Strong convergence theorems for finitely many nonexpansive mappings

Under the assumption that E$E$ is a reflexive Banach space whose norm is uniformly Gêteaux differentiable and which has a weakly continuous duality mapping _Jφ$J_{\phi }$ with gauge function φ$\phi$, Ceng–Cubiotti–Yao [Strong convergence theorems for finitely many nonexpansive mappings and applications, Nonlinear Analysis 67 (2007) 1464–1473] introduced a new iterative scheme for a finite commuting family of nonexpansive mappings, and proved strong convergence theorems about this iteration. In this paper, only under the hypothesis that E$E$ is a reflexive Banach space which has a weakly continuous duality mapping _Jφ$J_{\phi }$ with gauge function φ$\phi$, and several control conditions about the iterative coefficient are removed, we present a short and simple proof of the above theorem.     

Cone-separation and star-shaped separability with applications

Let X$X$ be a (real) Banach space, A$A$ be a subset of X$X$ and x∉A$x\notin A$. We present cone-separation in terms of separation by a collection of linear functionals defined on X$X$ and obtain necessary and sufficient conditions for cone-separability A$A$ and x$x$. Also, we give characterizations for star-shaped separability. Finally, as an application of separability, we characterize best approximation problem by elements of star-shaped sets.     

Nadler’s type principle with high order of convergence

Recently Proinov [P.D. Proinov, A generalization of the Banach contraction principle with high order of convergence of successive approximations, Nonlinear Analysis (2006), doi:10.1016/j.na.2006.09.008] generalized Banach contraction principle with high order of convergence. We extend some results of Proinov to the case of multi-valued maps from a complete metric space X$X$ into the space of all nonempty proximinal closed subsets of X$X$. Our results not only generalize Nadler’s fixed-point theorem (in the case when T$T$ is a mapping from a complete metric space X$X$ into the space of all nonempty proximinal closed subsets of X$X$) but also gives high order of convergence. As an application, we obtain an existence theorem for first-order initial value problem.     

Asymptotic average shadowing property on compact metric spaces

We prove that if for a continuous map f$f$ on a compact metric space X$X$, the chain recurrent set, R(f)$R\left(f\right)$ has more than one chain component, then f$f$ does not satisfy the asymptotic average shadowing property. We also show that if a continuous map f$f$ on a compact metric space X$X$ has the asymptotic average shadowing property and if A$A$ is an attractor for f$f$, then A$A$ is the single attractor for f$f$ and we have A=R(f)$A=R\left(f\right)$. We also study diffeomorphisms with asymptotic average shadowing property and prove that if M$M$ is a compact manifold which is not finite with dimM=2$dimM=2$, then the C¹$C^1$ interior of the set of all C¹$C^1$ diffeomorphisms with the asymptotic average shadowing property is characterized by the set of Ω$\Omega$-stable diffeomorphisms.     

Fixed point theorem for generalized Φ -pseudocontractive mappings

Let E$E$ be a real Banach space, C$C$ be a nonempty closed convex subset of E$E$ and T:C→C$T:C\to C$ be a continuous generalized Φ$\Phi$-pseudocontractive mapping. It is proved that T$T$ has a unique fixed point in C$C$.     

Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation

We develop a notion of nonlinear expectation–G$G$-expectation–generated by a nonlinear heat equation with infinitesimal generator G$G$. We first study multi-dimensional G$G$-normal distributions. With this nonlinear distribution we can introduce our G$G$-expectation under which the canonical process is a multi-dimensional G$G$-Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Itô’s type with respect to our G$G$-Brownian motion, and derive the related Itô’s formula. We have also obtained the existence and uniqueness of stochastic differential equations under our G$G$-expectation.     

Oscillation of harmonic functions for subordinate Brownian motion and its applications

In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the following form of relative Fatou theorem for such subordinate Brownian motion X$X$ in a bounded κ$\kappa$-fat open set; if u$u$ is a positive harmonic function with respect to X$X$ in a bounded κ$\kappa$-fat open set D$D$ and h$h$ is a positive harmonic function in D$D$ vanishing on D^c$D^c$, then the non-tangential limit of u/h$u/h$ exists almost everywhere with respect to the Martin-representing measure of h$h$.     

On topological entropy of commuting tree maps

Let T$T$ be a tree with s$s$ ends and f,g$f,g$ be continuous maps from T$T$ to T$T$ with f°g=g°f$f°g=g°f$. In this note we show that if there exists a positive integer m≥2$m\ge 2$ such that gcd(m,l)=1$gcd(m,l)=1$ for any 2≤l≤s$2\le l\le s$ and f,g$f,g$ share a periodic point which is a km$km$-periodic point of f$f$ for some positive integer k$k$, then the topological entropy of f°g$f°g$ is positive.     

Iterative approximation of solutions of nonlinear equations of Hammerstein type

Suppose X$X$ is a real q$q$-uniformly smooth Banach space and F,K:X→X$F,K:X\to X$ are Lipschitz ?$?$-strongly accretive maps with D(K)=F(X)=X$D\left(K\right)=F\left(X\right)=X$. Let u^∗$u^{\ast }$ denote the unique solution of the Hammerstein equation u+KFu=0$u+KFu=0$. An iteration process recently introduced by Chidume and Zegeye is shown to converge strongly to u^∗$u^{\ast }$. No invertibility assumption is imposed on K$K$ and the operators K$K$ and F$F$ need not be defined on compact subsets of X$X$. Furthermore, our new technique of proof is of independent interest. Finally, some interesting open questions are included.     

