Since the counterexamples arise for φ1, φ2, in the language unexpanded by "−", the sense in which a generalized version of the earlier equivalence would not be reasonable is that it would not be conservative over the initial logic.
Dedekind also defined the concept of a Euclidean domain, a number system in which a generalized version of the Euclidean algorithm can be defined (as described below).
In [4, Theorem ], this result was proved in the setting of cone metric spaces, but in a generalized version -for a pair of self-mappings satisfying certain conditions.
Note the superscript T of ({E_{t}^{T}} x,y)) in (5) is a generalized version of main orientation.
When the above inequality is true with c="c(D ∈ 0,∞), then we say that the 3G theorem holds in D. In this paper, we establish a generalized version of 3G theorem when D is a bounded κ-fat open set, which includes a bounded John domain.
In particular, we establish a generalized version ofAlexandroff-Pasynkoff theorem.
In the year 1978, a generalized version of the theorem of Hyers for approximately linear mappings was given by Rassias [4].
(1.2) In 1994, based on a generalized version of Aubry-Mather theorem, Pei [18] obtained the existence of Aubry-Mather sets of equation (1.2) for any continuous 2π-periodic function (p(t)) if (A1): (frac {varphi(x)}{x}>varphi'(x)>0), for (|x|geq d>0); (A2): (varphi(x)=o(x)), (|varphi(x -xvarphi'(x -xvarphi'y), as (|x|to+infty); (|xvasphi"(x)|x|to+inftyere the constant (C>0).
In recent years, many researches focused their attention on the study of a generalized version in q-calculus of the well-known linear and positive operators [1 6].
