On the other hand, Schwarz **lemma** at the boundary is also an active topic in complex analysis, various interesting results have been obtained [–]. Before summarizing these results, it is necessary to give some elementary contents on the boundary ﬁxed **points** []. Let D denote the unit disk in C , H( D , D ) denote the class of holomorphic self-mappings of D , N denote the set of all positive integers. The boundary point ξ ∈ ∂ D is called a ﬁxed point of f ∈ H( D , D ) if

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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**.

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.

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

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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.

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.

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What we are intuitively doing with this labeling is picking a corner the point does not move closer to. We will eventually get a sequence of small 1-2-3 triangles converging to a point. If this point was not a **fixed** point, the continuity of f would suggest that as our triangles approach the point, all the corners would have the same labeling. This will be formalized as follows. First, we need to show that this labeling is in fact a Sperner labeling.

In this paper, we introduce the notion of cyclic R-contraction mapping and then study the existence of ﬁxed **points** for such mappings in the framework of metric spaces. Examples and application are presented to support the main result. Our result unify, complement, and generalize various comparable results in the existing literature.

OF CONTINUOUS FUNCTIONS
ALIASGHAR ALIKHANI-KOOPAEI
Received 15 May 2002 and in revised form 27 November 2002
It is known that two commuting continuous functions on an interval need not have a common ﬁxed point. However, it is not known if such two functions have a common periodic point. We had conjectured that two commuting continuous functions on an interval will typically have disjoint sets of periodic **points**. In this paper, we ﬁrst prove that S is a nowhere dense subset of [0,1] if and only if {f ∈ C([0, 1]) : F m (f )∩S = ∅} is a nowhere dense subset of C([0,1]). We also give some results about the common ﬁxed, periodic, and recurrent **points** of functions.

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Abstract: Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets are introduced by Lotfi.A.Zadeh [27] as an extension of the classical notion of sets. Intuitionistic fuzzy set can be utilized as a proper tool for representing hesitancy concerning both membership and non-membership of an element to a set.
Atanassov [1] introduced and studied the concept of Intuitionistic fuzzy sets. The idea of Hausdorff fuzzy metric space introduced by Rodriguez-Lopez and Romaguera [16]. In this paper, a new concept of intuitionistic fuzzy **fixed** point theorems in Hausdorff intuitionistic L-fuzzy metric spaces is introduced and some properties and theorems about **fixed** **points** in Hausdorff intuitionistic L-fuzzy metric space are discussed.

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2 the sequence {J n x} converges to a **fixed** point y ∗ of J;
3 y ∗ is the unique **fixed** point of J in the set Y {y ∈ X | dJ n 0 x, y < ∞};
4 dy, y ∗ ≤ 1/1 − Ldy, Jy for all y ∈ Y .
In 1996, Isac and Th. M. Rassias 23 were the first to provide applications of stability theory of functional equations for the proof of new **fixed** point theorems with applications. By using **fixed** point methods, the stability problems of several functional equations have been extensively investigated by a number of authors see 24–28.

We improve Angelov’s ﬁxed point theorems of -contractions and j-nonexpansive maps in uniform spaces and investigate their ﬁxed point sets using the concept of virtual stability. Some interesting examples and an application to the **solution** of a certain integral equation in locally convex spaces are also given.

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All the results we prove assumed a kind of continuity for the map T involved. This condition is meant to guarantee that we can pass the limit under the function and keep it. In [4], we will present more general ﬁxed **points** results which do require any type of continuity.

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Abstract
In this note, we establish and improve some results on **fixed** point theory in topological vector spaces. As a generalization of contraction maps, the concept of quasi contraction multivalued maps on a topological vector space will be defined. Further, it is shown that a quasi contraction and closed multivalued map on a topological vector space has a unique **fixed** point if it is bounded value.

[15] I. A. Rus, A. Petrus¸el, and A. Sˆınt˘am˘arian, “Data dependence of the **fixed** point set of some multivalued weakly Picard operators,” Nonlinear Analysis, vol. 52, no. 8, pp. 1947–1959, 2003. Cristian Chifu: Department of Business, Faculty of Business, Babes¸-Bolyai University Cluj-Napoca, Horea 7, 400174 Cluj-Napoca, Romania

In this paper, we introduce three iterative methods for ﬁnding a common element of the set of ﬁxed **points** for a continuous pseudo-contractive mapping and the **solution** set of a variational inequality problem governed by continuous monotone mappings. Strong convergence theorems for the proposed iterative methods are proved. Our results improve and extend some recent results in the literature.

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In this article, we introduce a new iterative scheme for finding a common element of the set of **fixed** **points** for a continuous pseudo-contractive mapping and the **solution** set of a variational inequality problem governed by continuous monotone mappings. Strong convergence for the proposed iterative scheme is proved. Our results improve and extend some recent results in the literature.

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a and b are minimal and maximal **points** of the interval X = [a, b] it follows that x 6= a, b, which means that a and b are extreme **points** of X .
EXAMPLE 1. If a, b, c are noncollinear **points** and X is the solid triangular region consisting of all convex combinations of these vectors, then the extreme **points** of X are a, b, and c. — First of all, this set is convex because **Lemma** 3 implies that a convex combination of convex combinations is again a convex combination. To prove the assertion about extreme **points**, note that if t a + u b + v c is a convex combination in which at least two coefficients are positive, then an argument like the inductive step of **Lemma** 3 implies that this convex combination is between two others, and therefore the only possible extreme **points** are the original vectors. Furthermore, if p = t x + (1 − t) y where x and y are convex combinations and 0 < t < 1, then one can check directly that at least two barycentric coordinates of p must be positive (this is a bit messy but totally elementary). Therefore a point that is not an extreme point cannot be one of a, b, c and hence these must be the extreme **points** of X .

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In recent years, researchers have studied the size of different sets related to the dynamics of self-maps of an interval. In this note we investigate the sets of **fixed** **points** and periodic **points** of continuously differentiable functions and show that typically such functions have a finite set of **fixed** **points** and a countable set of periodic **points**.
1. Introduction and Notation

Many authors have been using the Hausdorﬀ metric to obtain **fixed** point and coincidence point theorems for multimaps on a metric space. In most cases, the metric nature of the Hausdorﬀ metric is not used and the existence part of theorems can be proved without using the concept of Hausdorﬀ metric under much less stringent conditions on maps. The aim of this paper is to illustrate this and to obtain **fixed** point and coincidence point theorems for multimaps with not necessarily bounded images. Incidentally we obtain improvements over the results of Chang [3], Daﬀer et al. [6], Jachymski [9], Mizoguchi and Takahashi [12], and We¸grzyk [17] on the famous conjecture of Reich on multimaps (Conjecture 3.12).

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