A fixed point theorem for weakly Zamfirescu mappings

In [5], Zamfirescu (1972) gave a fixed point theorem that generalizes the classical fixed point theorems by Banach, Kannan, and Chatterjea. In this paper, we follow the ideas of Dugundji and Granas to extend Zamfirescu’s fixed point theorem to the class of weakly Zamfirescu maps. A continuation method for this class of maps is also given.     

Reich-type iterated function systems

In this paper, we introduce the concept of Reich-type iterated function system and prove the existence and uniqueness of the attractor of such a system. Moreover, we study the properties of the canonical projection from the code space onto the attractor of such a system. We also present an iterated function system consisting of continuous Reich contractions having more than one attractor.     

SOME NEW ITERATED FUNCTION SYSTEMS CONSISTING OF GENERALIZED CONTRACTIVE MAPPINGS

Iterated function systems (IFS) were introduced by Hutchinson in 1981 as a natural generalization of the well-known Banach contraction principle.In 2010,D.R.Sahu and A.Chakraborty introduced K-Iterated Function System using Kannan mapping which would cover a larger range of mappings.In this paper,following Hutchinson,D.R.Sahu and A.Chakraborty,we present some new iterated function systems by using the so-called generalized contractive mappings,which will also cover a large range of mappings.Our purpose is to prove the existence and uniqueness of attractors for such class of iterated function systems by virtue of a Banach-like fixed point theorem concerning generalized contractive mappings.     

Fixed Point Theorems of the Iterated Function Systems

In this paper, we present some fixed point theorems of iterated function systems consisting of α-ψ-contractive type mappings in Fractal space constituted by the compact subset of metric space and iterated function systems consisting of Banach contractive mappings in Fractal space constituted by the compact subset of generalized metric space, which is also extensively applied in topological dynamic system.     

A new class of contractive mappings

We introduce a new class of Picard operators which includes the class of enriched contractions, enriched Kannan mappings, enriched Chatterjea mappings, and prove some fixed point theorems for these mappings. Some examples will illustrate the generality of our results.

     

On planar zero-diagonal and zero-trace iterated maps

We call an iterated map zero-diagonal, if it has a zero-diagonal Jacobi matrix for all x,y. Similarly, zero-trace iterated maps are the maps with zero-trace Jacobi matrix. In this paper, we present some of the geometric and algebraic properties of zero-diagonal planar maps. However, the main focus of this paper is the analysis of the zero-trace planar maps by linear transforming them to a zero-diagonal ones. Some sufficient conditions for the transformation are obtained. Stability for non-hyperbolic fixed points, two types of codim-2 bifurcations, and the local/global invariant manifolds for zero-diagonal and zero-trace maps are investigated.     

Polygonal quasiconformal maps and grunsky inequalities

There is a deep connection between the Grunsky coefficient inequalities for univalent functions and related extremal quasiconformal maps. In this paper, we develop the technique based on the Grunsky inequalities and apply it to solving a problem concerning polygonal quasiconformal maps. Dedicated to Edger Reich on the occasion of his 75th Birthday.     

Edelstein-Suzuki-type fixed point results in metric and abstract metric spaces

Generalizations of the Edelstein-Suzuki theorem [T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal. TMA 71 (2009), 5313-5317], including versions of the Kannan, Chatterjea and Hardy-Rogers-type fixed point results for compact metric spaces, are proved. Also, abstract metric versions of these results are obtained. Examples are presented to distinguish our results from the existing ones.     

Invariance principles for iterated maps that contract on average

We consider iterated function schemes that contract on average. Using a transfer operator approach, we prove a version of the almost sure invariance principle. This allows the system to be modelled by a Brownian motion, up to some error term. It follows that many classical statistical properties hold for such systems, such as the weak invariance principle and the law of the iterated logarithm.

     

The Existence of the Attractor of Countable Iterated Function Systems

The theory of iterated function systems (IFS) and of infinite iterated function systems consisting of contraction mappings has been studied in the last decades. Some extensions of the spaces and the contractions concern many authors in fractal theory. In this paper there are described some results in that topic concerning the existence and uniqueness of nonempty compact set which is a set ”fixed point” of a countable iterated function system (CIFS). Moreover, some approximations of the attractor of a CIFS by the attractors of the partial IFSs are given.     

无穷迭代函数系统的遍历定理

度量空间的压缩映射的一个集合称为一个迭代函数系统．凝聚迭代函数系统可以被看成无穷迭代函数系统．研究了紧度量空间上的无穷迭代函数系统．利用Banach极限的特性和均匀压缩性,证明了紧度量空间上无穷迭代函数系统的随机迭代算法满足遍历性．于是,凝聚迭代函数系统的随机迭代算法也满足遍历性．     

Stability of flows associated to gradient vector fields and convergence of iterated transport maps

In this paper we address the problem of stability of flows associated to a sequence of vector fields under minimal regularity requirements on the limit vector field, that is supposed to be a gradient. We apply this stability result to show the convergence of iterated compositions of optimal transport maps arising in the implicit time discretization (with respect to the Wasserstein distance) of nonlinear evolution equations of a diffusion type. Finally, we use these convergence results to study the gradient flow of a particular class of polyconvex functionals recently considered by Gangbo, Evans and Savin. We solve some open problems raised in their paper and obtain existence and uniqueness of solutions under weaker regularity requirements and with no upper bound on the jacobian determinant of the initial datum.This work was partially supported by grants of M.I.U.R. and of IMATI-CNR, Pavia, Italy.     

