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Search Results: 1 - 10 of 607 matches for " Sumit Chandok "
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Common Fixed Points, Invariant Approximation and Generalized Weak Contractions
Sumit Chandok
International Journal of Mathematics and Mathematical Sciences , 2012, DOI: 10.1155/2012/102980
Abstract: Sufficient conditions for the existence of a common fixed point of generalized -weakly contractive noncommuting mappings are derived. As applications, some results on the set of best approximation for this class of mappings are obtained. The proved results generalize and extend various known results in the literature. 1. Introduction and Preliminaries It is well known that Banach’s fixed point theorem for contraction mappings is one of the pivotal result of analysis. Let be a metric space. A mapping is said to be contraction if there exists such that for all , If the metric space is complete, then the mapping satisfying (1.1) has a unique fixed point. A natural question is that whether we can find contractive conditions which will imply existence of fixed point in a complete metric space but will not imply continuity. Kannan [1, 2] proved the following result, giving an affirmative answer to the above question. Theorem 1.1. If , where is a complete metric space, satisfies where and , then has a unique fixed point. The mappings satisfying (1.2) are called Kannan type mappings. A similar type of contractive condition has been studied by Chatterjea [3] and he proved the following result. Theorem 1.2. If , where is a complete metric space, satisfies where and , then has a unique fixed point. In Theorems 1.1 and 1.2 there is no requirement of continuity of . A map is called a weakly contractive (see [4–6]) if for each , where is continuous and nondecreasing, if and only if and . If we take , , then a weakly contractive mapping is called contraction. A map is called -weakly contractive (see [7]) if for each , where is a self-mapping, is continuous and nondecreasing, if and only if and . If we take , , then a -weakly contractive mapping is called -contraction. Further, if identity mapping and , , then a -weakly contractive mapping is called contraction. A map is called a generalized weakly contractive (see [5]) if for each , where is continuous such that if and only if . If we take , , then inequality (1.6) reduces to (1.3). Choudhury [5] shows that generalized weakly contractive mappings are generalizations of contractive mappings given by Chatterjea (1.3), and it constitutes a strictly larger class of mappings than given by Chatterjea. A map is called a generalized -weakly contractive [8] if for each , where is a self-mapping, is continuous such that if and only if . If identity mapping, then generalized -weakly contractive mapping is generalized weakly contractive. For a nonempty subset of a metric space and , an element is said to be a best approximant to
Some common fixed point theorems for Ciric type contraction mappings
Sumit Chandok
Tamkang Journal of Mathematics , 2012, DOI: 10.5556/j.tkjm.43.2012.187-202
Abstract: Some common fixed point theorems for '{C}iri'{c} type contraction mappings have been obtained in convex metric spaces. As applications, invariant approximation results for these type of mappings are obtained. The proved results generalize, unify and extend some of the results of the literature.
Fixed point theorem for weakly Chatterjea-type cyclic contractions
Sumit Chandok and Mihai Postolache
Fixed Point Theory and Applications , 2013, DOI: 10.1186/1687-1812-2013-28
Abstract: MSC: 41A50, 47H10, 54H25.
