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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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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The necessary details of fractional Hadamard calculus are given in Kilbas  and Samko et al. . Here we present some definitions of Hadamard derivative and integral as given in . Definition 2.1. The Hadamard fractional integral of order α ∈ R + of function f (x), for all x > 1 is defined as,
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
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 , and in book of S.G. Samko et al. , here we present some definitions of Hadamard derivative and integral as given in [2, p.159-171].
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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(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  once a = 0 and α → 0 with the Katugampola fractional integral , 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)
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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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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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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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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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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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.  proved Hermite– Hadamard-type inequalities by using generalized k-fractional-integrals. Aldhaifallah et al.  used the (k, s)-fractional integral operator to generalize the inequalities for a fam- ily/class of n positive functions. Set et al.  studied Hermite–Hadamard-type inequalities for a generalized fractional integral operator for functions with convex absolute values of derivatives. Khan et al.  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  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  studied inequalities for a class of n functions by means of Saigo fractional calculus. Jarad et al.  presented a Gronwall-type inequality for the analysis of the fractional-order Atangana–Baleanu diﬀerential equation and in  for generalized fractional derivatives.
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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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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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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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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.
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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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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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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