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In this article, we investigate the convergence properties of a stochastic primal-dual splitting algorithm for solving structured monotone inclusions involving the sum of a cocoercive operator and a composite monotone operator. The proposed method is the stochastic extension to monotone inclusions of a proximal method studied in the literature for saddle point problems. It consists in a forward step determined by the stochastic evaluation of the cocoercive operator, a backward step in the dual variables involving the resolvent of the monotone operator, and an additional forward step using the stochastic evaluation of the cocoercive operator introduced in the first step. We prove weak almost sure convergence of the iterates by showing that the primal-dual sequence generated by the method is stochastic quasi-Fejér-monotone with respect to the set of zeros of the considered primal and dual inclusions. Additional results on ergodic convergence in expectation are considered for the special case of saddle point models. [ABSTRACT FROM AUTHOR]

In this article, we prove several new ergodic, weak, and strong convergence theorems for solutions to the following general second-order difference equationwhereAis a maximal monotone operator in a real Hilbert spaceHand {cn} and {θn} are positive real sequences. We do not assumeA−1(0) ≠ ∅, and we prove among other things that the existence of solutions is in fact equivalent to the zero set ofAbeing nonempty. These theorems provide new approximation results for zeroes of monotone operators, as well as significantly unify and extend previously known results by assuming much weaker conditions on the coefficients {cn} and {θn}. [ABSTRACT FROM AUTHOR]

We prove several new weak and strong ergodic theorems, as well as weak and strong convergence theorems for solutions to the following second order difference equation:whereAis a maximal monotone operator in a real Hilbert spaceH, and {cn} is a positive sequence of real numbers. We do not assume thatA−1(0) ≠ ∅, and we prove among other things that the existence of solutions is in fact equivalent to the zero set ofAbeing nonempty. These theorems provide new approximation results for zeros of monotone operators, as well as unify and extend previously known results in [2, 3, 7, 10, 18, 22, 23, 25] by considering much weaker conditions on the coefficients {cn}. In particular, our new strong ergodic theorem extends the results of [7] and [23, Theorem 3.3] for first-order difference equations, to the case of second order difference equations, and implies also a new strong convergence theorem. [ABSTRACT FROM PUBLISHER]

The purpose of this article is to prove a strong convergence result associated with a generalization of the method of alternating resolvents introduced by the authors in convergence of the method of alternating resolvents [4] under minimal assumptions on the control parameters involved. Thus, this article represents a significant improvement of the article mentioned above. [ABSTRACT FROM PUBLISHER]

We investigate asymptotic behavior of solutions to the following second order difference equation where A is a maximal monotone operator in a real Hilbert space H and {c i } is a positive real sequence. We show weak and strong convergence of solutions to an element of A −1(0), under appropriate assumptions on A and the sequence {c i }. Our results extend and improve previous results by Morosanu [21] and Mitidieri-Morosanu [18] (see also [19, pp. 156–168]) under more general assumptions on c i . [ABSTRACT FROM AUTHOR]

In this paper, we prove three strong convergence theorems of modified Mann iteration for relatively asymptotically nonexpansive mappings, relatively nonexpansive mappings, and maximal monotone operators in a Banach space, respectively. Our results extend and improve the recent ones announced by Nakajo, Takahashi, Matsushita, Kim, Martinez-Yanex, Xu, and some others. [ABSTRACT FROM AUTHOR]

We give a sufficient condition, weaker than the others known so far, that guarantees that the sum of two maximal monotone operators on a reflexive Banach space is maximal monotone. Then we give a weak constraint qualification assuring the Brézis-Haraux-type approximation of the range of the sum of the subdifferentials of two proper convex lower-semicontinuous functions in nonreflexive Banach spaces, extending and correcting an earlier result due to Riahi. [ABSTRACT FROM AUTHOR]