New iterations of <Emphasis Type="Italic">R</Emphasis>-order four with reduced computational cost

From a one-point iterative method of R-order at least three, we construct new two-point iterations to solve nonlinear equations in Banach spaces such that the computational cost is reduced, whereas the R-order of convergence is increased to at least four.      

Toward a unified theory for third R-order iterative methods for operators with unbounded second derivative

In this paper, we provide a semilocal convergence analysis for a family of Newton-like methods, which contains the best-known third-order iterative methods for solving a nonlinear equation F(x)=0 in Banach spaces. It is assumed that the operator F is twice Fréchet differentiable and F^″ satisfies a Lipschitz type condition but it is unbounded. By using majorant sequences, we provide sufficient convergence conditions to obtain cubic semilocal convergence. Results on existence and uniqueness of solutions, and error estimates are also given. Finally, a numerical example is provided.     

Minimization of the Tikhonov functional in Banach spaces smooth and convex of power type by steepest descent in the dual

For Tikhonov functionals of the form Ψ(x)=‖Ax−y‖ _Y ^r +α‖x‖ _X ^q we investigate a steepest descent method in the dual of the Banach space X. We show convergence rates for the proposed method and present numerical tests.     

On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces

We prove that ifC is a bounded closed convex subset of a uniformly convex Banach space,T:C→C is a nonlinear contraction, andS _n =(I+T+…+T ⁿ⁻¹ )/n, then lim _n ‖S _n (x)−TS _n (x)‖=0 uniformly inx inC. T also satisfies an inequality analogous to Zarantonello’s Hilbert space inequality. which permits the study of the structure of the weak ω-limit set of an orbit. These results are valid forB-convex spaces if some additional condition is imposed on the mapping. Partially supported by NSF Grant MCS-7802305A01.     

OnL 1-vorticity for 2-D incompressible flow

We prove the existence of a classical weak solution for the 2-D incompressible Euler equations with initial vorticity ω₀=ω ₀ ^′ + ω ₀ ^″ , where ω ₀ ^′ is inL ¹(R ²)⌢H ⁻¹(R ²), compactly supported, and ω ₀ ^″ is a compactly supported positive Radon measure inH ⁻¹(R ²).     

The Ulm method under mild differentiability conditions

The Ulm method is considered to approximate a solution of a nonlinear operator equation F(x) = 0. We study the convergence of this method when F′ is ω-conditioned and prove that the R-order of convergence is at least 1 + p if ω is quasi-homogeneous of type ω(tz)≤ t ^p ω(z), for z > 0, tϵ[0,1] and pϵ[0,1]. Preparation of this paper was partly supported by the Ministry of Education and Science (MTM 2005-03091).     

Stability of a Cubically Convergent Method for Generalized Equations

In Geoffroy et al, Acceleration of convergence in Dontchev's iterative method for solving variational inclusions Serdica Math. J. 29 (2003), pp. 45–54] we showed the convergence of a cubic method for solving generalized equations of the form 0 ∈ f(x) +- G(x) where f is a function and G stands for a set-valued map. We investigate here the stability of such a method with respect to some perturbations. More precisely, we consider the perturbed equation y ∈ f(x) +- G(x) and we show that the pseudo-Lipschitzness of the map (f +- G)⁻¹ is closely tied to the uniformity of our method in the sense that the attraction region does not depend on small perturbations of the parameter y. Finally, we provide an enhanced version of the convergence theorem established by Geoffroy, et al.     

Optimal properties of contraction semigroups in banach spaces

Suppose that(T _t )_t>0 is aC ₀ semi-group of contractions on a Banach spaceX, such that there exists a vectorx∈X, ‖x‖=1 verifyingJ ⁻¹(Jx)={x}, whereJ is the duality mapping fromX toP(X ^*). If |<T _t x,f>|→1, whent→+∞ for somef∈X ^*, ‖f‖≤1 thenx is an eigenvector of the generatorA, associated with a purcly imaginary eigenvalue. Because of Lin's example [L], the hypothesis onx∈X is the best possible. If the hypothesisJ ⁻¹(Jx)={x} is not verified, we can prove that ifJx is a singleton and ifJ ⁻¹(Jx) is weakly compact, then if |<T _t x, f>|→1, whent→+∞ for somef∈X ^*, ‖f‖≤1, there existsy∈J ⁻¹(Jx) such thaty is an eigenvector of the generatorA, associated with a purely imaginary eigenvalue. We give also a counter-example in the case whereX is one of the spaces ℓ₁ orL ₁.     

Local convergence of some iterative methods for generalized equations

We study generalized equations of the following form:

(render)

0f(x)+g(x)+F(x),

where f is Fréchet differentiable in a neighborhood of a solution x^* of (*) and g is Fréchet differentiable at x^* and where F is a set-valued map acting in Banach spaces. We prove the existence of a sequence (x_k) satisfying

which is super-linearly convergent to a solution of (*). We also present other versions of this iterative procedure that have superlinear and quadratic convergence, respectively.     

Difference estimates for continuous and discrete operator semigroups

For suitable bounded operator semigroups (e ^tA ) _t≥0 in a Banach space, we characterize the estimate ‖Ae ^tA ‖≤c/F(t) for large t, where F is a function satisfying a sublinear growth condition. The characterizations are by holomorphy estimates on the semigroup, and by estimates on powers of the resolvent. We give similar characterizations of the difference estimate ‖T ⁿ −T ⁿ⁺¹‖≤c/F(n) for a power-bounded linear operator T, when F(n) grows faster than n ^1/2 for large n.     

