The Fixed Point Set of Real Multi-Valued Contraction Mappings

1972 ◽  
Vol 15 (4) ◽  
pp. 507-511 ◽  
Author(s):  
Arthur S. Finbow
Keyword(s):  
Fixed Point ◽  
Real Number ◽  
Contraction Mapping ◽  
Metric Spaces ◽  
Lipschitz Constant ◽  
Hausdorff Metric ◽  
Lipschitz Mapping ◽  
Fixed Point Set ◽  
Point Set ◽  
Image Position

Let (X, d1) and (Y, d2) be metric spaces. A mapping f:X→Y is said to be a Lipschitz mapping if there exists a real number λ such thatfor each x,y∊X. We call λ a Lipschitz constant for f. If λ∊[0, 1), f is called a contraction mapping. Throughout this note CB(Y) denotes the set of closed and bounded subsets of Y equipped with the Hausdorff metric induced by d2.

scholarly journals Fixed Point of Interpolative Rus–Reich–Ćirić Contraction Mapping on Rectangular Quasi-Partial b-Metric Space

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 32
Author(s):  
Pragati Gautam ◽  
Luis Manuel Sánchez Ruiz ◽  
Swapnil Verma
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Real Number ◽  
Metric Space ◽  
Contraction Mapping ◽  
Metric Spaces ◽  
Rectangular Metric Space ◽  
New Type ◽  
Symmetry Requirement ◽  
Definition Of

The purpose of this study is to introduce a new type of extended metric space, i.e., the rectangular quasi-partial b-metric space, which means a relaxation of the symmetry requirement of metric spaces, by including a real number s in the definition of the rectangular metric space defined by Branciari. Here, we obtain a fixed point theorem for interpolative Rus–Reich–Ćirić contraction mappings in the realm of rectangular quasi-partial b-metric spaces. Furthermore, an example is also illustrated to present the applicability of our result.

scholarly journals On convergence theorems for single-valued and multi-valued mappings in p-uniformly convex metric spaces

2021 ◽  
Vol 37 (3) ◽  
pp. 513-527
Author(s):  
JENJIRA PUIWONG ◽  
◽  
SATIT SAEJUNG ◽  
◽  
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Iterative Sequence ◽  
Uniformly Convex ◽  
Fixed Point Set ◽  
Convergence Theorems ◽  
Convex Metric Spaces ◽  
Strict Fixed Point ◽  
Point Set

We prove ∆-convergence and strong convergence theorems of an iterative sequence generated by the Ishikawa’s method to a fixed point of a single-valued quasi-nonexpansive mappings in p-uniformly convex metric spaces without assuming the metric convexity assumption. As a consequence of our single-valued version, we obtain a result for multi-valued mappings by showing that every multi-valued quasi-nonexpansive mapping taking compact values admits a quasi-nonexpansive selection whose fixed-point set of the selection is equal to the strict fixed-point set of the multi-valued mapping. In particular, we immediately obtain all of the convergence theorems of Laokul and Panyanak [Laokul, T.; Panyanak, B. A generalization of the (CN) inequality and its applications. Carpathian J. Math. 36 (2020), no. 1, 81–90] and we show that some of their assumptions are superfluous.

Examples and Questions in the Theory of Fixed Point Sets

1979 ◽  
Vol 31 (5) ◽  
pp. 1017-1032 ◽  
Author(s):  
John R. Martin ◽  
Sam B. Nadler
Keyword(s):  
Fixed Point ◽  
Continuous Function ◽  
Banach Spaces ◽  
Metric Spaces ◽  
Compact Manifolds ◽  
Invariance Property ◽  
Point Sets ◽  
Fixed Point Set ◽  
Point Set ◽  
Fixed Point Sets

All spaces considered in this paper will be metric spaces. A subset A of a space X is called a fixed point set of X if there is a map (i.e., continuous function) ƒ: X → X such that ƒ(x) = x if and only if x ∈ A. In [22] L. E. Ward, Jr. defines a space X to have the complete invariance property (CIP) provided that each of the nonempty closed subsets of X is a fixed point set of X. The problem of determining fixed point sets of spaces has been investigated in [14] through [20] and [22]. Some spaces known to have CIP are n-cells[15], dendrites [20], convex subsets of Banach spaces [22], compact manifolds without boundary [16], and a class of polyhedra which includes all compact triangulable manifolds with or without boundary [18].

scholarly journals Fixed points of Osilike-Berinde-G-nonexpansive mappings in metric spaces endowed with graphs

2021 ◽  
Vol 37 (2) ◽  
pp. 311-323
Author(s):  
A. KAEWKHAO ◽  
C. KLANGPRAPHAN ◽  
B. PANYANAK
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Nonexpansive Mapping ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Fixed Point Set ◽  
Quasinonexpansive Mapping ◽  
Ishikawa Iteration Process ◽  
Point Set

