Sentence examples for further this theorem from inspiring English sources

We will extend further this theorem.

Further, this theorem reduces the condition s 2 q < 1 to the condition s q < 1. Theorem 5.2 Let ( X, d ) be a b-metric space and T be a weak Jungck -contractive mapping such that φ is subadditive.

Further, Theorem 1.1, Theorem 1.2, and Theorem 1.3 gave strong convergence theorems of this problem under the uniform convex assumption on (h_{1}) or (g_{1}).

In this paper, we shall further improve this theorem by using the weighted Ricci curvature condition and remove the term of the S-curvature.

A further proof of this theorem repeats the proof of Lemma 5.3 (we have instead of ).

Kimura [21] obtained the further generalization of this theorem by using the Bregman projection; see also [22].

Further, the theorem is modified to the shift sampling in function spaces by using the Zak transform.

Further, this, together with Theorem 2 ii), shows that (z^{k}_{j}/lambda^{k}_{j}rightarrow 1) for (jin I_).

Some further results based on this theorem are discussed by Yang [10, 11].

Further, Theorem 2.5 contains [[7], Theorem 2] and [[14], Theorem 3] which are single-valued generalizations of the Caristi's fixed point theorem.

Theorem 3.7 is a further extension of Theorem 4.2 in [7] (here Theorem 1.5).