• Title/Summary/Keyword: Riemann-Liouville fractional integral

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RIEMANN-LIOUVILLE FRACTIONAL FUNDAMENTAL THEOREM OF CALCULUS AND RIEMANN-LIOUVILLE FRACTIONAL POLYA TYPE INTEGRAL INEQUALITY AND ITS EXTENSION TO CHOQUET INTEGRAL SETTING

  • Anastassiou, George A.
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.6
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    • pp.1423-1433
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    • 2019
  • Here we present the right and left Riemann-Liouville fractional fundamental theorems of fractional calculus without any initial conditions for the first time. Then we establish a Riemann-Liouville fractional Polya type integral inequality with the help of generalised right and left Riemann-Liouville fractional derivatives. The amazing fact here is that we do not need any boundary conditions as the classical Polya integral inequality requires. We extend our Polya inequality to Choquet integral setting.

GENERALIZING HARDY TYPE INEQUALITIES VIA k-RIEMANN-LIOUVILLE FRACTIONAL INTEGRAL OPERATORS INVOLVING TWO ORDERS

  • Benaissa, Bouharket
    • Honam Mathematical Journal
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    • v.44 no.2
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    • pp.271-280
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    • 2022
  • In this study, We have applied the right operator k-Riemann-Liouville is involving two orders α and β with a positive parameter p > 0, further, the left operator k-Riemann-Liouville is used with the negative parameter p < 0 to introduce a new version related to Hardy-type inequalities. These inequalities are given and reversed for the cases 0 < p < 1 and p < 0. We then improved and generalized various consequences in the framework of Hardy-type fractional integral inequalities.

Certain Inequalities Involving Pathway Fractional Integral Operators

  • Choi, Junesang;Agarwal, Praveen
    • Kyungpook Mathematical Journal
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    • v.56 no.4
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    • pp.1161-1168
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    • 2016
  • Belarbi and Dahmani [3], recently, using the Riemann-Liouville fractional integral, presented some interesting integral inequalities for the Chebyshev functional in the case of two synchronous functions. Subsequently, Dahmani et al. [5] and Sulaiman [17], provided some fractional integral inequalities. Here, motivated essentially by Belarbi and Dahmani's work [3], we aim at establishing certain (presumably) new inequalities associated with pathway fractional integral operators by using synchronous functions which are involved in the Chebychev functional. Relevant connections of the results presented here with those involving Riemann-Liouville fractional integrals are also pointed out.

RIEMANN-LIOUVILLE FRACTIONAL VERSIONS OF HADAMARD INEQUALITY FOR STRONGLY (α, m)-CONVEX FUNCTIONS

  • Farid, Ghulam;Akbar, Saira Bano;Rathour, Laxmi;Mishra, Lakshmi Narayan
    • Korean Journal of Mathematics
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    • v.29 no.4
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    • pp.687-704
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    • 2021
  • The refinement of an inequality provides better convergence of one quantity towards the other one. We have established the refinements of Hadamard inequalities for Riemann-Liouville fractional integrals via strongly (α, m)-convex functions. In particular, we obtain two refinements of the classical Hadamard inequality. By using some known integral identities we also give refinements of error bounds of some fractional Hadamard inequalities.

Existence and Uniqueness of Solutions of Fractional Differential Equations with Deviating Arguments under Integral Boundary Conditions

  • Dhaigude, Dnyanoba;Rizqan, Bakr
    • Kyungpook Mathematical Journal
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    • v.59 no.1
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    • pp.191-202
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    • 2019
  • The aim of this paper is to develop a monotone iterative technique by introducing upper and lower solutions to Riemann-Liouville fractional differential equations with deviating arguments and integral boundary conditions. As an application of this technique, existence and uniqueness results are obtained.

CERTAIN FRACTIONAL INTEGRAL INEQUALITIES INVOLVING HYPERGEOMETRIC OPERATORS

  • Choi, Junesang;Agarwal, Praveen
    • East Asian mathematical journal
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    • v.30 no.3
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    • pp.283-291
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    • 2014
  • A remarkably large number of inequalities involving the fractional integral operators have been investigated in the literature by many authors. Very recently, Baleanu et al. [2] gave certain interesting fractional integral inequalities involving the Gauss hypergeometric functions. Using the same fractional integral operator, in this paper, we present some (presumably) new fractional integral inequalities whose special cases are shown to yield corresponding inequalities associated with Saigo, Erd$\acute{e}$lyi-Kober and Riemann-Liouville type fractional integral operators. Relevant connections of the results presented here with those earlier ones are also pointed out.

CERTAIN FRACTIONAL INTEGRAL INEQUALITIES ASSOCIATED WITH PATHWAY FRACTIONAL INTEGRAL OPERATORS

  • Agarwal, Praveen;Choi, Junesang
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.1
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    • pp.181-193
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    • 2016
  • During the past two decades or so, fractional integral inequalities have proved to be one of the most powerful and far-reaching tools for the development of many branches of pure and applied mathematics. Very recently, many authors have presented some generalized inequalities involving the fractional integral operators. Here, using the pathway fractional integral operator, we give some presumably new and potentially useful fractional integral inequalities whose special cases are shown to yield corresponding inequalities associated with Riemann-Liouville type fractional integral operators. Relevant connections of the results presented here with those earlier ones are also pointed out.

SOME COMPOSITION FORMULAS OF JACOBI TYPE ORTHOGONAL POLYNOMIALS

  • Malik, Pradeep;Mondal, Saiful R.
    • Communications of the Korean Mathematical Society
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    • v.32 no.3
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    • pp.677-688
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    • 2017
  • The composition of Jacobi type finite classes of the classical orthogonal polynomials with two generalized Riemann-Liouville fractional derivatives are considered. The outcomes are expressed in terms of generalized Wright function or generalized hypergeometric function. Similar composition formulas are also obtained by considering the generalized Riemann-Liouville and $Erd{\acute{e}}yi-Kober$ fractional integral operators.

ON SOME WEIGHTED HARDY-TYPE INEQUALITIES INVOLVING EXTENDED RIEMANN-LIOUVILLE FRACTIONAL CALCULUS OPERATORS

  • Iqbal, Sajid;Pecaric, Josip;Samraiz, Muhammad;Tehmeena, Hassan;Tomovski, Zivorad
    • Communications of the Korean Mathematical Society
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    • v.35 no.1
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    • pp.161-184
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    • 2020
  • In this article, we establish some new weighted Hardy-type inequalities involving some variants of extended Riemann-Liouville fractional derivative operators, using convex and increasing functions. As special cases of the main results, we obtain the results of [18,19]. We also prove the boundedness of the k-fractional integral operator on Lp[a, b].

RESULTS ON THE HADAMARD-SIMPSON'S INEQUALITIES

  • Asraa Abd Jaleel Husien
    • Nonlinear Functional Analysis and Applications
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    • v.29 no.1
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    • pp.47-56
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    • 2024
  • It is well known that inequalities enable us to analyze and solve complex problems with precision and efficiency. The inequalities provide powerful tools for establishing bounds, optimizing solutions, and deepening our understanding of mathematical concepts, paving the way for advancements in areas such as optimization, analysis, and probability theory. In this paper, we present some properties for Hadamard-Simpsons type inequalities in the classic integral and Riemann-Liouville fractional integral. We use the convexity of the given function and its first derivative.