More on stability of almost surjective <Emphasis Type="Italic">ε</Emphasis>-isometries of Banach spaces

Let X and Y be two Banach spaces, and f: X → Y be a standard ε-isometry for some ε ≥ 0. In this paper, by using a recent theorem established by Cheng et al. (2013–2015), we show a sufficient condition guaranteeing the following sharp stability inequality of f: There is a surjective linear operator T: Y → X of norm one so that

$$\left\| {Tf(x) - x} \right\| \leqslant 2\varepsilon , for all x \in X.$$

As its application, we prove the following statements are equivalent for a standard ε-isometry f: X → Y:

1. (i)
   lim inf _t→∞ dist(ty, f(X))/|t| < 1/2, for all y ∈ S _Y ;
    
2. (ii)
   \(\tau(f)\equiv sup_{y\epsilon S_{Y}}\) lim inf _t→∞dist(ty, f(X))/|t| = 0;
    
3. (iii)
   there is a surjective linear isometry U: X → Y so that

   $$\left\| {f(x) - Ux} \right\| \leqslant 2\varepsilon , for all x \in X.$$
    

This gives an affirmative answer to a question proposed by Vestfrid (2004, 2015).     

The Schröder-Bernstein property for weakly minimal theories

For a countable, weakly minimal theory T, we show that the Schröder-Bernstein property (any two elementarily bi-embeddable models are isomorphic) is equivalent to each of the following:

1. 1.
   For any U-rank-1 type q ∈ S(acl ^eq (?)) and any automorphism f of the monster model C, there is some n < ω such that f ⁿ (q) is not almost orthogonal to q ? f(q) ? … ? f ^n?1(q)
    
2. 2.
   T has no infinite collection of models which are pairwise elementarily bi-embeddable but pairwise nonisomorphic.
    

We conclude that for countable, weakly minimal theories, the Schröder-Bernstein property is absolute between transitve models of ZFC.     

Decomposer and associative functional equations

In the previous researches [2,3] b-integer and b-decimal parts of real numbers were introduced and studied by M.H. Hooshmand. The b-parts real functions have many interesting number theoretic explanations, analytic and algebraic properties, and satisfy the functional equation f (f(x) + y - f(y)) = f(x). These functions have led him to a more general topic in semigroups and groups (even in an arbitrary set with a binary operation [4] and the following functional equations have been introduced: Associative equations:

f(xf(yz))=f(f(xy)z),f(xf(yz))=f(f(xy)z)=f(xyz)

. Decomposer equations:

f(f^*(x)f(y))=f(y),f(f(x)f_*(y))=f(x)

.Strong decomposer equations:

f(f^*(x)y)=f(y),f(xf_*(y))=f(x)

.Canceler equations:

f(f(x)y)=f(xy),f(xf(y))=f(xy),f(xf(y)z)=f(xyz)

, where f^*(x) f(x) = f (x) f_* (x) = x. In this paper we solve them and introduce the general solution of the decomposer and strong decomposer equations in the sets with a binary operation and semigroups respectively and also associative equations in arbitrary groups. Moreover we state some equivalent equations to them and study the relations between the above equations. Finally we prove that the associative equations and the system of strong decomposer and canceler equations do not have any nontrivial solutions in the simple groups.     

Lower bounds for the merit factors of trigonometric polynomials from Littlewood classes

With the notation ,

we prove the following result.Theorem 1. Assume that p is a trigonometric polynomial of degree at most n with real coefficients that satisfies

||p||_L2(K)An^1/2 and ||p′||_L2(K)Bn^3/2.

Then

M₄(p)−M₂(p)M₂(p)

with

We also prove that

and

M₂(p)−M₁(p)10⁻³¹M₂(p)

for every , where denotes the collection of all trigonometric polynomials of the form

     

On the sectionwise connectedness of a contingent

Let X be a real normed space and let f: ? → X be a continuous mapping. Let T _f (t ₀) be the contingent of the graph G(f) at a point (t ₀, f(t ₀)) and let S ⁺ ? (0,∞) × X be the “right” unit hemisphere centered at (0, 0 _X ). We show that

1. 1.
   If dimX < ∞ and the dilation D(f, t ₀) of f at t ₀ is finite then T _f (t ₀) ∩ S ⁺ is compact and connected. The result holds for \(T_f (t_0 ) \cap \overline {S^ + } \) even with infinite dilation in the case f: [0,∞) → X.
    
