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In this paper, we introduce the notion of ϕ -nonexpansive sequences and maps in a Banach space E, and prove an ergodic theorem, a fixed point theorem, and some weak and strong convergence theorems for such sequences and maps in E, with rather mild assumptions on the Banach space E, weaker than those assumed by previous authors for nonexpansive maps. These results give a partial answer to the still open problem raised in 1981 by Djafari Rouhani for nonexpansive sequences in a Banach space. [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]

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]

This article is concerned with a class of nonsmooth constrained convex optimization in a real Hilbert space. Coupling with the penalty method, we propose an automatic system (AS) and a nonautomatic system (NS) modeled by differential inclusions. Under a suitable assumption on the feasible region and a proper condition on the objective and constrained functions, some valuable convergence properties of (AS) are obtained. In order to obtain strong convergence result in general cases, based on evolution differential inclusion, we propose a nonautomatic system (NS). When the control item ε(t) of (NS) satisfies some basal conditions, global and unique existence of the solution, finite time convergence to the feasible region and slow solution choice are obtained. Moreover, under different conditions of ε(t), we give some strong convergence results of (NS). Furthermore, we end the article by numerical experiments to illustrate the efficiency and good performance of the proposed systems in this article. [ABSTRACT FROM AUTHOR]