2-uniformly Smooth Banach Space Research Articles

The Equivalence of Mann and Implicit Mann Iterations for Uniformly Pseudocontractive Mappings in Uniformly Smooth Banach Spaces

In this paper, suppose is an arbitrary uniformly smooth real Banach space, and is a nonempty closed convex subset of . Let be a generalized Lipschitzian and uniformly pseudocontractive self-map with . Suppose that , are defined by Mann iteration and implicit Mann iteration respectively, with the iterative parameter satisfying certain conditions. Then the above two iterations that converge strongly to fixed point of are equivalent.

A New General Iterative Method for Solution of a New General System of Variational Inclusions for Nonexpansive Semigroups in Banach Spaces

We introduce a new general system of variational inclusions in Banach spaces and propose a new iterative scheme for finding common element of the set of solutions of the variational inclusion with set‐valued maximal monotone mapping and Lipschitzian relaxed cocoercive mapping and the set of fixed point of nonexpansive semigroups in a uniformly convex and 2‐uniformly smooth Banach space. Furthermore, strong convergence theorems are established under some certain control conditions. As applications, finding a common solution for a system of variational inequality problems and minimization problems is given.

A Generalized Nonlinear Random Equations with Random Fuzzy Mappings in Uniformly Smooth Banach Spaces

We introduce and study the general nonlinear random -accretive equations with random fuzzy mappings. By using the resolvent technique for the -accretive operators, we prove the existence theorems and convergence theorems of the generalized random iterative algorithm for this nonlinear random equations with random fuzzy mappings in -uniformly smooth Banach spaces. Our result in this paper improves and generalizes some known corresponding results in the literature.

Approximation of Solutions to a System of Variational Inclusions in Banach Spaces

The purpose of this paper is to introduce an iterative method for finding solutions of a general system of variational inclusions with inverse-strongly accretive mappings. Strong convergence theorems are established in uniformly convex and 2-uniformly smooth Banach spaces.

A General Iterative Process for Solving a System of Variational Inclusions in Banach Spaces

The purpose of this paper is to introduce a general iterative method for finding solutions of a general system of variational inclusions with Lipschitzian relaxed cocoercive mappings. Strong convergence theorems are established in strictly convex and 2-uniformly smooth Banach spaces. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of strict pseudo-contraction mappings.

Convergence theorems for strict pseudo-contractions in [formula omitted]-uniformly smooth Banach spaces

In this paper, we discuss some properties of iterates generated by a strict pseudo-contraction or a finite family of strict pseudo-contractions in a real q -uniformly smooth Banach space. We prove the weak convergence and strong convergence of pseudo-contractive mappings. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007) 336–349], Acedo and Xu [G.L. Acedo, H.K. Xu, Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 67 (2007) 2258–2271] and Zhou [H. Zhou, Convergence theorems for λ -strict pseudo-contractions in 2-uniformly smooth Banach spaces, Nonlinear Anal. 69 (2008) 3160–3173].

A New System of Nonlinear Fuzzy Variational Inclusions Involving -Accretive Mappings in Uniformly Smooth Banach Spaces

A new system of nonlinear fuzzy variational inclusions involving -accretive mappings in uniformly smooth Banach spaces is introduced and studied many fuzzy variational and variational inequality (inclusion) problems as special cases of this system. By using the resolvent operator technique associated with -accretive operator due to Lan et al. and Nadler's fixed points theorem for set-valued mappings, an existence theorem of solutions for this system of fuzzy variational inclusions is proved. We also construct some new iterative algorithms for the solutions of this system of nonlinear fuzzy variational inclusions in uniformly smooth Banach spaces and discuss the convergence of the sequences generated by the algorithms in uniformly smooth Banach spaces. Our results extend, improve, and unify many known results in the recent literatures.

Cyclic algorithm for common fixed points of finite family of strictly pseudocontractive mappings of Browder–Petryshyn type

Let E be a real 2-uniformly smooth Banach space which is also uniformly convex (for example: L p or l p , 2 ≤ p < ∞ ) and K a nonempty closed convex subset of E . Let T 1 , T 2 , … , T N : K → K be strictly pseudocontractive mappings of K into K in the sense of Browder and Petryshyn with ∩ i = 1 N F ( T i ) ≠ 0̸ , where F ( T i ) = { x ∈ K : T i x = x } . Let { α n } n = 1 ∞ be a real sequence in [0,1] satisfying the condition 0 < a ≤ ( 1 − α n ) ≤ b < 2 λ c , ∀ n ≥ 1 and for some constants a , b , λ ∈ ( 0 , 1 ) and c ≥ 1 . Let { x n } be the sequence generated from an arbitrary x 1 ∈ K by x n + 1 = α n x n + ( 1 − α n ) T [ n ] x n , n ≥ 1 where T [ n ] = T i , i = n ( mod N ) , 1 ≤ i ≤ N . Weak and strong convergence theorems for the iterative approximation of common fixed points of the family { T i } i = 1 N are proved using the iterative sequence { x n } n = 1 ∞ . Furthermore, if E is an arbitrary real Banach space, it is shown that lim inf n → ∞ ‖ x n − T i x n ‖ = 0 for all i = 1 , 2 , … , N ; and a necessary and sufficient condition that guarantees the strong convergence of { x n } to a common fixed point of the family { T i } is given.

