# Ismat BegLahore School of Economics · Mathematics and Statistics

Ismat Beg

PhD

Order; Fixed point theory; Fuzzy sets and their applications; Multi-criteria decision theory; Deep learning.

## About

302

Publications

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Introduction

Ismat Beg is Professor at Lahore School of Economics. He is also Higher Education Commission (HEC) Distinguished National Professor. He has vast experience of teaching and research. His field of interest and specialization is versatile in nature. It covers many areas of Mathematics, Economics, Decision Theory, Computer Science, Social Sciences and Engineering. He is a Fellow of Pakistan Academy of Sciences, and Institute of Mathematics and its Applications (U K).

Additional affiliations

April 2013 - present

July 2011 - April 2013

**University of Central Punjab**

Position

- Managing Director

July 2009 - present

**Institute of Advanced Scientific Research**

Position

- Member

Education

September 1978 - January 1982

January 1971 - May 1973

September 1966 - May 1970

**Goverment Zamindar Postgraduate College, Gujrat**

Field of study

- Science

## Publications

Publications (302)

Some results on fixed point of asymptotically regular multivalued mapping are obtained in metric spaces. The structure of common fixed points and coincidence points of a pair of compatible multivalued mappings is also discussed. Our work generalizes known results of Aubin and Siegel, Dube, Dube and Singh, Hardy and Rogers, Hu, Iseki, Jungck, Kaneko...

We propose a new set of axioms that a value in the interval [0, 1] should satisfy to be a degree or a measure of similarity between fuzzy subsets of a given universe. The relevance of the new axioms with previous axioms and categories of similarity measures for fuzzy sets is also studied.

Let (X,⪯) be a partially ordered set and d be a complete metric on X. Let F,G be two set-valued mappings on X. We obtained sufficient conditions for the existence of common fixed point of F and G satisfying an implicit relation in partially ordered set X.

We prove the existence of coincidence point and common fixed point for mappings satisfying generalized weak contractive condition. As an application, related results on invariant approximation are derived. Our results generalize various known results in the literature.

The aim of this paper is to introduce the concept of polytopic fuzzy sets as an extension of spherical fuzzy sets, picture fuzzy sets, and-rung orthopair fuzzy sets. We define some basic operations and aggregation operators on polytopic fuzzy sets. Based on these operators, we present three new methods to deal with the multiple-attribute decision-m...

IIn this manuscript, best S^{JS}-approximation point theorems for a class of proximal contractive mappings namely proximal S^{JS}-quasi-contraction mappings and proximal S^{JS}-Z-contraction mappings have been proved on a S^{JS}-metric space endowed with a partial ordering . Several supporting examples are given to strengthen the hypothesis of our...

In this paper we introduce the concept of ordered uniform convexity in ordered convex metric spaces and study some properties of order uniform convexity. Finally as application we connect our results with existence of fixed points for monotone non-expansive mappings defined on these spaces.

In e-price bid auctions, we construct an auction model using regret decision theory to explain how a bidder decides his bidding price. The fundamental distinction between regret theory and other decision theories under uncertainty is clarified in this article. The suggested regret decision theory is scenario-based rather than the other theories. Th...

Sufficient conditions for existence of common fixed point on complex partial b-metric spaces are obtained. Our results generalize and extend several well-known results. In the end we explore applications of our key results to solve a system of Urysohn type integral equations.

This paper introduces the concept of k-ordered proximal contractions and then study best proximity point results for these mappings. An example is given to show accuracy and significance of our claims.

In the present paper, we introduce new types of convergence of a sequence in left dislocated and right dislocated metric spaces. Also, we generalize Banach contraction principle in these newly defined generalized metric spaces.

We prove Ekeland's variational principle in S JS-metric spaces. A generalization of Caristi fixed point theorem on S JS-metric spaces is obtained as a consequence.

In this paper, we introduce the concept of generalized F-proximal contraction mappings and prove some best proximity point theorems for a non-self mapping in a complete metric space. Then some of the well known results in the existing literature are generalized/extended using these newly obtained results. An example is being given to demonstrate us...

The aim of this article is to introduce a new notion of ordered convex metric spaces and study some basic properties of these spaces. Several characterizations of these spaces are proven that allow making geometric interpretations of the new concepts.
1. Introduction
Menger [1] initiated the study of convexity in metric spaces which was further de...