Some Common Fixed Point Results for Rational Type Contraction Mappings in Complex Valued Metric Spaces
Sumit Chandok,Deepak Kumar
Journal of Operators , 2013, DOI: 10.1155/2013/813707
Abstract: We prove some common fixed point theorems for two pairs of weakly compatible mappings satisfying a rational type contractive condition in the framework of complex valued metric spaces. The proved results generalize and extend some of the known results in the literature. 1. Introduction and Preliminaries The famous Banach contraction principle states that if is a complete metric space and is a contraction mapping (i.e., for all , where is a nonnegative number such that ), then has a unique fixed point. This principle is one of the cornerstones in the development of nonlinear analysis. Fixed point theorems have applications not only in the different branches of mathematics, but also in economics, chemistry, biology, computer science, engineering, and others. Due to its importance, generalizations of Banach’s contraction principle have been investigated heavily by several authors. Fixed point and common fixed point theorems for different types of nonlinear contractive mappings have been investigated extensively by various researchers (see [1–35] and references cited therein). Recently, Azam et al. [1] introduced the complex valued metric space, which is more general than the well-known metric spaces. Many researchers have obtained fixed point, common fixed point, coupled fixed point, and coupled common fixed point results in partially ordered metric spaces, complex valued metric spaces, and other spaces. In this paper, we prove some common fixed point theorems for two pairs of weakly mappings satisfying a contractive condition of rational type in the framework of complex valued metric spaces. The proved results generalize and extend some of the results in the literature. To begin with, we recall some basic definitions, notations, and results. The following definitions of Azam et al. [1] are needed in the sequel. Let be the set of complex numbers, and let . Define a partial order on as follows: It follows that if one of the following conditions is satisfied:(1) , and ;(2) , and ;(3) , and ;(4) , and .In particular, we will write if and one of (1), (2), and (3) is satisfied, and we will write if only (3) is satisfied. Note. We obtained that the following statements hold:(i) and ,??for all ;(ii) ;(iii) and imply . Definition 1. Let be a nonempty set. Suppose that the mapping satisfies the following conditions:(i) for all and if and only if ;(ii) for all ;(iii) for all .Then, is called a complex valued metric on , and is called a complex valued metric space. Example 2. Let . Define a mapping by where . Then, is a complex valued metric space. A point is called
Invariant Points and -Simultaneous Approximation
Sumit Chandok,T. D. Narang
International Journal of Mathematics and Mathematical Sciences , 2011, DOI: 10.1155/2011/579819
Abstract: We generalize and extend Brosowski-Meinardus type results on invariant points from the set of best approximation to the set of -simultaneous approximation. As a consequence some results on -approximation and best approximation are also deduced. The results proved in this paper generalize and extend some of the known results on the subject. 1. Introduction and Preliminaries Fixed point theory has gained impetus, due to its wide range of applicability, to resolve diverse problems emanating from the theory of nonlinear differential equations, theory of nonlinear integral equations, game theory, mathematical economics, control theory, and so forth. For example, in theoretical economics, such as general equilibrium theory, a situation arises where one needs to know whether the solution to a system of equations necessarily exists; or, more specifically, under what conditions will a solution necessarily exist. The mathematical analysis of this question usually relies on fixed point theorems. Hence finding necessary and sufficient conditions for the existence of fixed points is an interesting aspect. Fixed point theorems have been used in many instances in best approximation theory. It is pertinent to say that in best approximation theory, it is viable, meaningful, and potentially productive to know whether some useful properties of the function being approximated is inherited by the approximating function. The idea of applying fixed point theorems to approximation theory was initiated by Meinardus [1]. Meinardus introduced the notion of invariant approximation in normed linear spaces. Brosowski [2] proved the following theorem on invariant approximation using fixed point theory by generalizing the result of Meinardus [1]. Theorem 1.1. Let be a linear and nonexpansive operator on a normed linear space . Let be a -invariant subset of and a -invariant point. If the set of best -approximants to is nonempty, compact, and convex, then it contains a -invariant point. Subsequently, various generalizations of Brosowski's results appeared in the literature. Singh [3] observed that the linearity of the operator and convexity of the set in Theorem 1.1 can be relaxed and proved the following. Theorem 1.2. Let be a nonexpansive self-mapping on a normed linear space . Let be a -invariant subset of and a -invariant point. If the set is nonempty, compact, and star shaped, then it contains a -invariant point. Singh [4] further showed that Theorem 1.2 remains valid if is assumed to be nonexpansive only on . Since then, many results have been obtained in this direction (see
Fixed Points of Quasi-Nonexpansive Mappings and Best Approximation
T. D. Narang,Sumit Chandok
Sel?uk Journal of Applied Mathematics , 2009,
Abstract: Using fixed point theory, B.Brosowski [Mathematica (Cluj) 11 (1969), 195-220] proved that if T is a nonexpansive linear operator on a normed linear space X, C a T-invariant subset of X and x a T-invariant point, then the set PC(x) of best C-approximant to x contains a T-invariant point if PC(x) is non-empty, compact and convex. Subsequently, many generalizations of the Brosowski’s result have appeared. In this paper, we also prove some extensions of the results of Brosowski and others for quasi-nonexpansive mappings when the underlying spaces are metric linear spaces or convex metric spaces.
