The Locally And Globally Small Riemann Sums And Henstock Integral Of Fuzzy-number-valued Function

This research work focusses mainly on locally and globally small Riemann sums and Henstock integral of fuzzy-number-valued functions in E~nand set-valued functions.Yet another area the research work focusses to define and discuss the Henstock-Stieltjes integral of set-valued functions and fuzzy-number-valued functions in E~n.The thesis consists of seven chapters.Chapter 1 presents some basic definitions and results related with fuzzy numbers,fuzzy-number-valued functions and set-valued functions and their integrals which are used in the subsequent chapters.In chapter 2,we shall characterize McShane integral and Henstock integral of fuzzy-number-valued functions by Riemann-type integral of fuzzy-number-valued functions with a small Riemann sum on a small set,and the results show that McShane integral(Henstock integrals)of fuzzy-number-valued functions could be represented by almost Riemann integral(McShane integral)with a small Riemann sum on a small set,respectively.In chapter 3,we first define and discuss the locally small Riemann sums(LSRS)for fuzzy-number-valued functions.In addition the necessary and sufficient conditions have been obtained for a fuzzy-number-valued function which has(LSRS),i.e.,if a fuzzy-number-valued function is Henstock integrable on[a,b]then it has(LSRS)and the converse is always true.Secondly,the globally small Riemann sums(GSRS)for fuzzy-number-valued functions is defined and discussed,the necessary and sufficient conditions have been given for a fuzzy-number-valued function which has(GSRS),i.e.,if a fuzzy-number-valued function is Henstock integrable on[a,b]then it has(GSRS)and the converse is always true.Finally,by Egorov,s Theorem,we obtain the dominated convergence theorem for globally small Riemann sums(GSRS)of fuzzy-number-valued functions.Chapter 4 deals with locally and globally small Riemann sums which are defined in set-valued functions.First,we introduce the locally small Riemann sums(LSRS)for set-valued functions.Next,the globally small Riemann sums(GSRS)for set-valued functions is defined and discussed.Finally,we present two main theorems:(1)If the set-valued functions is Henstock integrable on[a,b]then it has(LSRS)and the converse is always true.(2)If the set-valued functions is Henstock integrable on[a,b]then it has(GSRS)and the converse is always true.In chapter 5,the notions of locally and globally small Riemann sums modifications with respect to a fuzzy-number-valued functions in E~nare introduced and studied.The basic properties and characterizations are presented.In particular,it is prove that a fuzzy-number-valued functions in E~nis Henstock integrable on[a,b]if and only if it has(LSRS),and also it is prove that a fuzzy-number-valued functions in E~nis Henstock integrable on[a,b]if and only if it has(GSRS).In chapter 6,we introduce the concept of Henstock-Stieltjes integrability for set-valued functions and investigate some properties.Furthermore,we propose the concept of weak equi-integrability for sequences of set Henstock-Stieltjes integrable functions.Under this concept,we prove two convergence theorems for sequences of the set Henstock-Stieltjes integrable functions.In chapter 7,the concept of fuzzy Henstock-Stieltjes integral for fuzzy-number-valued functions in E~nis presented,several necessary and sufficient conditions of in-tegrability for the fuzzy-number-valued functions in E~nare given by means of this concept.