Sentence examples for also valid in a from inspiring English sources

The principle is also valid in a space satisfying Opial's condition.

Our Theorem 3.1 is also valid in a wide class of general Banach spaces while Theorem 1.1 is valid in Banach spaces having weakly sequentially duality mappings.

If T is a nonexpansive mapping with a fixed point and the control sequence { α n } is chosen so that ∑ n = 0 ∞ α n ( 1 − α n ) = ∞, then the sequence { x n } generated by the normal Mann iterative algorithm converges weakly to a fixed point of T (this is also valid in a uniformly convex Banach space with the Fréchet differentiable norm).

If is a nonexpansive mapping with a fixed point and the control sequence is chosen so that, then the sequence defined by (1.8) converges weakly to a fixed point of ( this is also valid in a uniformly convex Banach space with the Fréchet differentiable norm [6] ).

If T is a nonexpansive mapping with a fixed point and the control sequence { α n } is chosen so that ∑ n = 0 ∞ α n ( 1 − α n ) = + ∞, then the sequence { x n } generated by original Mann's iterative process (1.1) converges weakly to a fixed point of T (this is also valid in a uniformly convex Banach space with a Frechet differentiable norm [6]).

If is a nonexpansive mapping with a fixed point and the control sequence is chosen so that then the sequence generated by normal Mann's iterative process (1.3) converges weakly to a fixed point of (this is also valid in a uniformly convex Banach space with the Fréchet differentiable norm [4]).

For example, if a periodic pattern (langle (T^2V emptyset W), (U), (1), (3), (7rangle ) is valid in an audit trail (A) then a periodic pattern (langle (T^2V emptyset W), (U), (1), (3), (4rangle ) is also valid in an audit trail (A).

For example, if a periodic pattern (langle emptyset emptyset (T^2V emptyset ), (U), (1), (3), (7rangle ) is valid in an audit trail (A) then a periodic pattern (langle (T^2V)), (U), (3), (3), (9rangle ) is also valid in an audit trail (A).

For example, if a periodic pattern (langle (T^2V emptyset W), (U), (1), (3), (7rangle ) is valid in an audit trail (A) then the periodic patterns (langle Temptyset W), (U), (1), (3), (7rangle ) is also valid in an audit trail (A).

For example, if the periodic patterns (langle (T^2V emptyset W), (U), (1), (3), (7rangle ) and (langle (T^2V emptyset W), (U), (10), (3), (13rangle ) are valid in an audit trail (A) then a periodic patterns (langle (T^2V emptyset W), (U), (1), (3), (13rangle ) is also valid in an audit trail (A).

For example, if the periodic patterns (langle (T^2V emptyset W), (U), (1), (4), (5rangle ) and (langle (T^2V emptyset W), (U), (3), (4), (7rangle ) are valid in an audit trail (A) then a periodic pattern (langle (T^2V emptyset W), (U), (1), (2), (7rangle ) is also valid in an audit trail (A).