Integral Operators Research Articles

The essential spectrum of the volume integral operator in electromagnetic scattering by a homogeneous obstacle with Lipschitz boundary and regularization

The essential spectrum of the volume integral operator in electromagnetic scattering by a homogeneous obstacle with Lipschitz boundary and regularization

New Integral Operator for Analytic Functions

Let Apn be the class of functions fz given by fz=zp+ap+nzp+n+ap+n+1zp+n+1+⋯ which are analytic in the open unit disk U. For fz∈Apn, new integral operators O−jfz and Ojfzj=0,1,2,⋯. are considered. The operators O−jfz and Ojfz satisfy OjO−jfz=O−jOjfz=fz and O−j∗Ojfz=Oj∗O−jfz=f∗fz for the convolution ∗ of O−jfz and Ojfz. In the present paper, the dominants for both operators O−jfz and Ojfz and subordinations for O−jfz and Ojfz are discussed. Also, new subclass Tpγm,δ,ρ;m,j concerning with m different boundary points is defined and discussed. Moreover, some interesting problems of Tpγm,δ,ρ;m,j associated with Ojfz are obtained. Furthermore, some interesting examples for our results are considered.

Construction of free quasi-idempotent differential Rota-Baxter algebras by Gröbner-Shirshov bases

Differential operators and integral operators are linked together by the first fundamental theorem of calculus. Based on this principle, the notion of a differential Rota-Baxter algebra was proposed by Guo and Keigher. Recently, the subject has attracted more attention since it is associated with many areas in mathematics, such as integro-differential algebras. This paper considers differential Rota-Baxter algebras in the quasi-idempotent operator context. We establish a Gröbner-Shirshov basis for free commutative quasi-idempotent differential algebras (resp. Rota-Baxter algebras, resp. differential Rota-Baxter algebras). This provides a linear basis of a free object in each of the three corresponding categories by the Composition-Diamond lemma. Communicated by P. Kolesnikov

On multiplicative Hermite–Hadamard- and Newton-type inequalities for multiplicatively (P,m)-convex functions

We develop a fresh family of functions, called as multiplicatively (P,m)-convex functions. In this direction, we study the properties of such type functions, and establish integer-order inequalities of Hermite–Hadamard type. After that, we introduce the multiplicative k-Riemann–Liouville fractional integrals and discuss their ⁎integrability, continuity as well as commutativity. Based on the proposed integral operators and the fact that the function is multiplicatively (P,m)-convex or (P,m)-convex, we derive some k-Riemann–Liouville fractional Hermite–Hadamard- and Newton-type inequalities. In the meantime, we provide certain examples together with their graphical descriptions, from which support the correctness of the inequalities obtained here. Some applications in special means are provided as well.

New exploration on approximate controllability of fractional neutral‐type delay stochastic differential inclusions with non‐instantaneous impulse

This paper aims to derive a new set of sufficient conditions for the existence and approximate controllability of neutral‐type fractional stochastic integrodifferential inclusions with infinite delay and non‐instantaneous impulse in a separable Hilbert space using the Atangana–Baleanu Caputo fractional derivative. We investigate the existence of a mild solution for the Atangana–Baleanu Caputo fractional neutral‐type delay integrodifferential stochastic system while taking into account the non‐instantaneous impulses. For this purpose, the Atangana–Baleanu Caputo fractional neutral‐type impulsive delay stochastic system is transferred into an equivalent fixed point problem via an integral operator, and then, the Bohnenblust–Karlin fixed point approach is applied. Further, the approximate controllability results of the proposed nonlinear stochastic impulsive control system are established under the consideration that the corresponding linear system is approximately controllable. The set of sufficient conditions is established by using the concepts of stochastic analysis, fractional calculus, fixed point technique, semigroup theory of bounded linear operators, and the theory of multivalued maps. To illustrate the abstract results, we provide an example at the end of the paper.

Relations between product and flag Triebel-Lizorkin spaces

Nagel, Ricci and Stein proved that product kernels are finite sums of flag kernels in the Euclidean space. We show that the product Triebel-Lizorkin space is the intersection of two flag Triebel-Lizorkin spaces. This extends a main result in [Chang D-C, Han Y, Wu X. Relations between product and flag Hardy spaces. J Geom Anal. 2021;31(7):6601–6623]. As an application, we provide a new proof of the boundedness of product singular integral operators on product Triebel-Lizorkin spaces.

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In this work, we present methods for constructing representations of polynomial covariance type commutation relations AB=BF(A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$AB=BF(A)$$\\end{document} by linear integral operators in Banach spaces Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p$$\\end{document}. We derive necessary and sufficient conditions on the kernel functions for the integral operators to satisfy the covariance type commutation relation for general polynomials F, as well as for important cases, when F is arbitrary affine or quadratic polynomial, or arbitrary monomial of any degree. Using the obtained general conditions on the kernels, we construct concrete examples of representations of the covariance type commutation relations by integral operators on Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p$$\\end{document}. Also, we derive useful general reordering formulas for the integral operators representing the covariance type commutation relations, in terms of the kernel functions.

