Sentence examples for further our theorem from inspiring English sources

Further, our theorem has been validated through Corollary 6.3, which is a result of Bor [14].

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}).

Simulations further confirm our theorems and visualize the significant improvement of the preconditioned fused Lasso estimator over the vanilla FLSA in exact pattern recovery.

If we assume G is such that E ( G ) : = X × X, then clearly G is connected and our Theorem 2.5(2) improves Nadler's theorem, and further if F is single-valued, then we improve the Banach contraction theorem.

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).

We will extend further this theorem.

Motivated by [16, Theorem ] and [14, Theorem ], we further extend Theorem 3.2 as follows.

Further, applying Theorem 1, we derive the following.

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

Recently Qi et al. [10] further improved Theorem 1.2 as below.