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. In this chapter the concepts of Cauchy sequences and the Cauchy convergence criteria has been presented. Some generalized summability methods as Cesaro summability and strong p-Cesaro summability of multiple sequences of fuzzy numbers have also been introduced. Some connections between the statistical convergence and the new generalized summability methods are discussed. Finally, a stronger generalization of statistical convergence for multiple sequences has been introduced. The objective of Chapter 3 is to define the notions of lacunary statistical limit point and lacunary statistical cluster points for sequences of fuzzy numbers with help of a lacunary sequence as these notions help in predicting the behavior of such a sequence which is not lacunary statistical convergent. The sets of lacunary statistical limit points and lacunary statistical cluster points have also been defined and some inclusion relation between these sets and the sets of ordinary statistical limit points and cluster points are obtained in this chapter. Also, a condition for which these sets coincide with the sets of ordinary statistical limit points and cluster points has been obtained. Further, the lacunary statistical limit and cluster points for sequences of fuzzy numbers with the help of difference operator has been studied in this chapter. The goal of Chapter 4 is to introduce the notions: lacunary statistical limit inferior, lacunary statistical limit superior, lacunary statistical boundedness and lacunary statistical core for sequences of fuzzy numbers. Some inclusion relations between these notions have been obtained. It has also been proved that if a sequence of fuzzy numbers is lacunary statistical convergent, then lacunary statistical limit inferior and lacunary statistical limit superior coincide with the lacunary statistical limit of the sequence. Subsequently, certain connections between lacunary statistical core and statistical core of a sequence of fuzzy numbers have been obtained. Finally, some new extensions of the above notions have been obtained by using the difference operator. xviii In Chapter 5, concepts of λ-statistical limit inferior, λ-statistical limit superior, λ-statistical bounded sequence and λ-statistical core for sequences of fuzzy numbers have been defined where λ = (λn) be a non-decreasing sequence of positive numbers tending to ∞ with λn+1 ≤ λn + 1, λ1 = 1. In addition, for β ∈ (0, 1], concepts of λ-statistical limit points, λ-statistical cluster points, λ-statistical limit inferior and λ- statistical limit superior of order β are also introduced for sequences of fuzzy numbers. Chapter 6 deals with the concepts of α-statistical convergence and statistical convergence for sequences in fuzzy neighborhood spaces which turn out to be a very useful subcategory of fuzzy topological spaces. Some properties of α-statistical convergence and statistical convergence in these spaces have been developed. Subsequently, certain generalizations of statistical convergence in fuzzy neighborhood spaces have been extended and some relevant connections with α-statistical convergence and statistical convergence have been discussed.