The solution of stochastic evolution equation with the fractional derivative

In this paper, we consider the stochastic evolution equation driven by the Gaussian noise with white time and colored space, where the noise coefficient is the Marchaud fractional derivative. The key idea is that we transform our model into a stochastic space-fractional equation by taking the Marchaud fractional derivative, and then use Chaos expansion to prove the mild solution. There are three main results in this paper. First, we apply Chaos expansion to obtain the existence, uniqueness and Lyapunove exponent of the solution of the transformed equation. Second, we prove that there exists an unique mild solution of the original equation, the approach is taking the fractional integral operator into the transformed equation. Finally, we explore Hölder continuity of the mild solution.

Subordinations Results on a q-Derivative Differential Operator

In this research paper, we utilize the q-derivative concept to formulate specific differential and integral operators denoted as Rqn,m,λ, Fqn,m,λ and Gqn,m,λ. These operators are introduced with the aim of generalizing the class of Ruscheweyh operators within the set of univalent functions. We extract certain properties and characteristics of the set of differential subordinations employing specific techniques. By utilizing the newly defined operators, this paper goes on to establish subclasses of analytic functions defined on an open unit disc. Additionally, we delve into the convexity properties of the two recently introduced q-integral operators, Fqn,m,λ and Gqn,m,λ. Special cases of the primary findings are also discussed.

Lp Bounds for Rough Maximal Operators along Surfaces of Revolution on Product Domains

In this paper, we study the boundedness of rough Maximal integral operators along surfaces of revolution on product domains. For several classes of surfaces, we establish appropriate Lp bounds of these Maximal operators under the assumption Ω∈Lq(Sm−1×Sn−1) for some q>1, and then we employ these bounds along with Yano’s extrapolation argument to obtain the Lp boundedness of the aforementioned integral operators under a weaker condition in which Ω belongs to either the space Bq(0,2τ′−1)(Sm−1×Sn−1) or to the space L(logL)2/τ′(Sm−×Sn−1). Our results extend and improve many previously known results.

Research on Application of Fractional Calculus Operator in Image Underlying Processing

Fractional calculus extends traditional, integer-based calculus to include non-integer orders, offering a powerful tool for a range of engineering applications, including image processing. This work delves into the utility of fractional calculus in two crucial aspects of image processing: image enhancement and denoising. We explore the foundational theories of fractional calculus together with its amplitude–frequency characteristics. Our focus is on the effectiveness of fractional differential operators in enhancing image features and reducing noise. Experimental results reveal that fractional calculus offers unique benefits for image enhancement and denoising. Specifically, fractional-order differential operators outperform their integer-order counterparts in accentuating details such as weak edges and strong textures in images. Moreover, fractional integral operators excel in denoising images, not only improving the signal-to-noise ratio but also better preserving essential features such as edges and textures when compared to traditional denoising techniques. Our empirical results affirm the effectiveness of the fractional-order calculus-based image-processing approach in yielding optimal results for low-level image processing.

Multilinear Fourier integral operators on modulation spaces

Abstract In this article, we study properties of multilinear Fourier integral operators on weighted modulation spaces. In particular, using the theory of Gabor frames, we study boundedness of multilinear Fourier integral operators on products of weighted modulation spaces. Further, we investigate the periodic multilinear Fourier integral operator. Finally, we study continuity of bilinear pseudo-differential operators on modulation spaces for certain symbol classes, namely 𝐒𝐆 {\mathbf{SG}} -class.

Diffuse orthogonally additive operators

A regular orthogonally additive operator is a diffuse operator if it is disjoint from all operators in the band generated by the disjointness preserving operators. We present a criterion for principal lateral projections in an order complete vector lattice $E$ to be disjoint. We also state a criterion for a regular orthogonally additive operator to be diffuse. A criterion for the regularity of an integral Urysohn operator acting on ideal spaces of measurable functions is also presented. This criterion is used to show that an integral operator is diffuse. Examples of vector lattices are considered in which the sets of diffuse operators consist only of the zero element. The general form of an order projection operator onto the band generated by the disjointness preserving operators is found. Bibliography: 47 titles.

