Sentence examples for means defined on from inspiring English sources

where { μ n } n = 1 ∞ is a sequence of left strong regular invariant means defined on an appropriate invariant subspace of l ∞ ( S ).

The purpose of this paper is to study the viscosity iterative schemes for approximating a fixed point of an asymptotically nonexpansive semigroup on a compact convex subset of a smooth Banach space with respect to a sequence { μ i, n } i = 1, n = 1 m, ∞ of strongly asymptotic invariant means defined on an appropriate space of bounded real valued functions of the semigroup.

for an asymptotically nonexpansive semigroup φ = { T s : s ∈ S } on a compact convex subset C of a smooth Banach space E with respect to a finite family of left regular sequences { μ i, n } i = 1, n = 1 m, ∞ of invariant means defined on an appropriate invariant subspace of l ∞ ( S ).

Let X be a left invariant φ-stable subspace of L ∞ containing 1, let { μ n } n = 1 ∞ be a sequence of left strong regular invariant means defined on X such that lim n → ∞ ∥ μ n + 1 − μ n ∥ = 0, and let { c n } n = 1 ∞ be a sequence defined by c n = sup x, y ∈ C ( ∥ T μ n x − T μ n y ∥ − ∥ x − y ∥ ), n ≥ 1.

In this paper, we approximate a fixed point of the semigroup φ = { T s : s ∈ S } of Lipschitzian mappings from a nonempty compact convex subset C of a smooth Banach space E into C with a uniform Lipschitzian condition and with respect to a finite family of sequences { μ i, n } i = 1, n = 1 m, ∞ of left strong regular invariant means defined on an appropriate invariant subspace of l ∞ ( S ).

Let X be a left invariant φ-stable subspace of L ∞ containing 1, { μ n } n = 1 ∞ is a sequence of left strong regular invariant means defined on X such that lim n → ∞ ∥ μ n + 1 − μ n ∥ = 0 and { c n } n = 1 ∞ be the sequence defined by c n = sup x, y ∈ C ( ∥ T μ n x − T μ n y ∥ − ∥ x − y ∥ ), n ≥ 1.

In the following theorem we will prove that a weighted mean defined on a ring can be extended to its quotient field denoted as (operatorname {Quot}(R)).

Let I be an interval, (R subset mathbb {R}) be a ring, (mathscr{M}) be a weighted mean defined on I over R. Then there exists a unique mean (widetilde{mathscr{M}}) defined on I over (operatorname {Quot}(R)) such that widetilde{mathscr{M}}vert_{bigcup_{n=1}^{+infty} I^{n} times W_{n}(R)} =mathscr{M}.

Sometimes, this means defining core values.

If is convex on, then the integral arithmetic mean defined in (1.3) is convex on.

Few years later, Wulbert, in [2], proved that the integral arithmetic mean defined in (1.3) is convex on if is convex on.