Strong convergence theorems of <Emphasis Type="Italic">k</Emphasis>-strict pseudo-contractions in Hilbert spaces

Let K be a nonempty closed convex subset of a real Hilbert space H such that K ± K ⊂ K, T: K → H a k-strict pseudo-contraction for some 0 ⩽ k < 1 such that F(T) = {x ∈ K: x = Tx} ≠ $ \not 0 $ \not 0 . Consider the following iterative algorithm given by

(渐近)非扩张映象的不动点的迭代逼近

Let E be a uniformly convex Banach space which satisfies Opial‘s condition or has aFrechet differentiable norm,and C be a bounded closed convex subset of E. If T： C→C is(asymptotically)nonexpansive,then the modified Ishikawa iteration process defined by     

Approximating common fixed points of asymptotically nonexpansive mappings by composite algorithm in Banach spaces

Let E be a uniformly convex Banach space and K a nonempty convex closed subset which is also a nonexpansive retract of E. Let T ₁, T ₂ and T ₃: K → E be asymptotically nonexpansive mappings with {k _n }, {l _n } and {j _n }. [1, ∞) such that Σ _n=1 ^∞ (k _n − 1) < ∞, Σ _n=1 ^∞ (l _n − 1) < ∞ and Σ _n=1 ^∞ (j _n − 1) < ∞, respectively and F nonempty, where F = {x ∈ K: T _1x = T _2x = T ₃ x} = x} denotes the common fixed points set of T ₁, _{T 2} and T ₃. Let {α _n }, {α′ _n } and {α″ _n } be real sequences in (0, 1) and ∈ ≤ {α _n }, {α′ _n }, {α″ _n } ≤ 1 − ∈ for all n ∈ N and some ∈ > 0. Starting from arbitrary x ₁ ∈ K define the sequence {x _n } by

Viscosity approximation methods for monotone mappings and a countable family of nonexpansive mappings

We use viscosity approximation methods to obtain strong convergence to common fixed points of monotone mappings and a countable family of nonexpansive mappings. Let C be a nonempty closed convex subset of a Hilbert space H and P _C is a metric projection. We consider the iteration process {x _n } of C defined by x ₁ = x ∈ C is arbitrary and

Some Inequalities for Tree Martingales

In this paper we study tree martingales and proved that if 1≤α,β〈∞,1≤p〈∞ then for every predictable tree martingale f=（ft,t∞T）and E[σ^（P）（f）]〈∞,E[S^（P）（f）]〈∞,it holds that ‖（St^（p）（f）,t∈T）‖M^α∞≤Cαβ‖f‖p^αβ,‖（σt^（p）（f）,t∈T）‖M^α,β‖f‖P^αβ,where Cαβ depends only on α and β.     

非Lipschitz条件下严格伪压缩映射不动点的逼近

Abstract. Without the Lipschitz assumption and boundedness of K in arbitrary Banach spaces,the Ishikawa iteration     

New iterations with errors for approximating common fixed points for two generalized asymptotically quasi-nonexpansive nonself-mappings

Let X be a real uniformly convex Banach space and C a nonempty closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T ₁, T ₂: C → X be two uniformly L-Lipschitzian, generalized asymptotically quasi-nonexpansive non-self-mappings of C satisfying condition A′ with sequences {k _n ⁽ⁱ⁾ } and {δ _n ⁽ⁱ⁾ } ? [1, ∞),, i = 1, 2, respectively such that Σ _n=1 ^∞ (k _n ⁽ⁱ⁾ ? 1) < ∞, Σ _n=1 ⁽ⁱ⁾ δ _n ⁽ⁱ⁾ < ∞, and F = F(T ₁) ∩ F(T ₂) ≠ ?. For an arbitrary x ₁ ∈ C, let {x _n } be the sequence in C defined by $$ \begin{gathered} y_n = P\left( {\left( {1 - \beta _n - \gamma _n } \right)x_n + \beta _n T_2 \left( {PT_2 } \right)^{n - 1} x_n + \gamma _n v_n } \right), \hfill \\ x_{n + 1} = P\left( {\left( {1 - \alpha _n - \lambda _n } \right)y_n + \alpha _n T_1 \left( {PT_1 } \right)^{n - 1} x_n + \lambda _n u_n } \right), n \geqslant 1, \hfill \\ \end{gathered} $$ where {α _n }, {β _n }, {γ _n } and {λ _n } are appropriate real sequences in [0, 1) such that Σ _n=1 ^∞ ] γ _n < ∞, Σ _n=1 ^∞ λ _n < ∞, and {u _n }, }v _n } are bounded sequences in C. Then {x _n } and {y _n } converge strongly to a common fixed point of T ₁ and T ₂ under suitable conditions.     

