We study the relationships between Gateaux, Weak Hadamard and Fréchet differentiability and their bornologies for Lipschitz and for convex functions. AMS Subject Classification. Primary: 46A17, 46G05, 58C20. Secondary: 46B20.

Suppose n points are scattered uniformly at random in the unit square [0, 1]. Question: How many of these points can possibly lie on some curve of length λ? Answer, proved here: OP (λ · √ n). We consider a general class of such questions; in each case, we are given a class Γ of curves in the square, and we ask: in a cloud of n uniform random points, how many can lie on some curve γ ∈ Γ? Classes...

This paper deals with the class of continuous-time Descriptor Systems with continuous or differentiable uniformly bounded time-varying state delays. An improved delay-dependent stability and stabilization conditions are established without using model transformation and bounding technique for cross terms. Linear matrix inequality (LMI)based algorithm to design a state feedback control that stab...

‎In this paper‎, ‎we shall establish some extended Simpson-type inequalities‎ ‎for differentiable convex functions and differentiable concave functions‎ ‎which are connected with Hermite-Hadamard inequality‎. ‎Some error estimates‎ ‎for the midpoint‎, ‎trapezoidal and Simpson formula are also given‎.

0 |f(s)|ds ≤ |t| ≤ 1. Therefore the image of Bp is (uniformly) bounded. By Arzela-Ascoli, V : L p[0, 1]→ C[0, 1] is compact. The preceeding argument does not go through when V acts on L1[0, 1]. In this case equicontinuity fails, as is demonstrated by the following family {fn} ⊂ B1: fn(s) = n1[0,1/n](s). This suffices to preclude compactness of V ; in particular, V fn has no Cauchy subsequence. ...

Let E be a real Banach space with a uniformly Gâteaux differentiable norm and which possesses uniform normal structure, K a nonempty bounded closed convex subset of E, {Ti}i=1 a finite family of asymptotically nonexpansive self-mappings on K with common sequence {kn}∞n=1 ⊂ [1,∞), {tn}, {sn} be two sequences in (0, 1) such that sn + tn = 1 (n ≥ 1) and f be a contraction on K . Under suitable con...

The nearest correlation matrix problem is to find a correlation matrix which is closest to a given symmetric matrix in the Frobenius norm. The well-studied dual approach is to reformulate this problem as an unconstrained continuously differentiable convex optimization problem. Gradient methods and quasi-Newton methods such as BFGS have been used directly to obtain globally convergent methods. S...

The nearest correlation matrix problem is to find a correlation matrix which is closest to a given symmetric matrix under the Frobenius norm. The well studied dual approach is to reformulate this problem as an unconstrained continuously differentiable convex optimization problem. Gradient methods and quasi-Newton methods like BFGS have been used directly to obtain globally convergent methods. S...

This section collects the standard terminology and results used in the sequel (see [5]). Let I = [a,b] be an interval of real numbers. C1(I ,I), the set of all continuously differentiable functions from I into I , is a closed subset of the Banach Space C1(I ,R) of all continuously differentiable functions from I into R with the norm ‖ · ‖c1 defined by ‖φ‖c1 = ‖φ‖c0 +‖φ′‖c0 , φ∈ C1(I ,R) where ‖...