New characterizations of Hajlasz-Sobolev spaces on metric spaces

This paper introduces the fractional Sobolev spaces on spaces of homogeneous type,includingmetric spaces and fractals. These Sobolev spaces include the well-known Hajfasz-Sobolev spaces as specialmodels.The author establishes varions chaaracterizations of(sharp)maximal functions for these spaces.Asapplications,the author identifies the fractional Sobolev spaces with some Lipscitz-type spaces.Moreover;some embedding theorems are also given.     

New characterizations of Hajłasz-Sobolev spaces on metric spaces

This paper introduces the fractional Sobolev spaces on spaces of homogeneous type, including metric spaces and fractals. These Sobolev spaces include the well-known Hajłasz-Sobolev spaces as special models. The author establishes various characterizations of (sharp) maximal functions for these spaces. As applications, the author identifies the fractional Sobolev spaces with some Lipscitz-type spaces. Moreover, some embedding theorems are also given.     

Halpern’s type iterations with perturbations in Hilbert spaces: equilibrium solutions and fixed points

In this paper, we consider an iteration process of Halpern’s type for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points for a quasi-nonexpansive mapping with perturbation in a Hilbert space and then prove a strong convergence theorem for such iterations. Using this result, we obtain new strong convergence theorems in a Hilbert space. In particular, we solve partially an open problem posed by Kurokawa and Takahashi (Nonlinear Anal 73:1562–1568, 2010) concerning Halpern’s iterations.     

An iterative method of fixed points for closed and quasi-strict pseudo-contractions in Banach spaces

The purpose of this article is to propose a modified hybrid projection algorithm and prove a strong convergence theorem for closed and quasi-strict pseudo-contractions. Its results hold in reflexive, strictly convex and smooth Banach spaces with the property (K). The results of this paper improve and extend the corresponding results of Matsushita and Takahashi (J. Approx. Theory 134:257–266, 2005), Qin and Su (Nonlinear Anal. 67:1958–1965, 2007), Marino and Xu (J. Math. Anal. Appl. 329:336–346, 2007) and others.     

Some new Triebel-Lizorkin spaces on spaces of homogeneous type and their frame characterizations

Let(X,p,μ)d,θ be a space of homogeneous type,(?) ∈(0,θ],|s|＜(?) andmax{d/(d＋(?)),d/(d＋s＋(?))}＜q≤∞.The author introduces the new Triebel-Lizorkin spaces (?)_∞q~s(X) and establishes the framecharacterizations of these spaces by first establishing a Plancherel-P(?)lya-type inequalityrelated to the norm of the spaces (?)_∞q~s(X).The frame characterizations of the Besovspace (?)_pq~s(X) with|s|＜(?),max{d/(d＋(?)),d/(d＋s＋(?))}＜p≤∞ and 0＜q≤∞and the Triebel-Lizorkin space (?)_pq~s(X)with|s|＜(?),max{d/(d＋(?)),d/(d＋s＋(?))}＜p＜∞ and max{d/(d＋(?)),d/(d＋s＋(?))}＜q≤∞ are also presented.Moreover,the au-thor introduces the new TriebeI-Lizorkin spaces b(?)_∞q~s(X) and H(?)_∞q~s(X) associated to agiven para-accretive function b.The relation between the space b(?)_∞q~s(X) and the spaceH(?)_∞q~s(X) is also presented.The author further proves that if s＝0 and q＝2,thenH(?)_∞q~s(X)＝(?)_∞q~s(X),which also gives a new characterization of the space BMO(X),since (?)_∞q~s(X)＝BMO(X).     

Denjoy-Wolff theorems for Hilbert’s and Thompson’s metric spaces

We study the dynamics of fixed point free mappings on the interior of a normal, closed cone in a Banach space that are nonexpansive with respect to Hilbert’s metric or Thompson’s metric. We establish several Denjoy-Wolff type theorems which confirm conjectures by Karlsson and Nussbaum for an important class of nonexpansive mappings. We also extend and put into a broader perspective results by Gaubert and Vigeral concerning the linear escape rate of such nonexpansive mappings.     

Dugundji’s theorem for cone metric spaces

In this work, we will prove the Dugundji extension theorem for the cone metric space. It is heavily reliant on the paracompactness of the cone topology that is proved by Ayse Sönmez in the paper Sönmez (2010) [11].     

