Set-valued dynamic systems and random iterations of set-valued weaker contractions in uniform spaces

Set-valued weaker contractions in uniform, locally convex and metric spaces are defined and dynamic systems of such weaker contractions are studied. Conditions guaranteeing the convergence of generalized sequences of random iterations and iterations and the existence and uniqueness of endpoints of set-valued weaker contractions are established. Our definitions and results are new for set-valued maps in uniform, locally convex and metric spaces and even for single-valued maps.     

Endpoints of set-valued dynamical systems of asymptotic contractions of Meir–Keeler type and strict contractions in uniform spaces

In this paper, we introduce the concepts of the set-valued dynamical systems of asymptotic contractions of Meir–Keeler type and set-valued dynamical systems of strict contractions in uniform spaces and we present a method which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints of these contractions and the convergence to these endpoints of all generalized sequences of iterations of these contractions. The result, concerning the investigations of problems of the set-valued asymptotic fixed point theory, include some well-known results of Meir and Keeler, Kirk and Suzuki concerning the asymptotic fixed point theory of single-valued maps in metric spaces. The result, concerning set-valued strict contractions (in which the contractive coefficient is not constant), is different from the result of Yuan concerning the existence of endpoints of Tarafdar–Vyborny generalized contractions (in which the contractive coefficient is constant) in bounded metric spaces and provides some examples of Tarafdar–Yuan topological contractions in compact uniform spaces. Definitions and results presented here are new for set-valued dynamical systems in uniform, locally convex and metric spaces and even for single-valued maps. Examples show a fundamental difference between our results and the well-known ones.     

Quasi-asymptotic contractions,set-valued dynamic systems,uniqueness of endpoints and generalized pseudodistances in uniform spaces

For set-valued dynamic systems in uniform spaces we introduce the concept of quasi-asymptotic contractions with respect to some generalized pseudodistances, describe a method which we use to establish general conditions guaranteeing the existence and uniqueness of endpoints (stationary points) of these contractions and exhibit conditions such that for each starting point each generalized sequence of iterations (in particular, each dynamic process) converges and the limit is an endpoint. The definition, result, ideas and techniques are new for set-valued dynamic systems in uniform, locally convex and metric spaces and even for single-valued maps.     

The uniqueness of endpoints for set-valued dynamical systems of contractions of Meir–Keeler type in uniform spaces

In this paper, the concept of the set-valued dynamical systems of contractions of Meir–Keeler type in uniform spaces is introduced and conditions guaranteeing the existence and uniqueness of endpoints of these contractions and the convergence to these endpoints of all generalized sequences of iterations of these contractions are established. The definition and the result presented here are new for set-valued dynamical systems in uniform, locally convex and metric spaces and even for single-valued maps. Examples show a fundamental difference between our result and the well-known ones.     

The uniqueness of endpoints for set-valued dynamical systems of contractions of Meir–Keeler type in uniform spaces

In this paper, the concept of the set-valued dynamical systems of contractions of Meir–Keeler type in uniform spaces is introduced and conditions guaranteeing the existence and uniqueness of endpoints of these contractions and the convergence to these endpoints of all generalized sequences of iterations of these contractions are established. The definition and the result presented here are new for set-valued dynamical systems in uniform, locally convex and metric spaces and even for single-valued maps. Examples show a fundamental difference between our result and the well-known ones.     

Fixed point theorems for set-valued contractions

In this paper, we obtain some fixed point theorems for new set-valued contractions in complete metric spaces. Then by using these results and the scalarization method, we present some fixed point theorems for set-valued contractions in complete cone metric spaces without the normality assumption. We also present some examples to support our results.     

Endpoint theory for set-valued nonlinear asymptotic contractions with respect to generalized pseudodistances in uniform spaces

In uniform spaces, inspired by ideas of Banach, Tarafdar and Yuan, we introduce the concepts of generalized pseudodistances and generalized gauge maps, for set-valued dynamic systems we define various nonlinear asymptotic contractions and contractions with respect to these pseudodistances and gauges, provide conditions on the iterates of these set-valued dynamic systems and present a method which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints (stationary points) of these set-valued dynamic systems and conditions that each generalized sequence of iterations (in particular, each dynamic process) converges and the limit of a generalized sequence of iterations is an endpoint. The definitions, the results and the method are new for set-valued dynamic systems in uniform, locally convex and metric spaces and even for single-valued maps. The paper includes a number of various examples which show a fundamental difference between our results and those existing in the literature.     

Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces

Investigations concerning the existence of dynamic processes convergent to fixed points of set-valued nonlinear contractions in cone metric spaces are initiated. The conditions guaranteeing the existence and uniqueness of fixed points of such contractions are established. Our theorems generalize recent results obtained by Huang and Zhang [L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive maps, J. Math. Anal. Appl. 332 (2007) 1467–1475] for cone metric spaces and by Klim and Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (1) (2007) 132–139] for metric spaces. The examples and remarks provided show an essential difference between our results and those mentioned above.     

Study on chaos induced by turbulent maps in noncompact sets

This paper is concerned with chaos induced by strictly turbulent maps in noncompact sets of complete metric spaces. Two criteria of chaos for such types of maps are established, and then a criterion of chaos, characterized by snap-back repellers in complete metric spaces, is obtained. All the maps presented in this paper are proved to be chaotic either in the sense of both Li–Yorke and Wiggins or in the sense of both Li–Yorke and Devaney. The results weaken the assumptions in some existing criteria of chaos. Several illustrative examples are provided with computer simulation.     

