scholarly journals Some fixed point theorems on non-convex sets

2017 ◽  
Vol 18 (2) ◽  
pp. 377
Author(s):  
Mohanasundaram Radhakrishnan ◽  
S. Rajesh ◽  
Sushama Agrawal
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Convex Sets ◽  
Compact Set ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Weakly Compact Set

<span style="color: #000000;">In this paper, we prove that if </span><span style="color: #008000;">$K$</span><span style="color: #000000;"> is a </span><span style="text-decoration: underline; color: #000000;">nonempty</span><span style="color: #000000;"> weakly compact set in a </span><span style="text-decoration: underline; color: #000000;">Banach</span><span style="color: #000000;"> space </span><span style="color: #008000;">$X$</span><span style="color: #000000;">, </span><span style="color: #008000;">$T:K\to K$</span><span style="color: #000000;"> is a </span><span style="text-decoration: underline; color: #000000;">nonexpansive</span><span style="color: #000000;"> map satisfying </span><span style="color: #008000;">$\frac{x+Tx}{2}\in K$</span><span style="color: #000000;"> for all </span><span style="color: #008000;">$x\in K$</span><span style="color: #000000;"> and if </span><span style="color: #008000;">$X$</span><span style="color: #000000;"> is </span><span style="color: #008000;">$3-$</span><span style="color: #000000;">uniformly convex or </span><span style="color: #008000;">$X$</span><span style="color: #000000;"> has the </span><span style="text-decoration: underline; color: #000000;">Opial</span><span style="color: #000000;"> property, then </span><span style="color: #008000;">$T$</span><span style="color: #000000;"> has a fixed point in </span><span style="color: #008000;">$K.$ <br /></span>

scholarly journals Fixed point theorems for nonexpansive mappings on nonconvex sets in UCED Banach spaces

2002 ◽  
Vol 31 (4) ◽  
pp. 251-257 ◽  
Author(s):  
Wei-Shih Du ◽  
Young-Ye Huang ◽  
Chi-Lin Yen
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Compact Convex ◽  
Nonexpansive Mappings ◽  
Finite Union ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Nonconvex Sets ◽  
Firmly Nonexpansive ◽  
Asymptotically Regular

It is shown that every asymptotically regular orλ-firmly nonexpansive mappingT:C→Chas a fixed point wheneverCis a finite union of nonempty weakly compact convex subsets of a Banach spaceXwhich is uniformly convex in every direction. Furthermore, if{T i}i∈Iis any compatible family of strongly nonexpansive self-mappings on such aCand the graphs ofT i,i∈I, have a nonempty intersection, thenT i,i∈I, have a common fixed point inC.

Common Fixed Points in Modular Spaces

2018 ◽  
pp. 502 ◽  
Author(s):  
Salwa Salman Abed ◽  
Karrar Emad Abdul Sada
Keyword(s):  
Fixed Point ◽  
Weak Convergence ◽  
Fixed Point Theorems ◽  
Active Role ◽  
Common Fixed Points ◽  
Weakly Compact ◽  
Expansive Mapping ◽  
Modular Spaces ◽  
Opial’S Property ◽  
Common Fixed Point Theorems

     In this paper,there are   new considerations about the dual of a modular spaces and weak convergence. Two common fixed point theorems for a -non-expansive mapping defined on a star-shaped weakly compact subset are proved,  Here the conditions of affineness, demi-closedness and Opial's property play an active role in the proving our results.  

scholarly journals Some related fixed point theorems for multivalued mappings on two metric spaces

2020 ◽  
Vol 12 (2) ◽  
pp. 392-400
Author(s):  
Ö. Biçer ◽  
M. Olgun ◽  
T. Alyildiz ◽  
I. Altun
Keyword(s):  
Fixed Point ◽  
Classical Result ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Multivalued Mappings ◽  
Compact Set ◽  
Complete Metric Spaces ◽  
Bounded Set ◽  
Contraction Type ◽  
Definition Of

The definition of related mappings was introduced by Fisher in 1981. He proved some theorems about the existence of fixed points of single valued mappings defined on two complete metric spaces and relations between these mappings. In this paper, we present some related fixed point results for multivalued mappings on two complete metric spaces. First we give a classical result which is an extension of the main result of Fisher to the multivalued case. Then considering the recent technique of Wardowski, we provide two related fixed point results for both compact set valued and closed bounded set valued mappings via $F$-contraction type conditions.

scholarly journals A New Version of Schauder and Petryshyn Type Fixed Point Theorems in S-Modular Function Spaces

Symmetry ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 15
Author(s):  
Maryam Ramezani ◽  
Hamid Baghani ◽  
Ozgur Ege ◽  
Manuel De la Sen
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Fixed Point Theorems ◽  
Nonlinear Integral Equation ◽  
Convex Sets ◽  
Nonexpansive Mappings ◽  
Modular Function ◽  
Modular Space ◽  
Ordinate Axis ◽  
Modular Function Spaces

In this paper, using the conditions of Taleb-Hanebaly’s theorem in a modular space where the modular is s-convex and symmetric with respect to the ordinate axis, we prove a new generalized modular version of the Schauder and Petryshyn fixed point theorems for nonexpansive mappings in s-convex sets. Our results can be applied to a nonlinear integral equation in Musielak-Orlicz space L p where 0 < p ≤ 1 and 0 < s ≤ p .

A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi

1976 ◽  
Vol 19 (1) ◽  
pp. 7-12 ◽  
Author(s):  
Joseph Bogin
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Compact Convex ◽  
Continuous Mapping ◽  
Normal Structure ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Image Position

In [7], Goebel, Kirk and Shimi proved the following:Theorem. Let X be a uniformly convex Banach space, K a nonempty bounded closed and convex subset of X, and F:K→K a continuous mapping satisfying for each x, y∈K:(1)where ai≥0 and Then F has a fixed point in K.In this paper we shall prove that this theorem remains true in any Banach space X, provided that K is a nonempty, weakly compact convex subset of X and has normal structure (see Definition 1 below).

scholarly journals Convergence of Ishikawa iterates of generalized nonexpansive mappings

1997 ◽  
Vol 20 (3) ◽  
pp. 517-520 ◽  
Author(s):  
M. K. Ghosh ◽  
L. Debnath
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
Uniformly Convex ◽  
Strictly Convex ◽  
Strictly Convex Banach Spaces

This paper is concerned with the convergence of Ishikawa iterates of generalized nonexpansive mappings in both uniformly convex and strictly convex Banach spaces. Several fixed point theorems are discussed.

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.

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