We introduce the idea of S^{JS}-metric spaces which is a generalization of S-metric spaces. Next we study the properties of S^{JS}-metric spaces and prove several theorems. We also deal with abstract S^{JS}-topological spaces induced by S^{JS}-metric and obtain several classical results including Cantor's intersection theorem in this setting.

This chapter interest with interconnection among the latterly introduced neutrosophic cubic sets and Önite state machines. The concept of neutrosophic cubic Önite state machines (Neutrosophic cubic FSM), subsystem of neutrosophic cubic FSM, Cartesian composition (direct product, P-(R-) union, P-(R)-intersection) of two subsystems of neutrosophic cu...

In this paper, based on embedding approach three numerical methods namely Richardson, Gauss-Seidel, and successive over relaxation (SOR) have been developed to solve bipolar neutrosophic system of linear equations. To check the accuracy of these newly developed schemes an example with exact and iterative solution is given. c ⃝ 2021 IAUCTB. All righ...

We obtain sufficient conditions for existence of fixed points of integral type contractive mappings on S^{JS}- metric spaces. We also study common fixed point and couple fixed point of integral type mappings and construct examples to support our results.

In the paper we obtain sufficient conditions for the existence of common fixed point for a pair of contractive type mappings in bicomplex valued metric spaces.

In this paper, we introduce the concept of a generalized convex metric space as a generalization of a convex metric space, which is due to Takahashi, and give the iterative scheme due to Xiao et al.. We also establish strong convergence of this scheme to a unique common fixed point of a finite family of asymptotically quasi-nonexpansive mappings. O...

In this paper, we introduce the concept of generalized orthogonal -Suzuki contraction mapping and prove some fixed point theorems on orthogonal -metric spaces. Our results generalize and extend some of the well-known results in the existing literature. As an application of our results, we show the existence of a unique solution of the first-order o...

A new contraction condition for multivalued maps in metric spaces is introduced and then, based on this new condition, we prove two fixed point theorems for such contractions. The new condition uses the altering distance technique and a Pompeiu type metric on the family of nonempty and closed subsets of a metric space. Our results essentially compl...

In this paper, we obtain sufficient conditions for existence of fixed points for a sequence of L-fuzzy mappings in a non-Archimedean ordered modified intuitionistic fuzzy metric space. We use contractive conditions of implicit relation. Further, as an application, we also generalize our usual contractive conditions into integral contractive conditi...

In this paper we extend fuzzy analytic hierarchy process into neutrosophic cubic environment. The neutrosophic cubic analytic hierarchy process can be used to manage more complex problems when the decision makers has a number of uncertainty, assigning preferences values to the considered object. We also define the concept of triangular neutrosophic...

The aim of this study is to propose a novel multi criteria group decision making method using analytic hierarchy process and hesitant probabilistic fuzzy linguistic set to model uncertainties due to both fuzziness and randomness. Proposed multi criteria group decision making method first uses analytic hierarchy process to construct decision prefere...

An important property of metric spaces is the existence and uniqueness of completion. In this paper, we prove existence and uniqueness of completion of a complex valued strong b-metric space.

In this paper we prove existence of fixed point theorems for Z-contractive map, Geraghty type contractive map and interpolative Hardy-Rogers type contrac-tive mapping in the setting of S JS metric spaces with two metrics. Examples are constructed to high light the significance of newly obtained results.

This paper presents operational laws along with their cosine measure for the numbers whose base is an interval value and study their properties. Consequent upon these definitions and properties neutrosophic cubic weighted exponential averaging and dual neutrosophic cubic weighted exponential averaging aggregation operators are defined. A multi attr...

We introduce the notion of sequentially compactness on S JS-metric spaces and study the properties of sequentially compact S JS-metric spaces. As an application, we obtain the results on fixed points of mappings defined on sequentially compact S JS-metric spaces.

Classical models of decision-making do not incorporate for the role of influence and honesty that affects the process. This paper develops on the theory of influence in social network analysis. We study the role of influence and honesty of individual experts on collective outcomes. It is assumed that experts have the tendency to improve their initi...

The aim of this paper is to introduce the notion of truthfulness in an influence based decision making model. An expert may submit his opinions truthfully or he may dismantle the original situation by undermining the actual opinion, such a decision maker is called an evasive decision maker or an almost truthful decision maker in this paper. It is a...

In this paper we introduce the concept of R-continuity and R-closedness for a pair of multivalued mappings. A new class of (R, α)-generalized rational multivalued contraction mappings is defined. After that we establish the existence of common fixed point of such mappings in the framework of b-metric spaces endowed with an arbitrary binary relation...

