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In the present paper, we obtain three results on the existence of a fixed point for nonexpansive mappings. Two of them are generalizations of the result for F-contraction, while third one is a generalization of a recent result for set-valued contractions. [ABSTRACT FROM AUTHOR]

In the present paper, under certain assumptions, we establish the convergence of iterates for self-mappings of complete metric spaces with graphs which are of a contractive type. The class of mappings considered in the paper contains the so-called cyclical mappings introduced by W. A. Kirk, P. S. Srinivasan and P. Veeramani in 2003 and the class of monotone nonexpansive operators. Our results hold in the case of a symmetric graph. [ABSTRACT FROM AUTHOR]

It is known that a strict contraction on complete metric spaces with graphs possesses a fixed point. In the present paper we show that this property holds for mappings on complete metric spaces with graphs which are merely contractive. We also obtain some stability results. [ABSTRACT FROM AUTHOR]

In 1988 N.A. Assad showed that a nonself nonlinear contraction taking a closed subset of a complete metrically convex space into the space so that the boundary of this subset is mapped back into the subset itself has a unique fixed point. In the present paper we extend this result by replacing the complete metrically convex space with a complete metric space which is a length space. [ABSTRACT FROM AUTHOR]

In this work, we study an iterative process induced by a contractive type set-valued mapping in a complete metric space and show its convergence, taking into account computational errors. [ABSTRACT FROM AUTHOR]

It is known that a strict contraction on a complete metric space with a graph possesses a fixed point. In the present paper, we show that this property holds for single valued and set-valued self-mappings of metric spaces with graphs that are of the contractive type. We also show the convergence of iterates of these mappings to fixed points. In particular, our results are true for metric spaces with symmetric graphs. [ABSTRACT FROM AUTHOR]

In this work, we investigate a minimization problem with a convex objective function on a domain, which is the solution set of a common fixed point problem with a finite family of nonexpansive mappings. Our algorithm is a combination of a projected subgradient algorithm and string-averaging projection method with variable strings and variable weights. This algorithm generates a sequence of iterates which are approximate solutions of the corresponding fixed point problem. Additionally, either this sequence also has a minimizing subsequence for our optimization problem or the sequence is strictly Fejer monotone regarding the approximate solution set of the common fixed point problem. [ABSTRACT FROM AUTHOR]

In this work we show that if iterates of a nonexpansive self-mapping of a complete metric with a graph converge uniformly on a subset of the space, then this convergence is stable under the presence of small computational errors. [ABSTRACT FROM AUTHOR]

In our joint work with Dan Butnariu (2006) we established the stability of the convergence of iterates of a nonexpansive mapping on a complete metric space in the presence of summable computational errors. In a recent paper of ours, we extended this result to inexact iterates of nonexpansive mappings on complete metric spaces with graphs under a certain assumption on the iterates. In the present paper we obtain an analogous result by removing that assumption on the iterates and replacing it with an additional assumption on the graph. [ABSTRACT FROM AUTHOR]

In 2007, in our joint work with D. Butnariu and S. Reich, we proved that if for a self-mapping of a complete metric that is uniformly continuous on bounded sets all its iterates converge uniformly on bounded sets, then this convergence is stable under the presence of small errors. In the present paper, we obtain an extension of this result for self-mappings of a metric space with a graph. [ABSTRACT FROM AUTHOR]

In this paper, we consider a discrete-time optimal control problem related to the model of Robinson, Solow and Srinivasan. We analyze this optimal control problem without concavity assumptions on a non-concave utility function which represents the preferences of the planner and establish the existence of good programs and optimal programs which are Stiglitz production programs. [ABSTRACT FROM AUTHOR]

In his recent research, M. K. Tam (2018) considered a framework for the analysis of iterative algorithms which can be described in terms of a structured set-valued operator. At each point in the ambient space, the value of the operator can be expressed as a finite union of values of single-valued para-contracting operators. He showed that the associated fixed point iteration is locally convergent around strong fixed points. In the present paper we generalize the result of Tam and show the global convergence of his algorithm for an arbitrary starting point. An analogous result is also proven for the Krasnosel'ski–Mann iterations. [ABSTRACT FROM AUTHOR]

In this work we consider a contraction mapping of Perov type which maps a closed subset of a generalized complete metric space into the space. We show the existence of a unique fixed point which attracts all (inexact) iterates of the mapping uniformly on bounded sets. [ABSTRACT FROM AUTHOR]

In 2006, together with D. Butnariu, we showed that if all iterates of a nonexpansive self-mapping of a complete metric space converge, then all its inexact iterates with summable computational errors converge too. In a recent paper of ours, we have extended this result to uniformly locally nonexpansive self-mappings of a complete metric space. In the present paper, we establish analogous results for uniformly locally nonexpansive mappings which take a nonempty closed subset of a complete metric space into the space. In the particular case of a Banach space, if the operator is symmetric, then the set of all limit points of its iterates is also symmetric. [ABSTRACT FROM AUTHOR]

In a 1961 paper by E. Rakotch it was shown that a uniformly locally contractive mapping has a fixed point. In the present paper we show that for such a mapping, the fixed point problem is well posed and that inexact iterates of such a mapping converge to its unique fixed point, uniformly on bounded sets. Using the porosity notion, we also show that most uniformly locally nonexpansive mappings are, in fact, uniformly locally contractive. [ABSTRACT FROM AUTHOR]

