Kadec-Klee Property Research Articles

On the convergence of hybrid projection algorithms for asymptotically quasi-ϕ-nonexpansive mappings

Using the Mann iteration in an infinite-dimensional Hilbert space, only weak convergence theorems are obtained even for nonexpansive mappings. The purpose of this paper is to modify the Mann iteration and prove the strong convergence theorems without any compactness assumption for asymptotically quasi- ϕ -nonexpansive mappings. Moreover, strong convergence theorems are also established in a uniformly smooth and strictly convex Banach space with the Kadec–Klee property.

Strong convergence of shrinking projection methods for quasi- [formula omitted]-nonexpansive mappings and equilibrium problems

The purpose of this paper is to consider the convergence of a shrinking projection method for a finite family of quasi- ϕ -nonexpansive mappings and an equilibrium problem. Strong convergence theorems are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec–Klee property.

Modified Block Iterative Algorithm for Solving Convex Feasibility Problems in Banach Spaces

The purpose of this paper is to use the modified block iterative method to propose an algorithm for solving the convex feasibility problems for an infinite family of quasi- -asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces with Kadec-Klee property. The results presented in the paper improve and extend some recent results.

On a Hybrid Method for Generalized Mixed Equilibrium Problem and Fixed Point Problem of a Family of Quasi-"Equation missing" -Asymptotically Nonexpansive Mappings in Banach Spaces

AbstractWe prove a strong convergence theorem by using a hybrid method for finding a common element of the set of solutions for generalized mixed equilibrium problems, the set of fixed points of a family of quasi-"Equation missing"-asymptotically nonexpansive mappings in strictly convex reflexive Banach space with the Kadec-Klee property and, a Fréchet differentiable norm under weaker conditions. The method of the proof is different from, S. Takahashi and W. Takahashi that by (2008) and that by Takahashi and Zembayashi (2008) and see references. It also shows that the type of projection used in the iterative method is independent of the properties of the mappings. The results presented in the paper improve and extend some recent results.

Strong solvability in Orlicz spaces

The main objective of the authors is to characterize strong solvability of optimization problems where convergence of the values to the optimum already implies norm-convergence of the approximations to the minimal solution. It turns out that strong solvability can be geometrically characterized by the local uniform convexity of the corresponding convex functional (local uniform convexity being appropriately defined). For bounded functionals we establish that in reflexive Banach spaces strong solvability is characterized by the Frechet-differentiability of the convex conjugate. These results are based in part on a paper of Asplund and Rockafellar on the duality of A-differentiability and B-convexity of conjugate pairs of convex functions, where B is the polar of A. Before we apply these results to Orlicz spaces, we turn to E-spaces introduced by Fan and Glicksberg. Using the properties of E-spaces we can show that for finite not purely atomic measures Frechet differentiability of an Orlicz space already implies its reflexivity. The main theorem gives — in 17 equivalent statements — a characterization of strong solvability, local uniform convexity, and Frechet differentiability of the dual space, in case LΦ is reflexive. It is remarkable that all these properties can also be equivalently expressed by the differentiability of Φ or the strict convexity of Ψ. In particular, LΦ is an E-space, if LΦ is reflexive and Φ is strictly convex. We discuss applications that refer to • Tychonov-regularization: local uniformly convex regularisations are sufficient to ensure convergence. As we have given a complete description of local uniform convexity in Orlicz spaces we can state such regularizing functionals explicitly. • Ritz method: it is well known that the Ritz procedure generates a minimizing sequence. Actual convergence of the minimal solutions on each subspace is achieved if the original problem is strongly solvable. • Greedy algorithms: the convergence proof makes use of the Kadec-Klee property of E-spaces.

Strongly extreme points in Orlicz spaces equipped with the [formula omitted]-Amemiya norm