Soft pedalling is a real world problem and it is used to understate the intensity of an issue at hand. Influence models are currently studied by researchers working in the field of social network analysis but they do not incorporate for soft pedalling. The aim of this work is to study the impact of truthfulness of each expert on the final outcome....

In this article, we develop a diminishing hesitant 2-tuple averaging operator (DH2TA) for hesitant 2-tuple linguistic arguments. DH2TA work in the way that it aggregate all hesitant 2-tuple linguistic elements and during the aggregation process it also controls the hesitation in translation of the resultant aggregated linguistic term. We develop a...

There are many different techniques used so for in decision making problems. Some researchers used fuzzy sets, some used soft sets and some used combination of fuzzy sets with soft sets in order to handle the imprecise information. On the other hand, matrices play very important role in handling such problems. Different researchers used fuzzy matri...

In this paper, we introduce weakly increasing F-contractions on an ordered partial metric space and establish a common fixed point theorem for weakly increasing F-contractions. This result generalizes some recent results on F-contractions. We give an example showing the significance of our main theorem. We show that the main result can be applied t...

We obtain necessary conditions for the existence of coincidence point and common fixed point for contractive mappings in cone metric spaces. An application to the stability of J-iterative procedure for mappings having coincidence point in cone metric spaces is also given.

In this article, hesitant 2-tuple linguistic arguments are used to evaluate the group decision making problems which have inter dependent or inter active attributes. Operational laws are developed for hesitant 2-tuple linguistic elements and based on these operational laws hesitant 2- tuple weighted averaging operator and generalized hesitant 2- tu...

New requirements and challenges arise in judgment aggregation, due to the complexity of judgment making process and the necessity of dealing with huge amounts of vague and uncertain information and alternatives. In this paper we propose an interactive fuzzy judgment aggregation method for consensus to deal with the situation where judges are partia...

In this article, we propose a method to deal with incomplete interval-valued hesitant fuzzy preference relations. For this purpose, an additive transitivity inspired technique for interval-valued hesitant fuzzy preference relations is formulated which assists in estimating missing preferences. First of all, we introduce a condition for decision mak...

We introduce a new concept of α-fuzzy H-contractive mapping which is essentially weaker than the class of fuzzy contractive mapping and stronger than the concept of α-φ-fuzzy contractive mapping. For this type of contractions, the existence and uniqueness of fixed point in fuzzy M-complete metric spaces is also established.

The aim of the present paper is to introduce the concept of joint common limit range property ((JCLR) property) for single–valued and set–valued maps in non–Archimedean fuzzy metric spaces. We also list some examples to show the difference between (CLR) property and (JCLR) property. Further, we establish common fixed point theorems using implicit r...

In this paper, we model a system of fuzzy soft differential equations to analyze the behavior over the time of an individual depending on their companion's actions under any particular situation against some decision. The Bonferroni mean (BM) is a very useful tool for group decision making problems when arguments are interrelated to each other as B...

Notion of almost partial Hausdorff metric is introduced and a generalization of well known Nadler's fixed point theorem for multi-valued mappings on weak partial metric spaces using almost partial Hausdorff metric is obtained. A homotopy result is derived as an application .

Uncertainties due to randomness and fuzziness comprehensively exist in control and decision support systems. In the present study, we introduce notion of occurring probability of possible values into hesitant fuzzy linguistic element (HFLE) and define hesitant probabilistic fuzzy linguistic set (HPFLS) for ill structured and complex decision making...

We prove common fixed point theorems for weakly commuting and occasionally coincidentally idempotent L-fuzzy mappings in ordered b-metric spaces. We also obtain common fixed point for pair of mapping satisfy (JCLR) property. An application to integral type and usual contractive condition is given.

In this paper, we introduce an ultrapower approach to prove fixed point theorems for H⁺-nonexpansive multi-valued mappings in the setting of CAT(0) spaces and prove several hybrid fixed point results in CAT(0) spaces for families of single-valued nonexpansive or quasinonexpansive mappings and multi-valued upper semicontinuous, almost lower semicont...

In this paper, we first introduce a new approach to the classical fixed point theorems for H+-type nonexpansive
multivalued mappings in Banach spaces and obtain a generalization of the classical Nadler’s fixed point theorem. Based on this generalization of Nadler’s fixed point theorem, we study the invariant approximation and proved several new res...

In this paper, we prove fixed point theorems for mapping satisfying an implicit relation in ordered metric space. As application we obtain a homotopy result. Our results modify/extend several fixed point results in the literature.
Keywords: Fixed point; contraction; quasi contraction; Ćirić-Suzuki-contraction; implicit relation; ordered metric spac...

