scholarly journals k-Circle Formation and k-epf by Asynchronous Robots

Algorithms ◽  
2021 ◽  
Vol 14 (2) ◽  
pp. 62
Author(s):  
Subhash Bhagat ◽  
Bibhuti Das ◽  
Abhinav Chakraborty ◽  
Krishnendu Mukhopadhyaya
Keyword(s):  
Fixed Point ◽  
Pattern Formation ◽  
Positive Integer ◽  
Fixed Points ◽  
Distributed Algorithm ◽  
Euclidean Plane ◽  
Formation Problem ◽  
The Common ◽  
Circle Formation

For a given positive integer k, the k-circle formation problem asks a set of autonomous, asynchronous robots to form disjoint circles having k robots each at distinct locations, centered at a set of fixed points in the Euclidean plane. The robots are identical, anonymous, oblivious, and they operate in Look–Compute–Move cycles. This paper studies the k-circle formation problem and its relationship with the k-epf problem, a generalized version of the embedded pattern formation problem, which asks exactly k robots to reach and remain at each fixed point. First, the k-circle formation problem is studied in a setting where the robots have an agreement on the common direction and orientation of one of the axes. We have characterized all the configurations and the values of k, for which the k-circle formation problem is deterministically unsolvable in this setting. For the remaining configurations and the values of k, a deterministic distributed algorithm has been proposed, in order to solve the problem. It has been proved that for the initial configurations with distinct robot positions, if the k-circle formation problem is deterministically solvable then the k-epf problem is also deterministically solvable. It has been shown that by modifying the proposed algorithm, the k-epf problem can be solved deterministically.

Coincidence and fixed points for hybrid tangential maps

2010 ◽  
Vol 17 (2) ◽  
pp. 273-285
Author(s):  
Tayyab Kamran ◽  
Quanita Kiran
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Property ◽  
Fixed Point Theorems ◽  
Acta Math ◽  
Contractive Condition ◽  
The Common

Abstract In [Int. J. Math. Math. Sci. 2005: 3045–3055] by Liu et al. the common property (E.A) for two pairs of hybrid maps is defined. Recently, O'Regan and Shahzad [Acta Math. Sin. (Engl. Ser.) 23: 1601–1610, 2007] have introduced a very general contractive condition and obtained some fixed point results for hybrid maps. We introduce a new property for pairs of hybrid maps that contains the property (E.A) and obtain some coincidence and fixed point theorems that extend/generalize some results from the above-mentioned papers.

scholarly journals Iterative Schemes for Fixed Point Computation of Nonexpansive Mappings

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Rudong Chen
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Common Fixed Points ◽  
Minimum Norm ◽  
Practical Applications ◽  
Fixed Point Computation ◽  
The Common ◽  
Special Case

Fixed point (especially, the minimum norm fixed point) computation is an interesting topic due to its practical applications in natural science. The purpose of the paper is devoted to finding the common fixed points of an infinite family of nonexpansive mappings. We introduce an iterative algorithm and prove that suggested scheme converges strongly to the common fixed points of an infinite family of nonexpansive mappings under some mild conditions. As a special case, we can find the minimum norm common fixed point of an infinite family of nonexpansive mappings.

Fixed points and iterations of mean-type mappings

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Janusz Matkowski
Keyword(s):  
Fixed Point ◽  
Positive Integer ◽  
Fixed Points ◽  
Metric Space ◽  
Real Interval ◽  
Type Mapping ◽  
Continuous Map ◽  
The Mean ◽  
Fixed Positive Integer ◽  
Pythagorean Harmony

AbstractLet (X, d) be a metric space and T: X → X a continuous map. If the sequence (T n)n∈ℕ of iterates of T is pointwise convergent in X, then for any x ∈ X, the limit $$\mu _T (x) = \mathop {\lim }\limits_{n \to \infty } T^n (x)$$ is a fixed point of T. The problem of determining the form of µT leads to the invariance equation µT ○ T = µT, which is difficult to solve in general if the set of fixed points of T is not a singleton. We consider this problem assuming that X = I p, where I is a real interval, p ≥ 2 a fixed positive integer and T is the mean-type mapping M =(M 1,...,M p) of I p. In this paper we give the explicit forms of µM for some classes of mean-type mappings. In particular, the classical Pythagorean harmony proportion can be interpreted as an important invariance equality. Some applications are presented. We show that, in general, the mean-type mappings are not non-expansive.

scholarly journals On Multistep Iterative Scheme for Approximating the Common Fixed Points of Contractive-Like Operators

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
J. O. Olaleru ◽  
H. Akewe
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Common Fixed Point ◽  
Iterative Scheme ◽  
Common Fixed Points ◽  
Weakly Compatible ◽  
The Common ◽  
Unique Common Fixed Point

We introduce the Jungck-multistep iteration and show that it converges strongly to the unique common fixed point of a pair of weakly compatible generalized contractive-like operators defined on a Banach space. As corollaries, the results show that the Jungck-Mann, Jungck-Ishikawa, and Jungck-Noor iterations can also be used to approximate the common fixed points of such maps. The results are improvements, generalizations, and extensions of the work of Olatinwo and Imoru (2008), Olatinwo (2008). Consequently, several results in literature are generalized.

