New general fixed-point approach to compute the resolvent of composite operators

The forward-reflected-backward splitting method recently introduced for solving variational inclusion problems involves just one forward evaluation and one backward evaluation of the monotone operator and the maximal monotone operator, respectively, per iteration. This structure gives it some advantage over the earlier proposed methods. However, it only provides weak convergence, in general. Our aim in this paper is to improve the forward-reflected-backward splitting method in order to obtain strong convergence. To this end, we first study a regularized variational inclusion problem of finding the zero of the sum of two monotone operators. We then propose a regularized forward-reflected-backward splitting method for approximating a solution to the problem and prove the strong convergence of our iterative scheme under some suitable assumptions on the parameters. Moreover, we show that our algorithm has the bounded perturbation resilience property. Furthermore, we apply our results to convex minimization, split feasibility, split variational inclusion, and image deblurring problems, and illustrate the performance of our algorithm with several numerical examples.

Numerical approximation of the stochastic equation driven by the fractional noise

Convergence in uncertain linear systems

A theorem due to Fitzpatrick provides a representation of arbitrary maximal monotone operators by convex functions. This paper explores representability of arbitrary (nonnecessarily maximal) monotone operators by convex functions. In the finite-dimensional case, we identify the class of monotone operators that admit a convex representation as the one consisting of intersections of maximal monotone operators and characterize the monotone operators that have a unique maximal monotone extension.

We prove strong and weak convergence theorems for a new resolvent of maximal monotone operators in a Banach space and give an estimate of the convergence rate of the algorithm. Finally, we apply our convergence theorem to the convex minimization problem. The result present in this paper extend and improve the corresponding result of Ibaraki and Takahashi (2007), and Kim and Xu (2005).

Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization

One of the major problems in the theory of maximal monotone operators is to find a point in the solution set Zer( ), set of zeros of maximal monotone mapping . The problem of finding a zero of a maximal monotone in real Hilbert space has been investigated by many researchers. Rockafellar considered the proximal point algorithm and proved the weak convergence of this algorithm with the maximal monotone operator. Güler gave an example showing that Rockafellar’s proximal point algorithm does not converge strongly in an infinite-dimensional Hilbert space. In this paper, we consider an explicit method that is strong convergence in an infinite-dimensional Hilbert space and a simple variant of the hybrid steepest-descent method, introduced by Yamada. The strong convergence of this method is proved under some mild conditions. Finally, we give an application for the optimization problem and present some numerical experiments to illustrate the effectiveness of the proposed algorithm.

Under suitable extreme point conditions weak convergence can imply strong convergence inL 1-spaces [28, 31, 12, 26] Here a number of such results are generalized by means of a unifying, very general approach using Young measures. The required results from Young measure theory are derived in a new fashion, based on pointwise averages [6], from well-known results on weak convergence of probability measures.

It is well known that many problems in image recovery, signal processing, and machine learning can be modeled as finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. Many papers have studied forward-backward splitting methods for finding zeros of the sum of two monotone operators in Hilbert spaces. Most of the proposed splitting methods in the literature have been proposed for the sum of maximal monotone and inverse-strongly monotone operators in Hilbert spaces. In this paper, we consider splitting methods for finding zeros of the sum of maximal monotone operators and Lipschitz continuous monotone operators in Banach spaces. We obtain weak and strong convergence results for the zeros of the sum of maximal monotone and Lipschitz continuous monotone operators in Banach spaces. Many already studied problems in the literature can be considered as special cases of this paper.

This paper aims to obtain a strong convergence result for a Douglas–Rachford splitting method with inertial extrapolation step for finding a zero of the sum of two set-valued maximal monotone operators without any further assumption of uniform monotonicity on any of the involved maximal monotone operators. Furthermore, our proposed method is easy to implement and the inertial factor in our proposed method is a natural choice. Our method of proof is of independent interest. Finally, some numerical implementations are given to confirm the theoretical analysis.

In this article, motivated by Rockafellar’s proximal point algorithm in Hilbert spaces, we discuss various weak and strong convergence theorems for resolvents of accretive operators and maximal monotone operators which are connected with the proximal point algorithm. We first deal with proximal point algorithms in Hilbert spaces. Then, we consider weak and strong convergence theorems for resolvents of accretive operators in Banach spaces which generalize the results in Hilbert spaces. Further, we deal with weak and strong convergence theorems for three types of resolvents of maximal monotone operators in Banach spaces which are related to proximal point algorithms. Finally, in Section 7, we apply some results obtained in Banach spaces to the problem of finding minimizers of convex functions in Banach spaces.

In this paper, we present two iterative algorithms for approximating a solution of the split feasibility problem on zeros of a sum of monotone operators and fixed points of a finite family of nonexpansive mappings. Weak and strong convergence theorems are proved in the framework of Hilbert spaces under some mild conditions. We apply the obtained main result for the problem of finding a common zero of the sum of inverse strongly monotone operators and maximal monotone operators, for finding a common zero of a finite family of maximal monotone operators, for finding a solution of multiple sets split common null point problem, and for finding a solution of multiple sets split convex feasibility problem. Some applications of the main results are also provided.

We study infinitary term rewriting systems containing finitely many rules. For these, we show that if a weakly convergent reduction is not strongly convergent, it contains a term that reduces to itself in one step (but the step itself need not be part of the reduction). Using this result, we prove the starkly surprising result that for any orthogonal system with finitely many rules, the system is weakly normalizing under weak convergence if{f} it is strongly normalizing under weak convergence if{f} it is weakly normalizing under strong convergence if{f} it is strongly normalizing under strong convergence. As further corollaries, we derive a number of new results for weakly convergent rewriting: Systems with finitely many rules enjoy unique normal forms, and acyclic orthogonal systems are confluent. Our results suggest that it may be possible to recover some of the positive results for strongly convergent rewriting in the setting of weak convergence, if systems with finitely many rules are considered. Finally, we give a number of counterexamples showing failure of most of the results when infinite sets of rules are allowed.

Let H be a real Hilbert space and let A : D(A) ⊂ H → H be a (possibly multivalued) maximal monotone operator. We are concerned with the difference equation ∆un + cnAun+1 ∋ fn, n = 0, 1, ..., where (cn) ⊂ (0,+∞), ( fn) ⊂ H are p-periodic sequences for a positive integer p. We investigate the existence of periodic solutions to this equation as well as the weak or strong convergence of solutions to p-periodic solutions. The first result of this paper (Theorem 1) is a discrete analogue of the 1977 result by Baillon and Haraux (on the periodic forcing problem for the continuous counterpart of the above equation) and was essentially stated by Djafari Rouhani and Khatibzadeh in a recent paper [5]. Here we provide a simpler proof of this result that is based on old existing results due to Browder and Petryshyn [4] and Opial (see, e.g., [6], p.5). A strong convergence result is also given and some examples are discussed to illustrate the theoretical results.

New results on the asymptotic behavior of solutions to a class of second order nonhomogeneous difference equations