Sentence examples for basic notations and lemmas from inspiring English sources

To formulate our main results on the existence of nontrivial solutions for the system ((P_{lambda })), we present the following basic notations and lemmas.

To apply critical point theory to discuss the existence of solutions for problem (1.1), we present some basic notations and lemmas and construct a variational framework, which will be used in the proof of our main results.

In this section, we introduce the basic notations and a lemma of majorization, which will be used in Section 4. For a detailed description of majorization, the readers are referred to [8] and [9].

In the next section we introduce the basic notations and two basic lemmas.

In Section 2 we briefly give some basic notations, definitions and lemmas.

For convenience of the reader, we list some basic notations, definitions, and lemmas, which will be used in the subsequent sections.

In the next section, we generally introduce some basic notations and concepts, related properties under sub-linear expectations and preliminary lemmas that are useful to prove the main theorems.

Section 3 introduces basic notations and assumptions.

In Section 2, we recall some basic notations and concepts.

In Section 2, the basic notations and assumptions are introduced.

In Section 2, we collect some basic notations and definitions.