• Korean Journal of Mathematics
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• v.28 no.4
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• pp.955-971
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• 2020

In this study, some new inequalities of Hermite-Hadamard type for convex and co-ordinated convex functions via Riemann-Liouville fractional integrals are derived. It is also shown that the results obtained in this paper are the extension of some earlier ones.

• Korean Journal of Mathematics
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• v.29 no.1
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• pp.91-103
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• 2021

In this paper we have established inequalities for a unified integral operator by using convexity of composition of two functions. The obtained results are directly connected to bounds of various fractional and conformable integral operators which are already known in literature. A generalized Hadamard integral inequality is obtained which further leads to its various versions for associated fractional integrals. Further, some implicated results are discussed.

• Kyungpook Mathematical Journal
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• v.58 no.1
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• pp.55-66
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• 2018

By making use of the Riemann-Liouville fractional integrals, we establish further results on Chebyshev inequality. Other Steffensen integral results of the weighted Chebyshev functional are also proved. Some classical results of the paper:[ Steffensen's generalization of Chebyshev inequality. J. Math. Inequal., 9(1), (2015).] can be deduced as some special cases.

• Journal of the Korean Mathematical Society
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• v.41 no.5
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• pp.821-864
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• 2004

Both parametric and parameter-free necessary and sufficient optimality conditions are established for a class of nondiffer-entiable nonconvex optimal control problems with generalized fractional objective functions, linear dynamics, and nonlinear inequality constraints on both the state and control variables. Based on these optimality results, ten Wolfe-type parametric and parameter-free duality models are formulated and weak, strong, and strict converse duality theorems are proved. These duality results contain, as special cases, similar results for minmax fractional optimal control problems involving square roots of positive semi definite quadratic forms, and for optimal control problems with fractional, discrete max, and conventional objective functions, which are particular cases of the main problem considered in this paper. The duality models presented here contain various extensions of a number of existing duality formulations for convex control problems, and subsume continuous-time generalizations of a great variety of similar dual problems investigated previously in the area of finite-dimensional nonlinear programming.

• Korean Journal of Mathematics
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• v.27 no.2
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• pp.279-295
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• 2019

In this paper, using local fractional integrals on fractal sets $R^{\alpha}(0<{\alpha}{\leq}1)$ of real line numbers, we establish new some inequalities of Simpson's type based on generalized convexity.

• 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.

• 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.

• Korean Journal of Mathematics
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• v.32 no.2
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• pp.297-314
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• 2024

Fractional integral operators have been studied extensively in the last few decades by various mathematicians, because it plays a vital role in the developments of new inequalities. The main goal of the current study is to establish some new Milne-type inequalities by using the special type of fractional integral operator i.e Caputo Fabrizio operator. Additionally, generalization of these developed Milne-type inequalities for s-convex function are also given. Furthermore, applications to some special means, quadrature formula, and q-digamma functions are presented.

• The Pure and Applied Mathematics
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• v.31 no.3
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• pp.311-324
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• 2024

This article presents the above and below bounds for Midpoint and Trapezoid types inequalities for 𝜓-Hilfer fractional integrals with the assistance of the functions whose second derivatives are bounded. We also possess some extensions and generalizations of Hermite-Hadamard inequalities via 𝜓-Hilfer fractional integrals with the aid of the functions that have the conditions that will said.

• Journal of Applied and Pure Mathematics
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• v.4 no.1_2
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• pp.9-26
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• 2022

The purpose of this work is to study the existence and continuous dependence on neutral stochastic partial integrodifferential equations with impulsive effects, perturbed by a fractional Brownian motion with Hurst parameter $H{\in}({\frac{1}{2}},\;1)$. We use the theory of resolvent operators developed in Grimmer [19] to show the existence of mild solutions. Further, we establish a new impulsive-integral inequality to prove the exponential stability of mild solutions in the mean square moment. Finally, an example is presented to illustrate our obtained results.