Some Hermite-Hadamard Type Inequalities via Katugampola Fractional for pq-Convex on the Interval-Valued Coordinates

. In this paper, we established the Hermite-Hadamard inequalities via Katugampola fractional. Meanwhile, interval analysis is a particular case of set-interval analysis. We established the fractional inequalities and these results are an extension of a previous research. pq-convex; interval-valued.


Introduction
The classical Hermite-Hadamard inequalities such that where f : I → R is a convex function on the closed bounded interval I of R,and a, b ∈ I with a < b.
On the other hand, interval analysis is a particular case of set-valued analysis which is the study of sets in the spirit of mathematical analysis and general topology. It was introduced as an attempt to handle interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena. An old example of interval enclosure is Archimede's method which is related to the computation of the circumference of circle. In 1966, the first book related to interval analysis was given by Moore who is known as the first user of intervals in computational mathematics.
After this book, several scientists started to investigate theory and application of interval arithmetic.
Nowadays, because of its application, interval analysis is a useful tool in various areas related to uncertain data. We can see applications in computer graphics, experimental and computational physics, error analysis, robotics and many others.

Interval Calculus
A real valued interval X is bounded, closed subset of R and is defined by where X, X ∈ R and X ≤ X. The number X and X are called the left and right endpoints of interval X, respectively. When X = X = a, the interval X is said to be degenerate and we use the form X = a = [a, a] . Also we call X positive if X> 0 or negative if X < 0. The set of all closed intervals of R, the sets of all closed positive intervals of R and closed negative intervals of R is denoted by R I , R + I and R − I , respectively. The Pompeiu-Hausdorff distance between the intervals X and Y is defined by It is known that (R I , d) is a complete metric space. Now, we give the definitions of basic interval arithmetic operations for the intervals X and Y as follows: Scalar multiplication of the interval X is defined by The opposits of the interval X is The subtraction is given by In general, −X is not additive inverse for X, i.e. X − X = 0.

Use of monotonic functions
The definitons of operations lead to a number of algebraic properties which allows R I to be quasilinear space. They can be listed as follows (Additivity elemant) X + 0 = 0 + X = X for all X ∈ R I , (Associativity law) λ (µX) = (λµ) X for all X ∈ R I , and for all λ, µ ∈ R, (Second distributiviyu law) (λ + µ) X = λX + µX for all X ∈ R I , and for all λ, µ ∈ R.
But, this law holds in certain cases. If Y · Z > 0, then What's more, one of the set property is the inclusion ⊆ that is given by Considering together with arthmetic operations and inclusion, one has the following property which is called inclusion isotone of interval operations: Let be the addition, multiplication, subtraction or division. If X, Y, Z and T areintervals such that X ⊆ Y and Z ⊆ T, then the following relation is valid

Intgral of Interval-Valued Functions
In this section, the notion of integral is mentioned for interval-valued functions. Before the definition of integral, the necessary concepts will be given as the following: A function F is said to be an interval-valued function of t on [a, b] , if it assigns a nonempty interval A partition of [a, b] is any finite ordered subset P having the form: The mesh of a partition P defined by and let us define the sum where F : [a, b] → R I . We call S (F, P, δ) a Riemann sum of F corresponding to P ∈ P (δ, [a, b]) .
The collection of all functions that are (IR)-integrable on [a, b] will be denoted by IR ([a,b]) .
The following theorem gives relation between (IR) −integrable and Riemann integrable (R)integrable.
where R ([a,b]) denoted the all R-integrable functions.
It is seen easily that, if F (t) ⊆ G (t) for all t ∈ [a, b] , then And we denote the set of all δ-fine partition P of with choose arbitrary ξ i , η j and get for each P ∈ P (δ, ) . We denote by IR ( ) the set of all ID-integrable function on , and by , the set of all R-integrable and IR-integrable functions on [a, b] ,respectively.

Fractional Integrals
In [2], Katugampola introduced a new fractional which generalizes the Riemann-Liouville and the Hadamard fractional integrals into a single form as follow.
Let [a, b] ⊂ R be a finite interval. Then, the left-and right-side Katugampola fractional integralsod where a < x < b and ρ > 0, if the integral exists.

Theorem 4.2
Let α > 0 and ρ > 0. Then for x > a, . Similar results also hold for right-sided operators.
) . If f is also a convex function on [a, b] , then the following inequalities hold: where the fractional integral are considered for the function f (x ρ ) and evaluated at a and b, respectively.
In [28], Yaldiz established the new definitions and theorem related Katugampola fractional integrals for two variables functions: Similarly, we introduce the following integrals: . Then the following inequalities hold: with a < x < b and c < y < d.

Definition 4.6
Let I ⊂ (0, ∞) be a real interval and p ∈ R\ {0} . A function f : I → R is said to be a p-convex for all x, y ∈ I and t ∈ [0, 1] . If the inequality is reserved, then f is said to be p-concave.
In [5], Fang and Shi established the following inequlaity Theorem 4.7 Let f : I → R be a p-convex function and a, b ∈ I with a < b. If f ∈ L [a, b] , then we have In [27], Toplu et al. established the following inequality Theorem 4.8 Let f : I → R be a p-convex function, p > 0, α > 0 and a, b ∈ I with a < b. If f ∈ L [a, b] , then the following inequality for fractional integrals holds: In this paper, we can give a different version of the definition of the pq-convex function as below.
We recall the following special functions and inequalities.
(1)The Gamma Function: The Gamma Γ function is defined by for all complex numbers z with Re(z) > 0, respectively. The gamma function is a natural extension of the factorial from integers n to real (and complex) numbers z.

Main Result
Theorem 5.1 Let f : I × I → R be an interval-valued pq-convex function such that f (t) = f (t) , f (t) and the order p, q > 0, α, β > 0 and a, b, c, d ∈ I with a < b and c < d.
, then the following inequality for fractional integrals holds: Multiplying both sides of the inequality by t α−1 λ β−1 and then integrating the resulting inequality with respect to t over [0, 1] and with respect to λ over [0, 1] , then we obtain, Thus we have which completes the proof of the first inequality.
For the second inequalitiy, br using pq-convexity of f , we have By adding these inequalities, then, we have Multiplying both sides of the inequality by t α−1 λ β−1 , α > 0, β > 0 and then integrating the resulting inequality with respect to t over [0, 1] and with respect to λ over [0, 1] , then we similarly obtain,

Lemma 5.2
Let f : I × I → R be a partial differentiable function with 0 ≤ a < b and 0 ≤ c < d. Then the equality holds.
pr oof : Let By using integrating by part, we have, and similarly we get So that we combine I 1 − I 2 − I 3 + I 4 , we will get the equality.
P r oof : From Lemma by using the property of the modulus, the power mean inequality and the pq-convexity of ∂ 2 ∂t∂λ f m , then we have where by simple computation, we obtain,

Conclusion
In this work, the author established Hermite-Hadamard type inequalities via Katugampola fractional integral. Furthermore, the author extend the ineqalities on interval-valued coordinated. It is an interesting issue, and many researchers work to generalize the Ostrowski' inequalities, Chebyshev type inequalities and Opial-type inequalities on fuzzy interval-valued set. We hope to establish the general fractional integrals in their future research.
Author Contributions: The author contributed has read and agreed to the published version of the manuscript.