Approximation by nonlinear integral operators via summability process

In this paper, we study the approximation properties of nonlinear integral operators of convolution-type by using summability process. In the approximation, we investigate the convergence with respect to both the variation semi-norm and the classical supremum norm. We also compute the rate of approximation on some appropriate function classes. At the end of the paper, we construct a specific sequence of nonlinear operators, which verifies the summability process. Some graphical illustrations and numerical computations are also provided.     

On pointwise convergence of the family of Urysohn‐type integral operators

In this paper, we consider a family of nonlinear integral operators of Urysohn‐type and study the pointwise convergence of the family at characteristic points of L₁?function. The kernel K_λ(x,t,u(t)) depends on the positive parameter λ changing on the set of numbers with the accumulation point at infinity and K_λ(x,t,u(t)) is an entire analytic function of variable u, which is a bounded function belonging to L₁(R).     

Homogeneous operators and homogeneous integral operators

We introduce and study in a general setting the concept of homogeneity of an operator and, in particular, the notion of homogeneity of an integral operator. In the latter case, homogeneous kernels of such operators are also studied. The concept of homogeneity is associated with transformations of a measure—measure dilations, which are most natural in the context of our general research scheme. For the study of integral operators, the notions of weak and strong homogeneity of the kernel are introduced. The weak case is proved to generate a homogeneous operator in the sense of our definition, while the stronger condition corresponds to the most relevant specific examples—classes of homogeneous integral operators on various metric spaces—and allows us to obtain an explicit general form for the kernels of such operators. The examples given in the article—various specific cases—illustrate general statements and results given in the paper and at the same time are of interest in their own way.     

Fundamental solutions and Liouville type theorems for nonlinear integral operators

In this article we study basic properties for a class of nonlinear integral operators related to their fundamental solutions. Our goal is to establish Liouville type theorems: non-existence theorems for positive entire solutions for Iu?0 and for Iu+up?0, p>1.We prove the existence of fundamental solutions and use them, via comparison principle, to prove the theorems for entire solutions. The non-local nature of the operators poses various difficulties in the use of comparison techniques, since usual values of the functions at the boundary of the domain are replaced here by values in the complement of the domain. In particular, we are not able to prove the Hadamard Three Spheres Theorem, but we still obtain some of its consequences that are sufficient for the arguments.     

Voronovskaya‐type theorems for Urysohn type nonlinear Bernstein operators

The concern of this paper is to continue the investigation of convergence properties of nonlinear approximation operators, which are defined by Karsli. In details, the paper centers around Urysohn‐type nonlinear counterpart of the Bernstein operators. As a continuation of the study of Karsli, the present paper is devoted to obtain Voronovskaya‐type theorems for the Urysohn‐type nonlinear Bernstein operators.     

On nonlinear bivariate singular integral operators

In this paper, we give some pointwise convergence and Fatou type convergence theorems for a family of nonlinear bivariate singular integral operators in the following form: where m₁,m₂ ≥ 1 are fixed natural numbers, and ω ∈ Ω, Ω denotes a nonempty set of indices endowed with a topology. Here, denotes a family of kernel functions and f belongs to the space of Lebesgue integrable functions . Some numerical examples and graphical illustrations supporting the results are also given.     

Solvability of nonlinear integral equations and surjectivity of nonlinear Markov operators

In the present paper, we consider integral equations, which are associated with nonlinear Markov operators acting on an infinite-dimensional space. The solvability of these equations is examined by investigating nonlinear Markov operators. Notions of orthogonal preserving and surjective nonlinear Markov operators defined on infinite dimension are introduced, and their relations are studied, which will be used to prove the main results. We show that orthogonal preserving nonlinear Markov operators are not necessarily satisfied surjective property (unlike finite case). Thus, sufficient conditions for the operators to be surjective are described. Using these notions and results, we prove the solvability of Hammerstein equations in terms of surjective nonlinear Markov operators.     

Approximation of functions by certain nonlinear integral operators

We study approximation properties of certain nonlinear integral operators L _n * obtained by a modification of given operators L _n . The operators L _n;r and L _n;r * of r-times differentiable functions are also studied. We give theorems on approximation orders of functions by these operators in polynomial weight spaces.     

Weighted inequalities for integral operators with some homogeneous kernels

In this paper we study integral operators of the form

Linear and bilinear estimates for oscillatory integral operators related to restriction to hypersurfaces

We obtain bilinear estimates for oscillatory integral operators which are variable coefficient generalizations of bilinear restriction estimates for hypersurfaces. As applications, we improve the known estimates for oscillatory integrals.     

