# Simeon Reich's research while affiliated with Technion - Israel Institute of Technology and other places

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## Publications (425)

We introduce a new generalized cyclic iterative method for finding solutions of variational inequalities over the solution set of a split common fixed point problem with multiple output sets in a real Hilbert space.

The notion of well-posedness has drawn the attention of many researchers in non-linear analysis in connection with problems where the exact solution is unknown or may be costly to compute. Well-posedness guarantees the convergence of a sequence of approximate solutions obtained by iterative methods to the exact solution of the given problem. Motiva...

In this paper, we propose and study several strongly convergent versions of the forward-reflected-backward splitting method of Malitsky and Tam for finding a zero of the sum of two monotone operators in a real Hilbert space. Our proposed methods only require one forward evaluation of the single-valued operator and one backward evaluation of the set...

We propose two very simple methods, the first one with constant step sizes and the second one with self-adaptive step sizes, for finding a zero of the sum of two monotone operators in real reflexive Banach spaces. Our methods require only one evaluation of the single-valued operator at each iteration. Weak convergence results are obtained when the...

Abstract. In general Banach spaces, the metric projection map lacks the powerful properties it enjoys in Hilbert spaces. There are a few gen- eralized projections that have been proposed in order to resolve many of the deficiencies of the metric projection. However, such notions are pre- dominantly studied in Banach spaces with rich topological str...

In general Banach spaces, the metric projection map lacks the powerful properties it enjoys in Hilbert spaces. There are a few generalized projections that have been proposed in order to resolve many of the deficiencies of the metric projection. However, such notions are predominantly studied in Banach spaces with rich topological structures, such...

We propose two very simple methods, the first one with constant step sizes and the second one with self-adaptive step sizes, for finding a zero of the sum of two monotone operators in real reflexive Banach spaces. Our methods require only one evaluation of the single-valued operator at each iteration. Weak convergence results are obtained when the...

We study split feasibility and fixed point problems for Lipschitzian pseudocontractive and nonexpansive mappings in real Hilbert spaces. Using Tikhonov's regularization technique, we first propose an Ishikawa-type gradient-projection iterative scheme for approximating solutions to such problems and then carry out its convergence analysis. A weak co...

We study the method of cyclic projections when applied to closed and linear subspaces $M_i$, $i=1,\ldots,m$, of a real Hilbert space $\mathcal H$. We show that the average distance to individual sets enjoys a polynomial behaviour $o(k^{-1/2})$ along the trajectory of the generated iterates. Surprisingly, when the starting points are chosen from the...

We investigate typical properties of nonexpansive mappings on unbounded, closed and convex subsets of hyperbolic metric spaces. For a metric of uniform convergence on bounded sets, we show that the typical nonexpansive mapping is a contraction in the sense of Rakotch on every bounded subset and there is a bounded set which is mapped into itself by...

We study the existence and uniqueness of solutions to the inverse quasi-variational inequality problem. Motivated by the neural network approach to solving optimization problems such as variational inequality, monotone inclusion, and inverse variational problems, we consider a neural network associated with the inverse quasi-variational inequality...

We study variational inequalities and fixed point problems in real Hilbert spaces. A new algorithm is proposed for finding a common element of the solution set of a pseudo-monotone variational inequality and the fixed point set of a demicontractive mapping. The advantage of our algorithm is that it does not require prior information regarding the L...

In this work we focus on the convex feasibility problem (CFP) in Hilbert space. A specific method in this area that has gained a lot of interest in recent years is the Douglas-Rachford (DR) algorithm. This algorithm was originally introduced in 1956 for solving stationary and non-stationary heat equations. Then in 1979, Lions and Mercier adjusted a...

"The notion of porosity is well known in Optimization and Nonlinear Analysis. Its importance is brought out by the fact that the complement of a -porous subset of a complete pseudo-metric space is a residual set, while the existence of the latter is essential in many problems which apply the generic approach. Thus, under certain circumstances, some...

Extending the concept of a modular space, we introduce generalized metric spaces and prove a fixed point result for Rakotch type contractive operators which map a closed subset into the space.

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...

Various versions of inertial subgradient extragradient methods for solving variational inequalities have been and continue to be studied extensively in the literature. In many of the versions that were proposed and studied, the inertial factor, which speeds up the convergence of the method, is assumed to be less than 1, and in many cases, stringent...

We study the split common fixed point problem with multiple output sets in Hilbert spaces. In order to solve this problem, we propose a new algorithm and establish a strong convergence theorem for it. Moreover, using our method, we also remove the assumptions imposed on the norms of the transfer operators.

