Sentence examples for tools for the proof from inspiring English sources

Key tools for the proof are amenable traces and measured groupoids associated with generalized box spaces.

The condition and the limit relative category (see [3]) are useful tools for the proof of the main theorem.

Since and are strongly indefinite functinals, we use the notion of the condition and the limit relative category instead of the notion of condition and the relative category, which are the useful tools for the proofs of the main theorems.

This is the key tool for the proof of Theorem 1.1.

As the basic tool for the proof of our main theorem, we introduce the following theorem known as the global continuation theorem.

It is well known that Sperner's lemma became a simple tool for the proof of the existence of Brouwer fixed points.

In traditional game theory, fixed point theory in topological spaces or metric spaces has been an essential tool for the proof of the existence of Nash equilibria of noncooperative games, in which the payoff functions of the players take real values (see [1 6]).

The main tool for the proof of Theorem 1 is Amann's theorem on three fixed points (see, e.g., [[5], Theorem 7.F and Corollary 7.40]): Theorem 4 Let X be a real Banach space with an order cone having a nonempty interior.

First of all, let us introduce some mathematical tools for the following proof, which can be found in [23 26].

Remark 3.1 It is easy to see that the problem w ∈ M is essential, if and only if, the mapping S : M ⇉ X is lower semicontinuous at w. First of all, let us introduce some mathematical tools for the following proof.

The mathematical tool for this proof is the so-called consistently linearized eigenproblem in the frame of the Finite Element Method.