An L^0(F,R)-valued Function's Intermediate Value Theorem and Its Applications to Random Uniform Convexity

Let （Ω, F, P） be a probability space and L0（F,R） the algebra of equivalence classes of real- valued random variables on （Ω, F, P）. When L0（F,R） is endowed with the topology of convergence in probability, we prove an intermediate value theorem for a continuous local function from L0（F, R） to L0（F,R）. As applications of this theorem, we first give several useful expressions for modulus of random convexity, then we prove that a complete random normed module （S, ｜｜·｜｜） is random uniformly convex iff LP（S） is uniformly convex for each fixed positive number p such that 1 〈 p 〈 ＋∞.

On Uniform Convexity of Banach Spaces

This paper gives some relations and properties of several kinds of generalized convexity in Banach spaces. As a result, it proves that every kind of uniform convexity implies the Banach-Sakes property, and several notions of uniform convexity in literature are actually equivalent.

THE MODULUS OF CONVEXITY AND THE EQUIVALENT CONDITIONS OF UNIFORMLY NON-SQUARE SPACES

This paper shows that X is uniformly non-square iff P(η_o)】0 for someη_o∈(0,2).Thus X is super-reflexive if and only if for X there exists an equivalent normwhich meets the above condition.By virtue of the before mentioned result,the author derives the necessary,and suffi-cient conditions for X and l^p(X_j)being uniformly non-square,respectively,and gives acharacterization of finite-dimensional spaces which are uniformly non-square.

MARTINGALE REPRESENTATION IN NEARLY UNIFORMLY CONVEX SPACES

In this paper we give a martingale representation in nearly uniformly convex Banach spaces. Our result generalizes the representation theorem established by Landers and Rogge.

UNIFORMLY STARLIKE MAPPINGS AND UNIFORMLY CONVEX MAPPINGS ON THE UNIT BALL B^n

In this article, we extend the definition of uniformly starlike functions and uni- formly convex functions on the unit disk to the unit ball in C^n, give the discriminant criterions for them, and get some inequalities for them.

A New Class of Banach Spaces with Uniform Normal Structure

Let X be a Banach space, S(X) be the unit sphere of X, φ be a function: S(X)→ S(X *) such that φ(x)∈ x, and v φ(ε) =inf 1-12x+y: x,y∈S(X), and 〈φ(x), x-y 〉≥ε, 0≤ε≤2, where x is the set of norm 1 supporting functionals of S(X) at x. A geometric concept, modulus of V convexity V(ε)= sup {V φ(ε), for all φ: S(X)→S(X *)}, is introduced; the properties of V(ε) and the relationship between V(ε) and other geometric concepts are discussed. The main result is that V12>0 implies normal structure.

A RANDOM FIXED POINT ITERATION FOR THREE RANDOM OPERATORS ON UNIFORMLY CONVEX BANACH SPACES

In the present paper we introduce a random iteration scheme for three random operators defined on a closed and convex subset of a uniformly convex Banach space and prove its convergence to a common fixed point of three random operators. The result is also an extension of a known theorem in the corresponding non-random case.

Banach Space Valued Martingales with Respect to Complex Measure and Their Inequalities

Let d μ=ψ d ν be a complex valued measure where ν is a non negative measure and ψ is a complex valued function which satisfies b + p or b + ∞∩a 1 condition. We prove some basic martingale inequalities such as B G inequalities, weak ( p,p) and strong (p,p) type inequalities for Banach space valued martingale with respect to complex measure μ .

STRONG CONVERGENCE OF APPROXIMATED SEQUENCES FOR NONEXPANSIVE MAPPINGS IN BANACH SPACES

This paper studies the convergence of the sequence defined by x0 ∈ C, xn＋l =αnu＋（1-αn）Txn, n=0, 1,2,..., where 0 ≤αn ≤ 1, limn→∞ αn = 0, ∑n=0^∞ αn = ∞, and T is a nonexpansive mapping from a nonempty closed convex subset C of a Banach space X into itself. The iterative sequence {xn} converges strongly to a fixed point of T in the case when X is a uniformly convex Banach space with a uniformly Gateaux differentiable norm or a uniformly smooth Banach space only. The results presented in this paper extend and improve some recent results.

