P-uniformly Convex Banach Space Research Articles

Umbel convexity and the geometry of trees

For every p∈(0,∞), a new metric invariant called umbel p-convexity is introduced. The asymptotic notion of umbel convexity captures the geometry of countably branching trees, much in the same way as Markov convexity, the local invariant which inspired it, captures the geometry of bounded degree trees. Umbel convexity is used to provide a “Poincaré-type” metric characterization of the class of Banach spaces that admit an equivalent norm with Rolewicz's property (β). We explain how a relaxation of umbel p-convexity, called infrasup-umbel p-convexity, plays a role in obtaining compression rate bounds for coarse embeddings of countably branching trees. Local analogues of these invariants - fork p-convexity and infrasup-fork p-convexity - are introduced, and their relationship to Markov p-convexity and relaxations of the p-fork inequality is discussed. The metric invariants are estimated for a large class of Heisenberg groups, and in particular a parallelogram p-convexity inequality is proved for Heisenberg groups over p-uniformly convex Banach spaces. Finally, a new characterization of non-negative curvature is given.

A strongly convergent algorithm for solving multiple set split equality equilibrium and fixed point problems in Banach spaces

AbstractIn this article, using an Halpern extragradient method, we study a new iterative scheme for finding a common element of the set of solutions of multiple set split equality equilibrium problems consisting of pseudomonotone bifunctions and the set of fixed points for two finite families of Bregman quasi-nonexpansive mappings in the framework of p-uniformly convex Banach spaces, which are also uniformly smooth. For this purpose, we design an algorithm so that it does not depend on prior estimates of the Lipschitz-type constants for the pseudomonotone bifunctions. Furthermore, we present an application of our study for finding a common element of the set of solutions of multiple set split equality variational inequality problems and fixed point sets for two finite families of Bregman quasi-nonexpansive mappings. Finally, we conclude with two numerical experiments to support our proposed algorithm.

Generalized split feasibility problem for multi-valued Bregman quasi-nonexpansive mappings in Banach spaces

In this paper, the notion of generalized split feasibility problem (GSFP) is studied in p-uniformly convex Banach spaces. Some special cases of the GSFP are highlighted. A self-adaptive step-size iterative algorithm which converges strongly to solution of the GSFP is proved. The implementation of the method is demonstrated with two numerical examples. Our method does not require prior information of operator norms. Our results extend, improve and enrich recently announced related results in the literature.

A common solution of split equality monotone inclusion problem and split equality fixed point problem in real Banach spaces

In this paper, we introduce an iterative algorithm for approximating a common solution of Split Equality Monotone Inclusion Problem (SEMIP) and Split Equality Fixed Point Problem (SEFPP) in p-uniformly convex Banach spaces which are also uniformly smooth. Under standard and mild assumptions of monotonicity and right Bregman strongly condition of the SEMIP- and SEFPP-associated mappings, we establish the strong convergence of the scheme. Finally, we applied our result to study the convex minimization problem (CMP) and Equilibrium Problem. Our result complements and extends some recent results in literature.

Convergence analysis of an iterative method for solving multiple-set split feasibility problems in certain Banach spaces

Our purpose in this paper is to introduce an iterative scheme for solving multiple-set split feasiblity problems in p-uniformly convex Banach spaces which are also uniformly smooth using Bregman distance techniques. We further obtain a strong convergence result for approximating solutions of multiple-set split feasiblity problems in the framework of p-uniformly convex Banach spaces which are also uniformly smooth.

A Modified Iterative Algorithm for Split Feasibility Problems of Right Bregman Strongly Quasi-Nonexpansive Mappings in Banach Spaces with Applications

In this paper, we present a new iterative scheme for finding a common element of the solution set F of the split feasibility problem and the fixed point set F ( T ) of a right Bregman strongly quasi-nonexpansive mapping T in p-uniformly convex Banach spaces which are also uniformly smooth. We prove strong convergence theorem of the sequences generated by our scheme under some appropriate conditions in real p-uniformly convex and uniformly smooth Banach spaces. Furthermore, we give some examples and applications to illustrate our main results in this paper. Our results extend and improve the recent ones of some others in the literature.

