GENERALIZED k-FRACTIONAL CONFORMABLE INTEGRALS AND RELATED INEQUALITIES

In the paper, the authors introduce the generalized k-fractional conformable integrals, which are the k-analogues of the recently introduced fractional conformable integrals and can be reduced to other fractional integrals under specific values of the parameters involved, prove the existence of k-fractional conformable integrals, and generalize some integral inequalities to ones for generalized k-fractional conformable integrals.


Introduction
Fractional calculus is the study of derivatives and integrals of non-integer order and is the generalized form of classical derivatives and integrals.It is as dated as classical calculus, but it acquires more importance in recent two decades, this is due to its applications in various fields such as physics, biology, fluid dynamics, control theory, image processing, signal processing, and computer networking.See [5,6,7,8,9,10,11,12,15,16,21,22,24,45,50,51,52,53].In recent years, the research has been proceeded to generalize the existing inequalities through innovative ideas and approaches of fractional calculus.One of the trendiest approaches among researchers is the use of fractional integral operators.Due to their potentials to be expended for the existence of nontrivial and positive solutions of several classes of fractional differential equations, the integral inequalities involving fractional integrals are considerably important.
A large bulk of existing literature consists of generalizations of numerous inequalities via fractional integral operators and their applications [3,27,32,46,49].Mubeen and Iqbal [28] contributed the ongoing research by presenting the improved version of generalized Grüss type integral inequalities for k-Riemann-Liouville fractional integrals.Agarwal et al. [2] obtained certain Hermite-Hadamard type inequalities for generalized k-fractional integrals.Set et al. [39] presented an integral identity and generalized Hermite-Hadamard type inequalities for Riemann-Liouville fractional integral.Mubeen et al. [29] established integral inequalities of Ostrowski type for k-fractional Riemann-Liouville integrals.Sarikaya and Budak [37] utilized local fractional integrals to derive a generalized inequality.Khan et al. [25] produced some important generalized inequalities for a finite class of positive decreasing functions for fractional conformable integrals.Jleli et al. [18] determined a Hartman-Winter type inequality involving fractional derivative with respect to another function.In the papers [43,44,48] and closely related references therein, there are more information on this topic.
The main object of this paper is to develop a new notion "generalized k-fractional conformable integral" which is the generalized form of fractional operators reported in [17].Hereafter, we also generalize some integral inequalities given in [25] for a finite class of positive and decreasing functions to ones involving our newly introduced k-fractional conformable integrals.For details of those inequalities, their applications, and their stability, we refer readers to [23,26,41,42].

Notations
The notion of left and right fractional conformable derivatives for a differentiable function f , introduced by Abdeljawad [1], can be expressed as Correspondingly, left and right fractional conformable integrals for 0 < α < 1 can be represented by Let Γ(z) for (z) > 0 denote the classical gamma function.The left and right fractional conformable integral (LFCI and RFCI) operators of order β ∈ C for (β) > 0 can be defined [17] respectively by Díaz and Pariguan [13] generalized the classical Pochhammer symbol (λ) n , the classical gamma function Γ(z), and the classical beta function B(u, v) respectively as See also [30,31,33,34,35].It is not difficult to see that the k-gamma function Γ k (x) and the k-beta function B k (u, v) satisfy and
It is clear that P = P + + P − , where Since P is measurable on ∆ , we can write So, by Tonelli's theorem for iterated integrals [14, p. 147], the function can be proved in a similar manner.The proof of Theorem 3.1 is complete.

