Search

In this paper we prove several turnpike results for trajectories of discrete disperse dynamical systems introduced in 1980 by A. M. Rubinov, which have a prototype in mathematical economics. [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 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 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 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 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]

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]

We study the space of all continuous and increasing self-mappings of the real interval [0, 1] equipped with the topology of uniform convergence. In particular, we show that most such functions have at least two different fixed points. [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 study the method of cyclic projections for inconsistent convex feasibility problems in a Hilbert space under the presence of computational errors. We show that our algorithm generates a good approximate solution, if computational errors are bounded from above by a small positive constant. Our main goal is, for a known computational error, to find out what approximate solution can be obtained and how many iterates one needs for this. [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]

In this paper, we study turnpike properties for discrete disperse dynamical systems generated by set-valued mappings which were introduced by A. M. Rubinov in 1980. [ABSTRACT FROM AUTHOR]

Given a convex objective function on a Banach space, which is Lipschitz on bounded sets, we consider the class of regular vector fields introduced in our previous work on descent methods. We analyze the behaviour of the values of the objective function for two iterative processes generated by a regular vector field in the presence of computational errors and show that if the computational errors tend to zero, then the values of the objective function converge to its infimum. [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]

In this paper we consider a complete metric space of cyclical nonexpansive mappings acting on a union of two sets in a complete hyperbolic space with an empty intersection. Using Baire category approach, we show for most mappings the distance between iterates converges to the distance between these two sets. [ABSTRACT FROM AUTHOR]

We present several fixed point and convergence results for nonexpansive set-valued mappings which map a closed subset of a complete metric space into the space. [ABSTRACT FROM AUTHOR]

In this paper, we study three fixed point problems. The first one is a fixed point problem with a set-valued mapping, while the second and third problems are convex feasibility problems with infinitely many constraints solved by the subgradient projection algorithm. We show that an approximate solution is reached after a finite number of iterations under the presence of computational errors. [ABSTRACT FROM AUTHOR]

For a certain class of mean nonexpansive self-mappings of complete metric spaces, we show that all the iterates of each mapping in this class converge, uniformly on bounded subsets, to its unique fixed point. [ABSTRACT FROM AUTHOR]

Given a Lipschitz convex and coercive objective function on a Banach space, we revisit the class of regular vector fields introduced in our previous work on descent methods. Taking into account computational errors, we study the behaviour of the values of the objective function for the process generated by a regular vector field and show that if the computational errors are small enough, then the values of the objective function become close to its infimum. [ABSTRACT FROM AUTHOR]

Given a Lipschitz and convex objective function of an unconstrained optimization problem, defined on a Banach space, we revisit the class of regular vector fields which was introduced in our previous work on descent methods. We study, in particular, the asymptotic behavior of the sequence of values of the objective function for a certain inexact process generated by a regular vector field when the sequence of computational errors converges to zero and show that this sequence of values converges to the infimum of the given objective function of the unconstrained optimization problem. [ABSTRACT FROM AUTHOR]

We study the random weak ergodic property of infinite products of mappings acting on complete metric spaces. Our results describe an aspect of the asymptotic behaviour of random infinite products of such mappings. More precisely, we show that in appropriate spaces of sequences of operators there exists a subset, which is a countable intersection of open and everywhere dense sets, such that each sequence belonging to this subset has the random weak ergodic property. Then we show that several known results in the literature can be deduced from our general result. [ABSTRACT FROM AUTHOR]

For certain classes of contractive mappings in complete metric spaces, we establish the existence of a fixed point which attracts all inexact orbits. [ABSTRACT FROM AUTHOR]

In our previous work we have shown that if for any initial point there exists a trajectory of a nonexpansive set-valued mapping attracted by a given set, then this property is stable under small perturbations of the mapping. In the present paper we obtain several extensions of this result. [ABSTRACT FROM AUTHOR]

We study monotone nonexpansive self-mappings of a closed and convex cone in an ordered Banach space with particular emphasis on the asymptotic behavior of their inexact iterates. [ABSTRACT FROM AUTHOR]

In a previous paper of ours we used the notion of porosity to show that most of the nonexpansive self-mappings of bounded, closed and convex subsets of a Banach space are contractive and possess a unique fixed point which is the uniform limit of all iterates. In this paper we extend this result to nonexpansive self-mappings of closed and convex sets in a Banach space which are not necessarily bounded. As a matter of fact, it turns out that our results are true for all complete hyperbolic metric spaces. [ABSTRACT FROM AUTHOR]

