In this paper, by using the ideas of the second author, we establish several intimate connections among the most simple infimum and supremum completeness properties of a generalized ordered set. That is, an arbitrary set equipped with an arbitrary inequality relation. In particular, we obtain straightforward extensions of some basic theorems on partially ordered sets. Due to the equalities inf ...

We present a simple framework which enables us to consider the well-known sup-t-norm and inf-residuum products of relations as two particular cases of a single, more general type of product. We present basic properties of the framework and consequences for the theory of fuzzy relations. Informally, the paper implies that in many cases of fuzzy relational modeling, such as in solving fuzzy relat...

and Applied Analysis 3 Definition 2.1. Let F : K → 2Rn be a set-valued mapping. F is said to be i monotone on K if for each pair of points x, y ∈ K and for all x∗ ∈ F x and y∗ ∈ F y , 〈y∗ − x∗, y − x〉 ≥ 0, ii maximal monotone on K if, for any u ∈ K, 〈ξ − x∗, u − x〉 ≥ 0 for all x ∈ K and all x∗ ∈ F x implies ξ ∈ F u , iii quasimonotone on K if for each pair of points x, y ∈ K and for all x∗ ∈ F ...

We propose a 3D conforming-nonconforming mixed finite element for solving symmetric stress Stokes equations. The low-order conforming finite elements are not inf-sup stable. The low-order nonconforming finite elements do not satisfy the Korn inequality. The proposed finite element space consists of two conforming components and one nonconforming component. We prove that the discrete inf-sup con...

A generally nonconvex optimization problem with equality constraints is studied. The problem is introduced as an “inf sup” of a generalized augmented Lagrangian function. A dual problem is defined as the “sup inf’ of the same generalized augmented Lagrangian. Sufftcient conditions are derived for constructing the augmented Lagrangian function such that the extremal values of the primal and dual...

To solve variational indefinite problems, a celebrated tool is the Banach–Nečas– Babuška theory, which relies on the inf-sup condition. Here, we choose an alternate theory, T-coercivity. This theory relies on explicit inf-sup operators, both at the continuous and discrete levels. It is applied to solve Helmholtz-like problems in acoustics and electromagnetics. We provide simple proofs to solve ...

We show that the sup-t-norm and inf-residuum types of fuzzy relational equations, considered in the literature as two different types, are in fact two particular instances of a single, more general type of equations. We demonstrate that several pairs of corresponding results on the sup-t-norm and inf-residuum types of equations are simple consequences of single results regarding the more genera...

This paper is concerned with an abstract inf-sup problem generated by a bilinear Lagrangian and convex constraints. We study the conditions that guarantee no gap between related sup-inf problems. The key assumption introduced in generalizes well-known Babuška–Brezzi condition. It based on condition defined for cones function spaces. also apply regularization method convenient solving derive com...

We estimate the constants sup x∈(0,1) sup f∈C[0,1]\Π1 |Bn(f,x)−f(x)| ω2 f, x(1−x) n and inf x∈(0,1) sup f∈C[0,1]\Π1 |Bn(f,x)−f(x)| ω2 f, x(1−x) n , where Bn is the Bernstein operator of degree n and ω2 is the second order modulus of continuity. 2000 Mathematical Subject Classification: 41A36, 41A10, 41A25,