Intuitionistic fuzzy relations are used to construct hierarchical structures for the evaluation of vague complicated humanistic systems. A novel algorithm to develop partition trees at different levels according to different intuitionistic fuzzy triangular norm composition is presented. Examples are given to demonstrate the usefulness of the propos...

We propose a novel notion of fuzzy similarity measure between fuzzy sets by using fuzzy equivalence. Furthermore, a similarity base fuzzy relational clustering algorithm is formulated. Applications are given to illustrate the usefulness of the chosen fuzzy equivalence and similarity measure in a partition algorithm.

We obtain sufficient conditions for existence of random fixed point of Suzuki type random multifunctions and hemiconvex multifunctions. Our results generalize the known results in the literature.

Let (X,d,≼) be a partially ordered metric space and f, F be single and set valued mappings on X. We obtained sufficient conditions for existence of fixed point of mappings f and F on X with a metric transform.

In this article, incomplete hesitant fuzzy preference relations are under consideration. In order to estimate expressible missing preferences, a hesitant upper bound condition (hubc) is defined for decision makers presenting incomplete information. With the help of this condition, the estimated preference intensities lie inside the defined domain a...

A notion for distance between hesitant fuzzy data is given. Using this new distance notion, we propose the technique for order preference by similarity to ideal solution for hesitant fuzzy sets and a new approach in modelling uncertainties. An illustrative example is constructed to show the feasibility and practicality of the new method.

This paper introduces a novel extension of soft rough fuzzy set so-called modified soft rough fuzzy set model in which new lower and upper approximation operators are presented together their related properties that are also investigated. Eventually it is shown that these new models of approximations are finer than previous ones developed by using...

In this paper we first introduce a new approach to the classical fixed point theorems for H⁺-type nonexpansive multivalued mappings in Banach spaces by reformulating the arguments in an ultrapower context which helps to illuminate many underlying ideas and obtained a generalization of classical Nadler's fixed point theorem. Secondly using this gene...

We prove existence of common fixed point of L-fuzzy mappings on non-Archimedean ordered fuzzy metric spaces by using integral type and contractive conditions. Examples are also given to illustrate significance of these results.

In this study we introduce the concept of denser property in fuzzy membership function used in neutrosophic sets. We present several new definitions and studied their properties. Defuzzification methods over neutrosophic triangular dense fuzzy sets and neutrsophic triangular intuitionistic dense fuzzy sets are then given. Finally practical applicab...

Arrow (1963) established that a group cannot always reach logically consistent collective outcome. Subsequently many developments like premise based, conclusion based and distance based methods have emerged in literature to reach group consistency. This study is focused on the judgment aggregation in fuzzy logic based setting with novel involvement...

In this paper, we introduce a modified soft fuzzy rough set model. The lower and upper approximation operators are presented and their related properties are investigated. It is shown that these new models of approximations are finer than already known in the literature.

This article deals with a new novel defuzzification method for the dense fuzzy sets. In our study, we first define the dense fuzzy set for triangular fuzzy numbers. Then new defuzzification methods have been formulated with crisp convergence tests. Cauchy sequence has been utilised for better illustrations. We show the usefulness and the global jus...

Convex hesitant fuzzy sets are define as an extension of convex fuzzy sets. Also level sets are defined for hesitant fuzzy sets and discussed with their convexity. We focus on aggregation functions for hesitant fuzzy elements. These aggregation functions are further extended for hesitant fuzzy sets as well as for the convex structures of these sets...

In this paper, we will define the notion of CLR property for hybrid pairs of L-fuzzy and crisp mappings in non-Archimedean fuzzy metric spaces, we establish some common fixed point theorems for L-fuzzy and crisp mappings in non-Archimedean fuzzy metric .

In this paper we obtain sufficient conditions for the existence of fixed point of T-isotone mappings and coincidence point of G-isotone mappings in partially ordered metric spaces.

In this article, we introduce the concept of ordered dualistic partial metric spaces and establish an order relation on quasi dualistic partial metric spaces. Later on, using this order relation, we prove �fixed point theorems for single and multivalued mappings. We support our results with some illustrative examples.

An interval valued preference relation is a preference structure that is used to
describe uncertainty in complex decision making problems. Retrieving complete
information from experts is improbable in real life scenarios. Discarding incomplete
information leads to loss of important data. In this paper, we introduce an
upper bound condition to deal...

