Intuitionistic Fuzzy Double Integrals and Their Fundamental Properties

Intuitionistic fuzzy set (IFS), as a significant generalization of the fuzzy set (FS), takes the membership function and the non-membership function into consideration simultaneously. The core components of an IFS are the ordered pairs which are called intuitionistic fuzzy numbers (IFNs). More recently, the intuitionistic fuzzy calculus (IFC) has been established, which is based on the basic operational laws of IFNs. As far as the IFC with several variables is concerned, its theory is still in the initial stage. The purpose of this paper is to systematically establish the theory of the intuitionistic fuzzy double integrals (IFDIs). To accomplish this, we first construct the IFDIs, and then assign them concrete values. To understand the IFDIs in depth, we investigate their fundamental properties in detail and offer simple proofs for them. Next, we give a counterexample to verify that in the IFC the first and second mean value theorems do not hold, and interpret the reason from the perspective of the geometric figure. In addition, we compare the results in the IFC with those in the classical calculus to demonstrate that they are thoroughly different despite their forms being sometimes similar. Finally, we offer an application to show the utility of the proposed IFDIs.