Strongly quasi-Hamiltonian-connected semicomplete multipartite digraphs

A semicomplete multipartite or semicomplete c$c$-partite digraph D$D$ is a biorientation of a c$c$-partite graph. A semicomplete multipartite digraph D$D$ is called strongly quasi-Hamiltonian-connected, if for any two distinct vertices x$x$ and y$y$ of D$D$, there is a path P$P$ from x$x$ to y$y$ such that P$P$ contains at least one vertex from each partite set of D$D$.     

Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory

In this paper, we introduce the concept of a Q$Q$-function defined on a quasi-metric space which generalizes the notion of a τ$\tau$-function and a w$w$-distance. We establish Ekeland-type variational principles in the setting of quasi-metric spaces with a Q$Q$-function. We also present an equilibrium version of the Ekeland-type variational principle in the setting of quasi-metric spaces with a Q$Q$-function. We prove some equivalences of our variational principles with Caristi–Kirk type fixed point theorems for multivalued maps, the Takahashi minimization theorem and some other related results. As applications of our results, we derive existence results for solutions of equilibrium problems and fixed point theorems for multivalued maps. We also extend the Nadler’s fixed point theorem for multivalued maps to a Q$Q$-function and in the setting of complete quasi-metric spaces. As a consequence, we prove the Banach contraction theorem for a Q$Q$-function and in the setting of complete quasi-metric spaces. The results of this paper extend and generalize many results appearing recently in the literature.     

On some geometric properties of generalized Orlicz–Lorentz function spaces

In this article we continue investigations concerning generalized Orlicz–Lorentz function spaces _Λφ${\Lambda }_{\phi }$ initiated in the papers (Foralewski, 2011)  and  (cf. also (Foralewski, 2008)  and ). First, it is shown that modular _?φ${?}_{\phi }$ is orthogonally subadditive. Next, monotonicity properties are considered. In order to get sufficient conditions for uniform monotonicity of the space _Λφ${\Lambda }_{\phi }$ a strong condition of _Δ2${\Delta }_2$ type and the notion of regularity of the generated Musielak–Orlicz function φ$\phi$ are introduced. Finally, criteria for non-squareness of _Λφ${\Lambda }_{\phi }$ are presented.     

Approximation of solutions of Hammerstein equations with bounded strongly accretive nonlinear operators

Suppose X$X$ is a real q$q$-uniformly smooth Banach space and F,K:X→X$F,K:X\to X$ are bounded strongly accretive maps with D(K)=F(X)=X$D\left(K\right)=F\left(X\right)=X$. Let u^∗$u^{\ast }$ denote the unique solution of the Hammerstein equation u+KFu=0$u+KFu=0$. A new explicit coupled iteration process is shown to converge strongly to u^∗$u^{\ast }$. No invertibility assumption is imposed on K$K$ and the operators K$K$ and F$F$ need not be defined on compact subsets of X$X$. Furthermore, our new technique of proof is of independent interest. Finally, some interesting open questions are included.     

Smale’s point estimate theory for Newton’s method on Lie groups

We introduce the notion of the (one-parameter subgroup) γ$\gamma$-condition for a map f$f$ from a Lie group to its Lie algebra and establish α$\alpha$-theory and γ$\gamma$-theory for Newton’s method for a map f$f$ satisfying this condition. Applications to analytic maps are provided, and Smale’s α$\alpha$-theory and γ$\gamma$-theory are extended and developed. Examples arising from initial value problems on Lie group are presented to illustrate applications of our results.     

Some results on general quadratic reflected BSDEs driven by a continuous martingale

We study the well-posedness of general reflected BSDEs driven by a continuous martingale, when the coefficient f$f$ of the driver has at most quadratic growth in the control variable Z$Z$, with a bounded terminal condition and a lower obstacle which is bounded above. We obtain the basic results in this setting: comparison and uniqueness, existence, stability. For the comparison theorem and the special comparison theorem for reflected BSDEs (which allows one to compare the increasing processes of two solutions), we give intrinsic proofs which do not rely on the comparison theorem for standard BSDEs. This allows to obtain the special comparison theorem under minimal assumptions. We obtain existence by using the fixed point theorem and then a series of perturbations, first in the case where f$f$ is Lipschitz in the primary variable Y$Y$, and then in the case where f$f$ can have slightly-superlinear growth and the case where f$f$ is monotonous in Y$Y$ with arbitrary growth. We also obtain a local Lipschitz estimate in BMO$BMO$ for the martingale part of the solution.     

Characterizations of hypersurfaces of a Danielewski type

Let k$k$ be a field of characteristic zero and R$R$ a factorial affine k$k$-domain. Let B$B$ be an affineR$R$-domain. In terms of locally nilpotent derivations, we give criteria for B$B$ to be R$R$-isomorphic to the residue ring of a polynomial ring R[_X1,_X2,Y]$R[X_1,X_2,Y]$ over R$R$ by the ideal (_X1X2−φ(Y))$(X_1{X}_2-\phi \left(Y\right))$ for φ(Y)∈R[Y]?R$\phi \left(Y\right)\in R\left[Y\right]?R$.