scholarly journals Viscosity Method for Hierarchical Fixed Point Problems with an Infinite Family of Nonexpansive Nonself-Mappings

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Yaqin Wang
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Fixed Points ◽  
Infinite Family ◽  
Common Fixed Points ◽  
Fixed Point Problems ◽  
Viscosity Method ◽  
Hierarchical Fixed Point ◽  
The Common ◽  
Hierarchical Fixed Point Problems

A viscosity method for hierarchical fixed point problems is presented to solve variational inequalities, where the involved mappings are nonexpansive nonself-mappings. Solutions are sought in the set of the common fixed points of an infinite family of nonexpansive nonself-mappings. The results generalize and improve the recent results announced by many other authors.

scholarly journals The convergence of Iteration Scheme to Fixed Points in Modular Spaces

2019 ◽  
pp. 2196-2201
Author(s):  
Meena Fouad Abduljabbar ◽  
Salwa Salman Abed
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Fixed Point ◽  
Limit Point ◽  
Contraction Mapping ◽  
Iteration Scheme ◽  
Multivalued Mappings ◽  
Modular Spaces ◽  
The Common

     The aim of this paper is to study the convergence of an iteration scheme for multi-valued mappings which defined on a subset of a complete convex real modular. There are two main results, in the first result, we show that the convergence with respect to a multi-valued contraction mapping to a fixed point. And, in the second result, we deal with two different schemes for two multivalued  mappings (one of them is a contraction and other has a fixed point) and then we show that the limit point of these two schemes is the same. Moreover, this limit will be the common fixed point the two mappings.

scholarly journals n-Tupled Fixed Points Theorem in Fuzzy Metric Spaces with Application

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
P. P. Murthy ◽  
Rashmi Kenvat
Keyword(s):  
Fixed Point ◽  
Positive Integer ◽  
Fixed Points ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Fuzzy Metric Space ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric ◽  
Contractive Type

We will introduce the concept ofn-tupled fixed points (for positive integern) in fuzzy metric space by mild modification of the concept ofn-tupled fixed points (for even positive intergern) introduced by Imdad et al. (2013) in metric spaces. As application of the above-mentioned concept, we will establish somen-tupled fixed point theorems for contractive type mappings in fuzzy metric space which extends the result of Roldán et al. (2013). Also we have given an application to solve a kind of Lipschitzian systems fornvariables and an integral system.

scholarly journals Retraction method in fixed point theory for monotone nonexpansive mappings

2016 ◽  
Vol 32 (3) ◽  
pp. 271-276
Author(s):  
M. R. ALFURAIDAN ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Fixed Point Theory ◽  
Nonexpansive Mappings ◽  
Point Theory ◽  
Common Fixed Points ◽  
The Common ◽  
Nonexpansive Retract

In this paper we study the properties of the common fixed points set of a commuting family of monotone nonexpansive mappings in Banach spaces endowed with a graph. In particular, we prove that under certain conditions, this set is a monotone nonexpansive retract.

scholarly journals Circle formation by asynchronous opaque robots on infinite grid

2021 ◽  
Vol 22 (1) ◽  
Author(s):  
Ranendu Adhikary ◽  
Manash Kumar Kundu ◽  
Buddhadeb Sau
Keyword(s):  
Communication System ◽  
Distributed Algorithm ◽  
Line Formation ◽  
Grid Environment ◽  
External Communication ◽  
Formation Problem ◽  
Infinite Grid ◽  
Circle Formation

This paper presents a distributed algorithm for circle formation problem under the infinite grid environment by asynchronous mobile opaque robots. Initially all the robots are acquiring distinct positions and they have to form a circle over the grid. Movements of the robots are restricted only along the grid lines. They do not share any global co-ordinate system. Robots are controlled by an asynchronous adversarial scheduler that operates in Look-Compute-Move cycles. The robots are indistinguishable by their nature, do not have any memory of their past configurations and previous actions. We consider the problem under luminous model, where robots communicate via lights, other than that they do not have any external communication system. Our protocol solves the  circle formation problem using seven colors. A subroutine of our algorithm also solves the line formation problem using three colors.

scholarly journals Equivalence results for implicit Jungck–Kirk type iterations

2018 ◽  
Vol 22 (1) ◽  
pp. 75-89
Author(s):  
H. Akewe ◽  
A. A. Mogbademu
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Fixed Points ◽  
Fixed Point Iteration ◽  
Linear Spaces ◽  
Mann Iteration ◽  
Normed Linear Spaces ◽  
Weakly Compatible ◽  
The Common

We show that the implicit Jungck–Kirk-multistep, implicit Jungck–Kirk–Noor, implicit Jungck–Kirk–Ishikawa, and implicit Jungck–Kirk–Mann iteration schemes are equivalently used to approximate the common fixed points of a pair of weakly compatible generalized contractive-like operators defined on normed linear spaces. Our results contribute to the existing results on the equivalence of fixed point iteration schemes by extending them to pairs of maps. An example to show the applicability of the main results is included.

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