This paper is a continuation of the paper [Y. Cui, L. Duan, H. Hudzik, M. Wisła, Basic theory of p -Amemiya norm in Orlicz spaces ( 1 ≤ p ≤ ∞ ) : Extreme points and rotundity in Orlicz spaces equipped with these norms, Nonlinear Anal. 69 (2008) 1796–1816] and an extension of the paper [Y. Cui, T. Wang, Strongly extreme points of Orlicz space, J. Math. 4 (1987) 335-340]. Criteria for the Kadec–Klee property with respect to the convergence in measure and for strongly extreme points in Orlicz spaces L Φ , p equipped with the p -Amemiya norm ( 1 ≤ p ≤ ∞ ) are given. Orlicz spaces with nonempty (and empty) set of strongly extreme points of the unit ball B ( L Φ , p ) are characterized. It is interesting to note that for some Orlicz functions Φ there is a critical number p 0 such that the problem of emptiness or nonemptiness of strongly extreme points of the unit ball B ( L Φ , p ) looks different for 1 ≤ p ≤ p 0 and for p 0 < p ≤ ∞ . Criteria for the midpoint local uniform rotundity of the spaces L Φ , p are also deduced. Theorems stated in this paper, even restricted to the case p ∈ { 1 , ∞ } , are more general than the results of the latter reference mentioned above because the assumption lim u → ∞ Φ ( u ) / u = ∞ was not used. The methods used in the proofs are universal for all 1 ≤ p ≤ ∞ . New notions of p -spherical convergence and H 2 , μ -points are introduced and considered. Moreover, Orlicz spaces L Φ , p which are linearly isomorphic or isometric to L ∞ are characterized as auxiliary techniques and facts for some proofs.

The fixed point property via dual space properties

A Banach space has the weak fixed point property if its dual space has a weak ∗ sequentially compact unit ball and the dual space satisfies the weak ∗ uniform Kadec–Klee property; and it has the fixed point property if there exists ε > 0 such that, for every infinite subset A of the unit sphere of the dual space, A ∪ ( − A ) fails to be ( 2 − ε ) -separated. In particular, E-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property.

On some geometric and topological properties of generalized Orlicz–Lorentz sequence spaces

Generalized Orlicz–Lorentz sequence spaces λφ generated by Musielak-Orlicz functions φ satisfying some growth and regularity conditions (see [28] and [33]) are investigated. A regularity condition δλ2 for φ is defined in such a way that it guarantees many positive topological and geometric properties of λφ. The problems of the Fatou property, the order continuity and the Kadec–Klee property with respect to the uniform convergence of the space λφ are considered. Moreover, some embeddings between λφ and their two subspaces are established and strict monotonicity as well as lower and upper local uniform monotonicities are characterized. Finally, necessary and sufficient conditions for rotundity of λφ, their subspaces of order continuous elements and finite dimensional subspaces are presented. This paper generalizes the results from [19], [4] and [17]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Approximating common fixed points of finite family of asymptotically nonexpansive non-self mappings

Let K be a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T1 , T2 , ... , TN : K ? E be N asymptotically nonexpansive nonself mappings with sequences {rin} such that ??(n=1) rin &lt; ?, for all 1 ? i ? N and n n=1 n F = ?N(i-1) F (Ti) ? ?. Let {?in}, {?in} and {?in} are sequences in [0, 1] with i=1 ?in + ?in + ?in = 1 for all i = 1, 2, ... , N . From arbitrary x1 ? K , define the sequence {xn} iteratively by (6), where {uin} are bounded sequences in K with ??(n=1) uin &lt; ?. (i) If the dual E*of E has the Kadec-Klee property, then {xn} converges weakly to a common fixed point x*? F ; (ii) if {T1 , T2 , ... , TN} satisfies condition (B), then {xn} converges strongly to a common fixed point x*? F. .

The uniform Kadec–Klee property for Orlicz–Lorentz spaces

AbstractWe give sufficient conditions, as well as some necessary conditions, for the Orlicz–Lorentz space Λϕ,ω to have the weak-star uniform Kadec–Klee property. These results generalize the characterization of the weak-star uniform Kadec–Klee property in the Lorentz space Λω = Lω,1 due to Dilworth and Hsu.

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

Abstract 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 Σn=1∞(k n − 1) &lt; ∞, Σn=1∞(l n − 1) &lt; ∞ and Σn=1∞(j n − 1) &lt; ∞, 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 ∈ &gt; 0. Starting from arbitrary x 1 ∈ K define the sequence {x n} by $$\left\{ \begin{gathered} z_n = P(\alpha ''_n T_3 (PT_3 )^{n - 1} x_n + (1 - \alpha ''_n )x_n ), \hfill \\ y_n = P(\alpha '_n T_2 (PT_2 )^{n - 1} z_n + (1 - \alpha '_n )x_n ), \hfill \\ x_{n + 1} = P(\alpha _n T_1 (PT_1 )^{n - 1} y_n + (1 - \alpha _n )x_n ). \hfill \\ \end{gathered} \right.$$ (i) If the dual E* of E has the Kadec-Klee property then {x n} converges weakly to a common fixed point p ∈ F; (ii) If T satisfies condition (A′) then {x n} converges strongly to a common fixed point p ∈ F.