Fixed point results for mappings satisfying (ψ φ)-weakly contractive condition in ordered partial metric spaces

In [18], Matthews introduced a new class of metric spaces, that is, the concept of partial metric spaces, or equivalently, weightable quasi-metrics, are investigated to generalize metric spaces (X, d), to develop and to introduce a new fixed point theory. In partial metric spaces, the self-distance for any point need not be equal to zero. In this paper, we study some results for single map satisfying (ψ,φ)-weakly contractive condition in partial metric spaces endowed with partial order. An example is given to support the useability of our results.     

A Weil–Petersson type metric on spaces of metric graphs

In this note, we discuss an analogue of the Weil–Petersson metric for spaces of metric graphs and some of its properties.     

A proof of dilworth’s decomposition theorem for partially ordered sets

A short proof of the following theorem is given: LetP be a finite partially ordered set. If the maximal number of elements in an independent subset ofP isk, thenP is the union ofk chains.     

Existence and approximation of fixed points of multivalued generalized α-nonexpansive mappings in Banach spaces

Numerical Algorithms - We consider the theoretical and numerical aspects of the quadrature rules associated with a sequence of polynomials generated by a special RII recurrence relation. We also...     

Abel?s summation formula in partially ordered groups

Abel?s partial summation formula has been used classically to obtain convergence tests for certain types of series of real or complex numbers. We generalize the formula and convergence tests to the setting of directed partially ordered topological groups, where the notion of a convergent series is replaced by that of a Cauchy multipliable sequence.     

Extraneous fixed points of Euler iteration and corresponding Sullivan’s basin

The phenomenon of “numerical extraneous roots” of Euler’s iteration has been found. By systematic searching, some polynomials and the corresponding initial values are given, which make the fixed points of Euler’s iteration not the roots of the polynomials. For those repelling extraneous fixed points, the adjoint dynamical types of Sullivan’s basins are also studied. Finally, the fractal pictures are produced.     

Hölder metric subregularity for multifunctions in type Banach spaces

Using the Borwein–Preiss variational principle and in terms of the proximal coderivative, we provide a new type of sufficient conditions for the Hölder metric subregularity and Hölder error bounds in a class of smooth Banach spaces. As an application, new characterizations for the tilt stability of Hölder minimizers are established.     

The weakest contractive conditions for Edelstein’s mappings to have a fixed point in complete metric spaces

In this paper, we give the answer to the following problem: Let (X, d) be a complete metric space and let T be a mapping on X satisfying \(d(Tx, Ty) < d(x, y)\) for any \(x, y \in X\) with \(x \ne y\). Then what are the weakest additional assumptions to imply the same conclusion as in the Banach contraction principle?     

New fixed point theorems for $${\alpha}$$-$${H\Theta}$$HΘ-contractions in ordered metric spaces

Recently, Jleli and Samet [J. Inequal. Appl. (2014), 2014:38] introduced and studied a new contraction to prove a generalization of the Banach contraction principle. In this paper, we introduce the concept of \({\alpha}\)-\({H\Theta}\)-contraction with respect to a general family of functions H and we establish Jleli–Samet-type fixed point results in metric and ordered metric spaces. As an application of our results we deduce Suzuki-type fixed point results for \({H\Theta}\)-contractions. We also derive certain fixed and periodic point results for orbitally continuous generalized \({\Theta}\)-contractions. Moreover, we present an illustrative example to highlight the obtained improvements.     

Extreme points and rotundity of Orlicz-Musielak sequence spaces

A.Kaminska and H.Hudzik[1-10] present a series of work concerning geometry ofsequence Orlicz-Musielak spaces. This paper continues their work, to give a character of extremepoints of the unit balls of sequence Orlicz-Musielak spaces epuipped with Luxemburg norm. Fromwhich a criterion of rotundity is obtained immediately.     

Multiplicity of fixed points and growth of ε-neighborhoods of orbits

We study the relationship between the multiplicity of a fixed point of a function g, and the dependence on ε of the length of ε-neighborhood of any orbit of g, tending to the fixed point. The relationship between these two notions was discovered in Elezovi?, ?ubrini? and ?upanovi? (2007) [5] in the differentiable case, and related to the box dimension of the orbit.Here, we generalize these results to non-differentiable cases introducing a new notion of critical Minkowski order. We study the space of functions having a development in a Chebyshev scale and use multiplicity with respect to this space of functions. With the new definition, we recover the relationship between multiplicity of fixed points and the dependence on ε of the length of ε-neighborhoods of orbits in non-differentiable cases.Applications include in particular Poincaré maps near homoclinic loops and hyperbolic 2-cycles, and Abelian integrals. This is a new approach to estimate the cyclicity, by computing the length of the ε-neighborhood of one orbit of the Poincaré map (for example numerically), and by comparing it to the appropriate scale.