# The fixed points of f and g on C

## Top PDF The fixed points of f and g on C:

### Results on common fixed points

≤ ϕ [i/k] δ(X) → 0 as i → ∞ . (2.8) That is, u = v. Therefore f and g have a unique common ﬁxed point. Note that f hu = hf u = hu = hgu = ghu for all h ∈ C fC g . By the uniqueness of common ﬁxed point of f and g, we have hu = u for all h ∈ C fC g . Since f , gC fC g , it follows that u is a unique common ﬁxed point of C fC g .

### Common fixed points of g-quasicontractions and related mappings in 0-complete partial metric spaces

Example  Consider the set X = {a, b, c} and the function p : X × X → R given by p(a, b) = p(b, c) = , p(a, c) =   , p(x, y) = p(y, x), p(a, a) = p(c, c) =   and p(b, b) = . Obviously, p is a partial metric on X, not being a metric (since p(x, x) =  for x = a and x = c). Deﬁne a selfmap f on X by

### $$C^{*}$$ Valued G contractions and fixed points

Motivated by the work of Jachymski, in this paper we extend and improve the result of Ma et al. by proving a ﬁxed point theorem for self-mappings on C ∗ -valued metric spaces satisfying the contractive condition for those pairs of elements from the metric space which form edges of a graph in the metric space. Our result generalizes and extends the main result of Jachymski and Ma et al. We also establish some examples to elaborate our new notions and to substantiate our result.

### 12. Common Fixed Points for Mappings Satisfying $\varphi$ and $F$-maps in $G$-Cone Metric Spaces

Abstract. The existence of points of coincidence and common fixed points for three self mappings satisfying generalized contractive conditions related to φ and F -maps in a G-cone metric space is proved. Our results extend and generalize several well-known comparable results in the existing literature.

### 9. Coincidence and fixed points in $G$-metric spaces

Abstract. The intent of this paper is to extend the notions of occasionally weakly compatible mappings, subcompatibility and subsequential continuity in framework of generalized metric spaces and prove some common fixed point theorems. We give some examples which demonstrate the validity of the hy- potheses and degree of generality of our results. Our results are independent of the continuity requirement of the involved mappings and completeness (or closedness) of the underlying space (or subspaces). Several known results are generalized in this note.

### Fixed points for G-contractions on uniform spaces endowed with a graph

In , the concepts of E -distance and S-completeness were introduced for uniform spaces in []. Recently in , Jachymski [] proved some ﬁxed point results in met- ric spaces endowed with a graph and generalized simultaneously the Banach contrac- tion principle from metric and partially ordered metric spaces. In , Bojor [] intro- duced (G, ϕ)-contractions and generalized Jachymski’s results. Finally, Nicolae et al. [] presented some ﬁxed point results for a new type of contractions using orbits and also for G-asymptotic contractions in metric spaces endowed with a graph.

### Nonexpansive mappings on complex C*-algebras and their fixed points

A normed space X is said to have the fixed point property, if for each nonexpansive mapping T : E −→ E on a nonempty bounded closed convex subset E of X has a fixed point. In this paper, we first show that if X is a locally compact Hausdorff space then the following are equivalent: (i) X is infinite set, (ii) C 0 (X) is infinite dimensional, (iii) C 0 (X) does not have the fixed point property.

### Fixed points of conditionally F-contractions in complete metric-like spaces

Recently, Wardowski [] introduced the notion of a F -contraction mapping and investi- gated the existence of ﬁxed points for such mappings. The results of Wardowski [] extend and unify several ﬁxed point results in the literature including the celebrated Banach con- traction mapping principle.

### Common coupled coincidence and coupled fixed points in $G$-metric spaces

Copyright 2012 c ⃝Wasfi Shatanawi, Mujahid Abbas, Hassen Aydi and Nedal Tahat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis- tribution, and reproduction in any medium, provided the original work is properly cited. Abstract

### Common fixed points of G-nonexpansive mappings on Banach spaces with a graph

In , Banach proved a remarkable and powerful result called the Banach contraction principle. Because of its fruitful applications, the principle has been generalized in many directions. The recent version of the theorem was given in Banach spaces endowed with a graph. In , Jachymski [] gave a generalization of the Banach contraction principle to mappings on a metric space endowed with a graph. In , Aleomraninejad et al. [] presented some iterative scheme results for G-contractive and G-nonexpansive mappings on graphs. In , Alfuraidan and Khamsi [] deﬁned the concept of G-monotone non- expansive multivalued mappings deﬁned on a metric space with a graph. In the same year, Alfuraidan [] gave a new deﬁnition of the G-contraction and obtained suﬃcient condi- tions for the existence of ﬁxed points for multivalued mappings on a metric space with a graph, and also in [], he proved the existence of a ﬁxed point of monotone nonexpansive mapping deﬁned in a Banach space endowed with a graph. Recently, Tiammee et al. [] proved Browder’s convergence theorem for G-nonexpansive mapping in a Banach space with a directed graph. They also proved the strong convergence of the Halpern iteration for a G-nonexpansive mapping.