Using iterative regularizations, we introduce a new proximal-like algorithm for solving split variational inclusion problems in Hilbert space and establish a strong convergence theorem for it. As applications, we also investigate the split feasibility and split optimization problems. Several numerical experiments are presented in support of our the...

The purpose of this paper is to introduce and analyze two inertial algorithms with self-adaptive stepsizes for solving variational inequalities in reflexive Banach spaces. Our algorithms are based on inertial hybrid and shrinking projection methods. Knowledge of the Lipschitz constant of the cost operator is not required. Under appropriate conditio...

We study the existence of extremal mild solutions to fractional delay integro-differential equations with non-instantaneous impulses by employing the monotone iterative technique coupled with the method of lower and upper solutions. Our existence results are established by appealing to two different theories. In our first result, the semigroup gene...

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.

We establish fixed point, stability and genericity theorems for strict contractions on complete metric spaces with graphs.

We study the finite convergence of iterative methods for solving convex feasibility problems. Our key assumptions are that the interior of the solution set is nonempty and that certain overrelaxation parameters converge to zero, but with a rate slower than any geometric sequence. Unlike other works in this area, which require divergent series of ov...

In this work, we investigate pseudomonotone variational inequality problems in a real Hilbert space and propose two projection-type methods with inertial terms for solving them. The first method does not require prior knowledge of the Lipschitz constant and the second one does not require the Lipschitz continuity of the mapping which governs the va...

We study the split monotone variational inclusion problem in two real Hilbert spaces. Combining the inertial and relaxation techniques with the proximal contraction algorithm, we propose two new methods for solving this problem without the usual co-coerciveness assumption on the associated operators.

We study the split common fixed point problem for Bregman relatively nonexpansive operators and the split feasibility problem with multiple output sets in real reflexive Banach spaces. Using Bregman distances, we propose several new cyclic projection algorithms for solving these problems.

We investigate the properties of the simultaneous projection method as applied to countably infinitely many closed and linear subspaces of a real Hilbert space. We establish the optimal error bound for linear convergence of this method, which we express in terms of the cosine of the Friedrichs angle computed in an infinite product space. In additio...

We propose a multi-time generalized Nash equilibrium problem and prove its equivalence with a multi-time quasi-variational inequality problem. Then, we establish the existence of equilibria. Furthermore, we demonstrate that our multi-time generalized Nash equilibrium problem can be applied to solving traffic network problems, the aim of which is to...

Using a product space approach, we propose and study several iterative methods for solving certain types of split equality problems in Hilbert spaces.

In this paper we establish the following general theorem. Let S be a semitopological reversible semigroup and let Y be a linear subspace of \({\ell }^\infty (S)\) which contains the constants and is left translation invariant. Suppose that Y has a left invariant mean. Let \((E, \Vert \cdot \Vert _E)\) be a uniformly convex Banach space and let C be...

To solve the split common null point problem with multiple output sets in Hilbert spaces, we introduce two new self-adaptive algorithms and prove strong convergence theorems for both of them.

We study the split feasibility problem with multiple output sets in Hilbert spaces. In order to solve this problem we introduce two iterative methods by using an optimization approach. Our iterative methods do not depend on the norm of the transfer operators.

"We analyze the asymptotic behavior of inexact infinite products 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."

We propose finitely convergent methods for solving convex feasibility problems defined over a possibly infinite pool of constraints. Following other works in this area, we assume that the interior of the solution set is nonempty and that certain overrelaxation parameters form a divergent series. We combine our methods with a very general class of d...

We study the finite convergence of iterative methods for solving convex feasibility problems. Our key assumptions are that the interior of the solution set is nonempty and that certain overrelaxation parameters converge to zero, but with a rate slower than any geometric sequence. Unlike other works in this area, which require divergent series of ov...

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.

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...

Using Bregman distances, we propose two extragradient-like methods for solving variational inequality problems with Lipschitz cost operators in a Hilbert space. Weak and strong convergence theorems for our algorithms are established when the cost operator is either monotone or pseudomonotone. The variable stepsizes are generated by the algorithms a...

We study the split feasibility problem with multiple output sets in Hilbert spaces. In order to solve this problem, we propose two new algorithms. We establish a weak convergence theorem for the first one and a strong convergence theorem for the second.

We study the recently introduced generalized split common null point problem in Hilbert spaces. In order to solve this problem, we propose two new parallel algorithms and establish strong convergence theorems for both of them. Our schemes combine the hybrid and shrinking projection methods with the proximal point algorithm.