A Specific Category of Harmonic Functions Characterized By A Generalized Komatu Operator in Conjunction With The (R-K) Integral Operator and Applications to Neutrosophic Complex Field

A Specific Category of Harmonic Functions Characterized By A Generalized Komatu Operator in Conjunction With The (R-K) Integral Operator and Applications to Neutrosophic Complex Field

Линейные интегральные операторы в пространствах непрерывных и существенно ограниченных вектор-функций

The well-established criterion for the action and boundedness of a linear integral operator K from the space L_∞ of essentially bounded functions to the space C of functions continuous on a compact set is extended to the case of functions taking values in Banach spaces. The study further shows that if the operator K is active and bounded in the space C, it is also active and bounded in the space L_∞, with the norms of K in C and L_∞ being identical. A precise expression for the general value of the norm of the operator K in these spaces, expressed in terms of its operator kernel, is provided. Addicionally, an example of an integral operator (for scalar functions) is given, active and bounded in each of the spaces C and L_∞, but not acting from L_∞ into C. Convenient conditions for checking the boundedness of the operator K in C and L_∞ are discussed. In the case of the Banach space Y of the image function values of K being finite-dimensional, these conditions are both necessary and sufficient. In the case of infinitedimensionality of Y, they are sufficient but not necessary (as proven). For "dim" Y<∞, unimprovable estimates for the norm of the operator K are provided in terms of a 1-absolutely summing constant π_1 (Y), determined by the geometric properties of the norm in Y. Specifically, it is defined as the supremum over finite sets of nonzero elements of Y of the ratio of the sum of the norms of these elements to the supremum (over functionals with unit norm) of the sums of absolute values of the functional on these elements.

Integral Operator-Based Characteristic Mode Theory for Conducting, Material, and Lossy Structures

Integral operator-based theory of characteristic modes (CMs) for conducting, material, and lossy structures is reviewed. CMs are defined as solutions of a generalized eigenvalue equation (GEE). These GEEs are presented for various surface and volume integral operators and material structures. Interpretation of the characteristic eigenvalues in terms of electromagnetic power is studied based on the Mie expansion and integral operator formalism. Orthogonality and diagonalizing properties of the CMs are summarized. Challenges related to dielectric-magnetic bodies, lossy structures and spurious modes are discussed, as well as differences between the surface and volume operator approaches.

Some well known inequalities on two dimensional convex mappings by means of Pseudo $ \mathcal{L-R} $ interval order relations via fractional integral operators having non-singular kernel

<abstract><p>Fractional calculus and convex inequalities combine to form a comprehensive mathematical framework for understanding and analyzing a variety of problems. This note develops Hermite-Hadamard, Fejér, and Pachpatte type integral inequalities within pseudo left-right order relations by applying fractional operators with non-singular kernels. Recently, results have been developed using classical Riemann integral operators in addition to various other partial order relations that have some defects that we explained in literature in order to demonstrate the unique characteristics of pseudo order relations. To verify the developed results, we constructed several interesting examples and provided a number of remarks that demonstrate that this type of fractional operator generalizes several previously published results when different things are set up. This work can lead to improvements in mathematical theory, computational methods, and applications across a wide range of disciplines.</p></abstract>

The dual of a space of compact operators

<abstract><p>Let $ X $ and $ Y $ be Banach spaces. We provide the representation of the dual space of compact operators $ K(X, Y) $ as a subspace of bounded linear operators $ \mathcal{L}(X, Y) $. The main results are: (1) If $ Y $ is separable, then the dual forms of $ K(X, Y) $ can be represented by the integral operator and the elements of $ C[0, 1] $. (2) If $ X^{**} $ has the weak Radon-Nikodym property, then the dual forms of $ K(X, Y) $ can be represented by the trace of some tensor products.</p></abstract>

HERMITE–HADAMARD TYPE INEQUALITIES FOR ℏ-CONVEX FUNCTION VIA FUZZY INTERVAL-VALUED FRACTIONAL q-INTEGRAL

Fractional [Formula: see text]-calculus is considered to be the fractional analogs of [Formula: see text]-calculus. In this paper, the fuzzy interval-valued Riemann–Liouville fractional (RLF) [Formula: see text]-integral operator is introduced. Also new fuzzy variants of Hermite–Hadamard (HH) type and HH–Fejér inequalities, involving [Formula: see text]-convex fuzzy interval-valued functions (FIVFs), are presented by making use of the RLF [Formula: see text]-integral. The results not only generalize existing findings in the literature but also lay a solid foundation for research on inequalities concerning FIVFs. Moreover, to verify our theoretical findings, numerical examples and imperative graphical illustrations are provided.

Fractional $ 3/8 $-Simpson type inequalities for differentiable convex functions

<abstract><p>The main objective of this study is to establish error estimates of the new parameterized quadrature rule similar to and covering the second Simpson formula. To do this, we start by introducing a new parameterized identity involving the right and left Riemann-Liouville integral operators. On the basis of this identity, we establish some fractional Simpson-type inequalities for functions whose absolute value of the first derivatives are s-convex in the second sense. Also, we examine the special cases $ m = 1/2 $ and $ m = 3/8 $, as well as the two cases $ s = 1 $ and $ \alpha = 1 $, which respectively represent the classical convexity and the classical integration. By applying the definition of convexity, we derive larger estimates that only used the extreme points. Finally, we provide applications to quadrature formulas, special means, and random variables.</p></abstract>