Hadamard fractional integral

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Nonlocal Hadamard fractional integral conditions for nonlinear Riemann-Liouville fractional differential equations

Nonlocal Hadamard fractional integral conditions for nonlinear Riemann-Liouville fractional differential equations

In the present paper we initiate the study of boundary value problems like (.)-(.), in which we combine Riemann-Liouville fractional differential equations subject to the Hadamard fractional integral boundary conditions. The key tool for this combination is Property . from [], p.. To the best of the authors’ knowledge this is the first paper dealing with the Riemann-Liouville fractional differential equation subject to Hadamard type integral boundary conditions.

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Certain inequalities via generalized proportional Hadamard fractional integral operators

Certain inequalities via generalized proportional Hadamard fractional integral operators

Theorem 3.1 Let f and h be two positive continuous functions on the interval [1, ∞) and f ≤ h on [1, ∞). If h f is decreasing and f is increasing on [1, ∞), then for a convex function Φ with Φ(0) = 0, the generalized proportional Hadamard fractional integral operator given by (2.7) satisfies the inequality

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Some new integral inequalities using Hadamard fractional integral operator

Some new integral inequalities using Hadamard fractional integral operator

The necessary details of fractional Hadamard calculus are given in Kilbas [14] and Samko et al. [15]. Here we present some definitions of Hadamard derivative and integral as given in [2]. Definition 2.1. The Hadamard fractional integral of order α ∈ R + of function f (x), for all x > 1 is defined as,

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Some new results using Hadamard fractional integral

Some new results using Hadamard fractional integral

From above definitions, we see the difference between Hadamard and Riemann-liouville fractional integrals as Kernel in the Hadamard integral has the from of ln( x t ) instead of the form of (x − t), which is involves in the Riemann- Liouville integral. The Hadamard derivative has the operator (x dx d ) n , while the Riemann-Liouville derivative has the operator ( d

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On some integral inequalities using Hadamard fractional integral

On some integral inequalities using Hadamard fractional integral

Recently many authors have studied integral inequalities on fractional calculus using Riemann-Liouville, Caputo derivative, see [3, 5, 6, 7, 8, 9, 10]. The necessary background details are given in the book A.A. Kilbas [1], and in book of S.G. Samko et al. [4], here we present some definitions of Hadamard derivative and integral as given in [2, p.159-171].

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Certain inequalities associated with Hadamard k fractional integral operators

Certain inequalities associated with Hadamard k fractional integral operators

Fractional calculus is a very helpful tool to perform differentiation and integration of real or complex number orders. This subject has earned much attention from researchers and mathematicians during the last few decades (see, e.g., [–]). Among a large number of the fractional integral operators developed, due to applications in many fields of sciences, the Riemann-Liouville fractional integral operator and Hadamard fractional integral op- erator have been extensively investigated.

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ON NEW CONFORMABLE FRACTIONAL INTEGRAL INEQUALITIES FOR PRODUCT OF DIFFERENT KINDS OF CONVEXITY

ON NEW CONFORMABLE FRACTIONAL INTEGRAL INEQUALITIES FOR PRODUCT OF DIFFERENT KINDS OF CONVEXITY

(b − t) 1−α dt. (4) The fractional integral in (3) coincides with the Riemann-Liouville fractional integral (1) when a = 0 and α = 1. It also coincides with the Hadamard fractional integral [9] once a = 0 and α → 0 with the Katugampola fractional integral [8], when a = 0. Similarly, Notice that, (Qf )(t) = f(a + b − t) then we have β a J α f (x) = Q( β J α b )f (x). Moreover (4)

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Integral inequalities with ‘maxima’ and their applications to Hadamard type fractional differential equations

Integral inequalities with ‘maxima’ and their applications to Hadamard type fractional differential equations

Another kind of fractional derivative that appears in the literature is the fractional derivative due to Hadamard, introduced in  [], which differs from the Riemann- Liouville and Caputo derivatives in the sense that the kernel of the integral contains a log- arithmic function of an arbitrary exponent. Details and properties of Hadamard fractional derivative and integral can be found in [–]. Recently in the literature there appeared some results on fractional integral inequalities using the Hadamard fractional integral; see [–].

