Convergence of the modified Mann’s iteration method for asymptotically strict pseudo-contractions

Let C$C$ be a closed convex subset of a real Hilbert space H$H$ and assume that T$T$ is an asymptotically κ$\kappa$-strict pseudo-contraction on C$C$ with a fixed point, for some 0≤κ<1$0\le \kappa <1$. Given an initial guess _x0∈C$x_0\in C$ and given also a real sequence {_αn}$\left\{{\alpha }_n\right\}$ in (0, 1), the modified Mann’s algorithm generates a sequence {_xn}$\left\{x_n\right\}$ via the formula: _xn+1=_αnxn+(1−_αn)Tⁿ_xn$x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)T^n{x}_n$, n≥0$n\ge 0$. It is proved that if the control sequence {_αn}$\left\{{\alpha }_n\right\}$ is chosen so that κ+δ<_αn<1−δ$\kappa +\delta <{\alpha }_n<1-\delta$ for some δ∈(0,1)$\delta \in (0,1)$, then {_xn}$\left\{x_n\right\}$ converges weakly to a fixed point of T$T$. We also modify this iteration method by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strongly convergent sequence.     

New general convergence theory for iterative processes and its applications to Newton–Kantorovich type theorems

Let T:D⊂X→X$T:D\subset X\to X$ be an iteration function in a complete metric space X$X$. In this paper we present some new general complete convergence theorems for the Picard iteration _xn+1=T_xn$x_{n+1}=T{x}_n$ with order of convergence at least r≥1$r\ge 1$. Each of these theorems contains a priori and a posteriori error estimates as well as some other estimates. A central role in the new theory is played by the notions of a function of initial conditions   of T$T$ and a convergence function   of T$T$. We study the convergence of the Picard iteration associated to T$T$ with respect to a function of initial conditions E:D→X$E:D\to X$. The initial conditions in our convergence results utilize only information at the starting point _x0$x_0$. More precisely, the initial conditions are given in the form E(_x0)∈J$E\left(x_0\right)\in J$, where J$J$ is an interval on _R+${\mathbb{R}}_{+}$ containing 0. The new convergence theory is applied to the Newton iteration in Banach spaces. We establish three complete ω$\omega$-versions of the famous semilocal Newton–Kantorovich theorem as well as a complete version of the famous semilocal α$\alpha$-theorem of Smale for analytic functions.     

Coalescence in the recent past in rapidly growing populations

In a rapidly growing population one expects that two individuals chosen at random from the n$n$th generation are unlikely to be closely related if n$n$ is large. In this paper it is shown that for a broad class of rapidly growing populations this is not the case. For a Galton–Watson branching process with an offspring distribution {_pj}$\left\{p_j\right\}$ such that _p0=0$p_0=0$ and ψ(x)=_{∑jpjI{j≥x}}$\psi \left(x\right)={\sum }_j{p}_j{I}_{\{j\ge x\}}$ is asymptotic to x^−αL(x)$x^{-\alpha }L\left(x\right)$ as x→∞$x\to \infty$ where L(⋅)$L(\cdot )$ is slowly varying at ∞$\infty$ and 0<α<1$0<\alpha <1$ (and hence the mean m=∑j_pj=∞$m=\sum j{p}_j=\infty$) it is shown that if _Xn$X_n$ is the generation number of the coalescence of the lines of descent backwards in time of two randomly chosen individuals from the n$n$th generation then n−_Xn$n-X_n$ converges in distribution to a proper distribution supported by N={1,2,3,…}$\mathbb{N}=\{1,2,3,\dots \}$. That is, in such a rapidly growing population coalescence occurs in the recent past rather than the remote past. We do show that if the offspring mean m$m$ satisfies 1<m≡∑j_pj<∞$1<m\equiv \sum j{p}_j<\infty$ and _p0=0$p_0=0$ then coalescence time _Xn$X_n$ does converge to a proper distribution as n→∞$n\to \infty$, i.e., coalescence does take place in the remote past.     

On a Mann type implicit iteration process for continuous pseudo-contractive mappings

Let K$K$ be a nonempty closed convex subset of a Banach space E$E$, T:K→K$T:K\to K$ a continuous pseudo-contractive mapping. Suppose that {_αn}$\left\{{\alpha }_n\right\}$ is a real sequence in [0,1]$[0,1]$ satisfying appropriate conditions; then for arbitrary _x0∈K$x_0\in K$, the Mann type implicit iteration process {_xn}$\left\{x_n\right\}$ given by _xn=_αnxn−1+(1−_αn)T_xn,n≥0$x_n={\alpha }_n{x}_{n-1}+(1-{\alpha }_n)T{x}_n,n\ge 0$, strongly and weakly converges to a fixed point of T$T$, respectively.     

