New Investigation on the Generalized K-Fractional Integral Operators

The main objective of this paper is to develop a novel framework to study a new fractional operator depending on a parameter K > 0, known as the generalized K-fractional integral operator. To ensure appropriate selection and with the discussion of special cases, it is shown that the generalized K-fractional integral operator generates other operators. Meanwhile, we derived notable generalizations of the reverse Minkowski inequality and some associated variants by utilizing generalized K-fractional integrals. Moreover, two novel results correlate with this inequality, and other variants associated with generalized K-fractional integrals are established. Additionally, this newly defined integral operator has the ability to be utilized for the evaluation of many numerical problems.


INTRODUCTION
Fractional calculus is truly considered to be a real-world framework, for example, a correspondence framework that comprises extravagant interfacing, has reliant parts that are utilized to achieve a bound-together objective of transmitting and getting signals, and can be portrayed by utilizing complex system models (see [1][2][3][4][5][6][7][8]). This framework is considered to be a mind-boggling system, and the units that create the whole framework are viewed as the hubs of the intricate system. An attractive characteristic of this field is that there are numerous fractional operators, and this permits researchers to choose the most appropriate operator for the sake of modeling the problem under investigation (see [9][10][11][12][13]). Besides, because of its simplicity in application, researchers have been paying greater interest to recently introduced fractional operators without singular kernels [2,14,15], after which many articles considering these kinds of fractional operators have been presented. These techniques had been developed by numerous mathematicians with a barely specific formulation, for instance, the Riemann-Liouville (RL), the Weyl, Erdelyi-Kober, Hadamard integrals, and the Liouville and Katugampola fractional operators (see [16][17][18]). On the other hand, there are numerous approaches to acquiring a generalization of classical fractional integrals. Many authors have introduced new fractional operators generated from general classical local derivatives (see [9,19,20]) and the references therein. Other authors have introduced a parameter and enunciated a generalization for fractional integrals on a selected space. These are called generalized K-fractional integrals. For such operators, we refer to Mubeen and Habibullah [21] and Singh et al. [22] and the works cited in them. Inspired by these developments, future research can bring revolutionary thinking to provide novelties and produce variants concerning such fractional operators. Fractional integral inequalities are an appropriate device for enhancing the qualitative and quantitative properties of differential equations. There has been a continuous growth of interest in several areas of science: mathematics, physics, engineering, amongst others, and particularly, initial value problems, linear transformation stability, integral-differential equations, and impulse equations [23][24][25][26][27][28][29][30].
The well-known integral inequality, as perceived in Dahmani [31], is referred to as the reverse Minkowski inequality. In Nisar et al. [32,33], the authors investigated numerous variants of extended gamma and confluent hypergeometric K-functions and also established Gronwall inequalities involving the generalized Riemann-Liouville and Hadamard K-fractional derivatives with applications. In Dahmani [25], Dahmani explored variants on intervals that are known as generalized (K, s)-fractional integral operators for positive continuously decreasing functions for a certain family of n(n ∈ N). In Chinchane and Pachpatte [34], the authors obtained Minkowski variants and other associated inequalities by employing Katugampola fractional integral operators. Recently, some generalizations of the reverse Minkowski and associated inequalities have been established via generalized K fractional conformable integrals by Mubeen et al. in [35]. Additionally, Hardy-type and reverse Minkowski inequalities are supplied by Bougoffa [36]. Aldhaifallah et al. [37], explored several variants by employing the (K, s)-fractional integral operator.
In the present paper, the authors introduce a parameter and enunciate a generalization for fractional integrals on a selected space, which we name generalized K-fractional integrals. Taking into account the novel ideas, we provide a new version for reverse Minkowski inequality in the frame of the generalized K-fractional integral operators and also provide some of its consequences that are advantageous to current research. New outcomes are introduced, and new theorems relating to generalized K-fractional integrals are derived that correlate with the earlier results.
The article is composed as follows. In the second section, we demonstrate the notations and primary definitions of our newly described generalized K-fractional integrals. Also, we present the results concerning reverse Minkowski inequality. In the third section, we advocate essential consequences such as the reverse Minkowski inequality via the generalized K-fractional integral. In the fourth section, we show the associated variants using this fractional integral.

PRELUDE
In this section, we demonstrate some important concepts from fractional calculus that play a major role in proving the results of the present paper. The essential points of interest are exhibited in the monograph by Kilbas et al. [20].
and for the case p = ∞ In particular, when (λ) = λ (1 ≤ p < ∞), the space Now, we present a new fractional operator that is known as the generalized K-fractional integral operator of a function in the sense of another function .
Remark 2.1. Several existing fractional operators are just special cases of (2.1) and (2.2).
, and let be an increasing Then, the one-sided generalized K-fractional integral operator of a function Q 1 in the sense of another function of order η > 0 is stated as: where Ŵ K is the K-Gamma function.
In Set et al. [41] proved the Hermite-Hadamard and reverse Minkowski inequalities for an RL-fractional integral. The subsequent consequences concerning the reverse Minkowski inequalities are the motivation of work finished to date concerning the classical integrals.
In Dahmani [31], introduced the subsequent reverse Minkowski inequalities involving the RLFI operators.

REVERSE MINKOWSKI INEQUALITY VIA GENERALIZED K-FRACTIONAL INTEGRALS
Throughout the paper, it is supposed that all functions are integrable in the Riemann sense. Also, this segment incorporates the essential contribution for obtaining the proof of the reverse Minkowski inequality via the newly described generalized Kfractional integrals defined in section (2.4).
Proof: Multiplying 3.4 and 3.6 results in Involving the Minkowski inequality, on the right side of (3.8), we get (3.9) From 3.9, we conclude that
Likewise, it can be composed as Repeating the same procedure as above, for (4.45), we have The desired inequality (4.36) is obtained from (4.46) and (4.47).

CONCLUSION
This article succinctly expresses the newly defined fractional integral operator. We characterize the strategy of generalized K-fractional integral operators for the generalization of reverse Minkowski inequalities. The outcomes presented in section 3 are the generalization of the existing work done by Dahmani [31] for the RL-fractional integral operator. Also, the consequences in section 3 under certain conditions are reduced to the special cases proved in Set al. [41]. The variants built in section 4 are the generalizations of the existing results derived in Sulaiman [42]. Additionally, our consequences will reduce to the classical results established by Sroysang [43]. Our consequences with this new integral operator have the capacities to be used for the assessment of numerous scientific issues as utilizations of the work, which incorporates existence and constancy for the fractional-order differential equations.