On the range of the derivative of Gâteaux-smooth functions on separable banach spaces

We prove that there exists a Lipschitz function froml ¹ into ℝ² which is Gateaux-differentiable at every point and such that for everyx, y εl ¹, the norm off′(x) −f′(y) is bigger than 1. On the other hand, for every Lipschitz and Gateaux-differentiable function from an arbitrary Banach spaceX into ℝ and for everyε > 0, there always exist two pointsx, y εX such that ‖f′(x) −f′(y)‖ is less thanε. We also construct, in every infinite dimensional separable Banach space, a real valued functionf onX, which is Gateaux-differentiable at every point, has bounded non-empty support, and with the properties thatf′ is norm to weak* continuous andf′(X) has an isolated pointa, and that necessarilya ε 0. This work has been initiated while the second-named author was visiting the University of Bordeaux. The second-named author is supported by grant AV 1019003, A1 019 205, GA CR 201 01 1198.     

A globally convergent inexact Newton method with a new choice for the forcing term

In inexact Newton methods for solving nonlinear systems of equations, an approximation to the step s _k of the Newton’s system J(x _k )s=−F(x _k ) is found. This means that s _k must satisfy a condition like ‖F(x _k )+J(x _k )s _k ‖≤η _k ‖F(x _k )‖ for a forcing term η _k ∈[0,1). Possible choices for η _k have already been presented. In this work, a new choice for η _k is proposed. The method is globalized using a robust backtracking strategy proposed by Birgin et al. (Numerical Algorithms 32:249–260, 2003), and its convergence properties are proved. Several numerical experiments with boundary value problems are presented. The numerical performance of the proposed algorithm is analyzed by the performance profile tool proposed by Dolan and Moré (Mathematical Programming Series A 91:201–213, 2002). The results obtained show a competitive inexact Newton method for solving academic and applied problems in several areas. Supported by FAPESP, CNPq, PRONEX-Optimization.     

Continuous operators on asymmetric normed spaces

If (X, p) and (Y, q) are two asymmetric normed spaces, the set LC(X, Y) of all continuous linear mappings from (X, p) to (Y, q) is not necessarily a linear space, it is a cone. If X and Y are two Banach lattices and p and q are, respectively, their associated asymmetric norms (p(x) = ‖⁺‖, q(y) = ‖y ⁺‖), we prove that the positive operators from X to Y are elements of the cone LC(X, Y). We also study the dual space of an asymmetric normed space and finally we give open mapping and closed graph type theorems in the framework of asymmetric normed spaces. The classical results for normed spaces follow as particular cases. The author acknowledges the support of the Ministerio de Educación y Ciencia of Spain and FEDER, under grant MTM2006-14925-C02-01 and Generalitat Valenciana under grant GV/2007/198.     

On a Global Complexity Bound of the Levenberg-Marquardt Method

In this paper, we investigate a global complexity bound of the Levenberg-Marquardt method (LMM) for the nonlinear least squares problem. The global complexity bound for an iterative method solving unconstrained minimization of φ is an upper bound to the number of iterations required to get an approximate solution, such that ‖∇φ(x)‖≤ε. We show that the global complexity bound of the LMM is O(ε ⁻²).     

Onp-absolutely summing constants of banach spaces

Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ _p(X) as inf{Σ _i ^=1/m |x*(x _i)|^p p Σ _i ^=1/m ‖x _i‖^p p]¹ p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x ₁,x ₂, …,x _m} ⊂X such that Σ _i ^=1/m ‖x _i‖>0. It follows immediately from [2] thatμ _p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ _p(X) for various spaces, and obtain some asymptotic estimates ofμ _p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.     

On an Asymptotic Behavior of Exponential Functional Equation

The stability problems of the exponential （functional） equation on a restricted domain will be investigated, and the results will be applied to the study of an asymptotic property of that equation. More precisely, the following asymptotic property is proved： Let X be a real （or complex） normed space. A mapping f ： X → C is exponential if and only if f（x ＋ y） - f（x）f（y） → 0 as ｜｜x｜｜ ＋ ｜｜y｜｜ → ∞ under some suitable conditions.     

ω-非常凸空间与ω-非常光滑空间

本文研究了ω-非常凸空间和ω-非常光滑空间的问题.利用局部自反原理和切片证明了ω-非常凸空间和ω-非常光滑空间的对偶关系,讨论了ω-非常凸空间和ω-非常光滑空间与其它凸性和光滑性的关系,给出了ω-非常凸空间与ω-非常光滑空间的若干特征刻画,所得结果完善了关于Banach空间凸性与光滑性理论的研究.     

Equations fonctionnelles et estimations de normes de matrices

We solve independently the equations 1/θ(x)θ(y)=ψ(x)−ψ(y)+φ(x−y)/θ(x−y) and 1/θ(x)θ(y)=σ(x)−σ(y)/θ(x−y)+τ(x)τ(y), τ(0)=0. In both cases we find θ²=aθ⁴+bθ²+c. We deduce estimates for the spectral radius of a matrix of type(1/θ(x _r −x _s )) (the accent meaning that the coefficients of the main diagonal are zero) and we study the case where thex _r are equidistant.

Dédié to à Monsieur le Professeur Otto Haupt à l'occasion de son cententiare avec les meilleurs voeux     

Concerning the order of approximation of periodic continuous functions by trigonometric interpolation polynomials

Letx _kn=2θk/n,k=0,1 …n−1 (n odd positive integer). LetR _n(x) be the unique trigonometric polynomial of order 2n satisfying the interpolatory conditions:R _n(x_kn)=f(x_kn),R _n ^(j)(x_kn)=0,j=1,2,4,k=0,1…,n−1. We setw ₂(t,f) as the second modulus of continuity off(x). Then we prove that |R _n(x)-f(x)|=0(nw₂(1/nf)). We also examine the question of lower estimate of ‖R _n-f‖. This generalizes an earlier work of the author.