In this paper we study the structure of approximate solutions of an autonomous discrete-time optimal control system with a compact metric space of states. These optimal control systems are discrete-time analogs of Bolza problems in the calculus of variations. We obtain the full description of approximate solutions of these problems on large intervals. This description shows that on large intervals the approximate solutions are determined mainly by our optimality criterion and are essentially independent of the choice of time intervals and data. [ABSTRACT FROM AUTHOR]

We consider constrained minimization problems and propose to replace the projection onto the entire feasible region, required in the projected subgradient method, by projections onto the individual sets whose intersection forms the entire feasible region. Specifically, we propose to perform such projections onto the individual sets in an algorithmic regime of a feasibility-seeking iterative projection method. For this purpose we use the recently developed family of dynamic string-averaging projection methods wherein iteration-index-dependent variable strings and variable weights are permitted. This gives rise to an algorithmic scheme that generalizes, from the algorithmic structural point of view, earlier work of Helou Neto and De Pierro, of Nedić, of Nurminski, and of Ramet al. [ABSTRACT FROM AUTHOR]

Abstract: We study turnpike properties of approximate solutions of nonautonomous discrete-time optimal control systems which are determined by sequences of lower semicontinuous objective functions. To have these properties means that the approximate solutions of the problems are determined mainly by the objective functions, and are essentially independent of the choice of intervals and endpoint conditions, except in regions close to the endpoints. We show that these turnpike properties are stable under small perturbations of the objective functions. [Copyright &y& Elsevier]

We study bounded linear regularity of finite sets of closed subspaces in a Hilbert space. In particular, we construct for each natural number $n\geq 3$ a set of $n$ closed subspaces of ${\ell }^{2} $ which has the bounded linear regularity property, while the bounded linear regularity property does not hold for each one of its nonempty, proper nonsingleton subsets. We also establish a related theorem regarding the bounded regularity property in metric spaces. [ABSTRACT FROM PUBLISHER]

Abstract: In the present paper we study convergence of subgradient projection algorithms for solving convex feasibility problems in a Hilbert space. Our goal is to obtain an approximate solution of the problem in the presence of computational errors. We show that our subgradient projection algorithm generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant. [Copyright &y& Elsevier]

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]

Abstract: In a Hilbert space, we study the convergence of the subgradient method to a common solution of a finite family of variational inequalities and of a finite family of fixed point problems under the presence of computational errors. Most results known in the literature establish the convergence of algorithms, when computational errors are summable. In the present paper, the convergence of the subgradient method is established for nonsummable computational errors. We show that the subgradient method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant. [Copyright &y& Elsevier]

In this article, we establish the existence of optimal solutions for a large class of nonconvex infinite horizon discrete-time optimal control problems. This class contains optimal control problems arising in economic dynamics which describe a model with nonconcave utility functions representing the preferences of the planner. [ABSTRACT FROM PUBLISHER]

In this paper we use the penalty approach in order to study constrained minimization problems in infinite-dimensional Asplund spaces with nonsmooth nonconvex mixed constraints. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we establish sufficient conditions for the exact penalty properties using the notion of the Mordukhovich basic subdifferential. [ABSTRACT FROM AUTHOR]

Abstract: In a Hilbert space, we study the convergence of a proximal point method to a common zero of a finite family of maximal monotone operators under the presence of computational errors. Most results known in the literature establish the convergence of proximal point methods, when computational errors are summable. In the present paper, the convergence of the method is established for nonsummable computational errors. We show that the proximal point method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant. [Copyright &y& Elsevier]

In the present paper we study convergence of projection algorithms for solving convex feasibility problems in a Hilbert space. Our goal is to obtain an ε-approximate solution of the problem in the presence of computational errors, where e is a given positive number. We show that our projection algorithm generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant. [ABSTRACT FROM AUTHOR]

Abstract: We study the existence and a turnpike property of solutions of a discrete-time control system with discounting and with a compact metric space of states. In our recent work for this discrete-time optimal control system without discounting we establish the turnpike property and show that it is stable under perturbations of an objective function. In the present paper we show that this turnpike property together with its stability also hold for the system with discounting. [Copyright &y& Elsevier]

Abstract: In this paper we study bilevel minimization problems. Using the implicit function theorem, variational analysis and exact penalty results we establish necessary optimality conditions for these problems. [Copyright &y& Elsevier]

We study the structure of approximate solutions of a discrete-time control system with a compact metric space of states which arises in economic dynamics. We are interested in turnpike properties of the approximate solutions which are independent of the length of the interval, for all sufficiently large intervals and are stable under perturbations of an objective function. [ABSTRACT FROM AUTHOR]