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A survey of results on Grüss-type inequalities associated with a variety of fractional integral and differential operators is presented. The fractional differential operators includes, Riemann-Liouville fractional integral operators, Riemann-Liouville fractional integrals of a function with respect to another function, Katugampola fractional integral operators, Hadamard's fractional integral operators, $ k $-fractional integral operators, Raina's fractional integral operators, tempered fractional integral operators, conformable fractional integrals operators, proportional fractional integrals operators, generalized Riemann-Liouville fractional integral operators, Caputo-Fabrizio fractional integrals operators, Saigo fractional integral operators, quantum integral operators, and Hilfer fractional differential operators.

A survey of results on Grüss-type inequalities associated with a variety of fractional integral and differential operators is presented. The fractional differential operators includes, Riemann-Liouville fractional integral operators, Riemann-Liouville fractional integrals of a function with respect to another function, Katugampola fractional integral operators, Hadamard's fractional integral operators, k-fractional integral operators, Raina's fractional integral operators, tempered fractional integral operators, conformable fractional integrals operators, proportional fractional integrals operators, generalized Riemann-Liouville fractional integral operators, Caputo-Fabrizio fractional integrals operators, Saigo fractional integral operators, quantum integral operators, and Hilfer fractional differential operators. [ABSTRACT FROM AUTHOR]

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.

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, M T -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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this study, we present the midpoint and trapezoid inequalities for an F-convex function in terms of Katugampola fractional integral operators. We obtained new results involving Katugampola-fractional integral operators for differentiable mapping ϕ whose second derivatives in the absolute values are F-convex. Also established connections between our results with several renowned results in literature. Results proved in this paper may stimulate further research in this area. [ABSTRACT FROM AUTHOR]

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, M T -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. [ABSTRACT FROM AUTHOR]

Some generalized Ostrowski-type integral inequalities for r-times differentiable functions whose absolute values are MT-convex have been discussed. Moreover, some applications on special bivariate means are obtained. [ABSTRACT FROM AUTHOR]

In this paper, using the definition of functions (h, m, s)-convex modified of second type, various extensions of the classic Hermite-Hadamard Inequality are obtained using Katugampola integrals. In addition, we show that several results known are particular cases of ours. [ABSTRACT FROM AUTHOR]

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.

In this paper, we employ new version of the Atangana-Baleanu integral operator namely ABK-fractional integrals to obtain two general integral identities complying second-order derivatives for a given function. Thus allowing us to derive new generalized Hermite-Hadamard type inequalities via ABK- fractional integrals. Moreover we give several new versions of Mid-point and Trapezoid type inequalities by employing Hölder, Young and Jensen inequalities.

This paper deals with some existence of random solutions and the Ulam stability for a class of Katugampola random fractional differential equations in Banach spaces. A random fixed point theorem is used for the existence of random solutions, and we prove that our problem is generalized Ulam-Hyers-Rassias stable. An illustrative example is presented in the last section.

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.

The main aim of this study is to reveal new generalized-Ostrowski-type inequalities using Katugampola fractional integral operator which generalizes Riemann-Liouville and Hadamard fractional integral operators into a single form. For this purpose, at first, a new fractional integral identity is generated by the researchers. Then, by using this identity, some inequalities for the class of functions whose certain powers of absolute values of derivatives are p-convex are derived. Some applications to special means for positive real numbers are also given. It is observed that the obtained inequalities are generalizations of some well known results.

The author introduced the new fractional integral operator called ABK-fractional integral and proved four identities for this type. By applying the established identities, some integral inequalities connected with the right hand side of the Hermite-Hadamard type inequalities for the ABK-fractional integrals are given. Various special cases have been identified. The ideas of this paper may stimulate further research in the field of integral inequalities. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we obtain the Hermite–Hadamard type inequalities for s-convex functions and m-convex functions via a generalized fractional integral, known as Katugampola fractional integral, which is the generalization of Riemann–Liouville fractional integral and Hadamard fractional integral. We show that through the Katugampola fractional integral we can find a Hermite–Hadamard inequality via the Riemann–Liouville fractional integral.

