Expansive Mappings Research Articles

Chaotic dynamical systems on symbolic spaces

We try to find chaotic dynamical systems by extending some simple continuous functions defined on ℚp, to functions defined on Aℤ, where A = {0,1,2, , p — 1}. A combination of the powers of the shift map with addition of a constant in ℚp, gives rise to chaotic dynamical systems, conjugate to powers of the shift map. By extending the addition in ℤp, to the whole of Aℤ, and combining with powers of the shift map, we get an expansive map with the same topological entropy as that of powers of the shift. We also obtain two more positively expansive maps with positive entropy.

Some Fixed Point Theorems for Expansive Mappings in Cone Pentagonal Metric Spaces

In this paper, we prove some xed point theorems for mappings satisfying expansive conditions in non-normal cone pentagonal metric spaces. Our results extend and improve the recent results announced by Patil and Salunke [Fixed point theorems for Expansion mappings in Cone rectangular metric spaces, Gen. Math. Notes, 29(1), (2015), 30-39], Shatanawi and Awawdeh, [Some xed and coincidence point theorems for Expansive maps in Cone metric spaces, Fixed Point Theory and Applications, 1(2012), 1-10], Huang, Zhu and Wen, [Fixed point theorems for Expanding mappings in Cone metric spaces, Math. Reports 14(64), 2(2012), 141-148], Kadelburg, Murthy and Radenovic, [Common xed points for Expansive mappings in Cone metric spaces, Int. J. Math. Anal, 5(27), (2011), 1309-1319], Aage and Salunke, [Some xed point theorems for Expansion onto mappings on Cone metric spaces, Acta Mathematica Sinica, 27(6), (2011), 1101-1106], Kumar and Garg, [Common xed points for Expansion mappings Theorems in metric spaces, Int. J. Contemp. Math. Sciences, 4(36), (2009), 1749-1758], and many others in the literature.

Past stability and expansivity for self-maps

In this article we consider Lyapunov stability of orbits of self-maps on metric spaces. The main interest is posed on the relationship between past stability and expansiveness. It is shown that an expansive map with past stable orbits must be expanding. Moreover, when there are not past stable orbits, then every point has a future stable set of large diameter.

Cone $$b_2$$ b 2 -metric spaces over Banach algebra with applications

In the present paper, we first introduce the concept of Cone \(b_2\)-metric space over Banach algebras which generalizes the notions of \(b_2\)-metric space and cone metric spaces over Banach algebra. Next, we define generalized Lipschitz and expansive maps in the new structure and establish the existence and uniqueness of fixed points for such mappings in Cone \(b_2\)-metric space over Banach algebra. The results presented here generalize and extend some recent results of Singh et al. (Comment Math 52(2):143–151, 2012) and Wang et al. (Math Japonica 29:631–636, 1984). Also, we illustrate the result by an appropriate example. Finally, an application to integral equations is given to demonstrate the effectiveness of our acquired results.

Polynomial entropy and expansivity

We study the polynomial entropy of homeomorphisms on compact metric spaces. We construct a homeomorphism on a compact metric space with vanishing polynomial entropy that it is not equicontinuous. Also we give examples with arbitrarily small polynomial entropy. Finally, we show that expansive homeomorphisms and positively expansive maps of compact metric spaces with infinitely many points have polynomial entropy greater than or equal to 1.

Fixed point theorems for expansive mappings in Gp-metric spaces

In the present paper, we adopt the concept of expansive mapping in the context of Gp-metric spaces in a similar manner expansive mapping in metric spaces. Furthermore, we obtain some results on fixed points of expansive type mappings. Also, we prove some common fixed point results for expansive mappings by using the notion of weak compatibility in Gp-metric space. Our results generalize some comparable results in metric spaces and partial metric spaces to Gp-metric spaces. Moreover, some examples are introduced in order to support our new results.

The A-cone metric space over Banach algebra with applications

In the present paper, we introduce the notion of A-cone metric spaces over Banach algebra as a generalization of A-metric spaces and cone metric spaces over Banach algebra. We also defined generalized Lipschitz and expansive maps in such maps and establish some fixed point theorems for such maps in the setting of the new space. As an application, we prove a theorem for integral equation. We provide illustrative example to verify our results. Our results generalize and unify some well-known results in the literature.

Unique Common Fixed Point in b2 Metric Spaces

We establish some common fixed and common coincidence point theorems for expansive type mappings in the setting of b2 metric space. Our results extend some known results in metric spaces to b2metric space. The research is meaningful and I recommend it to be published when the followings have been improved.

Common fixed point theorems for expansive mappings via an implicit relation

Common fixed point theorems for expansive mappings via an implicit relation

F-cone metric spaces over Banach algebra

AbstractThe paper deals with the achievement of introducing the notion of F-cone metric spaces over Banach algebra as a generalization of $N_{p}$ N p -cone metric space over Banach algebra and $N_{b}$ N b -cone metric space over Banach algebra and studying some of its topological properties. Also, here we define generalized Lipschitz and expansive maps for such spaces. Moreover, we investigate some fixed points for mappings satisfying such conditions in the new framework. Subsequently, as an application of our results, we provide an example. Our results generalize some well-known results in the literature.