We propose finitely convergent methods for solving convex feasibility problems defined over a possibly infinite pool of constraints. Following other works in this area, we assume that the interior of the solution set is nonempty and that certain overrelaxation parameters form a divergent series. We combine our methods with a very general class of d...

We investigate the properties of the simultaneous projection method as applied to countably infinitely many closed and linear subspaces of a real Hilbert space. We establish the optimal error bound for linear convergence of this method, which we express in terms of the cosine of the Friedrichs angle computed in an infinite product space. In additio...

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 pa...

We study the split common fixed point problem for Bregman relatively nonexpansive operators in real reflexive Banach spaces. Using Bregman distances, we propose two new projection algorithms for solving this problem.

We introduce a new class of nonlinear contractive mappings in Banach spaces, study their iterates and establish a fixed point theorem for them.

Given a nonexpansive mapping which maps a closed subset of a complete metric space into the space, we study the convergence of its inexact iterates to its fixed point set in the case where the errors are nonsummable. Previous results in this direction concerned nonexpansive self-mappings of the complete metric space and inexact iterates with summab...

We investigate connections between the geometry of linear subspaces and the convergence of the alternating projection method for linear projections. The aim of this article is twofold: in the first part, we show that even in Euclidean spaces the convergence of the alternating method is not determined by the principal angles between the subspaces in...

We investigate connections between the geometry of linear subspaces and the convergence of the alternating projection method for linear projections. The aim of this article is twofold: in the first part, we show that even in Euclidean spaces the convergence of the alternating method is not determined by the principal angles between the subspaces in...

We show that the typical nonexpansive mapping on a small enough subset of a CAT($\kappa$)-space is a contraction in the sense of Rakotch. By typical we mean that the set of nonexpansive mapppings without this property is a $\sigma$-porous set and therefore also of the first Baire category. Moreover, we exhibit metric spaces where strict contraction...

"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 p...

We study variational inequalities which are governed by a strongly monotone and Lipschitz continuous operator $F$ over a closed and convex set $S$. We assume that $S=C\cap A^{-1}(Q)$ is the nonempty solution set of a (multiple-set) split convex feasibility problem, where $C$ and $Q$ are both closed and convex subsets of two real Hilbert spaces $\ma...

We study variational inequalities which are governed by a strongly monotone and Lipschitz continuous operator $F$ over a closed and convex set $S$. We assume that $S=C\cap A^{-1}(Q)$ is the nonempty solution set of a (multiple-set) split convex feasibility problem, where $C$ and $Q$ are both closed and convex subsets of two real Hilbert spaces $\ma...

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.

We study the split common null point problem in two Hilbert spaces. In order to solve this problem, we propose a new algorithm and establish a strong convergence theorem for it. Our scheme combines the hybrid projection method with the proximal point algorithm.

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.

We study the multiple-set split common null point problem in two Hilbert spaces. In order to solve this problem, we propose two new parallel algorithms and establish strong convergence theorems for both of them. Our schemes combine the hybrid and shrinking projection methods with the proximal point algorithm.

In this paper we prove that every infinite-dimensional and separable Banach space (X,‖⋅‖X) admits an equivalent norm ‖⋅‖X,1 such that (X,‖⋅‖X,1) has both the Kadec-Klee and the Opial properties. This result also has a quantitative aspect and when combined with the properties of Schauder bases and the Day norm it constitutes a basic tool in the proo...

We introduce three new parallel algorithms for solving the split common fixed point problem in Hilbert spaces. Those iterative methods for solving split common fixed point problems which involve step sizes that depend on the norm of a given bounded linear operator are often not easy to implement because one has to compute the norm of this operator....

The problem of minimization of the sum of two convex functions has various theoretical and real-world applications. One of the popular methods for solving this problem is the proximal gradient method (proximal forward–backward algorithm). A very common assumption in the use of this method is that the gradient of the smooth term is globally Lipschit...

We study variational inequalities which are governed by a strongly monotone and Lipschitz continuous operator $F$ over a closed and convex set $S$. We assume that $S=C\cap A^{-1}(Q)$ is the nonempty solution set of a (multiple-set) split convex feasibility problem, where $C$ and $Q$ are both closed and convex subsets of two real Hilbert spaces $\ma...

We propose and study new iterative methods for solving the generalized split common null point problem in Hilbert spaces.

We propose finitely convergent methods for solving convex feasibility problems defined over a possibly infinite pool of constraints. Following other works in this area, we assume that the interior of the solution set is nonempty and that certain overrelaxation parameters form a divergent series. We combine our methods with a very general class of d...

