scholarly journals A Study on Some Fundamental Properties of Continuity and Differentiability of Functions of Soft Real Numbers

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Ramkrishna Thakur ◽  
S. K. Samanta
Keyword(s):  
Order Relation ◽  
Mean Value ◽  
Soft Set ◽  
Mean Value Theorem ◽  
Weierstrass Theorem ◽  
Real Numbers ◽  
Soft Sets ◽  
Fundamental Properties ◽  
New Type ◽  
Intermediate Value

We introduce a new type of functions from a soft set to a soft set and study their properties under soft real number setting. Firstly, we investigate some properties of soft real sets. Considering the partial order relation of soft real numbers, we introduce concept of soft intervals. Boundedness of soft real sets is defined, and the celebrated theorems like nested intervals theorem and Bolzano-Weierstrass theorem are extended in this setting. Next, we introduce the concepts of limit, continuity, and differentiability of functions of soft sets. It has been possible for us to study some fundamental results of continuity of functions of soft sets such as Bolzano’s theorem, intermediate value property, and fixed point theorem. Because the soft real numbers are not linearly ordered, several twists in the arguments are required for proving those results. In the context of differentiability of functions, some basic theorems like Rolle’s theorem and Lagrange’s mean value theorem are also extended in soft setting.

A new type of generalized picture fuzzy soft set and its application in decision making

2021 ◽  
pp. 1-17
Author(s):  
Hanchuan Lu ◽  
Ahmed Mostafa Khalil ◽  
W. Alharbi ◽  
M. A. El-Gayar
Keyword(s):  
Decision Making ◽  
Set Theory ◽  
Soft Set ◽  
Fuzzy Decision Making ◽  
Fuzzy Soft Set ◽  
Soft Sets ◽  
New Type ◽  
Fuzzy Soft Sets ◽  
Novel Concept ◽  
Fuzzy Parameter

 In this article, we propose a novel concept of the generalized picture fuzzy soft set by combining the picture fuzzy soft set and the fuzzy parameter set. For possible applications, we explain five kinds of operations (e.g., subset, equal, union, intersection, and complement) based on generalized picture fuzzy soft sets. Then, we establish several theoretical operations of generalized picture fuzzy soft sets. In addition, we present the new type by using the AND operation of the generalized picture fuzzy soft set for fuzzy decision-making and clarify its applicability with a numerical example. Finally, we give a comparison between the picture fuzzy soft set theory and the generalized picture fuzzy soft set theory. It is shown that our proposed (i.e., generalized picture fuzzy soft set theory) is viable and provide decision makers a more mathematical insight before making decisions on their options.

scholarly journals On a new generalization of Alzer's inequality

2000 ◽  
Vol 23 (12) ◽  
pp. 815-818 ◽  
Author(s):  
Feng Qi ◽  
Lokenath Debnath
Keyword(s):  
Lower Bound ◽  
Large Class ◽  
Mean Value ◽  
Mathematical Induction ◽  
Mean Value Theorem ◽  
Natural Numbers ◽  
Real Numbers ◽  
Positive Real

Let{an}n=1∞be an increasing sequence of positive real numbers. Under certain conditions of this sequence we use the mathematical induction and the Cauchy mean-value theorem to prove the following inequality:anan+m≤((1/n)∑i=1nair(1/(n+m))∑i=1n+mair)1/r, wherenandmare natural numbers andris a positive number. The lower bound is best possible. This inequality generalizes the Alzer's inequality (1993) in a new direction. It is shown that the above inequality holds for a large class of positive, increasing and logarithmically concave sequences.

A New Type of Mean Value Theorem

1966 ◽  
Vol 39 (5) ◽  
pp. 264-268 ◽  
Author(s):  
Donald H. Trahan
Keyword(s):  
Mean Value ◽  
Mean Value Theorem ◽  
New Type

A New Type of Mean Value Theorem

1966 ◽  
Vol 39 (5) ◽  
pp. 264 ◽  
Author(s):  
Donald H. Trahan
Keyword(s):  
Mean Value ◽  
Mean Value Theorem ◽  
New Type

scholarly journals Belief and Possibility Belief Interval-Valued N-Soft Set and Their Applications in Multi-Attribute Decision-Making Problems

Entropy ◽  
2021 ◽  
Vol 23 (11) ◽  
pp. 1498
Author(s):  
Shahbaz Ali ◽  
Muneeba Kousar ◽  
Qin Xin ◽  
Dragan Pamučar ◽  
Muhammad Shazib Hameed ◽  
...  
Keyword(s):  
Decision Making ◽  
Set Theory ◽  
Soft Set ◽  
Mathematical Tool ◽  
Soft Sets ◽  
Fundamental Properties ◽  
Research Article ◽  
Multi Attribute Decision Making ◽  
Algebraic Operations ◽  
Interval Valued

In this research article, we motivate and introduce the concept of possibility belief interval-valued N-soft sets. It has a great significance for enhancing the performance of decision-making procedures in many theories of uncertainty. The N-soft set theory is arising as an effective mathematical tool for dealing with precision and uncertainties more than the soft set theory. In this regard, we extend the concept of belief interval-valued soft set to possibility belief interval-valued N-soft set (by accumulating possibility and belief interval with N-soft set), and we also explain its practical calculations. To this objective, we defined related theoretical notions, for example, belief interval-valued N-soft set, possibility belief interval-valued N-soft set, their algebraic operations, and examined some of their fundamental properties. Furthermore, we developed two algorithms by using max-AND and min-OR operations of possibility belief interval-valued N-soft set for decision-making problems and also justify its applicability with numerical examples.

