Fractional Integral Operators Research Articles

Fractional operators with homogeneous kernel on the Calderón product of Morrey spaces

AbstractWe investigate fractional operators with homogeneous kernel in Morrey spaces. In particular, we prove that fractional integral operators and fractional maximal operators with homogeneous kernel are bounded from the Calderón product of Morrey spaces to certain Morrey spaces. Our results can be seen as a generalization of a recent result on the relation between the boundedness of (classical) fractional operators and interpolation of Morrey spaces. What is new about this paper is not only the passage from the classical fractional integral operators to the rough integral operators. Even the case of fractional integral operators, handled in earlier papers, is significantly simplified.

Boundedness of commutators of variable Marcinkiewicz fractional integral operator in grand variable Herz spaces

Let Sn−1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{S}^{n-1}$\\end{document} denote unit sphere in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{n}$\\end{document} equipped with the normalized Lebesgue measure. Let Φ∈Ls(Sn−1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Phi \\in L^{s}(\\mathbb{S}^{n-1})$\\end{document} be a homogeneous function of degree zero such that ∫Sn−1Φ(y′)dσ(y′)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\int _{\\mathbb{S}^{n-1}}\\Phi (y^{\\prime})d \\sigma (y^{\\prime})=0$\\end{document}, where y′=y/|y|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$y^{\\prime}=y/|y|$\\end{document} for any y≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$y\ eq 0$\\end{document}. The commutators of variable Marcinkiewicz fractional integral operator is defined as [b,μΦ]βm(f)(x)=(∫0∞|∫|x−y|≤sΦ(x−y)[b(x)−b(y)]m|x−y|n−1−β(x)f(y)dy|2dss3)12.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ [b,\\mu _{\\Phi}]^{m}_{\\beta }(f)(x )= \\left ( \\int \\limits _{0} ^{ \\infty }\\left |\\int \\limits _{|x -y | \\leq s} \\frac{\\Phi (x -y )[b(x )-b(y )]^{m}}{|x -y |^{n-1-\\beta (x )}}f(y )dy \\right |^{2} \\frac{ds}{s^{3}}\\right )^{\\frac{1}{2}}. $$\\end{document}In this paper, we obtain the boundedness of the commutators of the variable Marcinkiewicz fractional integral operator on grand variable Herz spaces K˙p(⋅)α(⋅),q),θ(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\dot{K} ^{\\alpha (\\cdot ), q),\ heta}_{ p(\\cdot )}(\\mathbb{R}^{n})}$\\end{document}.

New integral inequalities for synchronous functions via Atangana–Baleanu fractional integral operators

Chebyshev’s inequality is a well-known inequality in the field of inequality theory and yields remarkable results for functions that exhibit synchronous behavior. The study, designed to obtain Chebyshev type inequalities with the help of the Atangana–Baleanu fractional integral operator, which has become the most popular concept of fractional analysis in recent years, aims to obtain general forms for synchronous functions. In the process of obtaining results for synchronized functions within the scope of the study, functional tools such as Atangana–Baleanu fractional integral operator, various mathematical analysis operations and induction were used. The most striking aspect of the findings is that for some special values of the parameter of the fractional integral operator, some of the results can be reduced to the Chebyshev inequality. Compared to the studies in the literature, this study has a unique aspect in that the findings obtained with the help of the Atangana–Baleanu fractional integral operator contain general forms, as well as bringing together a popular concept of synchronous functions and fractional analysis.

Solvability of Quadratic Integral Equations of Urysohn Type Involving Hadamard Variable-Order Operator

This study investigates the existence of solutions to integral equations in the form of quadratic Urysohn type with Hadamard fractional variable order integral operator. Due to the lack of semigroup properties in variable-order fractional integrals, it becomes challenging to get the existence and uniqueness of corresponding integral equations, hence the problem is examined by employing the concepts of piecewise constant functions and generalized intervals to address this issue. In this context, the problem is reformulated as integral equations with constant orders to obtain the main results. Both the Schauder and Banach fixed point theorems are employed to prove the uniqueness results. In addition, an illustration is included in order to verify those results in the final step.

Inequalities for strongly s-convex functions via Atangana-Baleanu fractional integral operators

It is more convenient to use fractional derivatives and integrals to express and represent rapid changes than to use integer derivatives and integrals. For this reason, fractional analysis has been found worthy of study in many fields. In recent years, fractional derivatives and integrals have been discussed together with inequality theory and the studies have attracted attention. In this article, we discuss new Hermite-Hadamard type approximations for strongly convex functions with the help of Atangana-Baleanu fractional integral operators. Additionally, new upper bounds have been obtained using various auxiliary inequalities with the help of twice differentiable strongly convex functions.

