This technical note describes a monotone and continuous fixpoint operator to compute the answer sets of programs with aggregates. The fixpoint operator relies on the notion of aggregate solution. Under certain conditions, this operator behaves identically to the three-valued immediate consequence operator Φ P for aggregate programs, independently proposed in (Pelov 2004; Pelov et al. 2004). Thi...

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar’s constraint qualification holds. In this paper, we prove the maximal monotonicity of A + ∂f provided that A is a maximally monotone linear relation, and f is a proper lower semicontinuous convex function satisfying domA ∩ int dom...

In this note we provide regularity conditions of closedness type which guarantee some surjectivity results concerning the sum of two maximal monotone operators by using representative functions. The first regularity condition we give guarantees the surjectivity of the monotone operator S(· + p) + T (·), where p ∈ X and S and T are maximal monotone operators on the reflexive Banach space X . The...

Let E be a finite alphabet and let W be the set of words on Z. A subset A of W is said to be generable if it is the closure of a computable inductive operator F over W; if F is monotone, then A is monotone generable. (The basic facts about inductive operators and about RE sets needed in this paper will be given in section one.) Now any generable set is RE and any RE set is I-i reducible to a (m...

In [3] we investigate the problem of defining a well-founded and stable semantics for programs with aggregates. Our work is based on Approximation Theory [1] which is a general algebraic framework for approximating non-monotone operators on a complete lattice L by approximating operators on the bi-lattice L. The theory identifies basic properties of the approximating operators and gives a metho...

In this paper, we consider the numerical solution for the discretization of semilinear elliptic complementarity problems. A monotone algorithm is established based on the upper and lower solutions of the problem. It is proved that iterates, generated by the algorithm, are a pair of upper and lower solution iterates and converge monotonically from above and below, respectively, to the solution o...

and Applied Analysis 3 nonexpansive mapping such that F S ∩ A−10/ ∅. Let {xn} be a sequence generated by x0 x ∈ C and un J−1 ( βnJxn ( 1 − βn ) JSJrnxn ) , Cn { z ∈ C : φ z, un ≤ φ z, xn } , Qn {z ∈ C : 〈xn − z, Jx0 − Jxn〉 ≥ 0}, xn 1 ΠCn∩Qnx0 1.6 for all n ∈ N ∪ {0}, where J is the duality mapping on E, {βn} ⊂ 0, 1 , and {rn} ⊂ a,∞ for some a > 0. If lim infn→∞ 1 − βn > 0, then {xn} converges s...

This technical note describes a monotone and continuous fixpoint operator to compute the answer sets of programs with aggregates. The fixpoint operator relies on the notion of aggregate solution. Under certain conditions, this operator behaves identically to the three-valued immediate consequence operator Φ P for aggregate programs, independently proposed in (Pelov 2004; Pelov et al. 2004). Thi...

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We provide a general regularization method for monotone variational inequalities, where the regularizer is a Lipschitz continuous and strongly monotone operator. We also introduce an...

A version of the Dynamical Systems Gradient Method for solving ill-posed nonlinear monotone operator equations is studied in this paper. A discrepancy principle is proposed and justified. A numerical experiment was carried out with the new stopping rule. Numerical experiments show that the proposed stopping rule is efficient. Equations with monotone operators are of interest in many application...