The relaxed Newton method derivative: Its dynamics and non-linear properties

The dynamic behaviour of the one-dimensional family of maps f(x)=_c2[(a−1)x+_c1]^{−λ/(α−1)}$f\left(x\right)=c_2{[(a-1)x+c_1]}^{-\lambda /(\alpha -1)}$ is examined, for representative values of the control parameters a,_c1$a,c_1$, _c2$c_2$ and λ$\lambda$. The maps under consideration are of special interest, since they are solutions of the relaxed Newton method derivative being equal to a constant a$a$. The maps f(x)$f\left(x\right)$ are also proved to be solutions of a non-linear differential equation with outstanding applications in the field of power electronics. The recurrent form of these maps, after excessive iterations, shows, in an _xn$x_n$ versus λ$\lambda$ plot, an initial exponential decay followed by a bifurcation. The value of λ$\lambda$ at which this bifurcation takes place depends on the values of the parameters a,_c1$a,c_1$ and _c2$c_2$. This corresponds to a switch to an oscillatory behaviour with amplitudes of f(x)$f\left(x\right)$ undergoing a period doubling. For values of a$a$ higher than 1 and at higher values of λ$\lambda$ a reverse bifurcation occurs. The corresponding branches converge and a bleb is formed for values of the parameter _c1$c_1$ between 1 and 1.20. This behaviour is confirmed by calculating the corresponding Lyapunov exponents.     

On Mann iteration in Hilbert spaces

Let K$K$ be a compact convex subset of a real Hilbert space H$H$; T:K→K$T:K\to K$ a hemicontractive map. Let {_αn}$\left\{{\alpha }_n\right\}$ be a real sequence in [0,1] satisfying appropriate conditions; then for arbitrary _x0∈K$x_0\in K$, the sequence {_xn}$\left\{x_n\right\}$ defined iteratively by _xn=_αnxn−1+(1−_αn)T_xn$x_n={\alpha }_n{x}_{n-1}+(1-{\alpha }_n)T{x}_n$, n≥1$n\ge 1$ converges strongly to a fixed point of T$T$.     

Regularity for the fractional Gelfand problem up to dimension 7

We study the problem (−Δ)^su=λe^u${(-\mathrm{\Delta })}^{s}u=\lambda e^u$ in a bounded domain Ω⊂Rⁿ$\Omega \subset {\mathbb{R}}^n$, where λ   is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result yields the boundedness of the extremal solution in dimensions n≤7$n\le 7$ for all s∈(0,1)$s\in (0,1)$ whenever Ω   is, for every i=1,...,n$i=1,...,n$, convex in the _xi$x_i$-direction and symmetric with respect to {_xi=0}$\{x_i=0\}$. The same holds if n=8$n=8$ and s?0.28206...$s?0.28206...$, or if n=9$n=9$ and s?0.63237...$s?0.63237...$. These results are new even in the unit ball Ω=_B1$\Omega =B_1$.     

On the removal lemma for linear systems over Abelian groups

In this paper we present an extension of the removal lemma to integer linear systems over abelian groups. We prove that, if the k$k$-determinantal of an integer (k×m)$(k\times m)$ matrix A$A$ is coprime with the order n$n$ of a group G$G$ and the number of solutions of the system Ax=b$Ax=b$ with _x1∈_X1,…,_xm∈_Xm$x_1\in X_1,\dots ,x_m\in X_m$ is o(n^m−k)$o\left(n^{m-k}\right)$, then we can eliminate o(n)$o\left(n\right)$ elements in each set to remove all these solutions.     

Holomorphic mappings preserving Minkowski functionals

We show that the equality _m1(f(x))=_m2(g(x))$m_1\left(f\left(x\right)\right)=m_2\left(g\left(x\right)\right)$ for x$x$ in a neighborhood of a point a$a$ remains valid for all x$x$ provided that f$f$ and g$g$ are open holomorphic maps, f(a)=g(a)=0$f\left(a\right)=g\left(a\right)=0$ and _m1,_m2$m_1,m_2$ are Minkowski functionals of bounded balanced domains. Moreover, a polynomial relation between f$f$ and g$g$ is obtained.     

Parametric inference for discretely observed multidimensional diffusions with small diffusion coefficient

We consider a multidimensional diffusion X$X$ with drift coefficient b(α,_Xt)$b(\alpha ,X_t)$ and diffusion coefficient ?σ(β,_Xt)$?\sigma (\beta ,X_t)$. The diffusion sample path is discretely observed at times _tk=kΔ$t_k=k\Delta$ for k=1…n$k=1\dots n$ on a fixed interval [0,T]$[0,T]$. We study minimum contrast estimators derived from the Gaussian process approximating X$X$ for small ?$?$. We obtain consistent and asymptotically normal estimators of α$\alpha$ for fixed Δ$\Delta$ and ?→0$?\to 0$ and of (α,β)$(\alpha ,\beta )$ for Δ→0$\Delta \to 0$ and ?→0$?\to 0$ without any condition linking ?$?$ and Δ$\Delta$. We compare the estimators obtained with various methods and for various magnitudes of Δ$\Delta$ and ?$?$ based on simulation studies. Finally, we investigate the interest of using such methods in an epidemiological framework.     

