Mann iteration $x_{n + 1} = (1 - α_{n}) x_{n} + α_{n} T x_{n}$ for a nonexpansive mapping $T : K \to K$ defined on some subset $K$ of the normed space $X$ . We present an innovative proof of the Ishikawa almost fixed point principle for nonexpansive mapping that reveals deeper aspects of the behavior of the process. This fact allows us, among other results, to derive convergence of the process under the assumption of existence of an accumulation point of ${x_{n}}$ .

Some approximate fixed point theorems without continuity of the operator using auxiliary functions

Sumit Chandok, Arslan Hojjat Ansari, Tulsi Dass Narang (2019)

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We introduce partial generalized convex contractions of order $4$ and rank $4$ using some auxiliary functions. We present some results on approximate fixed points and fixed points for such class of mappings having no continuity condition in $α$ -complete metric spaces and $μ$ -complete metric spaces. Also, as an application, some fixed point results in a metric space endowed with a binary relation and some approximate fixed point results in a metric space endowed with a graph have been obtained....

$P_{λ}$ -sets and skeletal mappings

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We prove that if the topology on the set Seq of all finite sequences of natural numbers is determined by $P_{λ}$ -filters and λ ≤ , then Seq is a $P_{λ}$ -set in its Čech-Stone compactification. This improves some results of Simon and of Juhász and Szymański. As a corollary we obtain a generalization of a result of Burke concerning skeletal maps and we partially answer a question of his.

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Given a probability space (Ω,,P) and a subset X of a normed space we consider functions f:X × Ω → X and investigate the speed of convergence of the sequence (fⁿ(x,·)) of the iterates $f ⁿ : X \times Ω^{ℕ} \to X$ defined by f¹(x,ω ) = f(x,ω₁), $f^{n + 1} (x, ω) = f (f ⁿ (x, ω), ω_{n + 1})$ .

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Let $C o n v_{H} (X)$ , $C o n v_{A W} (X)$ and $C o n v_{W} (X)$ be the spaces of all non-empty closed convex sets in a normed linear space X admitting the Hausdorff metric topology, the Attouch-Wets topology and the Wijsman topology, respectively. We show that every component of $C o n v_{H} (X)$ and the space $C o n v_{A W} (X)$ are AR. In case X is separable, $C o n v_{W} (X)$ is locally path-connected.

Singular arc-like continua

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CONTENTSIntroduction.......................................................................................................51. Preliminaries.................................................................................................6 A. Mappings....................................................................................................6 B. Arc-like continua.........................................................................................8 C. Pseudosuspensions...................................................................................8 D....

A generalized Kahane-Khinchin inequality

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The inequality $ʃ l o g | \sum a_{n} e^{2 π i φ_{n}} | d φ_{1} \dots d φ_{n} \geq C l o g (\sum | a_{n} {|^{2})}^{1 / 2}$ with an absolute constant C, and similar ones, are extended to the case of $a_{n}$ belonging to an arbitrary normed space X and an arbitrary compact group of unitary operators on X instead of the operators of multiplication by $e^{2 π i φ}$ .

On almost everywhere differentiability of the metric projection on closed sets in $l^{p} (ℝ^{n})$ , $2 < p < \infty$

Tord Sjödin (2018)

Czechoslovak Mathematical Journal

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Let $F$ be a closed subset of $ℝ^{n}$ and let $P (x)$ denote the metric projection (closest point mapping) of $x \in ℝ^{n}$ onto $F$ in $l^{p}$ -norm. A classical result of Asplund states that $P$ is (Fréchet) differentiable almost everywhere (a.e.) in $ℝ^{n}$ in the Euclidean case $p = 2$ . We consider the case $2 < p < \infty$ and prove that the $i$ th component $P_{i} (x)$ of $P (x)$ is differentiable a.e. if $P_{i} (x) \neq x_{i}$ and satisfies Hölder condition of order $1 / (p - 1)$ if $P_{i} (x) = x_{i}$ .