Dealing with uncertainty is always a challenging problem, and different tools have been proposed to deal with it. Fuzzy sets was presented to manage situations in which experts have some membership value to assess an alternative. The fuzzy linguistic approach has been applied successfully to many problems. The linguistic information expressed by me...

First we give notion of integral of intuitionistic fuzzy set and introduce intuitionistic fuzzy implicator and intuitionistic fuzzy inclusion measure. Then we propose a new measure of similarity between two intuitionistic fuzzy sets based on intuitionistic fuzzy inclusion measure. Examples are given to illustrate our notion and the application of t...

In this paper, the concept of a hesitant 2-tuple linguistic information model is introduced. It provides a linguistic and computational basis to manage the situations in which experts assess an alternative in linguistic term while feeling some hesitation to present its possible linguistic translations. A distance measure is defined between any two...

Dealing with uncertainty is a difficult task and different tools have been proposed in the literature to handle it. Hesitant fuzzy sets are highly useful in resolving situations where people hesitate when providing their preferences. In this paper, the concept of a hesitant fuzzy soft set is modified to manage the situations in which experts assess...

We introduce a method for aggregation of the experts' opinion given in the form of comparative linguistic expression. An algorithmic form of technique for order preference is proposed for group decision making. A simple example is given by using this method for the selection of the best alternative as well as ranking the alternatives from the best...

In this paper, we obtain some fixed point theorems for dominated mappings satisfying locally contractive conditions on a closed ball in a left K-sequentially O-complete ordered quasi-partial metric space and in a right K-sequentially O-complete ordered quasi-partial metric space, respectively. Our results improve several well-known results

In this paper, we obtain some fixed point theorems for dominated mappings satisfying locally contractive conditions on a closed ball in a left K-sequentially O-complete ordered quasi-partial metric space and in a right K-sequentially O-complete ordered quasi-partial metric space, respectively. Our results improve several well-known results.

The aim of this paper is to prove new common fixed point theorems on intuitionistic fuzzy metric spaces. Our main result, bring into play the concept of E.A. property, complete subspace and weakly compatible mappings. An example is furnished which demonstrates the validity of our main result. Efforts have also been invested to broaden our result on...

We proposed an aggregation operator which is used to aggregate decision makers' opinions in group decision making process. First, a Choquet integral-based distance between generalized interval valued trapezoidal fuzzy numbers is defined. Then combining the generalized interval valued trapezoidal fuzzy number aggregation operator with Choquet integr...

The aim of this paper is to propose a new approach for medical diagnosis by using trapezoidal valued intuitionistic fuzzy relations. First, we develop trapezoidal valued intuitionistic fuzzy relations and then use it to solve medical diagnosis decision making problem. We study Sanchez's method of medical diagnosis with the notion of trapezoidal val...

A modified dissimilarity measure for fuzzy data is proposed and is applied to real data with mixed feature variables of symbolic and fuzzy data. We also proposed an improved version of fuzzy relational clustering algorithm. Numerical examples and comparison are given between mixed variable fuzzy c-means and fuzzy relational clustering algorithm.

Sufficient conditions for the existence of fixed point for mappings satisfying locally contractive conditions on a closed ball in an ordered left K-sequentially as well as right K-sequentially complete dislocated quasi metric space have been obtained. The notion of dominated mappings is applied to approximate the unique solution to non linear funct...

We prove fixed point theorems for Suzuki type multi-functions on complete metric spaces. An example is constructed to illustrate that our results are new.

We propose a hybrid technique by merging fuzzy
version of the classical technique for order preference by similarity to ideal solution
and decision-making trial and evaluation laboratory technique for trapezoidal fuzzy
numbers, where interaction phenomena among the decision-making problem and
weights are taken into account. The feasibility of this...

In this article we prove the existence of common fixed points for Edelstein type locally contractive multivalued mappings in complete cone metric spaces, without assumption of normality of cone. We generalize/extend several remarkable and useful results in the existing literature.

A hierarchical structure is proposed for the performance evaluation of vague, complicated humanistic systems. An improved fuzzy clustering algorithm is developed to produce several partition trees with different levels and clusters according to different triangular norm compositions. Additionally a fuzzy clustering algorithm is given to produce a p...

We study the almost sure T-stability and strong convergence of two random iterative algorithms namely random Halpern iteration scheme and random Xu-Mann iteration scheme for a general class of random operators in a separable Banach space. Our results complement/generalize the known stability results in stochastic verse.