In this article we study graph-distance convergence of monotone operators. First, we prove a property that has been an open problem up to now: the limit of a sequence of graph-distance convergent maximal monotone operators in a Hilbert space is a maximal monotone operator. Next, we show that a sequence of maximal monotone operators converging in the same sense in a reflexive Banach space is uni...

In this paper we study some questions in connection with uniform rectifiability and the L boundedness of Calderón-Zygmund operators. We show that uniform rectifiability can be characterized in terms of some new adimensional coefficients which are related to the Jones’ β numbers. We also use these new coefficients to prove that n-dimensional CalderónZygmund operators with odd kernel of type C2 a...

A well-known model of nonstandard analysis is obtained by extending the structure of real numbers using an ultra power construction. A constructive approach due to Schmieden and Laugwitz uses instead a reduced power construction modulo a cofinite filter, but has the drawback that the transfer principle is weak. In this paper it is shown that this principle can be strengthened by employing Brouw...

Given an x0 ∈ Rn we study the infinite horizon problem of minimizing the expression ∫ T 0 f(t, x(t), x ′(t))dt as T grows to infinity where x : [0,∞) → Rn satisfies the initial condition x(0) = x0. We analyse the existence and the properties of approximate solutions for every prescribed initial value x0. We also establish that for every bounded set E ⊂ Rn the C([0, T ]) norms of approximate sol...

In this article, we study a class of Liénard equations x′′(t) + f(x(t))x′(t) + g1(x(t)) + g2(x(t− τ(t)) = e(t). Under some suitable conditions, we ensure that all solutions of the above Liénard equations are uniformly bounded. Our assumptions are less restrictive than those in [9]; thus we extend some previous results.

Let d ≥ 1 be fixed. Let F be a number field of degree d, and let E/F be an elliptic curve. Let E(F )tors be the torsion subgroup of E(F ). In 1996, Merel proved the uniform boundedness conjecture, i.e., there is a constant B(d), which depends on d but not on the chosen field F or on the curve E/F , such that the size of E(F )tors is bounded by B(d). Moreover, Merel gave a bound (exponential in ...

and Applied Analysis 3 The condition of right K-completeness for X, p leaves outside the scope of this theorem an important class of asymmetric normed spaces, the asymmetric normed spaces associated to normed lattices because these spaces are right K-complete only for the trivial case 13 . In this paper, we give a uniform boundedness type theorem in the setting of asymmetric normed spaces which...

Let K be a number field and let S be a finite set of places of K which contains all the Archimedean places. For any φ(z) ∈ K(z) of degree d ≥ 2 which is not a d-th power in K(z), Siegel’s theorem implies that the image set φ(K) contains only finitely many S-units. We conjecture that the number of such S-units is bounded by a function of |S| and d (independently of K, S and φ). We prove this con...

In this paper, the stability of translation-invariant spaces of distributions over locally compact groups is stated as boundedness of synthesis and projection operators. At first, a characterization of the stability of spline-type spaces is given, in the standard sense of the stability for shift-invariant spaces, that is, linear independence characterizes lower boundedness of the synthesis oper...