Importance of integrin receptors in the field of pharmaceutical & medical science  [PDF]
Sumit Goswami
Advances in Biological Chemistry (ABC) , 2013, DOI: 10.4236/abc.2013.32028
Abstract: Integrin receptors have remained as a key subject of interest in the pharmaceutical industry for the last few years. There are a total of 24 different types of integrin heterodimers. Each of these heterodimers plays important role in various biological processes that are inherent to different pathological conditions. As a result, integrin receptors have been extensively evaluated for their role in therapeutic targeting. There are different classes of inhibitors against integrin receptors and this review provides an overview on different classes of integrin inhibitors that are currently available. A number of review articles have been written on the possible application of integrin receptors in therapeutic targeting. Many of these articles have heavily emphasized on the importance of αvβ3 & αvβ5 receptors as major pharmaceutical target in cancer but little emphasis has been given on the importance of other integrin receptors, such as α5β1, αIIbβ3, α4β7, αvβ6 etc. While this review gives due importance to both αvβ3 & αvβ5 receptors and provides an historical perspective on how these two receptors have evolved as a potential target for cancer, significant emphasis has also been given on the other integrin receptors that have started enjoying the status of important drug target over the course of last few years. Effort has been maintained to discuss briefly on the key physiological basis of their importance as drug target. For example, involvement of αvβ3 in angiogenesis has made it a therapeutic target for the treatment of cancer. At the same time expression of this receptor on the surface of osteoclast has made it a target for the treatment of osteoporosis. Thus, emphasis has been given on discussing the role of the integrin receptors in different disease conditions followed by specific examples of drug molecules that have been trialed against these receptors. While hundreds of candidate molecules have been developed against different integrin receptors only a handful of them has been subject to phase-III clinical trial. That necessitates careful consideration of certain concerns that are associated with direct targeting of integrins and thus has also been an important goal of this review. In the last few years application of integrin receptors have extended beyond mere therapeutic targeting. Several integrin receptors are currently are studied for their potential of aiding at diagnostic imaging and drug delivery. In this review a brief overview has also been provided on how integrin are being
Adaptive Processing Gain Data Services in Cellular CDMA in Presence of Soft Handoff with Truncated ARQ  [PDF]
Dipta DAS, Sumit KUNDU
Int'l J. of Communications, Network and System Sciences (IJCNS) , 2009, DOI: 10.4236/ijcns.2009.22017
Abstract: An adaptive data transmission scheme based on variable spreading gain (VSG) is studied in cellular CDMA network in presence of soft handoff (HO). The processing gain is varied according to traffic intensity meet-ing a requirement on data bit error rate (BER). The overall performance improvement due to processing gain adaptation and soft HO is evaluated and compared with a fixed rate system. The influence of soft HO pa-rameters on rate adaptation and throughput and delay performance of data is indicated. Further truncated automatic repeat request (T-ARQ) is used in link layer to improve the performance of delay sensitive ser-vices. The joint impact of VSG based transmission in presence of soft handoff at physical layer and T-ARQ at link layer is evaluated. A variable packet size scheme is also studied to meet a constraint on packet loss.
Sequence trademarks in oncogene associated microRNAs
Sumit Sharma,Sumit Biswas
Bioinformation , 2011,
Abstract: The last decade was taken by storm when the existence of a class of small (~22nt long) non – coding RNA species, known as microRNAs (miRNAs) came into light. MicroRNAs are one of the most abundant groups of regulatory genes in multicellular organisms and play fundamental roles in many cellular processes. Among these, miRNAs have been shown to prevent cell division and drive terminal differentiation, thus playing a causal role in the generation or maintenance of cancerous tumours. The unique expression profiles of different miRNAs in various types and stages of cancer suggest their performance as novel biomarkers. This discussion focuses on miRNAs implicated in cancer-associated events and strives to re-establish their sequential features which may classify them to be oncogenic.
Some coupled common fixed point theorems for a pair of mappings satisfying a contractive condition of rational type without monotonicity
S. Chandok,M. S. Khan,K. P. R. Rao
International Journal of Mathematical Analysis , 2013,
Abstract: The purpose of this paper is to establish some coupled coincidence pointtheorems for a pair of mappings without mixed monotone property satisfying a contractivecondition of rational type in the framework of partially ordered metric spaces. Also, wepresent a result on the existence and uniqueness of coupled common fixed points. Theresults presented in the paper generalize and extend several well-known results in theliterature.
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