Integral inequalities for self-reciprocal polynomials

Let n ≥ 1 be an integer and let P _n be the class of polynomials P of degree at most n satisfying z ⁿ P(1/z) = P(z) for all z ∈ C. Moreover, let r be an integer with 1 ≤ r ≤ n. Then we have for all P ∈ P _n :

Some properties for a class of symmetric functions with applications

For x = (x ₁, x ₂, ..., x _n ) ∈ ℝ₊ ⁿ , the symmetric function ψ _n (x, r) is defined by $\psi _n (x,r) = \psi _n \left( {x_1 ,x_2 , \cdots ,x_n ;r} \right) = \sum\limits_{1 \leqslant i_1 < i_2 \cdots < i_r \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 + x_{i_j } }} {{x_{i_j } }}} } ,$\psi _n (x,r) = \psi _n \left( {x_1 ,x_2 , \cdots ,x_n ;r} \right) = \sum\limits_{1 \leqslant i_1 < i_2 \cdots < i_r \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 + x_{i_j } }} {{x_{i_j } }}} } ,     

Représentation intégrale de fonctionnelles convexes sur un espace de mesures

Résumé Partant d’un résultat abstrait de représentation intègrale pour une fonctionnelle convexe faiblement s.c.i. sur [M _b (Ω)] ^d (obtenu grace à [1]); on établit un résultat de convergence pour une suite de fonctionnelles du type μ _n εM _b ⁺ (Ω),f _n : Ω×R ^d →−∞, +∞] inté Quelques exemples motivés par des applications à la mécanique sont ensuite traités.

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Riassunto Partendo da un risultato astratto di rappresentazione integrale per un funzionale convesso debolmente s.c.i. su [M _b (Ω)] ^d (ottenuto grazie a [1]), si dimostra un risultato di convergenza per una successione di funzionali del tipo con μ _n εM _b ⁺ (Ω),f _n : Ω×R ^d →−∞, +∞] integrando normale convesso. Vengono inoltre trattati alcuni esempi motivati da applicazioni alla Meccanica.

     

Profile of Blow-up Solution to Hyperbolic System with Nonlocal Term

This paper is concerned with a nonlocal hyperbolic system as follows utt = △u ＋ （∫Ωvdx ）^p for x∈R^N,t〉0 ,utt = △u ＋ （∫Ωvdx ）^q for x∈R^N,t〉0 ,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N, where 1≤ N ≤3, p ≥1, q ≥ 1 and pq 〉 1. Here the initial values are compactly supported and Ω belong to R^N is a bounded open region. The blow-up curve, blow-up rate and profile of the solution are discussed.     

The general iterative methods for nonexpansive semigroups in Banach spaces

Let E be a real reflexive strictly convex Banach space which has uniformly Gâteaux differentiable norm. Let ${\mathcal{S} = \{T(s): 0 \leq s < \infty\}}$ be a nonexpansive semigroup on E such that ${Fix(\mathcal{S}) := \cap_{t\geq 0}Fix( T(t) ) \not= \emptyset}$ , and f is a contraction on E with coefficient 0 <  α <  1. Let F be δ-strongly accretive and λ-strictly pseudo-contractive with δ + λ >  1 and ${0 < \gamma < \min\left\{\frac{\delta}{\alpha}, \frac{1-\sqrt{ \frac{1-\delta}{\lambda} }}{\alpha} \right\} }$ . When the sequences of real numbers {α _n } and {t _n } satisfy some appropriate conditions, the three iterative processes given as follows : $${\left.\begin{array}{ll}{x_{n+1} = \alpha_n \gamma f(x_n) + (I - \alpha_n F)T(t_n)x_n,\quad n\geq 0,}\\ {y_{n+1} = \alpha_n \gamma f(T(t_n)y_n) + (I - \alpha_n F)T(t_n)y_n,\quad n\geq 0,}\end{array}\right.}$$ and $$ z_{n+1} = T(t_n)( \alpha_n \gamma f(z_n) + (I - \alpha_n F)z_n),\quad n\geq 0 $$ converge strongly to ${\tilde{x}}$ , where ${\tilde{x}}$ is the unique solution in ${Fix(\mathcal{S})}$ of the variational inequality $${ \langle (F - \gamma f)\tilde {x}, j(x - \tilde{x}) \rangle \geq 0,\quad x\in Fix(\mathcal{S}).}$$ Our results extend and improve corresponding ones of Li et al. (Nonlinear Anal 70:3065–3071, 2009) and Chen and He (Appl Math Lett 20:751–757, 2007) and many others.     

