On smoothness conditions and convergence rates in the CLT in Banach spaces

Summary In Banach spaces the rate of convergence in the Central Limit Theorem is of orderO(n^–1/2) for sets which have regular boundaries with respect to the given covariance structure and which are three times differentiable. We show that in infinite dimensional spaces it is impossible to weaken this differentiability condition in general, whereas in finite dimensional spaces the assumption of convexity suffices. Similar results hold for the expectation of smooth functionals.Research supported by SFB 343 at Bielefeld and by the Alexander von Humboldt Foundation and completed at the University of Bielefeld, FRGResearch supported by the SFB 343 at Bielefeld     

Effective rates of convergence for Lipschitzian pseudocontractive mappings in general Banach spaces

This paper gives an explicit and effective rate of convergence for an asymptotic regularity result ‖Tx_n−x_n‖→0 due to Chidume and Zegeye in 2004 [14] where (x_n) is a certain perturbed Krasnoselski-Mann iteration schema for Lipschitz pseudocontractive self-mappings T of closed and convex subsets of a real Banach space. We also give a qualitative strengthening of the theorem by Chidume and Zegeye, by weakening the assumption of the existence of a fixed point. For the bounded case, our bound is polynomial in the data involved.     

Semilocal convergence of a sixth-order method in Banach spaces

In this paper, we introduce a new iterative method of order six and study the semilocal convergence of the method by using the recurrence relations for solving nonlinear equations in Banach spaces. We prove an existence-uniqueness theorem and give a priori error bounds which demonstrates the R-order of the method to be six. Finally, we give some numerical applications to demonstrate our approach.     

Semilocal convergence of a continuation method in Banach spaces

This paper is concerned with the semilocal convergence of a continuation method between two third-order iterative methods, namely, the Halley’s and the convex acceleration of Newton’s method, also known as the Super-Halley’s method. This convergence analysis is discussed using the recurrence relations approach. This approach simplifies the analysis and leads to improved results. The convergence analysis is established under the assumption that the second Frëchet derivative satisfies Lipschitz continuity condition. An existence-uniqueness theorem is given. Also, a closed form of error bound is derived in terms of a real parameter α ∈ [0, 1]. Two numerical examples are worked out to demonstrate the efficacy of our approach. On comparing the existence and uniqueness region and error bounds for the solution obtained by our analysis with those obtained by using majorizing sequences [15], we observed that our analysis gives better results. Further, we have observed that for particular values of the α, our analysis reduces to those for the Halley’s method (α = 0) and the convex acceleration of Newton’s method (α = 1), respectively, with improved results.     

A third order method for fixed points in Banach spaces

This paper deals with a third order Stirling-like method used for finding fixed points of nonlinear operator equations in Banach spaces. The semilocal convergence of the method is established by using recurrence relations under the assumption that the first Fréchet derivative of the involved operator satisfies the Hölder continuity condition. A theorem is given to establish the error bounds and the existence and uniqueness regions for fixed points. The R-order of the method is also shown to be equal to at least (2p+1) for p∈(0,1]. The efficacy of our approach is shown by solving three nonlinear elementary scalar functions and two nonlinear integral equations by using both Stirling-like method and Newton-like method. It is observed that our convergence analysis is more effective and give better results.     

Semilocal convergence of a sixth-order Jarratt method in Banach spaces

In this paper, we study the semilocal convergence for a sixth-order variant of the Jarratt method for solving nonlinear equations in Banach spaces. The semilocal convergence of this method is established by using recurrence relations. We derive the recurrence relations for the method, and then prove an existence-uniqueness theorem, along with a priori error bounds which demonstrates the R-order of the method. Finally, we give some numerical applications to demonstrate our approach.     

Recurrence relations for semilocal convergence of a fifth-order method in Banach spaces

In this paper, we study the semilocal convergence for a fifth-order method for solving nonlinear equations in Banach spaces. The semilocal convergence of this method is established by using recurrence relations. We prove an existence-uniqueness theorem and give a priori error bounds which demonstrates the R-order of the method. As compared with the Jarratt method in Hernández and Salanova (Southwest J Pure Appl Math 1:29–40, 1999) and the Multi-super-Halley method in Wang et al. (Numer Algorithms 56:497–516, 2011), the differentiability conditions of the convergence of the method in this paper are mild and the R-order is improved. Finally, we give some numerical applications to demonstrate our approach.     

Recurrence relations for semilocal convergence of a Newton-like method in Banach spaces

The aim of this paper is to establish the semilocal convergence of a multipoint third order Newton-like method for solving F(x)=0 in Banach spaces by using recurrence relations. The convergence of this method is studied under the assumption that the second Fréchet derivative of F satisfies Hölder continuity condition. This continuity condition is milder than the usual Lipschitz continuity condition. A new family of recurrence relations are defined based on the two new constants which depend on the operator F. These recurrence relations give a priori error bounds for the method. Two numerical examples are worked out to demonstrate the applicability of the method in cases where the Lipschitz continuity condition over second derivative of F fails but Hölder continuity condition holds.     