FURTHER INVESTIGATION INTO APPROXIMATION OF A COMMON SOLUTION OF FIXED POINT PROBLEMS AND SPLIT FEASIBILITY PROBLEMS

The purpose of this paper is to study and analyze an iterative method for finding a common element of the solution set ~ of the split feasibility problem and the set F（T） of fixed points of a right Bregman strongly nonexpansive mapping T in the setting of p- uniformly convex Banach spaces which are also uniformly smooth. By combining Mann＇s iterative method and the Halpern＇s approximation method, we propose an iterative algorithm for finding an element of the set F（T）∩Ω moreover, we derive the strong convergence of the proposed algorithm under appropriate conditions and give numerical results to verify the efficiency and implementation of our method. Our results extend and complement many known related results in the literature.

ON THE EXISTENCE OF COMMON FIXED POINTS FOR A PAIR OF LIPSCHITZIAN MAPPINGS IN BANACH SPACES

The existence of common fixed points for a pair of Lipschitzian mappings in Banach spaces is proved. By using this result, some common fixed point theorems are also established for these mappings in Hilbert spaces, in L p spaces, in Hardy spaces H p, and in Sobolev spaces H r,p , for 1<p<+∞ and r≥0.

A Weak Convergence Theorem for A Finite Family of Asymptotically Nonexpansive Mappings

The purpose of this paper is to prove a new weak convergence theorem for a finite family of asymptotically nonexpansive mappings in uniformly convex Banach space.

Multiple Solutions for p-Laplacian Type Equations

We establish the existence and multiplicity of weak solutions for equations which involve a uniformly convex elliptic operator in divergence form(in particular, a p-Laplacian operator), while the nonlinearity has a(p- 1)-superlinear growth at infinity. Our result completes and extends the relevant results of recent papers. The argument in the proof of our main result relies on the Z2-symmetric version of mountain pass lemma.

Property A_(UB) of Metric Spaces under Decompositions of Finite Depth

Property AUB is the notion in metric geometry which has applications in higher index problems.In this paper,the permanence property of property AUB of metric spaces under large scale decompositions of finite depth is proved.

A one-step implicit iterative method for two finite families of asymptotically nonexpansive mappings in a hyperbolic space

We introduce a one-step implicit iterative method for two finite families of asymptotically nonexpansive mappings in a hyperbolic space and use it to approximate common fixed points of these families. The results presented in this paper are new in the setting of hyperbolic spaces. On top, these are generalizations of several results in literature from Banach spaces to hyperbolic spaces. At the end of the paper, we give an example to validate our results.

A REMARK ON CERTAIN DIFFERENTIAL INEQUALITIES INVOLVING p-VALENT FUNCTIONS

In the present paper, we study certain differential inequalities involving p-valent functions and obtain sufficient conditions for uniformly p-valent starlikeness and uniformly p-valent convexity. We also offer a correct version of some known criteria for uniformly p-valent starlike and uniformly p-valent convex functions.

Picard Approximation of Fixed Points of Nonexpansive Mappings

Let C be a bounded convex subset in a uniformly convex Banach space X, x 0, u n∈C , then x n+1 =S nx n, where S n=α n0 I+α n1 T+α n2 T 2+…+α nk T k+γ nu n, α ni ≥0, 0<α≤α n0 ≤b<1, ∑ki=0α ni +γ n=1, and n≥1. It is proved that x n converges to a fixed point on T if T is a nonexpansive mapping.

Convergence Theorems of Iterative Sequences for Asymptotically Non-Expansive Mapping in a Uniformly Convex Banach Space

In this paper, Jurgen's relative result is extended to a uniformly convex Banach space, and the convergence of iterative sequence in an uniformly convex Banach space for asymptotically non-expansive mapping is proved.

Non-expansive Mappings and Iterative Methods in Uniformly Convex Banach Spaces

In this article,we will investigate the properties of iterative sequence for non-expansive mappings and present several strong and weak convergence results of successive approximations to fixed points of non-expansive mappings in uniformly convex Banach spaces.The results presented in this article generalize and improve various ones concerned with constructive techniques for the fixed points of non-expansive mappings.

Coarse Embedding into Uniformly Convex Banach Spaces

In this paper, the author studies the coarse embedding into uniformly convex Banach spaces. The author proves that the property of coarse embedding into Banach spaces can be preserved under taking the union of the metric spaces under certain condi- tions. As an application, for a group G strongly relatively hyperbolic to a subgroup H, the author proves that B（n） = {g ∈ G／ │g│suЭe≤ n} admits a coarse embedding into a uniformly convex Banach space if H does.