2. 2.
   If dimX = ∞, then, given any compact set F ? S ⁺, there exists a Lipschitz mapping f: ? → X such that T _f (t ₀) ∩ S ⁺ = F.
    
3. 3.
   But if a closed set F ? S ⁺ has cardinality greater than that of the continuum then the relation T _f (t ₀) ∩ S ⁺ = F does not hold for any Lipschitz f: ? → X.
    

     

Existence of positive homoclinic solutions for damped differential equations

This paper is concerned with the existence of positive homoclinic solutions for the second-order differential equation

$$\begin{aligned} u^{\prime \prime }+cu^{\prime }-a(t)u+f(t,u)=0, \end{aligned}$$

where \(c\ge 0\) is a constant and the functions a and f are continuous and not necessarily periodic in t. Under other suitable assumptions on a and f, we obtain the existence of positive homoclinic solutions in both cases sub-quadratic and super-quadratic by using critical point theorems.

     

Holomorphic Jackson's theorems in polydiscs

The purpose of this article is to establish Jackson-type inequality in the polydiscs U^N of for holomorphic spaces X, such as Bergman-type spaces, Hardy spaces, polydisc algebra and Lipschitz spaces. Namely,

where is the deviation of the best approximation of fX by polynomials of degree at most k_j about the jth variable z_j with respect to the X-metric and is the corresponding modulus of continuity.     

Global behavior of a three-dimensional linear fractional system of difference equations

We investigate the global asymptotic behavior of solutions of the system of difference equations

where the parameters a, b, c, d, e, and f are in (0,∞) and the initial conditions x₀, y₀, and z₀ are arbitrary non-negative numbers. We obtain some global attractivity results for the positive equilibrium of this system for different values of the parameters.     

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.     

Stability for non-hyperbolic fixed points of scalar difference equations

We give some new criteria to determine the stability of a non-hyperbolic fixed point of the scalar difference equation

where and f is a sufficient smooth function. Our results are based on higher order derivative at a fixed point of .     

Existence of multiple positive solutions for nonlinear m-point boundary-value problems

In this paper, we afford some sufficient conditions to guarantee the existence of multiple positive solutions for the nonlinear m-point boundary-value problem for the one-dimensional p-Laplacian

(φ_p(u^′))^′+f(t,u)=0, t(0,1),

     

Positive solutions for one-dimensional -Laplacian boundary value problems with sign changing nonlinearity

This paper deals with the existence of positive solutions for the one-dimensional p-Laplacian

subject to the boundary value conditions:

where _p(s)=|s|^p−2s,p>1. We show that it has at least one or two positive solutions under some assumptions by applying the fixed point theorem. The interesting points are that the nonlinear term f is involved with the first-order derivative explicitly and f may change sign.     

On the nonlinear Hammerstein integral equations in Banach spaces and application to the boundary value problem of fractional order

In this paper, we study the existence of solutions of the operator equations p+λGfx=x in the Banach space C[I,E]. It is assumed the vector-valued function f is nonlinear Pettis-integrable. Some additional assumptions imposed on f are expressed in terms of a weak measure of noncompactness. To encompass the full scope of the paper, we investigate the existence of pseudo-solutions for the nonlinear boundary value problem of fractional type

under the Pettis integrability assumption imposed on f.     

Hardy–Sobolev critical elliptic equations with boundary singularities

Unlike the non-singular case s=0, or the case when 0 belongs to the interior of a domain Ω in (n3), we show that the value and the attainability of the best Hardy–Sobolev constant on a smooth domain Ω,

when 0<s<2, , and when 0 is on the boundary ∂Ω are closely related to the properties of the curvature of ∂Ω at 0. These conditions on the curvature are also relevant to the study of elliptic partial differential equations with singular potentials of the form:

where f is a lower order perturbative term at infinity and f(x,0)=0. We show that the positivity of the sectional curvature at 0 is relevant when dealing with Dirichlet boundary conditions, while the Neumann problems seem to require the positivity of the mean curvature at 0.     