Strong convergence theorems for maximal monotone mappings in Banach spaces

Let E be a uniformly convex and 2-uniformly smooth real Banach space with dual E ∗ . Let A : E ∗ → E be a Lipschitz continuous monotone mapping with A −1 ( 0 ) ≠ ∅ . For given u , x 1 ∈ E , let { x n } be generated by the algorithm x n + 1 : = β n u + ( 1 − β n ) ( x n − α n A J x n ) , n ⩾ 1 , where J is the normalized duality mapping from E into E ∗ and { λ n } and { θ n } are real sequences in ( 0 , 1 ) satisfying certain conditions. Then it is proved that, under some mild conditions, { x n } converges strongly to x ∗ ∈ E where J x ∗ ∈ A −1 ( 0 ) . Finally, we apply our convergence theorems to the convex minimization problems.

Convergence theorems for [formula omitted]-strict pseudo-contractions in 2-uniformly smooth Banach spaces

In this paper, we discuss some properties of iterates generated by a strict pseudo-contraction or a finite family of strict pseudo-contractions in a real 2-uniformly smooth Banach space. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 324 (2007) 336–349]. In order to get a strong convergence theorem, we modify the normal Mann’s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in C . This result extends a recent result of Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60] both from nonexpansive mappings to λ -strict pseudo-contractions and from Hilbert spaces to 2-uniformly smooth Banach spaces.

Strong Convergence Theorems for Fixed Points of Asymptotically Nonexpansive Mappings in Uniformly Smooth Banach Spaces

In this work, we establish strong convergence of the improved modified three-step iteration methods for asymptotically nonexpansive operators in uniformly smooth and convex Banach spaces. Our results generalize the previous results of Schu [10], Osilike and Aniagbosor [5], Xu and Noor [13], as well as Owojori and Imoru [7] among others.

Ishikawa Iterative Process in Uniformly Smooth Banach Spaces

Let E be a uniformly smooth Banach space, K be a nonempty closed convex subset of E, and suppose: T: K→K is a continuous Φ-strongly pseudocontractive operator with a bounded range. Using a new analytical method, under general cases, the Ishikawa iterative process {xn} converges strongly to the unique fixed point x* of the operator T were proved. The paper generalizes and extends a lot of recent corresponding results.

The Solution by Iteration of Nonlinear Equations in Uniformly Smooth Banach Spaces

LetEbe a uniformly smooth Banach space and letT:D(T)⊂E↦Ebe a strong pseudocontraction with an open domainD(T) inEand a fixed pointx*∈D(T). We establish the strong convergence of the Mann and Ishikawa iterative processes (with errors) to the fixed point ofT. Related results deal with the iterative solution of operator equations of the formsf∈Txandf∈x+λTx, λ>0, whenTis a set-valued strongly accretative operator. Our theorems include the cases in which the operatorTis defined only locally. Explicit error estimates are also given.

Error Bounds for Approximation Solutions to Nonlinear Equations of Strongly Accretive Operators in Uniformly Smooth Banach Spaces

LetXbe a uniformly smooth Banach space andT:X→Xa strongly accretive operator. In this paper, we give the error bounds for the approximation solutions of the nonlinear equationTx=fgenerated by both the Mann and the Ishikawa iteration process. On the other hand, letKbe a nonempty convex subset ofXandT:K→Ka strictly pseudocontractive mapping. The related results deal with the error bounds for the iterative approximation of the fixed point ofTgenerated by these two processes.

A Bound for the Modulus of Continuity for Metric Projections in a Uniformly Convex and Uniformly Smooth Banach Space

In 1979, Bjornestal obtained a local estimate for a modulus of uniform continuity of the metric projection operator on a closed subspace in a uniformly convex and uniformly smooth Banach spaceB. In the present paper we give the global version of this result for the projection operator on an arbitrary closed convex set inB.

An Alternating Procedure for Operators on Uniformly Convex and Uniformly Smooth Banach Spaces

An Alternating Procedure for Operators on Uniformly Convex and Uniformly Smooth Banach Spaces

On the Law of Large Numbers in 2-Uniformly Smooth Banach Spaces

In this paper we extend the Kolmogorov strong law of large numbers to random variables taking their values in a 2-uniformly smooth Banach space $(B, \| \|)$. In our result, the convergence of the classical series of variances is replaced by the convergence of the series having general term $\sup\{Ef^2(X_n)/n^2: \|f\|_{B'} \leq 1\}.$