Improving the rate of convergence of Newton methods on Banach spaces with a convergence structure and applications

In this note, we use inexact Newton-like methods to find solutions of nonlinear operator equations on Banach spaces with a convergence structure. Our technique involves the introduction of a generalized norm as an operator from a linear space into a partially ordered Banach space. In this way, the metric properties of the examined problem can be analyzed more precisely. Moreover, this approach allows us to derive from the same theorem, on the one hand, semilocal results of Kantorovich-type, and on the other hand, global results based on monotonicity considerations. By imposing very general Lipschitz-like conditions on the operators involved, on the one hand, we cover a wider range of problems, and on the other hand, by choosing our operators appropriately, we can find sharper error bounds on the distances involved than before. Furthermore, we show that special cases of our results reduce to the corresponding ones already in the literature. Finally, several examples are being provided where our results compare favorably with earlier ones.     

Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces

In this paper, we study a variant of the super-Halley method with fourth-order convergence for nonlinear equations in Banach spaces. We make an attempt to establish the semilocal convergence of this method by using recurrence relations. The recurrence relations for the method are derived and then an existence-uniqueness theorem is given to establish the R-order of the method to be four and a priori error bounds. Finally, some numerical applications are presented to demonstrate our approach.     

Convergence of a third order method for fixed points in Banach spaces

In this paper, the semilocal convergence of a third order Stirling-like method used to find fixed points of nonlinear operator equations in Banach spaces is established under the assumption that the first Fréchet derivative of the involved operator satisfies ??-continuity condition. It turns out that this convergence condition is weaker than the Lipschitz and the H?lder continuity conditions on first Fréchet derivative of the involved operator. The importance of our work lies in the fact that numerical examples can be given to show that our approach is successful even in cases where Lipschitz and H?lder continuity conditions on first Fréchet derivative fail. It also avoids the evaluation of second order Fréchet derivative which is difficult to compute at times. A priori error bounds along with the domains of existence and uniqueness of a fixed point are derived. The R-order of the method is shown to be equal to (2p?+?1) for p????(0,1]. Finally, two numerical examples involving nonlinear integral equations are worked out to show the efficacy of our approach.     

Weak convergence theorems for strict pseudo-contractions in Banach spaces

In this paper, it is shown that there is a gap in the paper [Chidume, C. E., Shahzad, N.: Weak convergence theorems for a finite family of strict pseudo-contractions. Nonlinear Anal., 72, 1257–1265(2010)], consequently, the main results of the paper do not hold in uniformly smooth Banach spaces. Meanwhile, it is also shown that the main results(Lemma 3.4, Theorems 3.5–3.6, 3.8–3.9) in the paper [Cholamjiak, P., Suantai, S.: Weak convergence theorems for a countable family of strict pseudo-contractions in Banach spaces. Fixed Point Theory Appl., 2010, Article ID 632137, 16 pages(2010)] do not hold in Lpfor p 3. Finally, some modified results are presented in the setting of uniformly smooth and uniformly convex Banach spaces which include Lpfor p ≥ 2 as special cases. Furthermore, our arguments are also different from the ones given by the authors above.     

Strong convergence theorems for nonexpansive semigroup in Banach spaces

Let K be a nonempty closed convex subset of a reflexive and strictly convex Banach space E with a uniformly Gâteaux differentiable norm, and a nonexpansive self-mappings semigroup of K, and a fixed contractive mapping. The strongly convergent theorems of the following implicit and explicit viscosity iterative schemes {xn} are proved.

xn=αnf(xn)+(1−αn)T(tn)xn,     

A convergence theorem for Newton-like methods in Banach spaces

Summary A convergence theorem for Newton-like methods in Banach spaces is given, which improves results of Rheinboldt [27], Dennis [4], Miel [15, 16] and Moret [18] and includes as a special case an updated (affine-invariant [6]) version of the Kantorovich theorem for the Newton method given in previous papers [35, 36]. Error bounds obtained in [34] are also improved. This paper unifies the study of finding sharp error bounds for Newton-like methods under Kantorovich type assumptions.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041 and by the Ministry of Education, Japan     

Strong convergence results for nonself multimaps in Banach spaces

We prove strong convergence theorems for multimaps under mild conditions, which include Browder's convergence theorem as well as Reich's convergence theorem. We thus provide a partial answer to Jung's question.

     

Integrals and Banach spaces for finite order distributions

Let B _c denote the real-valued functions continuous on the extended real line and vanishing at −∞. Let B _r denote the functions that are left continuous, have a right limit at each point and vanish at −∞. Define A _c ⁿ to be the space of tempered distributions that are the nth distributional derivative of a unique function in B _c . Similarly with A _r ⁿ from B _r . A type of integral is defined on distributions in A _c ⁿ and A _r ⁿ . The multipliers are iterated integrals of functions of bounded variation. For each n ∈ ℕ, the spaces A _c ⁿ and A _r ⁿ are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to B _c and B _r , respectively. Under the ordering in this lattice, if a distribution is integrable then its absolute value is integrable. The dual space is isometrically isomorphic to the functions of bounded variation. The space A _c ¹ is the completion of the L ¹ functions in the Alexiewicz norm. The space A _r ¹ contains all finite signed Borel measures. Many of the usual properties of integrals hold: H?lder inequality, second mean value theorem, continuity in norm, linear change of variables, a convergence theorem.     

Semilocal convergence of an eighth-order method in Banach spaces and its computational efficiency

The aim of this paper is to study the semilocal convergence of the eighth-order iterative method by using the recurrence relations for solving nonlinear equations in Banach spaces. The existence and uniqueness theorem has been proved along with priori error bounds. We have also presented the comparative study of the computational efficiency in case of R^m with some existing methods whose semilocal convergence analysis has been already discussed. Finally, numerical application on nonlinear integral equations is given to show our approach.