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Register a new userConvexity in G-metric spaces and approximation of fixed points by Mann iterative proces
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In this paper, we first define the concept of convexity in G-metric spaces. We then use Mann iterative process in this newly defined convex G-metric space to prove some convergence results for some classes of mappings. In this way, we can extend several existence results to those approximating fixed points. Our results are just new in the setting.
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Given a Lie group $G$ we study the class $M$ of proper metrizable $G$-spaces with metrizable orbit spaces, and show that any $G$-space $X in M$ admits a closed $G$-embedding into a convex $G$-subset $C$ of some locally convex linear $G$-space, such that $X$ has some $G$-neighborhood in $C$ which belongs to the class $M$. As corollaries we see that any $G$-ANE for $M$ has the $G$-homotopy type of some $G$-CW complex and that any $G$-ANR for $M$ is a $G$-ANE for $M$.
If $f:[a,b]to mathbb{R}$, with $a<b$, is continuous and such that $a$ and $b$ are mapped in opposite directions by $f$, then $f$ has a fixed point in $I$. Suppose that $f:mathbb{C}tomathbb{C}$ is map and $X$ is a continuum. We extend the above for certain continuous maps of dendrites $Xto D, Xsubset D$ and for positively oriented maps $f:Xto mathbb{C}, Xsubset mathbb{C}$ with the continuum $X$ not necessarily invariant. Then we show that in certain cases a holomorphic map $f:mathbb{C}tomathbb{C}$ must have a fixed point $a$ in a continuum $X$ so that either $ain mathrm{Int}(X)$ or $f$ exhibits rotation at $a$.
This paper introduce a new class of operators and contraction mapping for a cyclical map T on G-metric spaces and the approximately fixed point properties. Also,we prove two general lemmas regarding approximate fixed Point of cyclical contraction mapping on G-metric spaces. Using these results we prove several approximate fixed point theorems for a new class of operators on G-metric spaces (not necessarily complete). These results can be exploited to establish new approximate fixed point theorems for cyclical contraction maps. Further,there is a new class of cyclical operators and contraction mapping on G-metric space (not necessarily complete)which do not need to be continuous.Finally,examples are given to support the usability of our results.
The result of Boyce and Huneke gives rise to a 1-dimensional continuum, which is the intersection of a descending family of disks, that admits two commuting homeomorphisms without a common fixed point.
For a profinite group $G$, let $(text{-})^{hG}$, $(text{-})^{h_dG}$, and $(text{-})^{hG}$ denote continuous homotopy fixed points for profinite $G$-spectra, discrete $G$-spectra, and continuous $G$-spectra (coming from towers of discrete $G$-spectra), respectively. We establish some connections between the first two notions, and by using Postnikov towers, for $K vartriangleleft_c G$ (a closed normal subgroup), give various conditions for when the iterated homotopy fixed points $(X^{hK})^{hG/K}$ exist and are $X^{hG}$. For the Lubin-Tate spectrum $E_n$ and $G <_c G_n$, the extended Morava stabilizer group, our results show that $E_n^{hK}$ is a profinite $G/K$-spectrum with $(E_n^{hK})^{hG/K} simeq E_n^{hG}$, by an argument that possesses a certain technical simplicity not enjoyed by either the proof that $(E_n^{hK})^{hG/K} simeq E_n^{hG}$ or the Devinatz-Hopkins proof (which requires $|G/K| < infty$) of $(E_n^{dhK})^{h_dG/K} simeq E_n^{dhG}$, where $E_n^{dhK}$ is a construction that behaves like continuous homotopy fixed points. Also, we prove that (in general) the $G/K$-homotopy fixed point spectral sequence for $pi_ast((E_n^{hK})^{hG/K})$, with $E_2^{s,t} = H^s_c(G/K; pi_t(E_n^{hK}))$ (continuous cohomology), is isomorphic to both the strongly convergent Lyndon-Hochschild-Serre spectral sequence of Devinatz for $pi_ast(E_n^{dhG})$, with $E_2^{s,t} = H^s_c(G/K; pi_t(E_n^{dhK}))$, and the descent spectral sequence for $pi_ast((E_n^{hK})^{hG/K})$.
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