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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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.

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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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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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].

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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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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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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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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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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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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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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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**.

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.

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.

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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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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. (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.

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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