Multi-Dimensional Integral Transform with Fox Function in Kernel in Lebesgue-Type Spaces

This paper is devoted to the study of the multi-dimensional integral transform with the Fox H-function in the kernel in weighted spaces with integrable functions in the domain R+n with positive coordinates. Due to the generality of the Fox H-function, many special integral transforms have the form studied in this paper, including operators with such kernels as generalized hypergeometric functions, classical hypergeometric functions, Bessel and modified Bessel functions and so on. Moreover, most important fractional integral operators, such as the Riemann–Liouville type, are covered by the class under consideration. The mapping properties in Lebesgue-weighted spaces, such as the boundedness, the range and the representations of the considered transformation, are established. In special cases, it is applied to the specific integral transforms mentioned above. We use a modern technique based on the extensive use of the Mellin transform and its properties. Moreover, we generalize our own previous results from the one-dimensional case to the multi-dimensional one. The multi-dimensional case is more complex and needs more delicate techniques.

Results for Analytic Function Associated with Briot–Bouquet Differential Subordinations and Linear Fractional Integral Operators

Results for Analytic Function Associated with Briot–Bouquet Differential Subordinations and Linear Fractional Integral Operators

Erratum to: Generalized fractional maximal and integral operators on Orlicz and generalized Orlicz–Morrey spaces of the third kind

Erratum to: Generalized fractional maximal and integral operators on Orlicz and generalized Orlicz–Morrey spaces of the third kind

Novel Estimations of Hadamard-Type Integral Inequalities for Raina’s Fractional Operators

In the present paper, utilizing a wide class of fractional integral operators (namely the Raina fractional operator), we develop novel fractional integral inequalities of the Hermite–Hadamard type. With the help of the well-known Riemann–Liouville fractional operators, s-type convex functions are derived using the important results. We also note that some of the conclusions of this study are more reasonable than those found under certain specific conditions, e.g., s=1, λ=α, σ(0)=1, and w=0. In conclusion, the methodology described in this article is expected to stimulate further research in this area.

The Boundedness of Generalized Fractional Integral Operators on Small Morrey Spaces

The small Morrey space is the set of locally Lebesgue integrable functions with norm defined supremum over radius of ball . This paper aims to prove the boundedness properties of the generality of fractional integral operators in small Morrey spaces using Hedberg-type inequality. The first, in this paper will be discuss to prove Hedberg-type inequality on small Morrey spaces using dyadic decomposition, H lder inequality, and doubling condition. Furthermore, by using the inequality, it can be proven that the boundedness of generalized fractional integral operators on small Morrey spaces.

A two-weight boundedness criterion and its applications

In this paper, the authors establish a general (two-weight) boundedness criterion for a pair of functions, [Formula: see text], on [Formula: see text] in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz–)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón–Zygmund operators, Littlewood–Paley [Formula: see text]-functions, Lusin area functions, Littlewood–Paley [Formula: see text]-functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) boundedness criterion, the authors further obtain the (two-weight) boundedness of Riesz transforms, Littlewood–Paley [Formula: see text]-functions, and fractional integral operators associated with second-order divergence elliptic operators with complex bounded measurable coefficients on [Formula: see text] in the aforementioned function spaces.

A wavelet collocation method for fractional Black–Scholes equations by subdiffusive model

AbstractIn this investigation, we propose a numerical method based on the fractional‐order generalized Taylor wavelets (FGTW) for option pricing and the fractional Black–Scholes equations. This model studies option pricing when the underlying asset has subdiffusive dynamics. By applying the regularized beta function, we give an exact formula for the Riemann–Liouville fractional integral operator (RLFIO) of the FGTW. An error analysis of the numerical scheme for estimating solutions is performed. Finally, we conduct a variety of numerical experiments for several standard examples from the literature to assess the efficiency of the proposed method.

Bilinear strongly generalized fractional integrals and their commutators over non-homogeneous metric spaces

The aim of this paper is to establish the boundedness of the bilinear strongly generalized fractional integral operator T˜α and its commutator T˜α,b1,b2 generated by b1,b2∈RBMO˜(μ) and T˜α on non-homogeneous metric measure spaces. Under assumption that the dominating function λ and the Lebesgue measure function u(⋅,⋅) defined on X×(0,∞) satisfy certain conditions, the authors prove that the bilinear generalized fractional integral operator T˜α is bounded from product of Lebesgue spaces Lp1(μ)×Lp2(μ) into spaces Lq(μ), bounded from product of Morrey spaces Mq1p1(μ)×Mq2p2(μ) into spaces Mts(μ), and it is also bounded from product of spaces Lp1,u1(μ)×Lp2,u2(μ) into spaces Lq,u(Rn), where u1×u2=u, 1q=1p1+1p2−2α, 1t=1q1+1q2−2α and 1s=1p1+1p2−2α for 1<q1≤p1<1α and 1<q2≤p2<1α. Furthermore, the boundedness of the T˜α,b1,b2 on spaces Lp(μ), on spaces Mqp(μ) and on spaces Lp,u(μ) is obtained.