We study the existence of zeroes of mappings defined in Banach spaces. We obtain, in particular, an extension of the well-known Bolzano–Poincaré–Miranda theorem to infinite-dimensional Banach spaces. We also establish a result regarding the existence of periodic solutions to differential equations posed in an arbitrary Banach space.

We consider the split convex feasibility problem in a fixed point setting. Motivated by the well-known CQ-method of Byrne [Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 2002;18:441–453], we define an abstract Landweber transform which applies to more general operators than the metric projection. We...

We study the generic behavior of the method of successive approximations for set-valued mappings in Banach spaces. We consider, in particular, the case of those set-valued mappings which are defined by pairs of nonexpansive mappings and give a positive answer to a question raised by Francesco S. de Blasi.

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 th...

This paper presents a general and useful stability principle which, roughly speaking, says that given a uniformly continuous function defined on an arbitrary metric space, if we slightly change the constraint set over which the optimal (extreme) values of the function are sought, then these values vary slightly. Actually, this apparently new princi...

Let (X,‖⋅‖) be a uniformly convex Banach space and let C be a bounded closed and convex subset of X. Assume that C has nonempty interior and is locally uniformly rotund. Let T be a nonexpansive self-mapping of C. If T has no fixed point in the interior of C, then there exists a unique point x˜ on the boundary of C such that each sequence of iterate...

In this chapter we consider certain autonomous dynamical systems acting on the open unit ball of a complex Banach space. Our interest in such systems is based on the fact that if a dynamical system is differentiable with respect to time, then its derivative is a holomorphically dissipative mapping. Furthermore, different estimates on the numerical...

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

In this chapter we introduce and study the main topic of our book: holomorphic mappings, their numerical range, growth estimates of it and related material.

Ergodic approximations naturally associated with a given holomorphic mapping essentially determine the asymptotic behavior of the nonlinear mapping in a way similar to how the “big bang” seems to determine future developments. Most important among such approximations are those associated to fixed points of the given holomorphic mapping, stationary...

In this chapter we study conditions on a holomorphic mapping F :D →X which ensure that the set D∩F(D) contains a fixed point of F. A standard situation in this study is when F is a holomorphic self-mapping of D.

The crucial point in our subsequent considerations is the fact that for a holomorphic mapping, even in an infinite-dimensional space, one-sided boundedness of the numerical range already implies that the mapping has unit radius of boundedness. This allows us to study diverse local and global geometric properties and characteristics of holomorphic m...

In this chapter we mostly use standard concepts of operator theory which can be found, for example, in [38, 54, 111, 116, 117] and [240].

This book describes recent developments as well as some classical results regarding holomorphic mappings. The book starts with a brief survey of the theory of semigroups of linear operators including the Hille-Yosida and the Lumer-Phillips theorems. The numerical range and the spectrum of closed densely defined linear operators are then discussed i...

We consider the split convex feasibility problem in a fixed point setting. Motivated by the well-known CQ-method of Byrne (2002), we define an abstract Landweber transform which applies to more general operators than the metric projection. We call the result of this transform a Landweber operator. It turns out that the Landweber transform preserves...

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...

We establish linear convergence rates for a certain class of extrapolated fixed point algorithms which are based on dynamic string-averaging methods in a real Hilbert space. This applies, in particular, to the extrapolated simultaneous and cyclic cutter methods. Our analysis covers the cases of both metric and subgradient projections.

We study the existence of zeroes of mappings defined in Banach spaces. We obtain, in particular, an extension of the well-known Bolzano-Poincar\'e-Miranda theorem to infinite dimensional Banach spaces. We also establish a result regarding the existence of periodic solutions to differential equations posed in an arbitrary Banach space.

We establish linear convergence rates for a certain class of extrapolated fixed point algorithms which are based on dynamic string-averaging methods in a real Hilbert space. This applies, in particular, to the extrapolated simultaneous and cyclic cutter methods. Our analysis covers the cases of both metric and subgradient projections.

The problem of minimization of a separable convex objective function has various theoretical and real-world applications. One of the popular methods for solving this problem is the proximal gradient method (proximal forward-backward algorithm). A very common assumption in the use of this method is that the gradient of the smooth term in the objecti...

We first study the asymptotic behavior of infinite products of nonexpansive self-mappings of a star-shaped complete metric space which is not necessarily bounded. Using the notion of contractivity and the Baire category approach, we show that a typical (generic) sequence of nonexpansive mappings is contractive and generates infinite products which...