Dissipative differential inclusions,set-valued energy storage and supply rate maps,and stability of discontinuous feedback systems

In this paper, we develop dissipativity notions for dynamical systems with discontinuous vector fields. Specifically, we consider dynamical systems with Lebesgue measurable and locally essentially bounded vector fields characterized by differential inclusions involving Filippov set-valued maps specifying a set of directions for the system velocity and admitting Filippov solutions with absolutely continuous curves. In particular, we introduce a generalized definition of dissipativity for discontinuous dynamical systems in terms of set-valued supply rate maps and set-valued storage maps consisting of locally Lebesgue integrable supply rates and Lipschitz continuous storage functions, respectively. In addition, we introduce the notion of a set-valued available storage map and a set-valued required supply map, and show that if these maps have closed convex images they specialize to single-valued maps corresponding to the smallest available storage and the largest required supply of the differential inclusion, respectively. Furthermore, we show that all system storage functions are bounded from above by the largest required supply and bounded from below by the smallest available storage, and hence, a dissipative differential inclusion can deliver to its surroundings only a fraction of its generalized stored energy and can store only a fraction of the generalized work done to it. Moreover, extended Kalman–Yakubovich–Popov conditions, in terms of the discontinuous system dynamics, characterizing dissipativity via generalized Clarke gradients and locally Lipschitz continuous storage functions are derived. Finally, these results are then used to develop feedback interconnection stability results for discontinuous systems thereby providing a generalization of the small gain and positivity theorems to systems with discontinuous vector fields.     

Critical orbits of holomorphic maps on projective spaces

We study the dynamics of iterated holomorphic maps of a complex projective space onto itself. Relations between the Fatou set and the orbits of critical points are investigated. In particular, results concerning critically finite maps on the Riemann sphere are generalized to higher dimensional case.     

Infinite non-conformal iterated function systems

We consider a class of iterated function systems consisting of a countable infinity of non-conformal contractions, extending both the self-affine limit sets of Lalley and Gatzouras as well as the infinite iterated function systems of Mauldin and Urbański. Natural examples include the sets of points in the plane obtained by taking the binary expansion along the vertical and the continued fraction expansion along the horizontal and deleting certain pairs of digits. We prove that the Hausdorff dimension of the limit set is equal to the supremum of the dimensions of compactly supported ergodic measures, which are given by a Ledrappier and Young type formula. In addition we consider the multifractal analysis of Birkhoff averages for countable families of potentials. We obtain a conditional variational principle for the level sets.     

Chromatic sums of general maps on the sphere and the projective plane

In this paper, we study the chromatic sum functions of rooted general maps on the sphere and the projective plane. The chromatic sum function equations of such maps are obtained. From the chromatic sum equations of such maps, the enumerating function equations of rooted loopless maps, bipartite maps and Eulerian maps are also derived. Moreover, some explicit expressions of enumerating functions are also derived.     

Hausdorff dimension of self-similar sets with overlaps

We provide a simple formula to compute the Hausdorff dimension of the attractor of an overlapping iterated function system of contractive similarities satisfying a certain collection of assumptions. This formula is obtained by associating a non-overlapping infinite iterated function system to an iterated function system satisfying our assumptions and using the results of Moran to compute the Hausdorff dimension of the attractor of this infinite iterated function system, thus showing that the Hausforff dimension of the attractor of this infinite iterated function system agrees with that of the attractor of the original iterated function system. Our methods are applicable to some iterated function systems that do not satisfy the finite type condition recently introduced by Ngai and Wang.      

Extremal maps in Sobolev type inequalities: Some remarks

In this work we make some observations on the existence of extremal maps for sharp L²-Riemannian Sobolev type inequalities as Nash and logarithmic Sobolev ones. Among other results, we prove also that there exist smooth compact Riemannian manifolds with scalar curvature changing signal on which there exist extremal maps.     

Coincidence and fixed point theorems for nonlinear hybrid generalized contractions

In this paper we first prove some coincidence and fixed point theorems for nonlinear hybrid generalized contractions on metric spaces. Secondly, using the concept of an asymptotically regular sequence, we give some fixed point theorems for Kannan type multi-valued mappings on metric spaces. Our main results improve and extend several known results proved by other authors.     

GENERALIZATIONS OF THE SUZUKI AND KANNAN FIXED POINT THEOREMS IN G-CONE METRIC SPACES

In this paper we develop the Banach contraction principle and Kannan fixed point theorem on generalized cone metric spaces.We prove a version of Suzuki and Kannan type generalizations of fixed point theorems in generalized cone metric spaces.