Weak and strong convergence to common fixed points of non-self nonexpansive mappings

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T1, T2 and T3 : K → E be nonexpansive mappings with nonempty common fixed points set. Let {αn}, {βn}, {γn}, {α′n}, {β′n}, {γ′n}, {α″n}, {β″n} and {γ″n} be real sequences in [0, 1] such that αn+βn+ γn=α′n + β′n + γ′n = α″n + β″n + γ″n = 1, starting from arbitrary X1 e K, define the sequence {xn} by {zn = P(α″nT1xn + β″nXn + γ″nwn) Yn = P(α′nT2zn + β′nxn + γ′nvn) xn+1 = PαnT3yn + βnxn + γnun) with the restrictions Σn=1∞ γn < ∞, Σn=1∞γ′n < ∞, Σn=1∞ γ″n < ∞. (i) If the dual E* of E has the Kadec-Klee property, then weak convergence of a {xn} to some x* ∈ F(T1) ∩ F(T2) ∩ (T3) is proved; (ii) If T1, T2 and T3 satisfy condition (A′), then strong convergence of {xn} to some x* ∈ F(T1) ∩ F(T2) ∩ (T3) is obtained.

Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces

Let E be a real uniformly convex Banach space whose dual space E ∗ satisfies the Kadec–Klee property, K be a closed convex nonempty subset of E. Let T 1 , T 2 , … , T m : K → K be asymptotically nonexpansive mappings of K into E with sequences (respectively) { k i n } n = 1 ∞ satisfying k i n → 1 as n → ∞ , i = 1 , 2 , … , m , and ∑ n = 1 ∞ ( k i n − 1 ) < ∞ . For arbitrary ϵ ∈ ( 0 , 1 ) , let { α i n } n = 1 ∞ be a sequence in [ ϵ , 1 − ϵ ] , for each i ∈ { 1 , 2 , … , m } (respectively). Let { x n } be a sequence generated for m ⩾ 2 by { x 1 ∈ K , x n + 1 = ( 1 − α 1 n ) x n + α 1 n T 1 n y n + m − 2 , y n + m − 2 = ( 1 − α 2 n ) x n + α 2 n T 2 n y n + m − 3 , ⋮ y n = ( 1 − α m n ) x n + α m n T m n x n , n ⩾ 1 . Let ⋂ i = 1 m F ( T i ) ≠ ∅ . Then, { x n } converges weakly to a common fixed point of the family { T i } i = 1 m . Under some appropriate condition on the family { T i } i = 1 m , a strong convergence theorem is also proved.

Approximative compactness and full rotundity in Musielak–Orlicz spaces and Lorentz–Orlicz spaces

We prove that approximative compactness of a Banach space X is equivalent to the conjunction of reflexivity and the Kadec–Klee property of X . This means that approximative compactness coincides with the drop property defined by Rolewicz in Studia Math. 85 (1987), 25–35. Using this general result we find criteria for approximative compactness in the class of Musielak–Orlicz function and sequence spaces for both (the Luxemburg norm and the Amemiya norm) as well as critria for this property in the class of Lorentz–Orlicz spaces. Criteria for full rotundity of Musielak–Orlicz spaces are also presented in the case of the Luxemburg norm. An example of a reflexive strictly convex Köthe function space which is not approximatively compact and some remark concerning the compact faces property for Musielak–Orlicz spaces are given.

Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces

Let E be a real uniformly convex Banach space, K be a closed convex nonempty subset of E which is also a nonexpansive retract with retraction P. Let T 1 , T 2 , … , T m : K → E be asymptotically nonexpansive mappings of K into E with sequences (respectively) { k in } n = 1 ∞ satisfying k in → 1 as n → ∞ , i = 1 , 2 , … , m , and ∑ n = 1 ∞ ( k in − 1 ) < ∞ . Let { α in } n = 1 ∞ be a sequence in [ ϵ , 1 − ϵ ] , ϵ ∈ ( 0 , 1 ) , for each i ∈ { 1 , 2 , … , m } (respectively). Let { x n } be a sequence generated for m ⩾ 2 by { x 1 ∈ K , x n + 1 = P [ ( 1 − α 1 n ) x n + α 1 n T 1 ( P T 1 ) n − 1 y n + m − 2 ] , y n + m − 2 = P [ ( 1 − α 2 n ) x n + α 2 n T 2 ( P T 2 ) n − 1 y n + m − 3 ] , ⋮ y n = P [ ( 1 − α m n ) x n + α m n T m ( P T m ) n − 1 x n ] , n ⩾ 1 . Let ⋂ i = 1 m F ( T i ) ≠ ∅ . Strong and weak convergence of the sequence { x n } to a common fixed point of the family { T i } i = 1 m are proved. Furthermore, if T 1 , T 2 , … , T m are nonexpansive mappings and the dual E ∗ of E satisfies the Kadec–Klee property, weak convergence theorem is also proved.

