Katugampola Fractional Integrals Research Articles

Extension of Milne-type inequalities to Katugampola fractional integrals

This study explores the extension of Milne-type inequalities to the realm of Katugampola fractional integrals, aiming to broaden the analytical tools available in fractional calculus. By introducing a novel integral identity, we establish a series of Milne-type inequalities for functions possessing extended s-convex first-order derivatives. Subsequently, we present an illustrative example complete with graphical representations to validate our theoretical findings. The paper concludes with practical applications of these inequalities, demonstrating their potential impact across various fields of mathematical and applied sciences.

On the Katugampola fractional integral and dimensional analysis of the fractal basin boundary for a random dynamical system

In this paper, we mainly investigate the geometric based relationship between the Katugampola fractional calculus and a Weierstrass-type function whose graph can be characterized as a fractal basin boundary for a random dynamical system. Using the potential-theoretic approach with some classical analytical tools, we have derived some kinds of fractal dimensions of the graph of the Katugampola fractional integral of this fractal function. It has been shown that there is a linear relationship between the order of the Katugampola fractional integral and the fractal dimension of the graph of this generalized Weierstrass function. Numerical results have also been provided to corroborate such linear connection.

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.

NEW FRACTIONAL INTEGRAL INEQUALITIES FOR LR-ℏ-PREINVEX INTERVAL-VALUED FUNCTIONS

Based on the pseudo-order relation, we introduce the concept of left and right [Formula: see text]-preinvex interval-valued functions (LR-[Formula: see text]-PIVFs). Further, we establish the Hermite–Hadamard and Hermite–Hadamard–Fejér-type estimates for LR-[Formula: see text]-PIVFs using generalized fractional integrals. Finally, an example of interval-valued fractional integrals is provided to illustrate the validity of the results derived herein. Our results not only extend some existing inequalities for Hadamard, Riemann–Liouville, and Katugampola fractional integrals, but also provide new insights for future research on generalized convexity and IVFs, among others.

Ostrowski-Type Fractional Integral Inequalities: A Survey

This paper presents an extensive review of some recent results on fractional Ostrowski-type inequalities associated with a variety of convexities and different kinds of fractional integrals. We have taken into account the classical convex functions, quasi-convex functions, (ζ,m)-convex functions, s-convex functions, (s,r)-convex functions, strongly convex functions, harmonically convex functions, h-convex functions, Godunova-Levin-convex functions, MT-convex functions, P-convex functions, m-convex functions, (s,m)-convex functions, exponentially s-convex functions, (β,m)-convex functions, exponential-convex functions, ζ¯,β,γ,δ-convex functions, quasi-geometrically convex functions, s−e-convex functions and n-polynomial exponentially s-convex functions. Riemann–Liouville fractional integral, Katugampola fractional integral, k-Riemann–Liouville, Riemann–Liouville fractional integrals with respect to another function, Hadamard fractional integral, fractional integrals with exponential kernel and Atagana-Baleanu fractional integrals are included. Results for Ostrowski-Mercer-type inequalities, Ostrowski-type inequalities for preinvex functions, Ostrowski-type inequalities for Quantum-Calculus and Ostrowski-type inequalities of tensorial type are also presented.

Results on Katugampola Fractional Derivatives and Integrals

In this paper, we introduce and develop a new definitions for Katugampola derivative and Katugampola integral. In particular, we defined a (left) fractional derivative starting from a of a function f of order α∈(m-1, m] and a (right) fractional derivative terminating at b, where m ∈ N. Then, we give some proprieties in relation to these operators such as linearity, product rule, quotient rule, power rule, chain rule, and vanishing derivatives for constant functions.

A Comprehensive Review on the Fejér-Type Inequality Pertaining to Fractional Integral Operators

A review of the results on the fractional Fejér-type inequalities, associated with different families of convexities and different kinds of fractional integrals, is presented. In the numerous families of convexities, it includes classical convex functions, s-convex functions, quasi-convex functions, strongly convex functions, harmonically convex functions, harmonically quasi-convex functions, quasi-geometrically convex functions, p-convex functions, convexity with respect to strictly monotone function, co-ordinated-convex functions, (θ,h−m)−p-convex functions, and h-preinvex functions. Included in the fractional integral operators are Riemann–Liouville fractional integral, (k−p)-Riemann–Liouville, k-Riemann–Liouville fractional integral, Riemann–Liouville fractional integrals with respect to another function, the weighted fractional integrals of a function with respect to another function, fractional integral operators with the exponential kernel, Hadamard fractional integral, Raina fractional integral operator, conformable integrals, non-conformable fractional integral, and Katugampola fractional integral. Finally, Fejér-type fractional integral inequalities for invex functions and (p,q)-calculus are also included.

