Zeros Of Monotone Operators Research Articles

Proximal point methods with possible unbounded errors for monotone operators in Hadamard spaces

In this paper, we investigate and analyse the strong convergence of the sequence generated by an inexact proximal point method with possible unbounded errors for finding zeros of monotone operators in Hadamard spaces. We show that the boundedness of the generated sequence is equivalent to the zero set of the operator to be nonempty. In this case, we prove the strong convergence of the generated sequence to a zero of the operator. We also provide some applications of our main results and give a numerical example to show the performance of the proposed algorithm.

The zeros of monotone operators for the variational inclusion problem in Hilbert spaces

In this paper, we introduce a regularization method for solving the variational inclusion problem of the sum of two monotone operators in real Hilbert spaces. We suggest and analyze this method under some mild appropriate conditions imposed on the parameters, which allow us to obtain a short proof of another strong convergence theorem for this problem. We also apply our main result to the fixed point problem of the nonexpansive variational inequality problem, the common fixed point problem of nonexpansive strict pseudocontractions, the convex minimization problem, and the split feasibility problem. Finally, we provide numerical experiments to illustrate the convergence behavior and to show the effectiveness of the sequences constructed by the inertial technique.

Asymptotics of a General Second-Order Difference Equation and Approximation of Zeroes of Monotone Operators

ABSTRACTIn this article, we prove several new ergodic, weak, and strong convergence theorems for solutions to the following general second-order difference equationwhere A is a maximal monotone operator in a real Hilbert space H and {cn} and {θn} are positive real sequences. We do not assume A−1(0) ≠ ∅, and we prove among other things that the existence of solutions is in fact equivalent to the zero set of A being 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}.

Asymptotics of a Difference Equation and Zeroes of Monotone Operators

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: where A is a maximal monotone operator in a real Hilbert space H, and {c n } is a positive sequence of real numbers. We do not assume that A −1(0) ≠ ∅, and we prove among other things that the existence of solutions is in fact equivalent to the zero set of A being 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 {c n }. 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.

Iterative approximation of zeroes of monotone operators and system of generalized mixed equilibrium problems

In this paper, we construct a new iterative scheme by hybrid method for approximation of common element of set of zeroes of a finite family of α-inverse-strongly monotone operators and set of common solutions to a system of generalized mixed equilibrium problems in a 2-uniformly convex real Banach space which is also uniformly smooth. Then, we prove strong convergence of the scheme to a common element of the two sets.

A Proximal Point Method in Nonreflexive Banach Spaces

We propose an inexact version of the proximal point method and study its properties in nonreflexive Banach spaces which are duals of separable Banach spaces, both for the problem of minimizing convex functions and of finding zeroes of maximal monotone operators. By using surjectivity results for enlargements of maximal monotone operators, we prove existence of the iterates in both cases. Then we recover most of the convergence properties known to hold in reflexive and smooth Banach spaces for the convex optimization problem. When dealing with zeroes of monotone operators, our convergence result requests that the regularization parameters go to zero, as is the case for standard (non-proximal) regularization schemes.

On the existence of zeros of monotone operators in reflexive Banach spaces

This paper is devoted to the investigation on the existence of zeros of monotone operators in reflexive Banach spaces. We first present a sufficient condition under which single-valued monotone operators have zeros. The obtained theorem includes a previous result as a special case. A necessary and sufficient condition for the existence of zeros of maximal monotone operators is presented.