"In this paper, we introduce the notion of Osilike-Berinde-G-nonexpansive mappings in metric spaces and show that every Osilike-Berinde-G-nonexpansive mapping with nonempty fixed point set is a G-quasinonexpansive mapping. We also prove the demiclosed principle and apply it to obtain a fixed point theorem for Osilike-Berinde-G-nonexpansive mappings. Strong and \Delta-convergence theorems of the Ishikawa iteration process for G-quasinonexpansive mappings are also discussed."

scholarly journals On Some New Multivalued Results in the Metric Spaces of Perov’s Type

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 438
Author(s):  
Liliana Guran ◽  
Monica-Felicia Bota ◽  
Asim Naseem ◽  
Zoran D. Mitrović ◽  
Manuel de la Sen ◽  
...  
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Problem ◽  
Data Dependence ◽  
Point Problem ◽  
Contractive Condition ◽  
Well Posedness ◽  
Fixed Point Set ◽  
Generalized Metric Spaces ◽  
Point Set

The purpose of this paper is to present some new fixed point results in the generalized metric spaces of Perov’s sense under a contractive condition of Hardy–Rogers type. The data dependence of the fixed point set, the well-posedness of the fixed point problem and the Ulam–Hyers stability are also studied.

scholarly journals A Mean Ergodic Theorem for Affine Nonexpansive Mappings in Nonpositive Curvature Metric Spaces

2021 ◽  
Vol 29 (1) ◽  
pp. 111-125
Author(s):  
Hadi Khatibzadeh ◽  
Hadi Pouladi
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Ergodic Theorem ◽  
Tauberian Theorem ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Nonpositive Curvature ◽  
Fixed Point Set ◽  
Von Neumann ◽  
Point Set

Abstract In this paper, we consider the orbits of an affine nonexpansive mapping in Hadamard (nonpositive curvature metric) spaces and prove an ergodic theorem for the inductive mean, which extends the von Neumann linear ergodic theorem. The main result shows that the sequence given by the inductive means of iterations of an affine nonexpansive mapping with a nonempty fixed point set converges strongly to a fixed point of the mapping. A Tauberian theorem is also proved in order to ensure convergence of the iterations.

scholarly journals Fixed point results for multivalued hardy-rogers contractions in b-metric spaces

Filomat ◽  
2017 ◽  
Vol 31 (8) ◽  
pp. 2499-2507 ◽  
Author(s):  
Cristian Chifu ◽  
Gabriela Petruşel
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Problem ◽  
Data Dependence ◽  
Point Problem ◽  
Contractive Condition ◽  
Well Posedness ◽  
Fixed Point Set ◽  
Point Set

The purpose of this paper is to present some fixed point results in b-metric spaces using a contractive condition of Hardy-Rogers type with respect to the functional H. The data dependence of the fixed point set, the well-posedness of the fixed point problem, as well as, the Ulam-Hyres stability are also studied.

A Note on an Iterative Test of Edelstein(1)

1972 ◽  
Vol 15 (3) ◽  
pp. 381-386 ◽  
Author(s):  
Sam B. Nadler
Keyword(s):  
Real Number ◽  
Metric Spaces ◽  
Lipschitz Constant ◽  
Strict Inequality ◽  
Lipschitz Mapping

Let (X1, d1) and (X2, d2) be metric spaces. A mapping/: X1→X2 is said to be a Lipschitz mapping (with respect to d1 and d2) if and only if (*)d2(f(x), f(y))≤λ⋅ d1(x, y) for all x, y∈X1, where λ is a fixed real number. The constant λ is called a Lipschitz constant for f. If (*) is satisfied for λ=l, then f is called nonexpansive (see, for example, [21]) and if (*), again with λ=1, is replaced by a strict inequality for all x≠y, then f is called contractive [1]. If x∊X1 and X1=X2, then the sequence where f1(x)=f(x) and fn(x)=f(fn-1(x)) for each n>1, is called the sequence of iterates of f at x.

Limiting cases of Boardman's five halves theorem

2013 ◽  
Vol 56 (3) ◽  
pp. 723-732 ◽  
Author(s):  
Michael C. Crabb ◽  
Pedro L. Q. Pergher
Keyword(s):  
Fixed Point ◽  
Dimensional Manifold ◽  
Fixed Point Set ◽  
Minimal Dimension ◽  
Point Set ◽  
Image Position

AbstractThe famous five halves theorem of Boardman states that, if T: Mm → Mm is a smooth involution defined on a non-bounding closed smooth m-dimensional manifold Mm (m > 1) and ifis the fixed-point set of T, where Fj denotes the union of those components of F having dimension j, then 2m ≤ 5n. If the dimension m is written as m = 5k − c, where k ≥ 1 and 0 ≤ c < 5, the theorem states that the dimension n of the fixed submanifold is at least β(m), where β(m) = 2k if c = 0, 1, 2 and β(m) = 2k − 1 if c = 3, 4. In this paper, we give, for each m > 1, the equivariant cobordism classification of involutions (Mm, T), for which the fixed submanifold F attains the minimal dimension β(m).

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