Sentence examples for assumption imposed on from inspiring English sources

The basis is a homogenization/averaging scheme using a micro-randomness assumption imposed on a directional higher gradient interaction term.

Thus the assumption imposed on (B_{i}) in Theorem 10 is valid.

In 2007, Wang [10] introduced an explicit hybrid iteration method for nonexpansive mappings in Hilbert space; and then Osilike et al. [11] extended Wang's results to arbitrary Banach spaces without the strong monotonicity assumption imposed on the hybrid operator.

To obtain Painleve-Kuratowski lower convergence of the solution sets, we introduce a sequence of gap functions based on the nonlinear scalarization function introduced by Chen et al. in [14] and a key assumption imposed on the sequence of gap functions.

We prove that recent results of Wang (2007) concerning the iterative approximation of fixed points of nonexpansive mappings using a hybrid iteration method in Hilbert spaces can be extended to arbitrary Banach spaces without the strong monotonicity assumption imposed on the hybrid operator.

Compared with Theorem A (see [26], Theorem 1.1]), on one hand, the assumptions imposed on f are much weaker.

They obtained at least one or two positive solutions under some assumptions imposed on the nonlinearity of by applying Krasnoselskii fixed point theorem.

It is well-known that ∥T t)fT t g−T t)fg∥t tends to 0 under certain smoothness assumptions imposed on f and g.

With real rank zero added to the assumptions imposed on the C⁎-algebras, ideal-related K-theory is known to be strongly complete for four of these six spaces.

Under the assumptions imposed on Lemma  3.2.

Some additional assumptions imposed on were needed there.