A Comprehensive Review of the Hermite–Hadamard Inequality Pertaining to Fractional Integral Operators

In the frame of fractional calculus, the term convexity is primarily utilized to address several challenges in both pure and applied research. The main focus and objective of this review paper is to present Hermite–Hadamard (H-H)-type inequalities involving a variety of classes of convexities pertaining to fractional integral operators. Included in the various classes of convexities are classical convex functions, m-convex functions, r-convex functions, (α,m)-convex functions, (α,m)-geometrically convex functions, harmonically convex functions, harmonically symmetric functions, harmonically (θ,m)-convex functions, m-harmonic harmonically convex functions, (s,r)-convex functions, arithmetic–geometric convex functions, logarithmically convex functions, (α,m)-logarithmically convex functions, geometric–arithmetically s-convex functions, s-convex functions, Godunova–Levin-convex functions, differentiable ϕ-convex functions, MT-convex functions, (s,m)-convex functions, p-convex functions, h-convex functions, σ-convex functions, exponential-convex functions, exponential-type convex functions, refined exponential-type convex functions, n-polynomial convex functions, σ,s-convex functions, modified (p,h)-convex functions, co-ordinated-convex functions, relative-convex functions, quasi-convex functions, (α,h−m)−p-convex functions, and preinvex functions. Included in the fractional integral operators are Riemann–Liouville (R-L) fractional integral, Katugampola fractional integral, k-R-L fractional integral, (k,s)-R-L fractional integral, Caputo-Fabrizio (C-F) fractional integral, R-L fractional integrals of a function with respect to another function, Hadamard fractional integral, and Raina fractional integral operator.

Controllability results to non-instantaneous impulsive with infinite delay for generalized fractional differential equations

This paper discusses controllability results for active types with infinite-time delay of non-instantaneous impulsive fractional differential equations. The model is constructed based on the generalized Caputo (Caputo-Katugampola) fractional derivative and the control function with non-local Katugampola fractional integral as a boundary condition. Our principal results are established by giving some sufficient hypotheses, utilizing well-known fractional calculus truths and using Krasnoselskii’s fixed point theorem. The infinite time delay has been treated with the abstract phase space techniques and fulfilling the ensuing axioms due to Hale and Kato. It turns out that under some sufficient conditions, the problem has at least one controllable solution. An implementation of our theoretical results is demonstrated by a numerical example.

HERMITE–HADAMARD TYPE INEQUALITIES FOR KATUGAMPOLA FRACTIONAL INTEGRALS

In the paper, basing on the Katugampola fractional integrals ${}^\rho\mathcal{K}^\alpha_{a+}f$ and ${}^\rho\mathcal{K}^\alpha_{b-}f$ with $f\in\mathfrak{X}_c^p(a,b)$, the authors establish the Hermite-Hadamard type inequalities for convex functions, give their left estimates, and apply these newly-established inequalities to special means of real numbers. When $\rho\to1$, these results become the corresponding ones for the Riemann-Liouville fractional integrals.

New midpoint and trapezoidal-type inequalities for prequasiinvex functions via generalized fractional integrals

"In this work, we establish some new midpoint and trapezoidal type inequalities for prequasiinvex functions via the Katugampola fractional integrals. Some of the results obtained in this paper are generalizations of some earlier results in the literature."

Fractal dimensions of mixed Katugampola fractional integral associated with vector valued functions

The aim of this article is to study the fundamental properties of the mixed Katugampola fractional integral (K-integral) of vector-valued functions and fractal dimensional results. We show that the mixed K-integral preserves the basic properties such as boundedness, continuity, and bounded variation of vector-valued functions. We also estimate the Hausdorff dimension of the graph of the vector-valued function and the graph of the mixed K-integral on the rectangular region. Moreover, we prove that the upper bound of the box dimension of the graph of each coordinate function of mixed K-integral of vector-valued functions is 3−min{μ1,μ2}, where μ1 and μ2 are order of the fractional integral with 0<μ1<1,0<μ2<1. Moreover, we give an example of unbounded variational vector-valued functions. In the end, we discuss some problems for future direction.