Frobenius groups and retract rationality

Let k$k$ be any field, G$G$ be a finite group acting on the rational function field k(_xg:g∈G)$k(x_g:g\in G)$ by h⋅_xg=_xhg$h\cdot x_g=x_{hg}$ for any h,g∈G$h,g\in G$. Define k(G)=k(_xg:g∈G)^G$k\left(G\right)=k{(x_g:g\in G)}^G$. Noether’s problem asks whether k(G)$k\left(G\right)$ is rational (= purely transcendental) over k$k$. A weaker notion, retract rationality introduced by Saltman, is also very useful for the study of Noether’s problem. We prove that, if G$G$ is a Frobenius group with abelian Frobenius kernel, then k(G)$k\left(G\right)$ is retract k$k$-rational for any field k$k$ satisfying some mild conditions. As an application, we show that, for any algebraic number field k$k$, for any Frobenius group G$G$ with Frobenius complement isomorphic to S_L2(_F5)$S{L}_2\left({\mathbb{F}}_5\right)$, there is a Galois extension field K$K$ over k$k$ whose Galois group is isomorphic to G$G$, i.e. the inverse Galois problem is valid for the pair (G,k)$(G,k)$. The same result is true for any non-solvable Frobenius group if k(_ζ8)$k\left({\zeta }_8\right)$ is a cyclic extension of k$k$.     

An approximation method for strictly pseudocontractive mappings

Let K$K$ be a closed convex subset of a q$q$-uniformly smooth separable Banach space, T:K→K$T:K\to K$ a strictly pseudocontractive mapping, and f:K→K$f:K\to K$ an L$L$-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1)$t\in (0,1)$, let _xt$x_t$ be the unique fixed point of tf+(1-t)T$\mathit{tf}+(1-t)T$. We prove that if T$T$ has a fixed point, then {_xt}$\{x_t\}$ converges to a fixed point of T$T$ as t$t$ approaches to 0.     

Weak convergence of censored and reflected stable processes

It is shown that if a sequence of open n$n$-sets _Dk$D_k$ increases to an open n$n$-set D$D$ then reflected stable processes in _Dk$D_k$ converge weakly to the reflected stable process in D$D$ for every starting point x$x$ in D$D$. The same result holds for censored α$\alpha$-stable processes for every x$x$ in D$D$ if D$D$ and _Dk$D_k$ satisfy the uniform Hardy inequality. Using the method in the proof of the above results, we also prove the weak convergence of reflected Brownian motions in unbounded domains.     

On first-fit coloring of ladder-free posets

Bosek and Krawczyk exhibited an on-line algorithm for partitioning an on-line poset of width w$w$ into w^14lgw$w^{14lgw}$ chains. They also observed that the problem of on-line chain partitioning of general posets of width w$w$ could be reduced to First-Fit chain partitioning of 2w²+1$2w^2+1$-ladder-free posets of width w$w$, where an m$m$-ladder is the transitive closure of the union of two incomparable chains _x1≤?≤_xm$x_1\le ?\le x_m$, _y1≤?≤_ym$y_1\le ?\le y_m$ and the set of comparabilities {_x1≤_y1,…,_xm≤_ym}$\{x_1\le y_1,\dots ,x_m\le y_m\}$. Here, we provide a subexponential upper bound (in terms of w$w$ with m$m$ fixed) for the performance of First-Fit chain partitioning on m$m$-ladder-free posets, as well as an exact quadratic bound when m=2$m=2$, and an upper bound linear in m$m$ when w=2$w=2$. Using the Bosek–Krawczyk observation, this yields an on-line chain partitioning algorithm with a somewhat improved performance bound. More importantly, the algorithm and the proof of its performance bound are much simpler.     

Mosco convergence of the sets of fixed points for one-parameter nonexpansive semigroups

We consider the Mosco convergence of the sets of fixed points for one-parameter strongly continuous semigroups of nonexpansive mappings. One of our main results is the following: Let C$C$ be a closed convex subset of a Hilbert space E$E$. Let {T(t):t≥0}$\{T\left(t\right):t\ge 0\}$ be a strongly continuous semigroup of nonexpansive mappings on C$C$. The set of all fixed points of T(t)$T\left(t\right)$ is denoted by F(T(t))$F\left(T\left(t\right)\right)$ for each t≥0$t\ge 0$. Let τ$\tau$ be a nonnegative real number and let {_tn}$\left\{t_n\right\}$ be a sequence in R$\mathbb{R}$ satisfying τ+_tn≥0$\tau +t_n\ge 0$ and _tn≠0$t_n\ne 0$ for n∈N$n\in \mathbb{N}$, and _limntn=0${lim}_n{t}_n=0$. Then {F(T(τ+_tn))}$\left\{F\left(T(\tau +t_n)\right)\right\}$ converges to _?t≥0F(T(t))${?}_{t\ge 0}F\left(T\left(t\right)\right)$ in the sense of Mosco.