The fixed points of Case 1

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Fixed Points and Mappings

Fixed Points and Mappings

Commuting polynomials are of particular interest in this research area. An entire set of commuting polynomials is a set of polynomials which contains at least one polynomial of each degree and in which any two polynomials commute with each other. H. D. Block and H. P. Thielman [3] gave all the entire sets of commuting polynomials. In this research article we consider a special case of functions namely polynomials and its self compositions. As polynomials are continuous everywhere, we can extend the theorem of J. E. Maxfield and W. J. Mourant [17], which states that, if the fixed points of T ◦ T and T on [0, 1] are necessarily same then so are of T ◦ T ◦ T · · · ◦ T (n times) and T .
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Fixed points via a generalized local commutativity: the compact
case

Fixed points via a generalized local commutativity: the compact case

GERALD F. JUNGCK Received 19 July 2001 and in revised form 6 August 2002 In an earlier paper, the concept of semigroups of self-maps which are nearly com- mutative at a function g : X → X was introduced. We now continue the investi- gation, but with emphasis on the compact case. Fixed-point theorems for such semigroups are obtained in the setting of semimetric and metric spaces.

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

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

The classical Banach contraction principle is a very useful tool in nonlinear analysis with many applications to operatorial equations, fractal theory, optimization theory and other topics. Banach contraction principle was extended for singlevalued contraction on spaces endowed with vector-valued metrics by Perov [] and Perov and Kibenko []. For some other contributions to this topic, we also refer to [–], etc. The case of multivalued con- tractions on spaces endowed with vector-valued metrics is treated in [–], etc.

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Enumeration of fixed points of an involution on β(1, 0)-trees

Enumeration of fixed points of an involution on β(1, 0)-trees

Proof As mentioned above, we provide a sketch of a proof. The proof is based on induction on the size of a tree with the obvious base case, the one node tree going to itself. As for the inductive step, it is enough to show that h turns right-decompositions into (usual) decompositions. More concretely, we would like to prove the prop- erty of h that is shown schematically in Figure 5 (with b ≥ 1). Once this property is proved, the fact that h is an involution follows readily from the definition and the induction hypothesis. The property in Figure 5 is also proved by induction on the size of the tree.
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On common fixed points, periodic points, and recurrent points of continuous functions

On common fixed points, periodic points, and recurrent points of continuous functions

Conjecture 1.1 and raised the possibility of using a Baire-category argument to settle the problem. It is also not difficult to show that commuting continuous self-maps of the intervals must have common recurrent points. However, this is not true in general for the case of metric spaces. In [1], we also constructed an example of a pair of commuting continuous self-maps of a compact metric space with no common periodic points, which also provided an answer to a question raised in [2].

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FIXED POINTS ON T0-QPMS

FIXED POINTS ON T0-QPMS

Our purpose is to generalise some known fixed point results in the context of an ”asym- metric metric space”. Although the proofs follow closely the ones in the classical case, the results require a different type of assumptions. The terminology ”asymmetric metric space” could be confusing in the sense that metrics are symmetric, but it is just to empha- size on the fact that we start from metric spaces and remove the ”symmetry property”. The distance from a point x to a point y may be different from the distance from y to x. Therefore, this could look like an orientation on the space. Theses ”oriented distances” can be useful in practical application. For instance, in a hilly country, it makes a difference whether an auto-mobile climbs from a locality A to a locality B or goes down from B to A, considering the cost of transport.
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Moral Fixed Points and Conceptual Deficiency:

Moral Fixed Points and Conceptual Deficiency:

Given that, typically, error theorists do find such candidate fixed points intuitive, we may wonder whether a more accurate characterization is that they are not actually conceptually deficient, but are meta- conceptually deficient. Perhaps they are disposed to intuitively grasp such propositions for what they are, namely, fixed points but theoretical con- siderations force them to resist acknowledging them as such. If that is the case, error theorists are not conceptually deficient because their moral conceptual mastery is competent enough to reliably grasp moral concep- tual truths. But intuitive grasp is one thing, theoretical acknowledgement another. 11 This leads to a related point.
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Fixed points in countably Hilbert spaces

Fixed points in countably Hilbert spaces

Let K be a nonempty convex subset of a real normed linear space E. For strict contrac- tion self-mappings of K into itself, with a fixed point in K, a well-known iterative method ‘the celebrated Picard method’ has successfully been employed to approximate such fixed points. If, however, the domain of a mapping is a proper subset of E (and this is the case in several applications) and if it maps K into E, this iteration method may not be well de- fined. In this situation, for Hilbert spaces and uniformly convex uniformly smooth Banach spaces, this problem has been overcome by the introduction of the metric projection in the recursion formulas (see, e.g., [–]). The advantage of this is that if K is a nonempty closed convex subset of a Hilbert space H and P K : H → K is the metric projection of H onto K, then P K is nonexpansive. This fact characterizes Hilbert spaces and unfortunately is not available in general Banach spaces.
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Maximum number of fixed points in AND–OR–NOT networks

Maximum number of fixed points in AND–OR–NOT networks

Considering AND-OR-NOT-nets reaching the bounds is interesting. For exam- ple, in the loop-less case, AND-NOT-nets reaching the bounds are symmetric, contains only negative arcs, and a lot of “triangles” that is cycles of length 3. Thus, in the loop-less case, to reach the bound, a lot of negative cycles are necessary, and this is not very intuitive since negative cycles are mostly known to be unfavorable to fixed points. Now, when loops are allowed, AND-NOT- nets reaching the bound have no negative cycles. This shows that the influence of negative cycles on the number of fixed points is subtile, not yet well un- derstood while the influence of positive cycle is rather well understand: the number of fixed points is at most 2 τ p , where τ
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Superattracting fixed points of quasiregular mappings

