Sentence examples for main result see from inspiring English sources

Instead, for our main result (see Theorem 3.1), we need the stronger assumption as follows.

First we will give some definitions about time scales before presenting our main result (see [28, 29]).

The last one gives a monotonicity rule for the ratio of two Laplace transforms, which is crucial to proving our main result (see [34, Remark 3]).

Many authors undertook further investigations in this direction to obtain some generalizations and extensions of the above main result (see, e.g., [5 7]).

We want to prove a bound for the solution v of the above problem (see Lemma 3.1 below), which will be the primary technical tool in the proof of our main result (see the next Section).

We begin this section with the basic error bound for an integer derivative, and then we refer to fractional derivatives, which are important for the main result (see [22] for more details).

The following fact plays an important role in obtaining the main results (see e.g. [12]).

We end the introduction by showing the following lemma which is used to prove our main results (see [7]).

The main results (see Theorem 3.1 and Theorem 3.2) with the detailed proofs can be found in Sect.

In [8], the authors also mentioned some existing results can be considered as a particular case of their main results, see e.g. [11 19].

In this section, we recalled some fundamental definitions and lemmas which are required to demonstrate our main results (see [20 24, 32 35]).