Inequalities for generalized k-fractional conformable integrals
Fractional integral inequalities have been analyzed for many useful purposes.One of the most useful applications of such inequalities is the existence of nontrivial solutions of fractional differential equations.Many applications find in the literature for the existence of nontrivial solution eigenvalue problems by inequalities, see [32,49].Generalizing pre-existing inequalities by applying fractional integral operators is the most popular trend in the research field nowadays.
In this section, we present some k-analogues of inequalities in [40,46,47] for generalized k-fractional conformable integrals.
Proof.Under given conditions, we have Let us define a function Under given assumptions, the function β k J α a + (x, ρ, τ ) is positive for all τ ∈ (a, b].Integrating on both sides of the above equation (4.2) with respect to τ from a to x gives 0 Multiplying the relation (4.3) by and integrating on both sides with respect to ρ from a to x yield Dividing on both sides of relation (4.5) by results in (4.1).The proof of Theorem 4.1 is complete.
Proof.Multiplying on both sides of the relation (4.3) by and integrating on both sides with respect to ρ from a to x arrive at Dividing on both sides of (4.8) by leads to (4.6).The proof of Corollary 4.1 is complete.
Theorem 4.2.Let h(x) be a continuous increasing function and {g i , 1 ≤ i ≤ n} be a sequence of continuous positive decreasing functions on the interval . (4.9) Proof.Under given conditions, we have Let us define a function Multiplying on both sides of the relation (4.11) by (4.4) and integrating on both sides with respect to ρ from a to x give Dividing on both sides of (4.12) by leads to (4.9).The proof of Theorem 4.2 is complete.
Corollary 4.2.Let h(x) be a continuous increasing function and {g i , 1 ≤ i ≤ n} be a sequence of continuous positive decreasing functions on the interval Proof.Multiplying on both sides of the relation (4.11) by (4.7) and integrating on both sides with respect to ρ from a to x derive Dividing on both sides of (4.14) by reveals (4.13).The proof of Corollary 4.2 is complete.
Proof.Under given conditions, we have Let us define a function Multiplying the relation (4.17) by and integrating on both sides with respect to ρ from x to b produce Dividing on both sides of (4.19) by yields (4.15).The proof of Theorem 4.3 is complete.
Proof.Multiplying the relation (4.17) by and integrating on both sides with respect to ρ from x to b procure Dividing on both sides of (4.22) by demonstrates (4.20).The proof of Corollary 4.3 is complete.

.23)
Proof.Under given conditions, we have Let us define a function Dividing on both sides of (4.26) by leads to (4.23).The proof of Theorem 4.4 is complete.
Corollary 4.4.Let h(x) be a continuous increasing function and {g i , 1 ≤ i ≤ n} be a sequence of continuous positive decreasing functions on the interval g γi i g ξ p (x)

Conclusions
In this paper, we have presented the left and right k-fractional conformable integrals and generalized some important integral inequalities to ones for our newly introduced k-FCI operators related to a finite sequence of positive and decreasing functions.Our work produces k-analogues of many pre-existing results in the literature.Further, many special cases for other integral operators can be derived from our generalizations.The results obtained can be employed to confirm the existence of nontrivial solutions of fractional differential equations of different classes.The k-FCI operators in this paper are different from those introduced by Katugampola [20] as their kernels depend on the boundary points a and b and need a different convolution theory under conformable Laplace.Our k-fractional conformable integrals in this paper generalize well-known fractional integral operators such as Caputo integral operators [36, p. 44], Riemann-Liouville integral operators [36, p. 44], Hadamard integral operators [4], and their k-analogues.

Definition 3 . 1 .
Let f be a continuous function on a finite real interval [a, b].Then the generalized left and right k-fractional conformable integrals (k-FCI) of order β ∈ C for (β) > 0 are respectively defined as (4.10)    Accordingly, the function β k J α a + (x, ρ, τ ) is positive for all τ ∈ (a, b].Integrating on both sides of the above equation (4.10) with respect to τ from a to x shows 0 (4.16) Consequently, the function β k J α b − (x, ρ, τ ) is positive for all τ ∈ (a, b].Integrating on both sides of the above equation (4.16) with respect to τ from x to b gives 0

Theorem 4 . 4 .
Let h(x) be a continuous increasing function and {g i , 1 ≤ i ≤ n} be a sequence of continuous positive decreasing functions on the interval [a, b].Let a < x ≤ b, η > 0, ξ ≥ γ p > 0 for 1 ≤ p ≤ n.Then the right k-FCI operator (4.24) Thus, the function β k J α b − (x, ρ, τ ) is positive for all τ ∈ (a, b].Integrating on both sides of the above equation (4.24) with respect to τ from x to b results in 0

(4. 25 )
Multiplying the relation (4.25) by (4.18) and integrating on both sides with respect to ρ from x to b yield 0