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It is known that some particular self-mappings of the closed unit ball Bl2 of l2 with no fixed points cannot be nonexpansive with respect to any renorming of l2. We give here a short proof of this ...

We define a class of nonlinear mappings which is properly larger than the class of nonexpansive mappings. We also give a fixed point theorem for this new class of mappings.

Fractals and multivalued fractals play an important role in biology, quantum mechanics, computer graphics, dynamical systems, astronomy and astrophysics, geophysics, etc. Especially, there are important consequences of the iterated function (or multifunction) systems in several topics of applied sciences [see for example: El Naschie MS. Iterated function systems and the two-slit experiment of quantum mechanics. Chaos, Solitons & Fractals 1994;4:1965–8; Iovane G. Cantorian spacetime and Hilbert space: Part I-Foundations. Chaos, Solitons & Fractals 2006;28:857–78; Iovane G. Cantorian space-time and Hilbert space: Part II-Relevant consequences. Chaos, Solitons & Fractals 2006;29:1–22; Fedeli A. On chaotic set-valued discrete dynamical systems. Chaos, Solitons & Fractals 2005;23:13814; Shi Y, Chen G. Chaos of discrete dynamical systems in complete metric spaces. Chaos, Solitons & Fractals 2004;22:55571]. The purpose of this paper is twofold. First, some existence and uniqueness results for the self-similar sets of a mixed iterated function systems are given. Then, using the concept of well-posed fixed point problem, the well-posedness of the self-similarity problem for some classes of iterated multifunction systems is also studied. Well-posedness is closely related to the approximation of the solution of a fixed point equation, which is an important aspect of the construction of the fractals using the so-called pre-fractals.

We give some sufficient conditions for normal structure in terms of the von Neumann-Jordan constant, the James constant and the weak orthogonality coefficient introduced by B. Sims. In the rest of the paper, the von Neumann-Jordan constant and the James constant for the Bynum space ℓ 2 , ∞ \ell _{2,\infty } are computed, and are used to show that our results are sharp.

It is shown that two well-known uniformly fixed point free lipschitzian semigroups of mappings have minimal Lipschitz constant on the positive part of the unit ball of ℓ 2 \ell _2 . This implies that a question raised by T. Kuczumow has a negative answer.

A closed convex subset of c0 has the fixed point property (fpp) if every nonexpansive self mapping of it has a fixed point. All nonempty weak compact convex subsets of c0 are known to have the fpp. We show that closed convex subsets with a nonempty interior and nonempty convex subsets which are compact in a topology slightly coarser than the weak topology may fail to have the fpp.

A famous open question in metric Fixed Point Theory is whether every Banach space which is isomorphic to the Hilbert space l2 has the fixed point property for nonexpansive mappings. We give an overview about the state of the advances towards their solution. Open image in new window

We give a sufficient condition for normal structure more general than the well known ɛ0(X) < 1. Moreover we obtain sufficient conditions for the fixed point property for some B-convex Banach spaces.

In this paper we present a fixed point theorem for contractive type multivalued operators in cone metric spaces by using the concept of c-distance.