A Survey on Gruss-Type Inequalities by Means of Different Generalized Fractional Integrals

In this survey paper, we review a generalization of Gruss - type inequality by means of two different operators, one through K - fractional integral and other through Katugampola fractional integral. We state some related theorems, corollaries and their proofs. We study how results are interrelated?&#x0D;

Trapezoidal Error Approximation by the Way of Integral Inequalities for Katugampola Fractional Integral Operator

Trapezoidal Error Approximation by the Way of Integral Inequalities for Katugampola Fractional Integral Operator

HERMITE–HADAMARD–FEJÉR-TYPE INEQUALITIES VIA KATUGAMPOLA FRACTIONAL INTEGRALS FOR S-CONVEX FUNCTIONS IN THE SECOND SENSE

The Hermite–Hadamard–Fejér-type inequality is a powerful tool for studying lower and upper estimations for the integral average of convex function. In this paper, we adopt Hölder’s inequality to establish Hermite–Hadamard–Fejér-type inequalities via Katugampola fractional integrals for the function [Formula: see text], where [Formula: see text] is an s-convex function on [Formula: see text] and [Formula: see text] is symmetric with respect to [Formula: see text]. Our results are generalizations of some earlier results. At the end of the paper, illustrative examples about Hermite–Hadamard–Fejér-type inequalities are given to support our results.

Box dimension of mixed Katugampola fractional integral of two-dimensional continuous functions

The goal of this article is to study the box dimension of the mixed Katugampola fractional integral of two-dimensional continuous functions on $$[0,1]\times [0,1]$$ . We prove that the box dimension of the mixed Katugampola fractional integral having fractional order $$(\alpha =(\alpha _1,\alpha _2);~ \alpha _1>0, \alpha _2>0)$$ of two-dimensional continuous functions on $$[0,1]\times [0,1]$$ is still two. Moreover, the results are also established for the mixed Hadamard fractional integral. Our new results are to improve the existing studies. We pose also some open problems for further research.

New Result for the Analysis of Katugampola Fractional-Order Systems—Application to Identification Problems

In the last few years, a new class of fractional-order (FO) systems, known as Katugampola FO systems, has been introduced. This class is noteworthy to investigate, as it presents a generalization of the well-known Caputo fractional-order systems. In this paper, a novel lemma for the analysis of a function with a bounded Katugampola fractional integral is presented and proven. The Caputo–Katugampola fractional derivative concept, which involves two parameters 0 &lt; α &lt; 1 and ρ &gt; 0, was used. Then, using the demonstrated barbalat-like lemma, two identification problems, namely, the “Fractional Error Model 1” and the “Fractional Error Model 1 with parameter constraints”, were studied and solved. Numerical simulations were carried out to validate our theoretical results.

FRACTAL DIMENSIONS OF KATUGAMPOLA FRACTIONAL INTEGRAL OF CONTINUOUS FUNCTIONS SATISFYING HÖLDER CONDITION

This paper mainly investigates fractal dimensions of Katugampola fractional integral of continuous functions. It is proven that there exist some interesting connections between the order of Katugampola fractional integral and the upper box dimension of continuous functions satisfied Hölder condition.

Simpson type Katugampola fractional integral inequalities via Harmonic convex functions

In this paper, the new lemma of Simpson type Katugampola fractional integral equality for harmonically convex functions is attained. With the help of this equality, some new results related to Simpson-like type Katugampola fractional integral inequalities are obtained. Then, some conclusions are given for some special cases of Katugampola fractional integrals when \rho\to 1.

New Hermite-Hadamard type inequalities for m and (α, m)-convex functions on the coordinates via generalized fractional integrals

In this paper, we obtained a new Hermite-Hadamard type inequality for functions of two independent variables that are m-convex on the coordinates via some generalized Katugampola type fractional integrals. We also established a new identity involving the second order mixed partial derivatives of functions of two independent variables via the generalized Katugampola fractional integrals. Using the identity, we established some new Hermite-Hadamard type inequalities for functions whose second order mixed partial derivatives in absolute value at some powers are (α, m)-convex on the coordinates. Our results are extensions of some earlier results in the literature for functions of two variables.