Sentence examples for gauge functions of from inspiring English sources

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).

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.

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

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.

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.

Lemma 2.6 Every gauge function of order r ≥ 1 defined by (2.1) and (2.2) is a b-Bianchini-Grandolfi gauge function with coefficient s ≥ 1. Proof It is immediately follows from the first part of Lemma 2.4 and using the fact that P n ( r ) ≥ n for r ≥ 1 and n ≥ 0. □.

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]}).

A function ({varphicolon J to J}) is said to be a gauge function of order ({r ge1}) on J if it satisfies the following conditions: (i) φ is quasi-homogeneous of degree r on J;   (ii) ({varphi(t) le t}) for all ({t in J}).

Let r ≥ 1 ; a function φ : J → J is said to be a gauge function of order r on J if it satisfies the following conditions: (i) φ ( λ t ) ≤ λ r φ ( t ) for all λ ∈ ( 0, 1 ) and t ∈ J,   (ii) φ ( t ) < t for all t ∈ J ∖ { 0 }.  .