The hausdorff metric and measurable selections

We construct measurable selections for closed set-valued maps into arbitrary complete metric spaces. We do not need to make any separability assumptions. We view the set-valued maps as point-valued maps into the hyperspace and our measurability assumptions arethe usual kinds of measurability of point-valued maps in this setting. We also discuss relationship of these measurability conditions to the ones usually considered in the theory of measurable selections.     

Fixed Points for Directional Multi-Valued <Emphasis Type="Italic">k</Emphasis>(·)-Contractions

A fixed point theorem for directional multi-valued k(·)-contractions acting m a complete metric space is established which extends similar results both for k(·)-contractions and directional contractions. Such theorem enables to obtain fixed points theorems for the former class of set-valued maps from those valid for the latter one without metrical convexity or proximinality assumptions, thereby contributing to unify the current setting of the theory. Connections with several recent advances on this subject are also examinated.Mathematics Subject Classifications (2000): 47H10, 54H25     

Common fixed point theorems for four mappings through generalized altering distances in ordered metric spaces

Using the concepts introduced by Abbas et?al. (Appl Math Lett 24:1520?C1526, 2011), the purpose of this paper is to establish coincidence point and common fixed point results for four maps satisfying generalized weak contractions under generalized altering distance functions in ordered complete metric spaces.     

Coupled-expanding maps and one-sided symbolic dynamical systems

This paper studies relationships between coupled-expanding maps and one-sided symbolic dynamical systems. The concept of coupled-expanding map is extended to a more general one: coupled-expansion for a transitive matrix. It is found that the subshift for a transitive matrix is strictly coupled-expanding for the matrix in certain disjoint compact subsets; the topological conjugacy of a continuous map in its compact invariant set of a metric space to a subshift for a transitive matrix has a close relationship with that the map is strictly coupled-expanding for the matrix in some disjoint compact subsets. A certain relationship between strictly coupled-expanding maps for a transitive matrix in disjoint bounded and closed subsets of a complete metric space and their topological conjugacy to the subshift for the matrix is also obtained. Dynamical behaviors of subshifts for irreducible matrices are then studied and several equivalent statements to chaos are obtained; especially, chaos in the sense of Li–Yorke is equivalent to chaos in the sense of Devaney for the subshift, and is also equivalent to that the domain of the subshift is infinite. Based on these results, several new criteria of chaos for maps are finally established via strict coupled-expansions for irreducible transitive matrices in compact subsets of metric spaces and in bounded and closed subsets of complete metric spaces, respectively, where their conditions are weaker than those existing in the literature.     

Multivalued generalized nonlinear contractive maps and fixed points

We introduce some notions of generalized nonlinear contractive maps and prove some fixed point results for such maps. Consequently, several known fixed point results are either improved or generalized including the corresponding recent fixed point results of Ciric [L.B. Ciric, Multivalued nonlinear contraction mappings, Nonlinear Anal. 71 (2009) 2716-2723], Klim and Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (2007) 132-139], Feng and Liu [Y. Feng, S. Liu, Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103-112] and Mizoguchi and Takahashi [N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177-188].     

The Set-Valued Dynamic System and Its Applications to Pareto Optima

In this paper, we first study the existence of endpoints for set-valued dynamic systems which are either upper or lower semicontinuous in metric spaces. Then the existence, uniqueness and algorithms of endpoints for set-valued dynamic systems which are either generalized contractions (defined in metric spaces) or topological contractions (defined in topological spaces which do not necessarily have any metric). These results are then applied to derive the existence of Pareto optima for mappings which take values in ordered Banach spaces. Finally, the stability of (generalized) nucleolus sets is also established.     

Some results of Perov type in rectangular cone metric spaces

In 2000, Branciari replaced the triangle inequality by a more general one which today is known as the rectangular inequality and introduced the notion of generalized metric space or rectangular metric space. Subsequently Azam, Arshad, and Beg introduced the concept of rectangular cone metric space and proved fixed point results for Banach-type contractions in rectangular cone metric spaces. In this paper, we establish fixed point results for mappings that satisfy a contractive condition of Perov type in rectangular cone metric spaces.     

Conditions of regularity in cone metric spaces

In this paper we prove some fixed points results on cone metric spaces for maps satisfying general contractive type conditions. Among other things, we extend some results of Nguyen [11] from metric spaces to cone metric spaces. The example is included.     

Fixed point theorems for multi-valued contractions in complete metric spaces

In this paper the concept of a contraction for multi-valued mappings in a metric space is introduced and the existence theorems for fixed points of such contractions in a complete metric space are proved. Presented results generalize and improve the recent results of Y. Feng, S. Liu [Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103-112], D. Klim, D. Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (2007) 132-139] and several others. The method used in the proofs of our results is new and is simpler than methods used in the corresponding papers. Two examples are given to show that our results are genuine generalization of the results of Feng and Liu and Klim and Wardowski.     

Topological convexities, selections and fixed points

A convexity on a set X is a family of subsets of X which contains the whole space and the empty set as well as the singletons and which is closed under arbitrary intersections and updirected unions. A uniform convex space is a uniform topological space endowed with a convexity for which the convex hull operator is uniformly continuous. Uniform convex spaces with homotopically trivial polytopes (convex hulls of finite sets) are absolute extensors for the class of metric spaces; if they are completely metrizable then a continuous selection theorem à la Michael holds. Upper semicontinuous maps have approximate selections and fixed points, under the usual assumptions.