In this paper, we make research on Katugampola and Hadamard fractional integral of one-dimensional continuous functions on [ 0 , 1 ]. We proved that Katugampola fractional integral of bounded and continuous function still is bounded and continuous. Box dimension of any positive order Hadamard fractional integral of one-dimensional continuous functions is one. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the existence and uniqueness of solutions for a new class of nonlinear fractional differential equations involving Hilfer fractional operator with fractional integral boundary conditions. Our analysis relies on classical fixed point theorems and the Boyd-Wong nonlinear contraction. At the end, an illustrative example is presented. The boundary conditions introduced in this work are of quite general nature and can be reduce to many special cases by fixing the parameters involved in the conditions. [ABSTRACT FROM AUTHOR]

In this paper, we give new definitons related to Katugampola fractional integral for two variables functions. We are interested in giving the Hermite{Hadamard inequality for a rectangle in plane via convex functions on co-ordinates involving Katugampola fractional integral. [ABSTRACT FROM AUTHOR]

We focus our attention in this article on some recent results regarding Hardy-Hilbert's inequalities. We derive an equivalent form using katugampola Fractional Calculus and introduce new analogs to some Hardy-Hilbert's type inequality. Several special cases are also given. [ABSTRACT FROM AUTHOR]

The main purpose of this article is to present some new Hilbert-type fractional integral inequalities by using katugampola Fractional Calculus. The strict inequality is considered and the best possible constant is determined for general and some other special cases. [ABSTRACT FROM AUTHOR]

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In this paper, a new identity for the generalized fractional integral is defined. Using this identity we studied a new integral inequality for functions whose first derivatives in absolute value are convex. The new generalized Hermite-Hadamard inequality for generalized convex function on fractal sets involving Katugampola type fractional integral is established. This fractional integral generalizes Riemann-Liouville and Hadamard’s integral, which possess a symmetric property. We derive trapezoid and mid-point type inequalities connected to this generalized Hermite-Hadamard inequality.

In this paper, we study a boundary value problem consisting from a fractional differential inclusion of Riemann-Liouville Langevin type subject to Katugampola fractional integral conditions. Some new existence results for convex as well as non-convex multivalued maps are obtained by using standard fixed point theorems. Enlighten examples illustrating the obtained results are also presented. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to obtain the Hermite−Hadamard-type inequalities for exponentially s-convex functions via the Katugampola fractional integral. The Katugampola fractional integral is a generalization of Riemann−Liouville fractional integral and Hadamard fractional integral. Some special cases and applications to special means are also discussed.

In this paper we obtain the Hermite–Hadamard and Hermite–Hadamard–Fejér type inequalities for fractional integrals which generalize the two familiar fractional integrals namely, the Riemann–Liouville and the Hadamard fractional integrals into a single form. We prove that, in most cases, we obtain the Riemann–Liouville and the Hadamard equivalence just by taking limits when a parameter ρ → 1 and ρ → 0 + , respectively. [ABSTRACT FROM AUTHOR]

The objective of this paper is to present an approximation formula for the Katugampola fractional integral, that allows us to solve fractional problems with dependence on this type of fractional operator. The formula only depends on first-order derivatives, and thus converts the fractional problem into a standard one. With some examples, we show the accuracy of the method, and then we present the utility of the method by solving a fractional integral equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of solutions for a problem consisting of nonlinear Langevin equation of Riemann-Liouville type fractional derivatives with the nonlocal Katugampola fractional integral conditions. A variety of fixed point theorems are used such as Banach's fixed point theorem, Krasnoselskii's fixed point theorem, Leray-Schauder's nonlinear alternative, and Leray-Schauder degree theory. Enlightening examples of the obtained results are also presented. [ABSTRACT FROM AUTHOR]

In this paper, a new identity for the generalized fractional integral is defined. Using this identity we studied a new integral inequality for functions whose first derivatives in absolute value are convex. The new generalized Hermite-Hadamard inequality for generalized convex function on fractal sets involving Katugampola type fractional integral is established. This fractional integral generalizes Riemann-Liouville and Hadamard's integral, which possess a symmetric property. We derive trapezoid and mid-point type inequalities connected to this generalized Hermite-Hadamard inequality. [ABSTRACT FROM AUTHOR]