Hermite–Hadamard-Type Inequalities for Convex Functions via the Fractional Integrals with Exponential Kernel

In this paper, we focus on the existence of positive solutions for a class of weakly singular Hadamard-type fractional mixed periodic boundary value problems with a changing-sign singular perturbation. By using nonlinear analysis methods combining with some numerical techniques, we further discuss the effect of the perturbed term for the existence of solutions of the problem under the positive, negative, and changing-sign cases. The interesting points are that the nonlinearity can be singular at the second and third variables and be changing-sign.

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“…Thus, (27), (28), Lemma 1, (F1), and the Arzela-Ascoli theorem guarantee that the operator T : B → P is completely continuous.…”

Section: Resultsmentioning

confidence: 83%

“…In particular, the Hadamard derivative is a nonlocal fractional derivative with singular logarithmic kernel. So the study of Hadamard fractional differential equations is relatively difficult; see [26][27][28][29][30][31][32].…”

Section: Introductionmentioning

confidence: 99%

Positive Solutions for a Weakly Singular Hadamard-Type Fractional Differential Equation with Changing-Sign Nonlinearity

Zhang

Jiang

et al. 2020

Journal of Function Spaces

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“…In [34], Wu et al obtained an inequality of Hermite-Hadamard type involving twice differentiable convex mappings. They used the following lemma to prove their result.…”

Section: Lemma 13 Let U : I → R Be a Twice Differentiable Function On Imentioning

confidence: 99%

“…Using fractional integrals with an exponential kernel, another integral identity involving twice differentiable mapping was presented by Wu et al [34] as follows.…”

Section: Lemma 13 Let U : I → R Be a Twice Differentiable Function On Imentioning

confidence: 99%

Some parameterized inequalities by means of fractional integrals with exponential kernels and their applications

et al. 2020

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We use the definition of a new class of fractional integral operators, recently introduced by Ahmad et al. in [J. Comput. Appl. Math. 353:120-129, 2019], to establish a fractional-type integral identity with one parameter. We derive some parameterized integral inequalities for convex mappings based on this identity, and provide two examples to illustrate the investigated results as well. Moreover, we present applications of our findings to special means of real numbers, and error estimations for the quadrature formula in numerical analysis.

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“…Wu et al 17 proved the bound estimate for the left side of Hermite-Hadamard-type inequalities (1.4) as follows.…”

Section: Introductionmentioning

confidence: 99%

Some new inequalities for generalized h‐convex functions involving local fractional integral operators with Mittag‐Leffler kernel

Sun

2020

Math Methods in App Sciences

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In this paper, we firstly construct two local fractional integral operators with Mittag‐Leffler kernel on Yang's fractal sets. Then, two local fractional integral identities with the first‐ and second‐order derivatives are derived. With these as auxiliary tools, we establish some new Hermite‐Hadamard–type local fractional integral inequalities involving the local fractional integral operators with Mittag‐Leffler kernel for generalized h‐convex functions. In addition, we obtain some special inequalities when the parameter β and function h take special values. Finally, two examples are given to illustrate the application of the results.

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Hermite–Hadamard-Type Inequalities for Convex Functions via the Fractional Integrals with Exponential Kernel

Cited by 22 publications

References 23 publications

Positive Solutions for a Weakly Singular Hadamard-Type Fractional Differential Equation with Changing-Sign Nonlinearity

Positive Solutions for a Weakly Singular Hadamard-Type Fractional Differential Equation with Changing-Sign Nonlinearity

Some parameterized inequalities by means of fractional integrals with exponential kernels and their applications

Some new inequalities for generalized h‐convex functions involving local fractional integral operators with Mittag‐Leffler kernel

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