In answer set programming, knowledge involving sets of objects collectively is naturally represented by aggregates, which are rewritten into simpler forms known as monotone aggregates by current implementations. However, there is a complexity gap between general and monotone aggregates. In this paper, this gap is filled by means of a polynomial, faithful, and modular translation function, which...

and Applied Analysis 3 (b) η-strongly monotone if there exists a constant η > 0 such that ⟨Sx − Sy, x − y⟩ ≥ η 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 2 , ∀x, y ∈ C; (12) (c) α-inverse-strongly monotone (α-ism) if there exists a constant α > 0 such that ⟨Sx − Sy, x − y⟩ ≥ α 󵄩󵄩󵄩󵄩Sx − Sy 󵄩󵄩󵄩󵄩 2 , ∀x, y ∈ C. (13) Obviously, if S is α-inverse-strongly monotone, then it is monotone and (1/α)-Lipschitz continuous. It can be ...

In this article we study graph-distance convergence of monotone operators. First, we prove a property that has been an open problem up to now: the limit of a sequence of graph-distance convergent maximal monotone operators in a Hilbert space is a maximal monotone operator. Next, we show that a sequence of maximal monotone operators converging in the same sense in a reflexive Banach space is uni...

In this paper, we introduce two iterative algorithms for finding the solution of the sum of two monotone operators by using hybrid projection methods and shrinking projection methods. Under some suitable conditions, we prove strong convergence theorems of such sequences to the solution of the sum of an inverse-strongly monotone and a maximal monotone operator. Finally, we present a numerical re...

In this paper, the authors prove some existence results of solutions for a new class of generalized quasi-variational inequalities (GQVI) for pseudo-monotone type III operators and strongly pseudo-monotone type III operators defined on noncompact sets in locally convex Hausdorff topological vector spaces. In obtaining these results on GQVI for pseudo-monotone type III operators, we shall use Ch...

and Applied Analysis 3 where A n = α n I + A for all n ≥ 0. In particular, if V ≡ 0, then (11) reduces to the following iterative scheme: x 0 = x ∈ C chosen arbitrarily, y n = P C (x n − ] n A n x n ) , z n = β n x n + γ n P C (x n − ] n A n y n ) + σ n TP C (x n − ] n A n y n ) , x n+1 = P C [λ n (1 − δ n ) γSx n + (I − λ n μF) z n ] , ∀n ≥ 0. (12) Further, if S = V, then (11) reduces to the f...

and Applied Analysis 3 Example 8. LetX = Z = R, Y = R, and E = F = [0, +∞) ⊂ X and let G,C : E → 2, e : E → Y define as G (u) = {(U, V) : U − u ≤ V ≤ u − U, 0 ≤ U ≤ u} , ∀u ∈ F, C (x) = {(U, V) : 0 ≤ V ≤ ( 3 2 + x)U,U ≥ 0} , ∀x ∈ E, e (x) = (1, 1) , ∀x ∈ E, (7) respectively. We can see that ξ G (x, u) = (1 + (2/(1 + 2x)))u. Definition 9. LetD ⊂ Y be a cone,H : F → 2, and ς : F → R. ς is calledm...

In this paper, we introduce regularization methods for finding a point, being not only solution monotone variational inequality problem but also common zero an infinite family of inverse strongly non-self operators closed convex subset in real Hilbert space. these methods, finite number the is used at each iteration step. Applications to fixed point strictly pseudo-contractive and split feasibi...

and Applied Analysis 3 the uniform r-prox-regularity C is equivalent to the convexity of C. Moreover, it is clear that the class of uniformly prox-regular sets is sufficiently large to include the class p-convex sets, C1,1 submanifolds possibly with boundary of H, the images under a C1,1 diffeomorphism of convex sets, and many other nonconvex sets; see 6, 8 . Now, let us state the following fac...