Irreducible domains in Banach spaces

Every bounded domain in a complex Banach spaceE is biholomorphically equivalent to a finite product of irreducibles if and only ifE does not containc ₀. A quantitative version of this holds if and only ifE has finite cotype.     

A class of quasi-Newton generalized Steffensen methods on Banach spaces

We consider a class of generalized Steffensen iterations procedure for solving nonlinear equations on Banach spaces without any derivative. We establish the convergence under the Kantarovich–Ostrowski's conditions. The majorizing sequence will be a Newton's type sequence, thus the convergence can have better properties. Finally, a numerical comparation with the classical methods is presented.     

Matrix transformations on some sequence spaces related to strong Cesàro summability and boundedness

The spaces and introduced by Ayd?n and Ba?ar [C. Ayd?n, F. Ba?ar, Some new difference sequence spaces, Appl. Math. Comput. 157 (3) (2004) 677-693] can be considered as the matrix domains of a triangle in the sets of all sequences that are summable to zero, summable, and bounded by the Cesàro method of order 1. Here we define the sets of sequences which are the matrix domains of that triangle in the sets of all sequences that are summable, summable to zero, or bounded by the strong Cesàro method of order 1 with index p?1. We determine the β-duals of the new spaces and characterize matrix transformations on them into the sets of bounded, convergent and null sequences.     

Matrix summability and a generalized Gibbs phenomenon

Letf be a real-valued function sequence {f _k } that converges to on a deleted neighborhoodD of . If there is a subsequence {f _k(j) } and a number sequencex such that lim _j x _j = and either lim _j f _k(j) (x _j )>lim sup _x (x) or lim _j f _k(j) (x _j ) x (x), thenf is said to display theGibbs phenomenon at . IfA is a (real) summability matrix, thenAf is a function sequence given by(Af) _n (x)= _k=0 a _n,k f _k (x). IfAf displays the Gibbs phenomenon wheneverf does, thenA is said to beGP-preserving. By replacingf _k (x) withf _k (x _j )F _k,j , the Gibbs phenomenon is viewed as a property of the matrixF, andGP-preserving matrices are determined by properties of the matrix productAF. The general results give explicit conditions on the entries {a _n,k } that are necessary and/or sufficient forA to beGP-preserving. For example: if(x)0 thenF displaysGP iff lim _k,j F _k,j 0, and ifA isGP-preserving then lim _n,k A _n,k 0. IfA is a triangular matrix that is stronger than convergence, thenA is notGP-preserving. The general results are used to study the preservation of the Gibbs phenomenon by matrix methods of Nörlund, Hausdorff, and others.     

On generalized Hyers-Ulam stability of additive mapings on restricted domains of Banach spaces

Let \({(G,\cdot)}\) be a group (not necessarily Abelian) with unit \({e}\) and \({E}\) be a Banach space. In this paper, we show that there exist \({\alpha(p) > 0}\) for any \({0 < p < 1}\) and \({\beta(p,\varepsilon),\gamma(p,\varepsilon) > 0}\) for any \({0 < \varepsilon < \alpha(p)}\), such that for any surjective map \({f: G\rightarrow E}\) satisfying \({\big|\|f(x) + f(y)\|-\|f(xy) \|\big|\leq\varepsilon \|f(x)+f(y)\|^p}\) for all \({x,y\in G}\), there is a unique additive \({T:G\rightarrow E}\) such that \({\|f(x)-T(x)\|\leq\gamma(p,\varepsilon)\|f(x)\|^p}\) for all \({x\in G}\) satisfying \({\|f(x)\|\geq\beta(p,\varepsilon)}\). Moreover, we have \({\lim_{\varepsilon\rightharpoonup 0}\frac{\gamma(p,\varepsilon)}{\varepsilon} < \infty.}\)     

Mapping properties of matrix summability methods

Summary Conditions on a complex matrix A are given which are necessary and sufficient for A to have the property that if x belongs to the complex linear space generated by the set of all convergent real sequences having nonnegative kth differences, then Ax belongs to the complex linear space generated by the set of all convergent real sequences having nonnegative jth differences. Entrata in Redazione il 18 giugno 1970.     

Chebysheff-Halley-like methods in Banach spaces

Chebysheff-Halley methods are probably the best known cubically convergent iterative procedures for solving nonlinear equations. These methods however require an evaluation of the second Fréchet-derivative at each step which means a number of function evaluations proportional to the cube of the dimension of the space. To reduce the computational cost we replace the second Fréchet derivative with a fixed bounded bilinear operator. Using the majorant method and Newton-Kantorovich type hypotheses we provide sufficient conditions for the convergence of our method to a locally unique solution of a nonlinear equation in Banach space. Our method is shown to be faster than Newton’s method under the same computational cost. Finally we apply our results to solve nonlinear integral equations appearing in radiative transfer in connection with the problem of determination of the angular distribution of the radiant-flux emerging from a plane radiation field.     

Bloch spaces on bounded symmetric domains in complex Banach spaces

We give a definition of Bloch space on bounded symmetric domains in arbitrary complex Banach space and prove such function space is a Banach space. The properties such as boundedness, compactness and closed range of composition operators on such Bloch space are studied. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Extragradient methods for split feasibility problems and generalized equilibrium problems in Banach spaces

The purpose of this paper is the presentation of a new extragradient algorithm in 2‐uniformly convex real Banach spaces. We prove that the sequences generated by this algorithm converge strongly to a point in the solution set of split feasibility problem, which is also a common element of the solution set of a generalized equilibrium problem and fixed points of of two relatively nonexpansive mappings. We give a numerical example to investigate the behavior of the sequences generated by our algorithm.     

Discontinuous implicit generalized quasi-variational inequalities in Banach spaces

We consider the following implicit quasi-variational inequality problem: given two topological vector spaces E and F, two nonempty sets X E and C F, two multifunctions Γ : X → 2 ^X and Ф : X → 2 ^C , and a single-valued map ψ : , find a pair such that , Ф and for all . We prove an existence theorem in the setting of Banach spaces where no continuity or monotonicity assumption is required on the multifunction Ф. Our result extends to non-compact and infinite-dimensional setting a previous results of the authors (Theorem 3.2 of Cubbiotti and Yao [15] Math. Methods Oper. Res. 46, 213–228 (1997)). It also extends to the above problem a recent existence result established for the explicit case (C = E ^* and ).     

Note on the iterative methods for computing the generalized inverse over Banach spaces

In this paper, we construct iterative methods for computing the generalized inverse A over Banach spaces, and also for computing the generalized Drazin inverses a^d of Banach algebra element a. Moreover, we estimate the error bounds of the iterative methods for approximating A or a^d. Copyright © 2011 John Wiley & Sons, Ltd.     

On matrix transformations and sequence spaces

In this paper are given results on the spacesw _τ (μ) andc _τ (μ, μ′) the second one generalizing the well-known spacec _∞ (μ) of sequences that are strongly bounded. Then we deal with matrix transformations into these spaces. These results generalize those given in [7].