Integral inequalities for the Hilbert transform applied to a nonlocal transport equation

We prove several weighted inequalities involving the Hilbert transform of a function f(x) and its derivative. One of those inequalities,

is used to show finite time blow-up for a transport equation with nonlocal velocity.     

An Improvement of the Constants in a Reverse Integral Inequality

We present an upper bound for the ratio [formula], where f is a positive decreasing function satisfying

Full-size image

for all t (0, a]. Our result sharpens an inequality of L. Nania.     

Bounds on margin distributions in learning problems

Let be a probability space and let P_n be the empirical measure based on i.i.d. sample (X₁,…,X_n) from P. Let be a class of measurable real valued functions on For define F_f(t):=P{ft} and F_n,f(t):=P_n{ft}. Given γ(0,1], define _n,γ(δ):=1/(n^1−γ/2δ^γ). We show that if the L₂(P_n)-entropy of the class grows as ^−α for some α(0,2), then, for all and all δ(0,Δ_n), Δ_n=O(n^1/2),

and

where and c(σ)↓1 as σ↓0 (the above inequalities hold for any fixed σ(0,1] with a high probability). Also, define

Then for all

uniformly in and with probability 1 (for the above ratio is bounded away from 0 and from ∞). The results are motivated by recent developments in machine learning, where they are used to bound the generalization error of learning algorithms. We also prove some more general results of similar nature, show the sharpness of the conditions and discuss the applications in learning theory.     

On Schwarzian Triangle Functions,Automorphic Forms and a Generalization of Ramanujan’s Triple Differential Equations

Let G be the group of the fractional linear transformations generated by

$$T(\tau ) = \tau + \lambda ,S(\tau ) = \frac{{\tau \cos \frac{\pi }{n} + \sin \frac{\pi }{n}}}{{ - \tau \sin \frac{\pi }{n} + \cos \frac{\pi }{n}}};$$

where

$$\lambda = 2\frac{{\cos \frac{\pi }{m} + \cos \frac{\pi }{n}}}{{\sin \frac{\pi }{n}}};$$

m, n is a pair of integers with either n ≥ 2,m ≥ 3 or n ≥ 3,m ≥ 2; τ lies in the upper half plane H.

A fundamental set of functions f₀, f_i and f_∞ automorphic with respect to G will be constructed from the conformal mapping of the fundamental domain of G. We derive an analogue of Ramanujan’s triple differential equations associated with the group G and establish the connection of f₀, f_i and f_∞ with a family of hypergeometric functions.     

Simple proofs and extensions of a result of L. D. Pustylnikov on the nonautonomous Siegel theorem

We present simple proofs of a result of L.D. Pustylnikov extending to nonautonomous dynamics the Siegel theorem of linearization of analytic mappings. We show that if a sequence f _n of analytic mappings of C ^d has a common fixed point f _n (0) = 0, and the maps f _n converge to a linear mapping A∞ so fast that

$$\sum\limits_n {{{\left\| {{f_m} - {A_\infty }} \right\|}_{L\infty \left( B \right)}} < \infty } $$

$${A_\infty } = diag\left( {{e^{2\pi i{\omega _1}}},...,{e^{2\pi i{\omega _d}}}} \right)\omega = \left( {{\omega _1},...,{\omega _q}} \right) \in {\mathbb{R}^d},$$

then f _n is nonautonomously conjugate to the linearization. That is, there exists a sequence h _n of analytic mappings fixing the origin satisfying

$${h_{n + 1}} \circ {f_n} = {A_\infty }{h_n}.$$

The key point of the result is that the functions hn are defined in a large domain and they are bounded. We show that

$${\sum\nolimits_n {\left\| {{h_n} - Id} \right\|} _{L\infty (B)}} < \infty .$$

We also provide results when f _n converges to a nonlinearizable mapping f∞ or to a nonelliptic linear mapping. In the case that the mappings f _n preserve a geometric structure (e. g., symplectic, volume, contact, Poisson, etc.), we show that the hn can be chosen so that they preserve the same geometric structure as the f _n . We present five elementary proofs based on different methods and compare them. Notably, we consider the results in the light of scattering theory. We hope that including different methods can serve as an introduction to methods to study conjugacy equations.

     

On stability of a cubic functional equation in intuitionistic fuzzy normed spaces

In this paper, we determine some stability results concerning the cubic functional equation

f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x)