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Hadamard k fractional inequalities of Fejér type for GA s convex mappings and applications

Hadamard k fractional inequalities of Fejér type for GA s convex mappings and applications

Fractional calculus, as a very useful tool, shows its significance to implement differenti- ation and integration of real or complex number orders. This topic has attracted much at- tention from researchers during the last few decades. Among a lot of the fractional integral operators that appeared, because of applications in many fields of sciences, the Riemann– Liouville fractional integral operator and Hadamard fractional integral operator have been extensively studied.

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New generalized Hermite Hadamard type inequalities and applications to special means

New generalized Hermite Hadamard type inequalities and applications to special means

In this paper, Hermite-Hadamard type inequalities involving Hadamard fractional integrals for the functions satisfying monotonicity, convexity and s-e-condition are studied. Three classes of left-type Hadamard fractional integral identities including the first-order derivative are firstly established. Some interesting Hermite-Hadamard type integral inequalities involving Hadamard fractional integrals are also presented by using the established integral identities. Finally, some applications to special means of real numbers are given.

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Inequalities for \(\mathbb{B}\) convex functions via generalized fractional integral

Inequalities for \(\mathbb{B}\) convex functions via generalized fractional integral

Additionally, these hypotheses are valid in our results. Namely, if we get g(x) = x in (23), the inequality returns to (11). Similarly, getting g(x) = x in (24) gives inequality (10). Corollary 2 Hermite–Hadamard inequality for a B-convex function involving Hadamard fractional integral is obtained from inequalities (23) and (24).

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Fractional Hermite–Hadamard inequalities for \((s,m)\) convex or s concave functions

Fractional Hermite–Hadamard inequalities for \((s,m)\) convex or s concave functions

In this paper, we have obtained a new fractional integral identity, which played a key role in proving our main inequalities. Several Hermite–Hadamard type fractional integral in- equalities presented here, being very general, are pointed out to be specialized to yield some known results.

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Minkowski’s inequality for the AB fractional integral operator

Minkowski’s inequality for the AB fractional integral operator

Nowadays the fractional calculus has an important role in diverse scientific fields due to its several applications in dynamical problems including signals, hydrodynamics, dynamics, fluid, viscoelastic theory, biology, control theory, image processing, computer network- ing, and many others [1–5]. A large number of scientists have worked on generalizations of existing results including theorems, definitions, models, and many more. A generaliza- tion of classical inequalities by means of fractional-order integral operators is considered as an interesting subject area. For instance, recently, Agarwal et al. [6] proved Hermite– Hadamard-type inequalities by using generalized k-fractional-integrals. Aldhaifallah et al. [7] used the (k, s)-fractional integral operator to generalize the inequalities for a fam- ily/class of n positive functions. Set et al. [8] studied Hermite–Hadamard-type inequalities for a generalized fractional integral operator for functions with convex absolute values of derivatives. Khan et al. [9] produced the Minkowski inequality by using the Hahn integral operator. On the other hand, noninteger-order calculus, usually referred to as fractional calculus, is used to generalize integrals and derivatives, in particular, integrals involving inequalities. Recently, Dumitru and Arran [10] introduced a new formula for fractional derivatives and integrals by using the Mittag-Leffler kernel. More theoretical concepts re- garding fractional operators with Mittag-Leffler kernels (Atangana–Baleanu operators) and the higher-order case have been discussed in [11, 12], whereas the generalization to the generalized Mittag-Leffler kernels to gain a semigroup property have been recently ini- tiated in [13, 14]. Khan [15] studied inequalities for a class of n functions by means of Saigo fractional calculus. Jarad et al. [16] presented a Gronwall-type inequality for the analysis of the fractional-order Atangana–Baleanu differential equation and in [17] for generalized fractional derivatives.

