Fixed points for some non-obviously contractive operators

The use of K-normed spaces gives us the possibility to prove that a fixed point theorem due to B. Luo is equivalent to the Banach Contraction Principle. This confirms the conspiracy among fixed point theorems. Moreover the theorem of Lou is improved and extended to different contexts. A counterexample about the fixed points of the sum of a contraction and an integral operator is given. The usefulness of K-norm is tested on a Volterra integral equation as well.

     

Fixed points of discontinuous multivalued increasing operators in Banach spaces with applications

By using relatively weakly compactness conditions, we obtain existence theorems of fixed points for discontinuous multivalued increasing operators in Banach spaces. As an application, we utilize the results obtained in Section 2 to lead to the existence principles for integral inclusions of Hammerstein type multivalued maps.     

Fixed points of generalized e-concave (generalized e-convex) operators and their applications

In this paper, we present the definitions of generalized e-concave operators and generalized e-convex operators, which are the generalizations of e-concave operators and e-convex operators, respectively. Without compactness or continuity assumption of generalized e-concave operators and generalized e-convex operators, we have proved the existence, uniqueness and monotone iterative techniques of their fixed points. Our results are even new to e-concave operators and e-convex operators. Finally, we apply the results to the singular boundary value problems for second order differential equations.     

Fixed points for mixed monotone multivalued operators in Banach spaces with applications

In this paper, the existence and iterative approximation of fixed points for a class of systems of mixed monotone multivalued operator are discussed. We present some new fixed point theorems of mixed monotone operators and increasing operators which need not be continuous or satisfy a compactness condition. We also give some applications to differential inclusions with discontinuous right hand side in Banach spaces and to Hammerstein integral inclusions on RN.     

Lamperti-type operators on a weighted space of continuous functions

For a locally convex Hausdorff topological vector space and for a system of weights vanishing at infinity on a locally compact Hausdorff space , let be the weighted space of -valued continuous functions on with the locally convex topology derived from the seminorms which are weighted analogues of the supremum norm. A characterization of the orthogonality preserving (Lamperti-type) operators on is presented in this paper.

     

Fixed points of non-expansive mappings associated with invariant means in a Banach space

In this paper, we study the fixed point set of the non-expansive mapping _Tμ$T_{\mu }$ for a Banach space with uniformly Gâteaux differentiable norm when μ$\mu$ is a multiplicative left invariant mean on l^∞(S)$l^{\infty }\left(S\right)$. As an application, we establish nonlinear ergodic properties for an extremely amenable semigroup of non-expansive mappings in a Banach space with uniformly Gâteaux differentiable norm. Furthermore, we improve a recent result of Atsushiba and Takahashi [S. Atsushiba, W. Takahashi, Weak and strong convergence theorems for non-expansive semigroups in a Banach spaces satisfying Opial’s condition, Sci. Math. Jpn. (in press)] on the fixed point set of non-expansive mappings associated with a left invariant mean on a left amenable semigroup.     

On convergence of a sequence in complete metric spaces and its applications to some iterates of quasi-nonexpansive mappings

The purpose of this paper is to establish some theorems on convergence of a sequence in complete metric spaces. As applications, some results of Ghosh and Debnath [J. Math. Anal. Appl. 207 (1997) 96-103], Kirk [Ann. Univ. Mariae Curie-Sk?odowska Sect. A LI.2, 15 (1997) 167-178] and Petryshyn and Williamson [J. Math. Anal. Appl. 43 (1973) 459-497] are obtained from our results as special cases. Also, we give comments on some results in [J. Math. Anal. Appl. 207 (1997) 96-103, J. Math. Anal. Appl. 43 (1973) 459-497]. Some examples are introduced to support our comments.     

Fixed points for convex continuous mappings in topological vector spaces

We prove the following result. Let be a convex compact subset in a topological vector space, and a convex continuous mapping. (See Definition 1.1.) Then has a fixed point. Moreover, continuous mappings that can be approximated by convex continuous mappings also have the fixed point property.

     

Hybrid fixed points of multivalued operators in metric spaces with applications

This paper deals with the existence of hybrid fixed points involving two multivalued operators in a complete metric space. Our claim is also illustrated with applications to a class of integral inclusions for proving the existence results.     

Fixed points for fuzzy mappings

In this paper the problem of the existence and computation of fixed points for fuzzy mappings is approached. A fuzzy mapping R over a set X is defined to be a function attaching to each x in X a fuzzy subset R_χ of X. An element x of X is called fixed point of R iff its membership degree to R_χ is at least equal to the membership degree to R_χ of any y?X, i.e. R_χ(χ)? R_χ(y)(?y?X). Two existence theorems for fixed points of a fuzzy mapping are proved and an algorithm for computing approximations of such a fixed point is described. The convergence theorem of our algorithm is proved under the restrictive assumption that for any x in X, the membership function of R_χ has a ‘complementary function’. Examples of fuzzy mappings having this property are given, but the problem of proving general criteria for a function to have a complementary remain open.     

非Lipschitz渐近伪压缩映象不动点的迭代逼近

本文在任意实的Banach空间中研究了用具误差的修正的Ishikawa与Mann迭代程序来逼近非Lipschitz的渐近伪压缩映象不动点的强收敛性问题;在去掉条件"‖Tnxn-xn‖→0(n→∞)"及"{xn}有界"之下,证明了相关文献的结果仍然成立.所得结果改进和推广了最近一些人的最新结果.     

Fixed points of meromorphic functions and of their differences,divided differences and shifts

Let f(z) be a finite order meromorphic function and let c∈C\{0} be a constant.If f(z)has a Borel exceptional value a∈C,it is proved that max{τ(f(z)),τ(△_cf(z))}=max{τ(f(z)),τ(f(z+c))}=max{τ(△_cf(z)),τ(f(z+c))}=σ(f(z)).If f(z) has a Borel exceptional value b∈(C\{0})∪{∞},it is proved that max{τ(f(z)),τ(△cf(z)/f(z))}=max{τ(△cf(z)/f(z)),τ(f(z+c))}=σ(f(z)) unless f(z) takes a special form.Here τ(g(z)) denotes the exponent of convergence of fixed points of the meromorphic function g(z),and σ(g(z)) denotes the order of growth of g(z).     

Ergodic theorems for hybrid sequences in a Hilbert space with applications

In this paper, we introduce the notion of generalized hybrid sequences, extending the notion of nonexpansive sequences introduced and studied in our previous work Djafari Rouhani (1981, 1990, 1990, 1997, 2002, 2004, 2002)  , , , , ,  and , and prove ergodic and convergence theorems for such sequences in a Hilbert space H$H$. Subsequently, we apply our results to prove new fixed point theorems for generalized hybrid mappings, first introduced in Kocourek et al. (2010)  [14], Takahashi and Takeuchi (2011)  [20], defined on arbitrary nonempty subsets of H$H$.     

Fixed points of the bucket handle

If a homeomorphism on the bucket handle has an invariant composant, it has a fixed point in that composant. It follows that a homeomorphism on the bucket handle has at least two fixed points. Our methods apply to general Knaster continua.

     

Fixed points of dual quantum operations

Let ?A be a normal completely positive map on B(H) with Kraus operators . Denote M the subset of normal completely positive maps by . In this note, the relations between the fixed points of ?A and are investigated. We obtain that , where K(H) is the set of all compact operators on H and is the dual of ?A∈M. In addition, we show that the map is a bijection on M.