Simeon Reich’s research while affiliated with Technion – Israel Institute of Technology and other places

We study the product of two relaxed cutters having a common fixed point. We assume that one of the relaxation parameters is greater than two so that the corresponding relaxed cutter is no longer quasi-nonexpansive, but rather demicontractive. We show that if both of the operators are (weakly/linearly) regular, then under certain conditions, the resulting product inherits the same type of regularity. We then apply these results to proving convergence in the weak, norm and linear sense of algorithms that employ such products.

The generic convergence of infinite products of nonexpansive mappings was established in a 1999 paper of ours. In the present paper, such results are extended to infinite products of uniformly locally nonexpansive mappings. In particular, the convergence of infinite products of uniformly locally contractive mappings, as well as its stability, are proved. Moreover, the Baire category approach and the porosity notion are used to show that most sequences of uniformly locally nonexpansive mappings are, in fact, uniformly locally contractive.

We study the product of two relaxed cutters having a common fixed point. We assume that one of the relaxation parameters is greater than two so that the corresponding relaxed cutter is no longer quasi-nonexpansive, but rather demicontractive. We show that if both of the operators are (weakly/linearly) regular, then under certain conditions, the resulting product inherits the same type of regularity. We then apply these results to proving convergence in the weak, norm and linear sense of algorithms that employ such products.

This paper aims to introduce a novel self-adaptive explicit iterative algorithm for solving the split common solution problem with monotone operator equations in real Hilbert spaces. This new approach does not employ resolvent operators nor does it use the norms of the bounded linear operators (transfer mappings) from the source space to the image spaces.

The split common fixed point problem is an optimization challenge that involves finding an element within one fixed point set such that when transformed by a bounded linear operator, it belongs to another fixed point set. This problem falls under the category of inverse problems in mathematics. We present a novel self-adaptive algorithm based on double inertial steps for solving the split common fixed point problem for demicontractive mappings. We also establish a weak convergence theorem for our method. Furthermore, we also present some numerical experiments illustrating the convergence behavior and the efficiency of our proposed algorithm.

The paper concerns the split mixed variational inequality problem with multiple output sets. In order to find a solution to this problem, we introduce an efficient algorithm with multiple inertial steps. Using simple techniques of variational analysis and imposing some mild conditions on the control parameters, we establish strong convergence of the sequences generated by our proposed algorithm.

We first propose a reflected-forward–backward splitting algorithm with two inertial effects for solving monotone inclusions and then establish that the sequence of iterates it generates converges weakly in a real Hilbert space to a zero of the sum of a set-valued maximal monotone operator and a single-valued monotone Lipschitz continuous operator. The proposed algorithm involves only one forward evaluation of the single-valued operator and one backward evaluation of the set-valued operator at each iteration. One inertial parameter is non-negative while the other is non-positive. These features are absent in many other available inertial splitting algorithms in the literature. Finally, we discuss some problems in image restoration in connection with the implementation of our algorithm and compare it with some known related algorithms in the literature.

We introduce three new inertial shrinking projection algorithms with multiple inertial effects for solving split common solution problems with multiple output sets. We establish the convergence of the sequences generated by our proposed algorithms under some mild conditions on the control parameters. More precisely, we only require the boundedness of the coefficients of the inertial components. Moreover, our algorithms do not depend on the norms of the transfer mappings.

In this paper we study a class of split variational inclusion (SVI) and regularized split variational inclusion (RSVI) problems in real Hilbert spaces. We discuss various analytical properties of the net generated by the RSVI and establish the existence and uniqueness of the solution to the RSVI. Using analytical properties of this net and under certain assumptions on the parameters and mappings associated with the SVI, we establish the strong convergence of the sequence generated by our proposed iterative algorithm. We also deduce another iterative algorithm by taking the regularization parameters to be zero in our proposed algorithm. We establish the weak convergence of the sequence generated by our new algorithm under certain assumptions. Moreover, we discuss two special cases of the SVI, namely the split convex minimization and the split variational inequality problems, and give several numerical examples.