On approximation properties of non-convolution type nonlinear integral operators

In the present paper we state some approximation theorems concerning pointwise convergence and its rate for a class of non-convolution type nonlinear integral operators of the form:Tλ (f;x) = B A Kλ (t,x, f (t))dt , x ∈< a,b >, λ∈Λ. In particular, we obtain the pointwise convergence and its rate at some characteristic points x0 of f as (x,λ ) → (x0,λ0) in L1 < A,B >, where < a,b > and < A,B > are is an arbitrary intervals in R, Λ is a non-empty set of indices with a topology and λ0 an accumulation point of Λ in this topology. The results of the present paper generalize several ones obtained previously in the papers [19]-[23].     

On the monotonicity of some singular integral operators

The paper deals with the monotonicity of singular integral operators of the form where is the Cauchy singular integral operator on the interval (0,1) of the real axis and q is a power or logarithmic function. Under suitable assumptions, such singular integral operators are proved to be monotone and maximal monotone in spaces with power weights. Moreover, two related integral equations with weakly singular kernels of logarithmic type are studied. Copyright © 2012 John Wiley & Sons, Ltd.     

Sharp norm estimates for a class of integral operators

ABSTRACT

Norm comparison inequalities for two integral operators with radial kernels are established. Sharp norm estimates for operators with monotone and convex/concave kernels are obtained. Integral analogues of Bennett's estimates for summability matrices are given. The exact operator norms with power weights are also obtained for a class of integral operators with radial quasimonotone kernels.     

Commutators of fractional integral operators on Vanishing-Morrey spaces

In this note we prove a sufficient condition for commutators of fractional integral operators to belong to Vanishing Morrey spaces VL ^p,λ. In doing this we use an extension on Morrey spaces of an inequality by Fefferman and Stein concerning the sharp maximal function and the fractional maximal function and related Morrey inequalities.      

Boundedness properties for a class of integral operators including the index transforms and the operators with complex Gaussian kernels

In this paper we study the behavior of general integral operators on weighted L^p spaces. Particular cases include the main index transforms and the operators with complex Gaussian kernels. We also extend some previous results established in [E.R. Negrin, Proc. Amer. Math. Soc. 123 (1995) 1185-1190].     

Dominant eigenvalue problem for positive integral operators and its solution by Monte Carlo method

In this paper, a method of numerical solution to the dominant eigenvalue problem for positive integral operators is presented. This method is based on results of the theory of positive operators developed by Krein and Rutman. The problem is solved by Monte Carlo method constructing random variables in such a way that differences between results obtained and the exact ones would be arbitrarily small. Some numerical results are shown.     

Singular integral operators of near-product-type on the Heisenberg group

In this paper we establish Lp-boundedness (1<p<∞) for a class of singular convolution operators on the Heisenberg group whose kernels satisfy regularity and cancellation conditions adapted to the implicit (n+1)-parameter structure. The polyradial kernels of this type arose in [A.J. Fraser, An (n+1)-fold Marcinkiewicz multiplier theorem on the Heisenberg group, Bull. Austral. Math. Soc. 63 (2001) 35-58; A.J. Fraser, Convolution kernels of (n+1)-fold Marcinkiewicz multipliers on the Heisenberg group, Bull. Austral. Math. Soc. 64 (2001) 353-376] as the convolution kernels of (n+1)-fold Marcinkiewicz-type spectral multipliers m(L₁,…,Ln,iT) of the n-partial sub-Laplacians and the central derivative on the Heisenberg group. Thus they are in a natural way analogous to product-type Calderón-Zygmund convolution kernels on Rn. Here, as in [A.J. Fraser, An (n+1)-fold Marcinkiewicz multiplier theorem on the Heisenberg group, Bull. Austral. Math. Soc. 63 (2001) 35-58; A.J. Fraser, Convolution kernels of (n+1)-fold Marcinkiewicz multipliers on the Heisenberg group, Bull. Austral. Math. Soc. 64 (2001) 353-376], we extend to the (n+1)-parameter setting the methods and results of Müller, Ricci, and Stein in [D. Müller, F. Ricci, E.M. Stein, Marcinkiewicz multipliers and two-parameter structures on Heisenberg groups I, Invent. Math. 119 (1995) 199-233] for the two-parameter setting and multipliers m(L,iT) of the sub-Laplacian and the central derivative.