Let A + B be the pointwise (Minkowski) sum of two convex subsets A and B of a Banach space. Is it true that every continuous mapping h : X → A + B splits into a sum h = f + g of continuous mappings f : X → A and g : X → B? We study this question within a wider framework of splitting techniques of continuous selections. Existence of splittings is guaranteed by hereditary invertibility of linear ...

Let E be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping from E to E∗, C a nonempty closed convex subset of E which is also a sunny nonexpansive retract of E, and T : C→ E a non-expansive nonself-mapping with F(T) = ∅. In this paper, we study the strong convergence of two sequences generated by xn+1 = αnx + (1− αn)(1/n+ 1) ∑n j=0(PT) xn and yn+...

We introduce the notions of pointwise modulus of squareness and local modulus of squareness of a normed space X. This answers a question of C. Benitez, K. PrzesÃlawski and D. Yost about the definition of a sensible localization of the modulus of squareness. Geometrical properties of the norm of X (Fréchet smoothness, Gâteaux smoothness, local uniform convexity or strict convexity) are character...

The objective of this paper is to introduce a general scheme for deriving a posteriori error estimates by using duality theory of the calculus of variations. We consider variational problems of the form inf v∈V {F (v) +G(Λv)}, where F : V → R is a convex lower semicontinuous functional, G : Y → R is a uniformly convex functional, V and Y are reflexive Banach spaces, and Λ : V → Y is a bounded l...

The goal of the present paper is to investigate some characterization for generalized Struve functions of first kind to be in the new subclasses of β uniformly starlike and β uniformly convex functions of order α. Further we point out some consequences of our main results. Mathematics subject classification: 30C45.

In this paper, we consider some sufficient conditions for two integral operators to be starlike, close-to-convex and uniformly convex functions defined in the open unit disk. Mathematics subject classification (2000): 30C45.

The function Df , called the Bregman distance associated with f , is always well defined because ∂f(x) is nonempty and bounded, for all x ∈ X (see, e.g., [22]), so that the infimum in (1.1) cannot be −∞. It is easy to check that Df (x, y) ≥ 0 and that Df (x, x) = 0, for all x, y ∈ X. If f is strictly convex then Df (x, y) = 0 only when x = y. For t ∈ [0,∞) and z ∈ X let U(z, t) = {x ∈ X : ‖x− z...

This paper is concerned with the homogenization of the Dirichlet eigenvalue problem, posed in a bounded domain Ω ⊂ R, for a vectorial elliptic operator −∇·A(·)∇ with ε-periodic coefficients. We analyse the asymptotics of the eigenvalues λ when ε → 0, the mode k being fixed. A first-order asymptotic expansion is proved for λ in the case when Ω is either a smooth uniformly convex domain, or a con...

for some α (0 α< 1). We denote by p(α) the subclass of p consisting of functions which are p-valently convex of order α in U. In particular, we denote 1(0)= . A function f(z)∈ 1 is said to be uniformly convex inU if f(z) is in the class and has the property that the image arc f(γ) is convex for every circular arc γ contained in U with center at t ∈ U. We also denote by the subclass of 1 consist...

‎Let $\Omega_X$ be a bounded‎, ‎circular and strictly convex domain in a complex Banach space $X$‎, ‎and $\mathcal{H}(\Omega_X)$ be the space of all holomorphic functions from $\Omega_X$ to $\mathbb{C}$‎. ‎The growth space $\mathcal{A}^\nu(\Omega_X)$ consists of all $f\in\mathcal{H}(\Omega_X)$‎ ‎such that $$|f(x)|\leqslant C \nu(r_{\Omega_X}(x)),\quad x\in \Omega_X,$$‎ ‎for some constant $C>0$‎...