Sentence examples for the completeness of space from inspiring English sources

By using R-weak commutativity of type ( A g ) and non-compatible conditions of self-mapping pairs in generalized metric space, without the conditions for the completeness of space and the continuity of mappings, we establish some new common fixed point theorems for two self-mappings.

As applications of our results, we characterize the completeness of -space (cone metric spaces and quasimetric spaces are special cases of -space) and studying the Ekeland type variational principle for single variable vector-valued functions as well as for multivalued bifunctions in the setting of cone metric spaces.

Moreover; we will prove the completeness of the space Let be any Cauchy sequence in the space where Then, for a given there exists a positive integer such that.

In particular, we study the completeness of the space.

In particular, we study the completeness of this space.

It remains to prove the completeness of the space l ( λ 2, p ).

It remains to prove the completeness of the space ℓ ∞ ( N ).

By the completeness of the space E, we can assume that x n → q ∈ E as n → ∞.

Thus { x n } is a Cauchy sequence in X, and the completeness of the space ( X, p s ) implies that { x n } converges and so there exists x ¯ ∈ X such that lim n → ∞ x n = x ¯.

Thus, (2.4) shows that the sequence ( t n ( x 0 ) ) n ∈ N is Cauchy and, as a consequence of the completeness of the space, it converges to an element x ∗ ∈ X.

Subsequent results by Nadler Jr. [11], and others address mainly the problem of replacing the completeness of the space X by the existence of fixed points (which was ensured otherwise by the completeness of X) and various relaxations on the contraction constant k.