Sentence examples for gauge function of from inspiring English sources
The following is a sufficient condition for a gauge function of order r.
where φ is a gauge function of order r ≥ 1 on an interval J.
Lemma 2.4 Let φ be a gauge function of order r ≥ 1 on J.
He proposed an iterative scheme for a mapping satisfying a contractive condition which involves a gauge function of order r ≥ 1 and obtained error estimates as well.
Lemma 3.6 Suppose x 0 ∈ D is an initial orbital point of f and φ is a gauge function of order r ≥ 1.
In Section 3 we have established two convergence theorems in the setting of a b-metric such that the self-mapping satisfies a contraction condition involving a gauge function of order r ≥ 1.
Furthermore, taking into account such a crucial condition in order to calculate prior and posterior estimates we consider the gauge functions of the form φ ( t ) = t ϕ ( t ) s for all t ∈ J, (2.1).
Let ({T colon D subset X to X}) be an operator in a cone normed space ({ X,|cdot|)}) over a solid vector space ({(Y,preceq)}), and let ({E colon D to mathbb{R}_) be a function of the initial conditions of T with a strict gauge function φ of order ({r ge1}) on an interval J. Suppose that T is an iterated contraction at a point ξ with respect to E and with control function ϕ satisfying (2.7).
A gauge function φ of order r on J is said to be a strict gauge function if the inequality in (ii) holds strictly whenever ({t in J backslash{ 0 }}).
Theorem 3.7 Let f : D ⊂ X → X be an operator on a complete b-metric space ( X, d ) such that the b-metric is continuous and f satisfies (1.1) with a b-Bianchini-Grandolfi gauge function φ of order r ≥ 1 on an interval J with coefficient s ≥ 1.
Theorem 3.10 Let f : D ⊂ X → X be an operator on a complete b-metric space ( X, d ) such that the b-metric is continuous and let f satisfy (1.1) with a b-Bianchini-Grandolfi gauge function φ of order r ≥ 1 and a coefficient s on an interval J. Further, suppose that x 0 ∈ D is an initial orbital point of f, then the following statements hold true.
