Sentence examples for be a gauge function from inspiring English sources

Lemma 2.4 Let φ be a gauge function of order r ≥ 1 on J.

If (varphi: mathbb{R^rightarrowmathbb{R^) be a gauge function

Let ((X, d)) be a complete metric space, φ: (mathbb{R^rightarrowmathbb{R^) be a gauge function such that (varphi^{-1}({0})={0}), (varphi (t)>t), and (lim_{mrightarrowinfty}varphi^{m}(t)=+infty) for any (t>0).

Let ((X, mathscr{F}, Delta)) be a complete multidimensional Menger PM-space with Δ a continuous related t-norm of H-type, φ: (mathbb{R^rightarrowmathbb{R^) be a gauge function such that (varphi^{-1}({0})={0}), (varphi (t)>t), and (lim_{mrightarrowinfty}varphi^{m}(t)=+infty) for any (t>0).

Let ((X, mathscr{F}, Delta)) be a complete multidimensional Menger PM-space with Δ a continuous related t-norm of H-type, φ: (mathbb{R^rightarrowmathbb{R^) be a gauge function such that (varphi^{-1}({0})={0}), (varphi (t)< t), and (lim_{mrightarrowinfty}varphi^{m}(t)=0) for any (t>0).

Let ((X, mathscr{F}, Delta)) be a complete multidimensional Menger PM-space with Δ a continuous related t-norm of H-type, φ: (mathbb{R^rightarrowmathbb{R^) be a gauge function such that (varphi^{-1}({0})={0}), (varphi (t)< t), and (lim_{mrightarrow+infty}varphi^{m}(t)=0) for any (t>0).

where φ is a gauge function of order r ≥ 1 on an interval J.

In fact, since, and is a gauge function, then for, and (318).

Lemma 3.6 Suppose x 0 ∈ D is an initial orbital point of f and φ is a gauge function of order r ≥ 1.

If ({varphicolon J to mathbb{R}_) is a quasi-homogeneous function of degree ({r ge1}) on an interval J and ({R > 0}) is a fixed point of φ in J, then φ is a gauge function of order r on ({[0, R]}).

Furthermore, if φ is a gauge function of order r ≥ 1 defined by (2.1) and (2.2), then E ( x n ) ≤ E ( x 0 ) μ P n ( r ) and ϕ ( E ( x n ) ) ≤ s μ r n = ϕ ( x 0 ) μ r n − 1, where μ = ϕ ( E ( x 0 ) ) s and ϕ is nonnegative nondecreasing on J satisfying (2.1) and (2.2).