Iterative solution of split variational inclusion problem in a real Banach spaces

In this paper, we study split variational inclusion problem in real Banach spaces with a view to analyze an iterative method for obtaining a solution of the split variational inclusion problem in Banach spaces. We propose an Halpern type algorithm and with our algorithm, we state and prove a strong convergence theorem for the approximation of solution of split variational inclusion problem in the frame work of p-uniformly convex Banach spaces which are also uniformly smooth. Our results extend and complement many known related results in the literature.

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 Ω 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.

Iterative process for a strictly pseudo-contractive mapping in uniformly convex Banach spaces

This paper is concerned with a new method to prove the weak convergence of a strictly pseudo-contractive mapping in a p-uniformly convex Banach space with more relaxed restrictions on the parameters. Our results extend and improve the corresponding earlier results. MSC:41A65, 47H17, 47J20.

State Dependent Sweeping Process in p-Uniformly Smooth and q-Uniformly Convex Banach Spaces

We prove the existence of solutions for the state depending convex sweeping processes in p-uniformly smooth and q-uniformly convex Banach spaces via new recent results on proximal analysis in reflexive smooth Banach spaces.

Strong Convergence Theorems of Viscosity Iterative Methods for a Countable Family of Strict Pseudo-contractions in Banach Spaces

For a countable family of strictly pseudo-contractions, a strong convergence of viscosity iteration is shown in order to find a common fixed point of in either a p-uniformly convex Banach space which admits a weakly continuous duality mapping or a p-uniformly convex Banach space with uniformly Gâteaux differentiable norm. As applications, at the end of the paper we apply our results to the problem of finding a zero of accretive operators. The main result extends various results existing in the current literature.

Manifolds of semi-negative curvature

The notion of nonpositive curvature in Alexandrov's sense is extended to include p-uniformly convex Banach spaces. Infinite dimensional manifolds of semi-negative curvature with a p-uniformly convex tangent norm fall in this class on nonpositively curved spaces, and several well-known results, such as existence and uniqueness of best approximations from convex closed sets, or the Bruhat-Tits fixed point theorem, are shown to hold in this setting, without dimension restrictions. Homogeneous spaces G/K of Banach-Lie groups of semi-negative curvature are also studied, explicit estimates on the geodesic distance and sectional curvature are obtained. A characterization of convex homogeneous submanifolds is given in terms of the Banach-Lie algebras. A splitting theorem via convex expansive submanifolds is proven, inducing the corresponding splitting of the Banach-Lie group G. Finally, these notions are used to study the structure of the classical Banach-Lie groups of bounded linear operators acting on a Hilbert space, and the splittings induced by conditional expectations in such setting.

Mann iteration of weak convergence theorems in Banach space

In this paper, by using Mann’s iteration process we will establish several weak convergence theorems for approximating a fixed point of k-strictly pseudocontractive mappings with respect to p in p-uniformly convex Banach spaces. Our results answer partially the open question proposed by Marino and Xu, and extend Reich’s theorem from nonexpansive mappings to k-strict pseudocontractive mappings.

ON THE EXISTENCE OF FIXED POINTS FOR LIPSCHITZIAN SEMIGROUPS IN BANACH SPACES

Let C be a nonempty bounded subset of a p-uniformly convex Banach space X, and T={T(t):t∈S} be a Lipschitzian semigroup on C with , where Np is the normal structure coefficient of X. Suppose also there exists a nonempty bounded closed convex subset E of C with the following properties: (P1)x ∈ E implies ωw(x)⊂ E; (P2)T is asymptotically regular on E. The authors prove that there exists a z∈E such that T(s)z=z for all s ∈ S. Further, under the similar condition, the existence of fixed points of Lipschitzian semigroups in a uniformly convex Banach space is discussed.

