Intuitionistic extensions of fundamental theorems in fuzzyset theory and some applications to initial value problems

In this thesis, we have firstly generalized the fundamental definitions and theorems from fuzzy set theory to intuitionistic fuzzy set theory and then studied the existence and uniqueness of the solution of the first order intuitionistic initial value problems under Hukuhara and strongly generalized Hukuhara derivative concept and we have given an algorithm to find the solution of the second order linear differential equations with intuitionistic fuzzy initial values and forcing coefficients. First of all, the characterization theorems in fuzzy set theory are extended to the intuitionistic fuzzy environment and the properties of α and β cuts of intuitionistic fuzzy numbers are investigated. Then we have defied Minkowski sum and scalar multiplication of intuitionistic fuzzy numbers in terms of α and β cuts. And then, we have shown that the set of intuitionistic fuzzy numbers are closed with respect to Minkowski sum and scalar multiplication. With the aid of these operations, we have given some properties of Hukuhara difference and generalized Hukuhara difference for intuitionistic fuzzy numbers. We have defied D∞ metric on the set of intuitionistic fuzzy numbers and given some of its properties. And we have proved that the space of intuitionistic fuzzy numbers is complete with respect to D∞. Moreover, we have proved that the space of continuous intuitionistic fuzzy number valued functions is complete with respect to the metric Ds. Besides, the concepts of Hukuhara derivative, strongly generalized Hukuhara derivative and Aumann integral are investigated in the intuitionistic fuzzy environment by taking the properties of α and β cuts into consideration. With the help of these generalizations, the existence and uniqueness of the solution of the first order intuitionistic initial value problems under Hukuhara and strongly generalized Hukuhara derivative concept are studied and the algorithm to find the solution of second order linear differential equations with intuitionistic fuzzy initial values and forcing coefficients is given. Finally, numerical examples are given in line with these results.