Nonlinear Integral Operators Research Articles

A class of integral operators preserving subordination and superordination

In this article, we obtain the subordination- and superordination-preserving properties of certain nonlinear integral operators defined on the space of normalized analytic functions in the open unit disk. A sandwich-type theorem for these integral operators is also presented.

Subordination properties for a general class of integral operators involving meromorphically multivalent functions

The purpose of the present paper is to investigate some subordination- and superordination-preserving properties of certain nonlinear integral operators defined on the space of meromorphically p-valent functions in the punctured open unit disk. The sandwich-type theorems associated with these integral operators are established. Relevant connections of the various results presented here with those involving relatively simpler nonlinear integral operators are also indicated.

Modular filter convergence theorems for Urysohn integral operators and applications

We prove some versions of modular convergence theorems for nonlinear Urysohn-type integral operators with respect to filter convergence. We consider pointwise filter convergence of functions giving also some applications to linear and nonlinear Mellin operators. We show that our results are strict extensions of the classical ones.

On Fatou type convergence of higher derivatives of certain nonlinear singular integral operators

The present paper concerns with the Fatou type convergence properties of the \(r-th\) and \((r+1)-th\) derivatives of the nonlinear singular integral operators defined as
 \[
 \left( I_{\lambda}f\right) (x)=\int\limits_{a}^{b}K_{\lambda}(t-x,f(t))\,{\rm d}t,\,\,\,\,\,\,\,x\in\left( a,b\right) ,
 \]
 acting on functions defined on an arbitrary interval \(\left( a,b\right)
 ,\) where the kernel \(K_{\lambda}\) satisfies some suitable assumptions. The present study is a continuation and extension of the results established in the paper [7].

Approximation Results with Respect to Multidimensional φ-Variation for Nonlinear Integral Operators

In this paper we study approximation problems for functions belonging to BV φ-spaces (spaces of functions of bounded φ-variation) in multidimensional setting. In particular, using a multidimensional concept of φ-variation in the sense of Tonelli introduced in [4], we obtain estimates, convergence results and, by means of suitable Lipschitz classes, results about the order of approximation for a family of nonlinear convolution integral operators.

Some convergence results for nonlinear singular integral operators

AbstractIn this paper, we establish some pointwise convergence results for a family of certain nonlinear singular integral operators

A General family of integral operators and associated subordination and superordination properties of some special analytic function classes

In the present paper, we obtain the subordination- and superordination-preserving properties of a certain general family of nonlinear integral operators which are defined on the space of normalized analytic functions in the open unit disk. The sandwich-type results for these integral operators are also presented.

Modular Convergence Theorems for Integral Operators in the Context of Filter Exhaustiveness and Applications

We prove some modular convergence theorems for nonlinear Urysohn-type integral operators, applying filter convergence of sequences of functions. We give some applications to Mellin operators, including moment, Mellin-Poisson-Cauchy and Mellin-Gauss-Weierstrass operators. We show that our results are proper extensions of the classical ones, and we pose an open problem.

Regularity theory for parabolic nonlinear integral operators

This article is dedicated to the regularity theory for solutions to a class of nonlinear integral variational problems. Those problems are involved in nonlocal image and signal processing.

Nonlinear integral operators with homogeneous kernels: pointwise approximation theorems

Here we give some approximation theorems concerning pointwise convergence for nets of nonlinear integral operators of the form: where the kernel (K λ)λ∈Λ satisfies some general homogeneity assumptions. Here Λ is a nonempty set of indices provided with a topology.

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.

Errata Corrige to: "Approximation by means of nonlinear integral operators in the space of functions with bounded $\varphi-$variation"

Here the authors want to point out that the term $I_2,$ of Proposition 2 of the original paper ([2]), has to be estimated in a different way. Moreover, now the proof of Theorem 4 of the original paper holds with the new assumption (2) (involving $K_w.3)'$ mentioned below), instead of (6.2) of [2] (involving $K_w.3)$ of the original paper), while, since Theorem 3 (convergence theorem) can be proved with both assumptions $K_w.3)$ and $K_w.3)'$, we prefer here to use directly $K_w.3)'$, in analogy with condition (2). Let us notice that it is easy to see that the two conditions $K_w.3)$ and $K_w.3)'$ cannot be compared. Here we want also to point out that in the convergence theorem of [3] as well as in Lemma 2 of [3], a similar problem occurs and it is solved in the same way proving that $V_{\varphi}[\lambda (H_w \circ f-f)]{\mbox{$\rightarrow$}} 0$, as $w{\mbox{$\rightarrow$}} +\infty$ for sufficiently small $\lambda>0,$ using assumption $K_w.3)$ (see Remark below).

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: $$ T_\lambda \left( {f;x} \right) = \int\limits_A^B {K_\lambda \left( {t,x,f\left( t \right)} \right)dt , x \in } \left\langle {a,b} \right\rangle ,\lambda \in \Lambda $$. In particular, we obtain the pointwise convergence and its rate at some characteristic points x 0 of f as (x, λ) ar(x 0, λ0) in L 1 〈 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].

Erratum to: Convergence in Variation and Rate of Approximation for Nonlinear Integral Operators of Convolution Type

In this paper we obtain estimates, convergence results and rate of approximation for functions belonging to BV–spaces (spaces of functions with bounded variation) by means of nonlinear convolution integral operators. We treat both the periodic and the non-periodic case using, respectively, the classical Jordan variation and the multidimensional variation in the sense of Tonelli.

Approximation of functions by certain nonlinear integral operators

We study approximation properties of certain nonlinear integral operators Ln* obtained by a modification of given operators Ln. The operators Ln;r and Ln;r* of r-times differentiable functions are also studied.

Rate of convergence of nonlinear integral operators for functions of bounded variation

The aim of this paper is to study the behavior of the operators Tλ defined by $$ T_\lambda (f;x) = \int\limits_a^b {K_\lambda (t - x,f(t))dt,x \in .} $$ . Here we estimate the rate of convergence at a point x, which has a discontinuity of the first kind as λ → λ0. This study is an extension of the papers [9] and [13], which includes Bernstein operators. Beta operators, Picard operators, Philips operators, Durrmeyer operators, etc. as special cases.

Approximation and geometric properties of some nonlinear complex integral convolution operators

In this paper we obtain convergence, shape-preserving results and rate of approximation of analytic functions by some nonlinear complex integral convolution operators.

On the Approximation Properties of a Class of Convolution Type Nonlinear Singular Integral Operators

Abstract In the present paper we obtain both the pointwise convergence and the rate of pointwise convergence theorems of a class of operators defined by as (𝑥,λ) → (𝑥0,λ 0) in 𝐿1 〈𝑎,𝑏〉, where 〈𝑎,𝑏〉 is an arbitrary interval in 𝑅. Here λ ∈ Λ and Λ is a nonempty set of indices.

A Class of Nonlinear Integral Operators Preserving Double Subordinations

The purpose of the present paper is to investigate some subordination- and superordination-preserving properties of certain integral operators defined on the space of meromorphic functions in the punctured open unit disk. The sandwich-type theorem for these integral operators is also considered.

A class of nonlinear integral operators preserving subordination and superordination

In the present article, we obtain the subordination- and superordination-preserving properties of certain nonlinear integral operators defined on the space of normalized analytic functions in the open unit disk. A differential sandwich-type theorem for these integral operators is also presented.