Sentence examples for be a bifunction such from inspiring English sources

Let h : C × C → R be a bifunction such that.

Let F 2 : C × C → R be a bifunction such that.

Let C be a convex closed subset of a Hilbert space H. Let F 1 : C × C → R be a bifunction such that.

Let C be a nonempty, closed and convex subset of a Hilbert space H. Let Θ : C × C → R be a bifunction such that.

For any given (varepsilon>0 ), let (f : K times K rightarrowmathbb{R} ) be a bifunction such that (i) for any (xin K ), (f x,x) geq0 );   (ii) for every (xin K ), the set ({yin K : f x,y) + varepsilon< 0} ) is convex set;   (iii) for every (yin K ), (f cdot,y) ) is upper semicontinuous.

A mapping f : C → C is a contraction on C if there exists a constant η ∈ (0, 1) such that ||f(x) - f y)|| ≤ η||x - y|| for all x, y ∈ C. In addition, let D : C → H be a nonlinear mapping, φ : C → ℝ ∪ be a real-valued function and let F : C × C → ℝ be a bifunction such that C ∩ dom φ ≠ ∅, where ℝ is the set of real numbers and dom φ = {x ∈ C : φ(x) < +∞}.

A is a p × n matrix, b ∈ ℝ p, f : C × C → ℝ is a bifunction such that f x, x) = 0 for every x ∈ C. Throughout this article, we assume that: (A.1) intC = {x | Ax < b} is nonempty.

The typical form of equilibrium problems is formulated by the Ky Fan inequality as follows (see [1]): Find  x ∗ ∈ C  such that  f ( x ∗, y ) ≥ 0 for all  y ∈ C, where C is a nonempty closed convex subset of R n and f : C × C → R is a bifunction such that f ( x, x ) = 0 for all x ∈ C, shortly EP ( f, C ).

Theorem 4.4 Let F : C × C → R be a bifunction satisfying (A1 - A4) such that EP ( F ) is not empty.

Theorem 3.5 Let C be a nonempty, closed, and convex subset of H. Let F : C × C → R be a bifunction satisfying (A1 - A4) such that E P ( F ) ≠ ∅.

Let be a proper extended real-valued function and let be a bifunction of into such that, where is the set of real numbers and.