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Some new fractional integral inequalities for exponentially m convex functions via extended generalized Mittag Leffler function

Some new fractional integral inequalities for exponentially m convex functions via extended generalized Mittag Leffler function

Fractional analysis can be regarded as an expansion of classical analysis. Fractional anal- ysis has been studied by many scientists and they have expressed the fractional derivative and integral in different ways with different notations. Although the expressions between these different definitions can be transformed into each other, but these different defini- tions and expressions have different physical meanings. It is well known that the first frac- tional integral operator is the Riemann–Liouville fractional integral operator. Recently, some new definitions of the fractional derivative were given by many mathematicians, which are the natural extensions of the classical derivative. These new definitions drew attention with their variability to classical derivative.

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Generalized Hermite Hadamard type inequalities involving fractional integral operators

Generalized Hermite Hadamard type inequalities involving fractional integral operators

Motivated by the work in [–], firstly, we will prove a generalization of the identity given by Zhu et al. using generalized fractional integral operators. Then we will give some new Hermite-Hadamard type inequalities, which are generalizations of the results in [] to the case λ = α, σ () =  and w = . Our results can be viewed as a significant extension and generalization of the previously known results.

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New results for boundary value problems of Hadamard-type fractional differential inclusions and integral boundary conditions

New results for boundary value problems of Hadamard-type fractional differential inclusions and integral boundary conditions

In this paper, we study the existence of solutions for a fractional boundary value problem involving Hadamard-type fractional differential inclusions and integral boundary conditions. Our results include the cases for convex as well as non-convex valued maps and are based on standard fixed point theorems for multivalued maps. Some illustrative examples are also presented.

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ON SOME FRACTIONAL INTEGRAL INEQUALITIES OF HERMITE-HADAMARD TYPE FOR r-PREINVEX FUNCTIONS

ON SOME FRACTIONAL INTEGRAL INEQUALITIES OF HERMITE-HADAMARD TYPE FOR r-PREINVEX FUNCTIONS

This double inequality (1.1) is known in the literature as Hermite–Hadamard in- tegral inequality for convex functions. Both inequalities hold in the reversed di- rection if f is concave.The inequality (1.1) has been extended and generalized for various classes of convex functions via different approaches, see [4, 7, 10, 12]. For several recent result concerning the inequality (1.1) we refer the interested reader to [1 − 11, 13, 15 − 17, 19] and references cited therein.

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Fractional Exponentially m-Convex Functions and Inequalities

Fractional Exponentially m-Convex Functions and Inequalities

The advantages of fractional calculus have been described and pointed out in the last few decades by many authors. Fractional calculus is based on derivatives and integrals of fractional order, fractional differential equations and methods of their solution. The most celebrated inequality has been studied extensively since it is established, is the Hermite-Hadamard inequality not only established for classical integrals but also for fractional integrals, see [18, 20, 27, 29].

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K-fractional integral inequalities of Hadamard type for (h − m)−convex functions

K-fractional integral inequalities of Hadamard type for (h − m)−convex functions

In this study some of the general versions of Hadamard inequality are analyzed in fractional calculus. A generalization of convex functions; namely (h − m)−convex function is used to establish these results. Some identities have been established which are further utilized in the formation of Hadamard type inequalities. Furthermore, Hadamard type inequalities for product of two (h − m)−convex functions have been studied and connection with already published results is investigated.

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New conformable fractional integral inequalities of Hermite-Hadamard type for convex functions

New conformable fractional integral inequalities of Hermite-Hadamard type for convex functions

In this work, we have established new conformable fractional integral inequalities of Hermite–Hadamard type for convex functions using the. As a special case, if we substitute α = 1 into the general definition of conformable fractional integrals (Definition 2), we obtain the classical integrals. In view of this, we obtained some new inequalities of Hermite–Hadamard type for convex functions involving classical integrals.

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