On Generalized Convergence and Related Concepts for Sequences of Fuzzy Numbers

Abstract

The present thesis entitled On Generalized Convergence and related Concepts for Sequences of Fuzzy Numbers comprises certain investigations carried out by me at the School of Mathematics (SOM), Thapar University, Patiala, under the supervision of Dr. S. S. Bhatia, Professor, SOM and Dean of Academic Affairs, Thapar University, Patiala and Dr. Vijay Kaushik, Associate Professor, HCTM Technical Campus, Kaithal, Haryana. Modern analysis is mainly concerned in finding the limiting value of a sequence. H. Fast [45] provided the major breakthrough in the direction by generalizing the concept of ordinary convergence and called it statistical convergence. Fridy [49] accelerated developed of statistical convergence and presented statistical analogue of many concepts known in the theory of usual convergence. In recent years much attention has been paid to generalize the basic concepts of classical analysis in fuzzy setting and thus a modern theory of fuzzy analysis is developed. Theory of fuzzy numbers plays an important role in the development of fuzzy analysis. Consequently, in present thesis, we have made efforts to develop or extend certain generalized summablity methods to the sequences in fuzzy environment. The whole work in the thesis has been divided into six chapters. Chapter 1 is an introductory one in which we begin with notion of natural density and show how Fast used it to define statistical convergence. After making some remarks on the early development of statistical convergence, we give some interesting extensions of statistical convergence. Finally, in this chapter, we show that how the ideas of statistical convergence are extended to the sequences of fuzzy numbers. Further, a xvi xvii brief plan of the results presented in the subsequent chapters is given. In Chapter 2, the concept of statistical convergence has been extended to the sequences of fuzzy numbers having multiplicity greater than two and develop some of its basic structural properties.