Analytical properties, fractal dimensions and related inequalities of [formula omitted]-Riemann–Liouville fractional integrals

In this paper, we rigorously explore (k,h)-Riemann–Liouville fractional integrals applied to continuous functions defined on closed real intervals. Firstly, we delve into the analytical properties of (k,h)-Riemann–Liouville fractional integrals for continuous functions, examining aspects like boundedness, continuity, and bounded variation. Secondly, we derive an upper bound for the upper box dimensions of (k,h)-Riemann–Liouville fractional integral graphs for continuous functions defined on real closed intervals. Finally, we present several applications of (k,h)-Riemann–Liouville fractional integral operators, including the reverse analog Hölder inequality, the reverse analog Minkowski inequality, and Hermite–Hadamard-type inequalities.

On the generalization of Hermite-Hadamard type inequalities for [formula omitted]-convex function via fractional integrals

The main motivation in this article is to prove new integral identities and related results. In this paper, we deal with E`-convex function, Hermite-Hadamard type inequalities, and Katugampola fractional integrals. Firstly, we established the Hermite-Hadamard type inequalities for E`-convex function via Katugampola fractional integrals operators. After that we obtain several new identities and give new results of Holder's inequality, Jensen's integral inequality, Young's inequality, and Power mean inequality for E`-convex function by using Katugampola fractional integral operators. We make connections of our results with various results recognized in the literature.

Advancements in Fractional Integral Operators: Theory, Applications, and Computational Methods

Advancements in Fractional Integral Operators: Theory, Applications, and Computational Methods

A numerical study on fractional order financial system with chaotic and Lyapunov stability analysis

In the last few decades, academic research has focused more on financial problems and poverty levels. These are among the two major challenges of the modern world today. To understand the challenge of financial crisis and poverty in societies. This paper explains a deterministic financial system of non linear differential equations. Which aim to provide recommendations to address the twin challenges. In this fractional chaotic model, memory and chaos are portrayed as effects on a system. Financial systems of fractional order have three compartments: interest rate x(t), investment demand rate y(t) and price indexes rate z(t). For the first time, the Legendre wavelet approximation is applied to fractional order financial systems. A major objective of this process is to convert fractional differential equations (FDEs) into algebraic equations by using Legendre wavelets and their fractional integral operators. For residual analysis, convergence and stability analysis, we have used iteration methods and Legendre wavelets. This study indicates good results between the approximate and numerical solutions and that this method is efficient and accurate. Using the iteration method, we prove that the solution exists, while the contraction theory proves its uniqueness. The findings of this study will be extremely useful for future financial management. We have demonstrated the applicability and effectiveness of these methods by analysing numerical simulations for the fractional order financial model and the numerical simulation has been calculated by MATLAB programming.

Hyers–Ulam Stability of 2D-Convex Mappings and Some Related New Hermite–Hadamard, Pachpatte, and Fejér Type Integral Inequalities Using Novel Fractional Integral Operators via Totally Interval-Order Relations with Open Problem

The aim of this paper is to introduce a new type of two-dimensional convexity by using total-order relations. In the first part of this paper, we examine the Hyers–Ulam stability of two-dimensional convex mappings by using the sandwich theorem. Our next step involves the development of Hermite–Hadamard inequality, including its weighted and product forms, by using a novel type of fractional operator having non-singular kernels. Moreover, we develop several nontrivial examples and remarks to demonstrate the validity of our main results. Finally, we examine approximate convex mappings and have left an open problem regarding the best optimal constants for two-dimensional approximate convexity.

Ozaki-Type Bi-Close-to-Convex and Bi-Concave Functions Involving a Modified Caputo’s Fractional Operator Linked with a Three-Leaf Function

The focus of the present work is on the establishment and investigation of the coefficient estimates of two new subclasses of bi-close-to-convex functions and bi-concave functions; these are of an Ozaki type and involve a modified Caputo’s fractional operator that is associated with three-leaf functions in the open unit disc. The classes are defined using the notion of subordination based on the previously established fractional integral operators and classes of starlike functions associated with a three-leaf function. For functions in these classes, the Fekete-Szegö inequalities and the initial coefficients, |a2| and |a3|, are discussed. Several new implications of the findings are also highlighted as corollaries.

Estimates for $$p$$-adic fractional integral operators and their commutators on $$p$$-adic mixed central Morrey spaces and generalized mixed Morrey spaces

Estimates for $$p$$-adic fractional integral operators and their commutators on $$p$$-adic mixed central Morrey spaces and generalized mixed Morrey spaces