Superattracting fixed points of quasiregular mappings

1. Introduction 1.1. Background. One of the key features of holomorphic functions with respect to complex dynamics is their behaviour near xed points. There is a complete classication of the possible iterative behaviours that can arise near a xed point based on the value of the derivative at that point, see for example [13]. We say that a xed point z 0 of a holomorphic function f is superattracting if f 0 (z 0 ) = 0 . In this case, it follows from Böttcher's Theorem that there is a conjugation of f to a map z 7→ z d in a neighbourhood of z 0 . That is, there exists a conformal map B dened in a neighbourhood of z 0 such that B(z 0 ) = 0 and B(f (z)) = B(z) d . Hence B(f k (z)) = B(z) d k and from this it can be deduced that
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Extended  Criterion  for  Absence  of  Fixed  Points

Extended Criterion for Absence of Fixed Points

5 Conclusions It was shown that the absence of fixed points criterion works only in case if S-box is considered as a separate function. There are isomorphic repre- sentations of ciphers in which this criterion is not met. The new method of AES description allows to reconsider some of criteria for substitutions from the practical point of view. This may lead to a weakening of the cipher strength.

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Averaging and fixed points in Banach spaces.

Averaging and fixed points in Banach spaces.

Theorem 1.0.8 (Kirk’s Theorem, Version 2). Suppose (X, k·k) is reflexive and has normal structure. Then X has fpp(ne). Note that Kirk’s Theorem is a genuine extension of both Browder’s and G¨ ohde’s results since all uniformly convex spaces are reflexive and have normal structure. There are two key elements to Kirk’s proof, both of which quite naturally generalize the techniques developed by Browder in the uniformly convex setting. The first element of Kirk’s proof is that any nonexpansive mapping T : C → C, with C weakly compact, must have a minimal invariant set ; that is, by Zorn’s Lemma, there must exist a set K ⊆ C for which T (K) ⊆ K and if K 0 ⊆ K also satisfies T (K 0 ) ⊆ K 0 , it must follow that K 0 = K. The second key element of his proof is that, in the presence of normal structure (or weak normal structure, as the case may be), any minimal invariant set must be a singleton; that is, T must have at least one fixed point.
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1. Triple fixed points in ordered uniform spaces

1. Triple fixed points in ordered uniform spaces

Abstract. In this paper, we prove some tripled fixed point theorems for gen- eralized contractive mappings in uniform spaces and apply them to study the existences-uniqueness problem for a class of nonlinear integral equations of with unbounded deviations. We also give some examples to show that our results are effective.

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FIXED POINTS FOR PAIRS OF

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.

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On characterizations of fixed 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.

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

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.

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Fixed Points and Coupled Fixed Points in Hausdorff Intuitionistic L Fuzzy Metric Spaces

Fixed Points and Coupled Fixed Points in Hausdorff Intuitionistic L Fuzzy Metric Spaces

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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Fixed Points in Functional Inequalities

Fixed Points in Functional Inequalities

The stability problem of functional equations originated from a question of Ulam 1 concerning the stability of group homomorphisms. Hyers 2 gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki 3 for additive mappings and by Th. M. Rassias 4 for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias 4 has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by G˘avrut¸a 5 by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias’
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Results on common fixed points

Results on common fixed points

. (2.17) That is, f and g satisfy (1) and (2.11). But f and g have no common fixed point in X. Example 2.13 . Let (X, d), f , g, p, q, m, and n be as in Example 2.12. It is easy to check that the conditions of Theorem 2.5 and Corollary 2.8 are satisfied except for the continuity assumption. However f has no fixed point in X.

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Fixed points in uniform spaces

Fixed points in uniform spaces

In  [], Angelov introduced the notion of -contractions on Hausdorff uniform spaces, which simultaneously generalizes the well-known Banach contractions on metric spaces as well as γ -contractions [] on locally convex spaces, and he proved the existence of their fixed points under various conditions. Later in  [], he also extended the no- tion of -contractions to j-nonexpansive maps and gave some conditions to guarantee the existence of their fixed points. However, there is a minor flaw in his proof of Theorem  [] where the surjectivity of the map j is implicitly used without any prior assumption. Additionally, we observe that such a map j can be naturally replaced by a multi-valued map J to obtain a more general, yet interesting, notion of J-nonexpansiveness. Therefore, in this work, we aim to correct and simplify the proof of Theorem  [] as well as extend the notion of j-nonexpansive maps to J-nonexpansive maps and investigate the existence of their fixed points. Then we introduce J-contractions, a special kind of J-nonexpansive maps, that play the similar role as Banach contractions in yielding the uniqueness of fixed points. With the notion of J-contractions, we are able to recover results on -contractions proved in [] as well as present some new fixed point theorems in which one of them nat- urally leads to a new existence theorem for the solution of a certain integral equation in locally convex spaces. Finally, we prove that, under a mild condition, J-nonexpansive maps are always virtually stable in the sense of [] and hence their fixed point sets are retracts of their convergence sets. An example of a virtually stable J-nonexpansive map whose fixed point set is not convex is also given.
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