scholarly journals A Study on p-Cyclic Orbital Geraghty type Contractions

2018 ◽  
Vol 7 (4.10) ◽  
pp. 883 ◽  
Author(s):  
M. L.Suresh ◽  
T. Gunasekar ◽  
S. Karpagam ◽  
B. Zlatanov ◽  
. .
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Periodic Point ◽  
Metric Space ◽  
Unique Fixed Point ◽  
Best Proximity Point ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Space Setting

Consider a metric space  and the non empty sub sets, of X. A map called p-cyclic orbital Geraghty type of contraction is introduced.  Convergence of a unique fixed point and a best proximity point for this map is obtained in a uniformly convex Banach space setting.  Also, this best proximity point is the unique periodic point of such a map.  

Asymptotic centres of filters on uniformly rotund spaces

1976 ◽  
Vol 15 (1) ◽  
pp. 87-96
Author(s):  
John Staples
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Convex Banach Space ◽  
Bounded Sequence ◽  
Uniform Convexity ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Filter Base

The notion of asymptotic centre of a bounded sequence of points in a uniformly convex Banach space was introduced by Edelstein in order to prove, in a quasi-constructive way, fixed point theorems for nonexpansive and similar maps.Similar theorems have also been proved by, for example, adding a compactness hypothesis to the restrictions on the domain of the maps. In such proofs, which are generally less constructive, it may be possible to weaken the uniform convexity hypothesis.In this paper Edelstein's technique is extended by defining a notion of asymptotic centre for an arbitrary set of nonempty bounded subsets of a metric space. It is shown that when the metric space is uniformly rotund and complete, and when the set of bounded subsets is a filter base, this filter base has a unique asymptotic centre. This fact is used to derive, in a uniform way, several fixed point theorems for nonexpansive and similar maps, both single-valued and many-valued.Though related to known results, each of the fixed point theorems proved is either stronger than the corresponding known result, or has a compactness hypothesis replaced by the assumption of uniform convexity.

scholarly journals Approximating fixed points of nonexpansive type mappings

2001 ◽  
Vol 26 (3) ◽  
pp. 183-188
Author(s):  
Hemant K. Pathak ◽  
Mohammad S. Khan
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Unique Fixed Point ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex

In a uniformly convex Banach space, the convergence of Ishikawa iterates to a unique fixed point is proved for nonexpansive type mappings under certain conditions.

scholarly journals Fixed points of mappings satisfying semicontractivity conditions

1992 ◽  
Vol 53 (1) ◽  
pp. 25-38
Author(s):  
Jürgen Schu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Convex Subset ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Opial’S Condition ◽  
Asymptotically Nonexpansive

AbstractLet A be a subset of a Banach space E. A mapping T: A →A is called asymptoically semicontractive if there exists a mapping S: A×A→A and a sequence (kn) in [1, ∞] such that Tx=S(x, x) for all x ∈A while for each fixed x ∈A, S(., x) is asymptotically nonexpansive with sequence (kn) and S(x,.) is strongly compact. Among other things, it is proved that each asymptotically semicontractive self-mpping T of a closed bounded and convex subset A of a uniformly convex Banach space E which satisfies Opial's condition has a fixed point in A, provided s has a certain asymptoticregurity property.

scholarly journals Approximating fixed points by ishikawa iterates

1989 ◽  
Vol 40 (1) ◽  
pp. 113-117 ◽  
Author(s):  
M. Maiti ◽  
M.K. Ghosh
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex

In a uniformly convex Banach space the convergence of Ishikawa iterates to a fixed point is discussed for nonexpansive and generalised nonexpansive mappings.

scholarly journals Coarse infinite-dimensionality of hyperspaces of finite subsets

2021 ◽  
Author(s):  
Thomas Weighill ◽  
Takamitsu Yamauchi ◽  
Nicolò Zava
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Dimensional Properties ◽  
Bounded Geometry ◽  
Infinite Dimensional

AbstractWe consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property.

scholarly journals Monotone asymptotic pointwise contractions

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3291-3294 ◽  
Author(s):  
Dehaish Bin ◽  
Mohamed Khamsi
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Partial Order ◽  
Metric Space ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Monotone Mappings ◽  
Uniformly Convex ◽  
Convex Metric Space ◽  
Uniformly Convex Metric Space

In this work, we extend the fixed point result of Kirk and Xu for asymptotic pointwise nonexpansive mappings in a uniformly convex Banach space to monotone mappings defined in a hyperbolic uniformly convex metric space endowed with a partial order.

scholarly journals Best proximity point theorems for weak cyclic Kannan contractions

Filomat ◽  
2011 ◽  
Vol 25 (1) ◽  
pp. 145-154 ◽  
Author(s):  
Ancuţa Petric
Keyword(s):  
Banach Space ◽  
Best Proximity Point ◽  
Convex Banach Space ◽  
Existence Results ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Best Proximity Points ◽  
Kannan Contraction

In this paper we introduce the notion of weak cyclic Kannan contraction. We give some convergence and existence results for best proximity points for weak cyclic Kannan contractions in the setting of a uniformly convex Banach space.

scholarly journals Duality Fixed Point and Zero Point Theorems and Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Qingqing Cheng ◽  
Yongfu Su ◽  
Jingling Zhang
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Monotone Mapping ◽  
Zero Point ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Lipschitz Mapping ◽  
Uniformly Convex ◽  
Uniformly Smooth ◽  
Inverse Strongly Monotone

The following main results have been given. (1) LetEbe ap-uniformly convex Banach space and letT:E→E*be a(p-1)-L-Lipschitz mapping with condition0<(pL/c2)1/(p-1)<1. ThenThas a unique generalized duality fixed pointx*∈Eand (2) letEbe ap-uniformly convex Banach space and letT:E→E*be aq-α-inverse strongly monotone mapping with conditions1/p+1/q=1,0<(q/(q-1)c2)q-1<α. ThenThas a unique generalized duality fixed pointx*∈E. (3) LetEbe a2-uniformly smooth and uniformly convex Banach space with uniformly convex constantcand uniformly smooth constantband letT:E→E*be aL-lipschitz mapping with condition0<2b/c2<1. ThenThas a unique zero pointx*. These main results can be used for solving the relative variational inequalities and optimal problems and operator equations.

Fixed Point Theorems for Proximately Nonexpansive Semigroups

1986 ◽  
Vol 29 (2) ◽  
pp. 160-166
Author(s):  
Mo Tak Kiang ◽  
Kok-Keong Tan
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Common Fixed Point ◽  
Commutative Semigroup ◽  
Fixed Point Theorems ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Asymptotic Center ◽  
Nonexpansive Semigroups

AbstractA commutative semigroup G of continuous, selfmappings on (X, d) is called proximately nonexpansive on X if for every x in X and every (β > 0, there is a member g in G such that d(fg(x),fg(y)) ≤ (1 + β) d (x, y) for every f in G and y in X. For a uniformly convex Banach space it is shown that if G is a commutative semigroup of continuous selfmappings on X which is proximately nonexpansive, then a common fixed point exists if there is an x0 in X such that its orbit G(x0) is bounded. Furthermore, the asymptotic center of G(x0) is such a common fixed point.

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