FUZZY OPTIMIZATION OF REAL FUNCTIONS

2004 ◽  
Vol 12 (04) ◽  
pp. 471-497 ◽  
Author(s):  
MARK BURGIN
Keyword(s):  
Fuzzy Optimization ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Computational Mathematics ◽  
Differentiable Functions ◽  
Fuzzy Mean ◽  
Real Functions ◽  
Intermediate Value ◽  
Derivatives Of ◽  
Neoclassical Analysis

The main goal of this paper is to develop such means of analysis that allows us to reflect and model vagueness and uncertainty of our knowledge, which result from imprecision of measurement and inaccuracy of computation. To achieve this goal, we use here neoclassical analysis to problems of optimization. Neoclassical analysis extends the scope and results of the classical mathematical analysis by applying fuzzy concepts to conventional mathematical objects, such as functions, sequences, and derivatives. Basing on the theory of fuzzy limits, we construct a fuzzy extension for the classical theory of differentiation in the context of computational mathematics. It is done in the second part of this paper, going after introduction. Two kinds of fuzzy derivatives of real functions are considered: weak and strong ones. In addition, we introduce and study extended fuzzy derivatives, which may take infinite values. In the third part of this paper, fuzzy derivatives are applied to a study of maxima and minima of real functions. Different conditions for maxima and minima of real functions are obtained. Some of them are the same or at least similar to the conditions for the differentiable functions, while others differ in many aspects from those for the standard differentiable functions. Many classical results are obtained as direct corollaries of propositions for fuzzy derivatives, which are proved in this paper. Such results as the Fuzzy Intermediate Value theorem, Fuzzy Fermat's theorem, Fuzzy Rolle's theorem, and Fuzzy Mean Value theorem are proved. These results provide better theoretical base for computational methods of optimization.

scholarly journals A New Type-2 Soft Set: Type-2 Soft Graphs and Their Applications

2017 ◽  
Vol 2017 ◽  
pp. 1-17 ◽  
Author(s):  
Khizar Hayat ◽  
Muhammad Irfan Ali ◽  
Bing-Yuan Cao ◽  
Xiao-Peng Yang
Keyword(s):  
Graph Theory ◽  
Communication Networks ◽  
Essential Role ◽  
Soft Set ◽  
Soft Sets ◽  
Present Procedure ◽  
Graph Type ◽  
New Concepts ◽  
New Type

The correspondence between a vertex and its neighbors has an essential role in the structure of a graph. Type-2 soft sets are also based on the correspondence of primary parameters and underlying parameters. In this study, we present an application of type-2 soft sets in graph theory. We introduce vertex-neighbors based type-2 soft sets overX(set of all vertices of a graph) andE(set of all edges of a graph). Moreover, we introduce some type-2 soft operations in graphs by presenting several examples to demonstrate these new concepts. Finally, we describe an application of type-2 soft graphs in communication networks and present procedure as an algorithm.

On soft set-valued maps and integral inclusions

2021 ◽  
pp. 1-15
Author(s):  
Monairah Alansari ◽  
Shehu Shagari Mohammed ◽  
Akbar Azam
Keyword(s):  
Fixed Point ◽  
Set Theory ◽  
Fuzzy Set Theory ◽  
Fixed Point Theorems ◽  
Soft Set ◽  
Mathematical Tool ◽  
Multivalued Maps ◽  
Soft Sets ◽  
Special Cases ◽  
New Development

As an improvement of fuzzy set theory, the notion of soft set was initiated as a general mathematical tool for handling phenomena with nonstatistical uncertainties. Recently, a novel idea of set-valued maps whose range set lies in a family of soft sets was inaugurated as a significant refinement of fuzzy mappings and classical multifunctions as well as their corresponding fixed point theorems. Following this new development, in this paper, the concepts of e-continuity and E-continuity of soft set-valued maps and αe-admissibility for a pair of such maps are introduced. Thereafter, we present some generalized quasi-contractions and prove the existence of e-soft fixed points of a pair of the newly defined non-crisp multivalued maps. The hypotheses and usability of these results are supported by nontrivial examples and applications to a system of integral inclusions. The established concepts herein complement several fixed point theorems in the framework of point-to-set-valued maps in the comparable literature. A few of these special cases of our results are highlighted and discussed.

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