New integral inequalities and their applications to convex functions with a continuous Caputo fractional derivative

Ahmad, Bashir ¹ ; Jleli, Mohamed ² ; Samet, Bessem ²

¹ Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
² Department of Mathematics, College of Science, King Saud University, P. O. Box 2455, Riyadh, 11451, Saudi Arabia

Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 5, p. 658-671.

Résumé

We say that a function $f:[a,b]\to \mathbb{R}$ is $(\varphi,\delta)$-Lipschitzian, where $\delta\geq 0$ and $\varphi:[0,\infty)\to [0,\infty)$, if

$ |f(x)-f(y)|\leq \varphi(|x-y|)+\delta,\quad (x,y)\in [a,b]\times [a,b]. $

In this work, some Hadamard's type inequalities are established for the class of $(\varphi,\delta)$-Lipschitzian mappings. Moreover, some applications to convex functions with a continuous Caputo fractional derivative are also discussed.

Détail

DOI : 10.22436/jnsa.011.05.07

Classification : 26D10, 35A23, 26A33, 26A51
Keywords: \((\varphi, \delta)\)-Lipschitzian, Hadamard's type inequalities, convex function, Caputo fractional derivative, fractional mean value theorem

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