### FIXED POINTS FOR PAIRS OF

Call for Papers Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.

### On characterizations of fixed points

Abstract. We give some necessary and suﬃcient conditions for the existence of ﬁxed points of a family of self mappings of a metric space and we establish an equivalent condi- tion for the existence of ﬁxed points of a continuous compact mapping of a metric space. 2000 Mathematics Subject Classiﬁcation. 54H25.

### Fixed Points and Mappings

In this paper we give some important types of mappings related to the fixed point concept. We begin with the basic definition of mappings. Then we define self maps and commutative maps. We discuss about the existence of the fixed points of such mappings with examples. The main part of this research article deals mainly with the common fixed points of a class of polynomial functions. The polynomials considered here are self compositions of a given polynomial of degree n. We prove that if a polynomial and its first composition with itself have an identical set of fixed points, then the polynomial and its n th composition with itself also have an identical set of

### Common fixed points of ordered g-contractions in partially ordered metric spaces

The concept of ordered g-contraction is introduced, and some ﬁxed and common ﬁxed point theorems for g-nondecreasing ordered g-contraction mapping in partially ordered metric spaces are proved. We also show the uniqueness of the common ﬁxed point in the case of an ordered g-contraction mapping. The theorems presented are generalizations of very recent ﬁxed point theorems due to Golubovi´c et al. (Fixed Point Theory Appl. 2012:20, 2012).

### Demiclodeness and fixed points of g-asymptotically nonexpansive mapping in Banach spaces with graph

Abstract. Let C be a nonempty closed convex subset of a uniformly convex Banach space endowed with a tran- sitive directed graph G = (V (G),E (G)), such that V (G) = C and E (G) is convex. We introduce the definition of G-asymptotically nonexpansive self-mapping on C. It is shown that such mappings are G-demiclosed. Finally, we prove the weak and strong convergence of a sequence generated by a modified Noor iterative process to a com- mon fixed point of a finite family of G-asymptotically nonexpansive self-mappings defined on C with nonempty common fixed points set. Our results improve and generalize several recent results in the literature.

### Common Fixed Points for Generalized Weakly Contractive Mappings in G-Metric Spaces

Metric fixed point theory plays an important role in mathematics and applied sciences. Some generalizations of the usual notion of a metric space have been proposed by several authors. One such generalization is a G -metric space initiated by Mustafa and Sims [11]. Moreover, they presented several interesting and useful facts about G -metric spaces, illustrated with appropriate examples. Thereafter, a series of articles about G -metric spaces have been dedicated to the improvement of fixed point theory. In 1997, Alber and Guerre-Delabriere [2] introduced the concept of weakly contractive mappings in Hilbert spaces and proved some fixed point theorems in this setting. Rhoades [16] showed that most of the results claimed by Alber and Guerre-Delabriere [2] are also valid for any metric spaces. In a very recent paper [1], the author established some fixed point theorems for mappings satisfying generalized weakly contractive conditions in G -metric spaces. In this work, we obtain sufficient conditions for existence of points of coincidence and common fixed points for a pair of self mappings in G -metric spaces under weakly contractive conditions related to altering distance functions. Our results generalize and extend some results of [1], [4], [12], [13], [16], [20]. Finally, some examples are presented to illustrate our results.

### Behaviour of fixed and critical points of the (α, c) −family of iterative methods

Received: date / Accepted: date Abstract In this paper we study the dynamical behavior of the (α, c)-family of iterative methods for solving nonlinear equations, when we apply the fixed point operator associated to this family on quadratic polynomials. This is a family of third-order iterative root-finding methods depending on two param- eters; so, as we show throughout this paper, its dynamics is really interesting, but complicated.

### FIXED POINTS AND COMMON FIXED POINTS FOR FUNDAMENTALLY NONEXPANSIVE MAPPINGS ON BANACH SPACES

MOHAMMAD MOOSAEI Abstract. In this paper, we present some fixed point theorems for fundamentally nonexpansive mappings in Banach spaces and give one common fixed point theorem for a commutative family of demiclosed fundamentally nonexpansive mappings on a nonempty weakly compact convex subset of a strictly convex Banach space with the Opial condition and a uniformly convex in every direction Banach space, respectively; moreover, we show that the common fixed points set of such a family of mappings is closed and convex.

### Vector-valued metrics, fixed points and coupled fixed points for nonlinear operators

In the study of the ﬁxed points for an operator, it is sometimes useful to consider a more general concept, namely coupled ﬁxed points. The concept of coupled ﬁxed point for nonlinear operators was introduced and studied by Opoitsev (see [–]) and then, in , by Guo and Lakshmikantham (see []) in connection with coupled quasisolutions of an initial value problem for ordinary diﬀerential equations. Later, a new research direction for the theory of coupled ﬁxed points in ordered metric spaces was initiated by Gnana Bhaskar and Lakshmikantham in [] and by Lakshmikantham and Ćirić in []. Their approach is based on some contractive type conditions on the operator. For other results on coupled ﬁxed point theory, see [–], etc.