Positive Solution of Multi-Point Boundary Value Problems for the One-Dimensional p-Laplacian with Singularities

In this paper, we obtain positive solution to the following multi-point singular boundary value problem with p-Laplacian operator,{（ φp（u＇））＇＋q（t）f（t,u,u＇）=0,0〈t〈1,u（0）=∑i=1^nαiu（ξi）,u＇（1）=∑i=1^nβiu＇（ξi）,whereφp（s）=｜s｜^p-2s,p≥2;ξi∈（0,1）（i=1,2,…,n）,0≤αi,βi〈1（i=1,2,…n）,0≤∑i=1^nαi,∑i=1^nβi〈1,and q（t） may be singular at t=0,1,f（t,u,u＇）may be singular at u＇=0     

Quasi-convex Mappings on the Unit Polydisk in $\mathbb{C}^{n}$

In this paper, the sharp estimates of all homogeneous expansions for f are established, where f(z) = (f ₁(z), f ₂(z), …, f _n (z))′ is a k-fold symmetric quasi-convex mapping defined on the unit polydisk in ℂ ⁿ and

A note on the Hoffman-Wielandt theorem for generalized eigenvalue problems

Let {A, B} and {C, D} be diagonalizable pairs of order n, i.e., there exist invertible matrices P, Q and X, Ysuchthat A = P∧Q, B = PΩQ, C =XГY, D= X△Y, where
∧ = diag（α1, α2, …, αn）, Ω= diag（βl, β2, …βn）,
Г=diag（γ1,γ2,…,γn）, △=diag（δl,δ2,…,δn）.
Let ρ（（α,β）, （γ,δ））=｜αδ-βγ｜/√｜α｜^2＋｜β｜^2√｜γ｜^2＋｜δ｜^2.In this paper, it will be proved that there is a permutation τ of {1,2,... ,n} such that
n∑i=1[ρ（（αi,βi）,（γτ（i）,δτ（i）））]^2≤n[1-1/κ^2（Y）κ^2（Q）（1-d2F（Z,W）/n）],
where κ（Y） = ｜｜Y｜｜2｜｜Y^-1｜｜2,Z= （A,B）,W= （C, D） and dF（Z,W） = 1/√2｜｜Pz＊ -Pw＊｜｜F.     

On Sikkema-Kantorovič polynomials of order K

Let a_n≥0 and F(u)∈C [0,1], Sikkema constructed polynomials: , ifα _n ≡0, then B_n (0, F, x) are Bernstein polynomials. Let , we constructe new polynomials in this paper: Q _n ^(k) (α _n ,f(t))=d ^k /dx ^k B _n+k (α _n ,F _k (u),x), which are called Sikkema-Kantorovic polynomials of order k. Ifα _n ≡0, k=1, then Q_n ⁽¹⁾ (0, f(t), x) are Kantorovič polynomials P_n(f). Ifα _n =0, k=2, then Q_n ⁽²⁾, (0, f(t), x) are Kantorovič polynomials of second order (see Nagel). The main result is: Theorem 2. Let 1≤p≤∞, in order that for every f∈L^P [0, 1], , it is sufficient and necessary that , § 1. Let f(t) de a continuous function on [a, b], i. e., f∈C [a, b], we define^{[1–2],[8–10]}: . As usual, for the space L^p [a,b](1≤p<∞), we have and L[a, b]=l¹[a, b]. Letα _n ⩾0and F(u)∈C[0,1],Sikkema-Bernstein polynomials ^{[3] [4]}. The author expresses his thanks to Professor M. W. Müller of Dortmund University at West Germany for his supports.     

Rational combination of\left\{ {x^{\lambda _n } } \right\} are always dense inC [0,1]

The present paper shows that for any sequence of real numbers with infirite distinct elements {λ _n }, the rational combinations of are always dense in C_[0.1]     

Precise asymptotics in the law of the iterated logarithm*

Let X₁, X₂, ... be i.i.d. random variables with EX₁ = 0 and positive, finite variance σ², and set S_n = X₁ + ... + X_n. For any α > −1, β > −1/2 and for κ_n(ε) a function of ε and n such that κ_n(ε) log log n → λ as n ↑ ∞ and , we prove that

Multiple Boundary Concentrating Solutions to Dirichlet Problem of Hénon Equation

Abstract  Let Ω be the unit ball centered at the origin in . We study the following problem

Asymptotic number of solutions of some systems of diophantine inequalities

The problem of finding the asymptotic number of solutions of the system of inequalities $$\begin{gathered} \left\| {\alpha _i q} \right\|< q^{ - \sigma _i } (i = 1,...,n), \sigma _i > 0, \hfill \\ \sigma = \sum\nolimits_{i = 1}^n {\sigma _i< c(\alpha _1 ,...,\alpha _n ), q = 1,...,N,} \hfill \\ \end{gathered}$$ is solved under the assumption that for real numbers α₁,..., α_n, starting from some Q=max(q₁...,q_n) the inequality holds for any real λ≥0.