Qualitative an‎d deterministic analysis of some functional equations

1/26/2026 12:00:00 AM

In many scientific problems, fuzzy probability can be a useful solution for decision-making, an‎d many studies have been conducted in this field. However, the reality is that when faced with real problems, there is a need to consider various aspects of certainty, which is not fully met in the fuzzy approach. In this type of approach, only quality can be measured. In this thesis, quality, certainty, an‎d overlap of these two with respect to each other were measured to examine functional equations. Fuzzy sets were used to measure quality, probability distribution function was used to measure certainty, an‎d continuous t-norm function was used to examine overlap of these two with respect to each other. To formulate this approach, the generalized Z-number was introduced as a special matrix of the form diag(A,B,A*B), where A is a fuzzy set, B is a probability distribution function, an‎d A*B, where * is a continuous t-norm, indicating the overlap of quality an‎d certainty with respect to each other. Using the generalized Z-number, a new control function was introduced to check the quality, certainty, an‎d overlap stability of functional equations. The main goal of this thesis was to investigate a set of stability problems in functional equations, differential equations, an‎d fractional differential-integral systems with the concept of generalized Z-numbers. In order to achieve more accurate approximations an‎d richer analyses in terms of information, new classes of control functions were introduced, in whose structure criteria of quality, certainty an‎d overlap were included. This thesis is organized into five chapters, an‎d each chapter is dedicated to the development of a type of control function an‎d stability analysis in a different mathematical framework. In the first chapter, the basic concepts, definitions, an‎d theorems used throughout the thesis were presented. In the second chapter, a new class of control functions based on special aggregate functions was introduced. These control functions played a role in the stability an‎d approximation of triple homomorphisms. Using the alternative fixed point theorem, more precise estimates were obtained for the collective triple inequality of the s-functional an‎d the permutation of Banach triple derivatives. The results of this chapter were a step towards improving the estimates an‎d increasing the flexibility in stability analysis in advanced algebraic structures. The third chapter was devoted to the study of stochastic fractional Volterra differential-integral equations, equations in which stochastic phenomena were present. To analyze the stability of these devices, new control functions were introduced that incorporated quality an‎d certainty criteria simultaneously. Using the Hyers-Ulam an‎d Hyers-Ulam-Rassias stability methods, the stability properties of these equations were investigated. The theoretical results of this chapter have applicability in engineering modeling, physics, an‎d other fields of applied science. In the fourth chapter, the approximation theory for Apollonius-type functional equations was reviewed an‎d strengthened. It was shown that to achieve better approximations, it is necessary to simultaneously measure the quality an‎d certainty of the approximation. Fuzzy sets were used to measure the quality an‎d probability distribution functions to assess the certainty. Then, by using the concept of the generalized Z-number, a new control function was presented that enabled the analysis of the Hyers-Ulam-Rassias stability along with the measurement of the quality an‎d certainty of the approximation. In the fifth chapter, this information-driven approach was extended to two-point fractional iterative equations. In this chapter, the quality, certainty, an‎d overlap between these two criteria were examined. Based on the generalized Z-number, a new control function was defined to analyze the stability of Hyers-Ulam-Rassias for the desired fractional iterative equation while simultaneously examining their quality, certainty, an‎d the extent of their influence on each other. Overall, this thesis presents an innovative approach to studying stability in various structures of functional equations an‎d differential equations by combining fuzzy logic, probability distributions, generalized Z-numbers, an‎d fixed point theorems. The results of this research can pave the way for widespread applications in mathematics, engineering, computational modeling, an‎d interdisciplinary sciences.

Z-عدد تعميم يافته , پايداري هايرز-اولام-راسياس , معادلات تابعي , توابع خاص , انتگرال و مشتقات كسري , قضيه نقطه ثابت آلترناتيو , جايگشت همومورفيسم‌هاي سه گانه

Generalized Z-number , H-U-R stability , Functional equations , Special functions , Fractional integral an‎d derivative , Alternative fixed point theorem , Permuting tri-homomorphism

Azam Ahadi

Prof. Dr. Reza Saadati