ON GEOMETRIC AND TOPOLOGICAL PROPERTIES OF THE CLASSES OF HEREDITARILY $\ell_{p}$ BANACH SPACES

A class of hereditarily $\ell_{p}$ ($1 \leq p \lt \infty$) Banach sequence spaces is constructed and denoted by $X_{\alpha,p}$. Any constructed space is a dual space. We show that (i) the predual of any member $X$ of the class of $X_{\alpha,1}$ contains asymptotically isometric copies of $c_0$.(ii) Every infinite dimensional subspace of $X$ contains asymptotically isometric complemented copies of $\ell_{1}$, and consequently, the dual X$^*$ of X contains subspaces isometrically isomorphic to $C[0,1]^*$. (iii) Every member of the class of $X_{\alpha,p}$ ($1 \leq p \lt \infty$) fails the Dunford-Pettis property. (iv) We observe that all $X_{\alpha,p}$ spaces are Banach spaces without unconditional basis but all constructed spaces contain a subspace which is weakly sequentially complete with an unconditional basis which is weakly null sequence but not in norm. (v) All spaces have asymptotic-norming and Kadec-Klee property. The predual of any $X_{\alpha,p}$ is an Asplund space.

Approximating fixed points of non‐self asymptotically nonexpansivemappings in Banach spaces

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T : K → E be an asymptotically nonexpansive mapping with {kn}⊂[1, ∞) such that and F(T) is nonempty, where F(T) denotes the fixed points set of T. Let {αn}, {αn′}, and {αn′′} be real sequences in (0,1) and ε ≤ αn, αn′, αn′′≤1 − ε for all n ∈ ℕ and some ε &gt; 0. Starting from arbitrary x1 ∈ K, define the sequence {xn} by x1 ∈ K, zn = P(αn′′T(PT) n−1xn + (1 − αn′′)xn), yn = P(αn′T(PT) n−1zn + (1 − αn′)xn), xn+1 = P(αnT(PT) n−1yn + (1 − αn)xn). (i) If the dual E* of E has the Kadec-Klee property, then { xn} converges weakly to a fixed point p ∈ F(T); (ii) if T satisfies condition (A), then {xn} converges strongly to a fixed point p ∈ F(T).

Rotundity properties in Banach spaces via sublinear operators

Conditions for extreme points, strongly extreme points, LUR-points, SU-points, points in direction of which the space is uniformly rotund and points with the Kadec–Klee property with respect to the convergence μ -almost everywhere for the space D E ( S ) , generated by a Köthe space E and a sublinear operator S defined on a linear space X and with values in L 0 , are given. As corollaries, conditions for rotundity, midpoint local uniform rotundity, local uniform rotundity, uniform rotundity in every direction and the Kadec–Klee property for the convergence locally in measure of these spaces are deduced. Interpretations of the results in cases of Köthe–Bochner spaces, Cesáro–Orlicz spaces and K- interpolation spaces are presented.

Regularization of Monotone Variational Inequalities with Mosco Approximations of the Constraint Sets

In this paper we study the convergence and stability in reflexive, smooth and strictly convex Banach spaces of a regularization method for variational inequalities with data perturbations. We prove that, when applied to perturbed variational inequalities with monotone, demiclosed, convex valued operators satisfying certain conditions of asymptotic growth, the regularization method we consider produces sequences which converge weakly to the minimal-norm solution of the original variational inequality, provided that the perturbed constraint sets converge to the constraint set of the original inequality in the sense of a modified form of Mosco convergence of order ≥1. If the underlying Banach space has the Kadec–Klee property, then the sequence generated by that regularization method is strongly convergent.

Approximating fixed points of non-self nonexpansive mappings in Banach spaces

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T : K → E be a nonexpansive non-self map with F ( T ) := { x ∈ K : Tx = x } ≠ ∅ . Suppose { x n } is generated iteratively by x 1 ∈ K , x n + 1 = P ( ( 1 - α n ) x n + α n TP [ ( 1 - β n ) x n + β n Tx n ] ) , n ⩾ 1 , where { α n } and { β n } are real sequences in [ ε , 1 - ε ] for some ε ∈ ( 0 , 1 ) . (1) If the dual E * of E has the Kadec–Klee property, then weak convergence of { x n } to some x * ∈ F ( T ) is proved; (2) If T satisfies condition ( A ) , then strong convergence of { x n } to some x * ∈ F ( T ) is obtained.