Let α ∈ (0, 2) and consider the operator Lf(x) = ∫ [f(x+ h)− f(x)− 1(|h|≤1)∇f(x) · h] A(x, h) |h|d+α dh, where the ∇f(x) · h term is omitted if α < 1. We consider the martingale problem corresponding to the operator L and under mild conditions on the function A prove that there exists a unique solution.

Starting from the study of the Shepard nonlinear operator of maxprod type in [2], [3], in the recent monograph [4], Open Problem 5.5.4, pp. 324-326, the Baskakov max-prod type operator is introduced and the question of the approximation order by this operator is raised. The aim of this note is to obtain the order of uniform approximation Cω1(f ; 1 √ n ) (with the explicit constant C = 24) of an...

The Dynamical Systems Method (DSM) is justified for solving operator equations F (u) = f , where F is a nonlinear operator in a Hilbert space H. It is assumed that F is a global homeomorphism of H onto H, that F ∈ C loc, that is, it has a continuous with respect to u Fréchet derivative F ′(u), that the operator [F ′(u)]−1 exists for all u ∈ H and is bounded, ||[F ′(u)]−1|| ≤ m(u), where m(u) > ...

The operator Lμ : f 7→ ∫ f(x)−f(y) x−y dμ(y) is, for a compactly supported measure μ with an L density, a closed, densely defined operator on L(μ). We show that the operator Q = pLμ−qLμ has polynomial eigenfunctions if and only if μ is a free Meixner distribution. The only time Q has orthogonal polynomial eigenfunctions is if μ is a semicircular distribution. More generally, the only time the o...

Let H(B) denote the space of all holomorphic functions on the unit ball B ⊂ Cn. In this paper, we investigate the integral operator Tg( f )(z) = ∫ 1 0 f (tz) g(tz)(dt/t), f ∈ H(B), z ∈ B, where g ∈H(B) and g(z)=∑nj=1 zj(∂g/∂zj)(z) is the radial derivative of g. The operator can be considered as an extension of the Cesàro operator on the unit disk. The boundedness of the operator on a-Bloch spac...

Some generalizations of Banach's contraction principle, which is a fixed-point theorem for mapping in metric spaces, have developed rapidly recent years. the things that support development generalization are emergence mappings more general than and spaces spaces. The generalized Kannan type one mappings. Furthermore, some b-metric modular bring concept into theorems Kannan-type on been given. ...

It is well known that for multipliers f of the Drury-Arveson space H n, ‖f‖∞ does not dominate the operator norm of Mf . We show that in general ‖f‖∞ does not even dominate the essential norm of Mf . A consequence of this is that there exist multipliers f of H n for which Mf fails to be essentially hyponormal, i.e., if K is any compact, self-adjoint operator, then the inequality M∗ f Mf −MfM f ...

Let f be a function in an Orlicz space L and μ(f,L) be the set of all the best Φ-approximants to f, given a σ−lattice L. Weak type inequalities are proved for the maximal operator f∗ = supn |fn|, where fn is any selection of functions in μ(f,Ln), and Ln is an increasing sequence of σ-lattices. Strong inequalities are proved in an abstract set up which can be used for an operator as f∗.

In this paper, we establish some existence of fixed-point results for asymptotically regular multivalued mappings satisfying Kannan-type contractive condition without assuming compactness the underlying metric space or continuity mapping.