Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan.
Department of Mathematics, College of Education, University of Sulaimani, Sulaimani, Kurdistan Region, Iraq.
Math Biosci Eng. 2021 Jul 30;18(5):6552-6580. doi: 10.3934/mbe.2021325.
In this study, we introduce and study new fuzzy-interval integral is known as fuzzy-interval double integral, where the integrand is fuzzy-interval-valued functions (FIVFs). Also, some fundamental properties are also investigated. Moreover, we present a new class of convex fuzzy-interval-valued functions is known as coordinated convex fuzzy-interval-valued functions (coordinated convex FIVFs) through fuzzy order relation (FOR). The FOR (≼) and fuzzy inclusion relation (⊇) are two different concepts. With the help of fuzzy-interval double integral and FOR, we have proved that coordinated convex fuzzy-IVF establish a strong relationship between Hermite-Hadamard (HH-) and Hermite-Hadamard-Fejér (HH-Fejér) inequalities. With the support of this relation, we also derive some related HH-inequalities for the product of coordinated convex FIVFs. Some special cases are also discussed. Useful examples that verify the applicability of the theory developed in this study are presented. The concepts and techniques of this paper may be a starting point for further research in this area.
在本研究中，我们引入并研究了一种新的模糊区间积分，称为模糊区间双重积分，其中积分项是模糊区间值函数（FIVFs）。此外，还研究了一些基本性质。此外，我们通过模糊序关系（FOR）提出了一类新的凸模糊区间值函数，称为协调凸模糊区间值函数（协调凸 FIVFs）。模糊序关系（≼）和模糊包含关系（⊇）是两个不同的概念。借助模糊区间双重积分和模糊序关系，我们证明了协调凸模糊 IVF 建立了 Hermite-Hadamard（HH-）和 Hermite-Hadamard-Fejér（HH-Fejér）不等式之间的强关系。在这种关系的支持下，我们还为协调凸 FIVFs 的乘积导出了一些相关的 HH 不等式。还讨论了一些特殊情况。给出了有用的实例来验证本文所提出理论的适用性。本文的概念和技术可能为该领域的进一步研究提供一个起点。