In this paper, we study a fixed-point problem with a set-valued mapping by using an algorithm based on unions of nonexpansive mappings. We show that an approximate solution is reached after a finite number of iterations in the presence of computational errors. This result is an extension of the results known in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study a feasibility problem with infinitely many sets in a metric space. We present a novel algorithm and analyze its convergence. The algorithms used for the feasibility problem in the literature work for finite collections of sets and cannot be applied if the collection of sets is infinite. The main feature of these algorithms is that, for iterative steps, we need to calculate the values of all the operators belonging to our family of maps and even their sums with weighted coefficients. This is impossible if the family of maps is not finite. In the present paper, we introduce a new algorithm for solving feasibility problems with infinite families of sets and study its convergence. It turns out that our results hold for feasibility problems in a general metric space. [ABSTRACT FROM AUTHOR]

We analyze the asymptotic behavior of infinite products of non-linear operators which take a non-empty, closed subset of a complete metric space into the space, taking into account summable computational errors. Our results can be applied in methods for solving convex feasibility and optimization problems. [ABSTRACT FROM AUTHOR]

In our 2006 paper with D. Butnariu, it was shown that the convergence of iterates of a nonexpansive self-mapping of a complete metric space is stable in the presence of summable computational errors. In the present paper, we establish such results for monotone nonexpansive mappings. [ABSTRACT FROM AUTHOR]

In our recent paper we obtained an extension of a theorem of Tam (2018), which was established for iterates of a set-valued paracontracting operators in a finite-dimensional Euclidean space. This extension was obtained for exact iterates of maps in a metric space. In the present paper we prove two analogs of this result for inexact iterates under the presence of computational errors. In the first result, the errors are summable, while for the second one they converge to zero. In a particular case of a Banach space, if all the operators are symmetric, then the set of all limit points of iterates is symmetric too. [ABSTRACT FROM AUTHOR]

In this paper, we obtain two extensions of a recent result of Tam, which was proved for iterates of set-valued paracontracting operators in a finite-dimensional space. Our results are obtained for operators in an arbitrary metric space. In the first result, we study exact iterates of the set-valued mapping, while in the second one, we deal with its inexact iterates, taking computational errors into account. In a particular case of a Banach space, if all the operators are symmetric, then the set of all limit points of iterates is symmetric too. [ABSTRACT FROM AUTHOR]

We study the convergence of inexact iterates of nonexpansive mappings which take a nonempty closed subset of a complete metric space into the space in the case where the errors are sufficiently small. [ABSTRACT FROM AUTHOR]

In our 2014 work with M. Gabour, we introduced a metric space of generalized nonexpansive self-mappings of bounded and closed subsets of a Banach space and studied, using the Baire category approach, the asymptotic behavior of iterates of a generic operator belonging to this class. In the definition of a generalized nonexpansive mapping the norm is replaced by a general function which can be symmetric as a particular case. In this paper, we prove the convergence of infinite products of generalized nonexpansive self-mappings to a common fixed point in a generic setting. [ABSTRACT FROM AUTHOR]

We have recently established the existence of a unique fixed point for nonlinear contractive self-mappings of a closed subset of a Banach space, which is not necessarily bounded. In this paper, we extend this result to contractive mappings, which map a closed subset of a Banach space into the space. [ABSTRACT FROM AUTHOR]

In this paper, we study the turnpike phenomenon for trajectories of continuous-time dynamical systems generated by differential inclusions, which have a prototype in mathematical economics. In particular, we show that, if the differential inclusion has a certain symmetric property, the turnpike possesses the corresponding symmetric property. If we know a finite number of approximate trajectories of our system, then we know the turnpike and this information can be useful if we need to find new trajectories of our system or their approximations. [ABSTRACT FROM AUTHOR]

We establish fixed point, stability and genericity theorems for strict contractions on complete metric spaces with graphs. [ABSTRACT FROM AUTHOR]

We study the behavior of inexact products of uniformly continuous self-mappings of a complete metric space that is uniformly continuous and bounded on bounded sets. It is shown that previously established convergence theorems for products of non-expansive mappings continue to hold even under the presence of computational errors. [ABSTRACT FROM AUTHOR]

In the present work, we investigate a collection of symmetric minimization problems, which is identified with a complete metric space of lower semi-continuous and bounded from below functions. In our recent paper, we showed that for a generic objective function, the corresponding symmetric optimization problem possesses two solutions. In this paper, we strengthen this result using a porosity notion. We investigate the collection of all functions such that the corresponding optimization problem is well-posed and prove that its complement is a σ -porous set. [ABSTRACT FROM AUTHOR]

In this work, we obtain a weak version of the turnpike property of trajectories of perturbed discrete disperse dynamical systems, which have a prototype in mathematical economics. [ABSTRACT FROM AUTHOR]

In this paper we study a class of symmetric optimization problems which is identified with a space of objective functions, equipped with an appropriate complete metric. Using the Baire category approach, we show the existence of a subset of the space of functions, which is a countable intersection of open and everywhere dense sets, such that for every objective function from this intersection the corresponding symmetric optimization problem possesses a solution. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of trajectories of discrete disperse dynamical systems with a Lyapunov function which are generated by set-valued mappings. We establish a weak version of the turnpike property which holds for all trajectories of such dynamical systems which are of a sufficient length. This result is usually true for models of economic growth which are prototypes of our dynamical systems. [ABSTRACT FROM AUTHOR]

In our recent work we have introduced and studied a notion of a generalized nonexpansive mapping. In the definition of this notion the norm has been replaced by a general function satisfying certain conditions. For this new class of mappings, we have established the existence of unique fixed points and the convergence of iterates. In the present paper we construct an example of a generalized nonexpansive self-mapping of a bounded, closed and convex set in a Hilbert space, which is not nonexpansive in the classical sense. [ABSTRACT FROM AUTHOR]

In a recent work, we established the existence of a unique fixed point for nonlinear contractive self-mappings of a bounded and closed set in a Banach space. In the present paper we extend this result to the case of unbounded sets. [ABSTRACT FROM AUTHOR]