Fixed point theorems for generalized Lipschitzian semigroups

LetKbe a nonempty subset of ap-uniformly convex Banach spaceE,Ga left reversible semitopological semigroup, and𝒮={Tt:t∈G}a generalized Lipschitzian semigroup ofKinto itself, that is, fors∈G,‖Tsx−Tsy‖≤as‖x−y‖+bs(‖x−Tsx‖+‖y−Tsy‖)+cs(‖x−Tsy‖+‖y−Tsx‖), forx,y∈Kwhereas,bs,cs>0such that there exists at1∈Gsuch thatbs+cs<1for alls≽t1. It is proved that if there exists a closed subsetCofKsuch that⋂sco¯{Ttx:t≽s}⊂Cfor allx∈K, then𝒮with[(α+β)p(αp⋅2p−1−1)/(cp−2p−1βp)⋅Np]1/p<1has a common fixed point, whereα=lim sups(as+bs+cs)/(1-bs-cs)andβ=lim sups(2bs+2cs)/(1-bs-cs).

Random fixed point theorems for a certain class of mappings in banach spaces

Let (Ω, Σ) be a measurable space and C a nonempty bounded closed convex separable subset of p-uniformly convex Banach space E for some p > 1. We prove random fixed point theorems for a class of mappings T: Ω × C → C satisfying: for each x, y ∈ C, ω ∈ Ω and integer n ≥ 1, $$\left\| {T^\user1{n} (\omega ,\user1{x}) - T^\user1{n} (\omega ,\user1{x})} \right\|$$ $$ \geqslant \user1{a}(\omega ) \cdot \left\| {\user1{x} - \user1{y}} \right\| + \user1{b}(\omega )\left\{ {\left\| {\user1{x} - T^\user1{n} (\omega ,\user1{x})} \right\| + \left\| {\user1{y} - T^\user1{n} (\omega ,\user1{y})} \right\|} \right\}$$ $$ + \user1{c}(\omega )\left\{ {\left\| {\user1{x} - T^\user1{n} (\omega ,\user1{y})} \right\| + \left\| {\user1{y} - T^\user1{n} (\omega ,\user1{x})} \right\|} \right\},$$ where a, b, c: Ω → [0, ∞) are functions satisfying certain conditions and T n(ω, x) is the value at x of the n-th iterate of the mapping T(ω, ·). Further we establish for these mappings some random fixed point theorems in a Hilbert space, in L p spaces, in Hardy spaces H p and in Sobolev spaces H k,p for 1 < p < ∞ and k ≥ 0. As a consequence of our main result, we also extend the results of Xu [43] and randomize the corresponding deterministic ones of Casini and Maluta [5], Goebel and Kirk [13], Tan and Xu [37], and Xu [39, 41].

Fixed points of a pair of asymptotically regular mappings

In this paper, some theorems on fixed points of pair of asymptotically regular mappings in p-uniformly convex Banach space are proved. For these mappings some fixed point theorems in a Hilbert space, in Lp spaces, in Hardy spaces Hp and in Sobolev spaces Hpk, for 1<p<+∞ and k≥0 are also established. Thus, results of Gornicki[9, 10]. Kruppel[11, 12] and others are extended.

Fixed points of Lipschitzian semigroups in Banach spaces

We prove the following theorem: Let p > 1 and let E be a real p-uniformly convex Banach space, and C a nonempty bounded closed convex subset of E. If $T = {T_s: C → C: s ∈ G = [0,∞)}$ is a Lipschitzian semigroup such that $g = lim inf_{G ∋ α → ∞} inf_{G ∋

A fixed point theorem for nonexpansive semigroups in p-uniformly convex Banach spaces

We prove that if RUC(S) has a left invariant mean,ρ = {Ts:s ∈S} is a continuous representation ofS as nonexpansive mappings on a closed convex subsetC of a p-uniformly convex and p-uniformly smooth Banach space andC contains an element of bounded orbit, thenC contains a common fixed point forρ.

A Remark on Fixed Point Theorems for Lipschitzian Mappings

The aim of this note is to give the following theorem: Let p > 1 and let E be a p-uniformly convex Banach space, C a nonempty bounded closed convex subset of E, and A = [an,k]n,k ≥ 1 a strongly ergodic matrix. If T: C → C is a mapping such that g = lim infn → ∞infm = 0,1,2… ∑∞k = 1an,k · ||Tk + m||p < 1 + c, where c > 0 is some constant, then T has a fixed point in C.