On common fixed points in generalized Menger spaces

R. Vasuki [1] proved fixed point theorems for expansive mappings in Menger spaces. R. Gujetiya and et al [2] presented an extension of the main result of Vasuki, for four expansive mappings in Menger space. In this article, an important lemma is given to prove that the iteration sequence is Cauchy under suitable condition in Menger probabilistic G-metric space (shortly, MPGM-space). And then, used to obtain three common fixed point theorems for expansive type mappings.

Fixed Point Theorems for Generalized \((\alpha,\psi)\)-Expansive Mappings in Generalized Metric Spaces

The aim of this paper is to introduce new notion of generalized \((\alpha,\psi)\)-expansive mappings in generalized metric spaces and to study the existence of a fixed point for the mappings in this space. Our new notion complements the concept of generalized \((\alpha,\psi)\)-contractions on generalized metric spaces introduced recently by Aydi et al. (Journal of Inequalities and Applications 2014, 229 (2014)). The presented theorems extend, generalize and improve many existing results in the literature.

SEVERAL FIXED POINT THEOREMS FOR EXPANSIVE MAPPINGS IN MULTIPLICATIVE METRIC SPACES

In this paper, we prove common fixed point results for compatible mappings and its variants satisfying expansive mappings in multiplicative metric spaces. We also give examples in support of our results. AMS Subject Classification: 47H10, 54H25

COMMON FIXED POINTS OF $R$-WEAKLY COMMUTING MAPPINGS AND ITS VARIANTS FOR MULTIPLICATIVE EXPANSIVE MAPPINGS

In this paper, we introduce the notions of R-weakly commuting and its variants for multiplicative expansive mappings in multiplicative metric spaces and prove common fixed point theorems for these mappings. Moreover, we prove a common fixed point theorem for weakly reciprocally continuous mappings. We also provide examples in support of our results. AMS Subject Classification: 47H10, 54H25

Coincidence Points and Common Fixed Points for Expansive Type Mappings in $b$-Metric Spaces

The main purpose of this paper is to obtain sucient condi- tions for existence of points of coincidence and common xed points for a pair of self mappings satisfying some expansive type conditions in b- metric spaces. Finally, we investigate that the equivalence of one of these results in the context of cone b-metric spaces cannot be obtained by the techniques using scalarization function. Our results extend and generalize several well known comparable results in the existing literature.

On bijections, isometries and expansive maps

In this paper we show when a bijection on a set $X$ can be made either an isometry or an expansive map with respect to a non-discrete metric on $X$. As a corollary we obtain that any bijection on an infinite set can be made biLipschitz by a non-discrete metric.

Coincidence Points & Common Fixed Points for Multiplicative Expansive Type Mappings

Coincidence Points & Common Fixed Points for Multiplicative Expansive Type Mappings

On ParametricS-Metric Spaces and Fixed-Point Type Theorems for Expansive Mappings

We introduce the notion of a parametricS-metric space as generalization of a parametric metric space. Using some expansive mappings, we prove a fixed-point theorem on a parametricS-metric space. It is important to obtain new fixed-point theorems on a parametricS-metric space because there exist some parametricS-metrics which are not generated by any parametric metric. We expect that many mathematicians will study various fixed-point theorems using new expansive mappings (or contractive mappings) in a parametricS-metric space.

Operator approach to values of stochastic games with varying stage duration

We study the links between the values of stochastic games with varying stage duration $h$, the corresponding Shapley operators $\bf{T}$ and ${\bf{T}}\_h$and the solution of $\dot f\_t = ({\bf{T}} - Id )f\_t$. Considering general non expansive maps we establish two kinds of results, under both the discounted or the finite length framework, that apply to the class of "exact" stochastic games. First, for a fixed length or discount factor, the value converges as the stage duration go to 0. Second, the asymptotic behavior of the value as the length goes to infinity, or as the discount factor goes to 0, does not depend on the stage duration. In addition, these properties imply the existence of the value of the finite length or discounted continuous time game (associated to a continuous time jointly controlled Markov process), as the limit of the value of any time discretization with vanishing mesh.

An abelian group associated with topological dynamics

For a continuous surjection $T:X \to X$ on a compact metric space $X$ and a unimodular continuous weight on $X$, we consider a weighted composition operator $U_{T,w}$ on the Banach space $C(X)$ of complex-valued continuous functions on $X$ with the sup norm. The set ${\mathcal W}_{T}$ of all weights $w$, for which the operator $U_{T,w}$ has an eigenvalue with a unimodular eigenfunction, forms a topological abelian group. The group ${\mathcal W}_{T}$ admits a homomorphism $W_T$ to the first integral \v{C}ech cohomology of the space $X$. The image and the kernel of $W_T$ carry topological and ergodic aspects of the dynamics $T$. A concrete description of $\operatorname{Im}W_{T}$ and $\operatorname{Ker}W_{T}$ is given for positively expansive eventually-onto open maps (under an assumption on the induced homomorphism of the first \v{C}ech cohomology) and minimal rotations on tori.