Sentence examples for main theorems see from inspiring English sources

Besides the existence, the localization of solutions will be obtained in our main theorems (see Theorem 5.1 and Theorem 5.2).

The main theorems (see Theorems 3.8, 3.9) proved in Section 3.3 generalize (1.12), (1.11) for the case of matrix eigenvalue problems (1.1), (1.2).

We assume that the reader is familiar with the standard notations and results in Nevanlinna's value distribution theory of meromorphic functions such as the characteristic function, proximity function, counting function, the first and second main theorems (see, e.g., [1 4]).

From now on, we restrict ourselves to the model problem (2.7) which is the only non-standard step in the proof of the main theorem, see Remark 2.3.

The main tools of the paper are the following well-known fixed-point index theorems (see [2 4]).

It is assumed that the reader is familiar with the standard symbols and fundamental results of Nevanlinna theory such as the characteristic function T (r, f ), proximity function m r, f ), counting function N r, f ), the first and second main theorem etc.,(see [1, 2]).

Although their notion seems to be attractive, in this paper, we show how this concept can be reduced to the unidimensional notion of coincidence point, and how their main theorems can be seen as particular cases of existing results.

To prove our main results, we need the following theorem (see [29]).

Unfortunately, there are some significant deficiencies in the course of the proofs of their main theorems in [8] (see Remark 3.1 and Remark 3.3).

In Section 3, by using the symmetric mountain pass theorem (see [19]), we get the main result of this paper.

From the main theorem, we can see that if (|D|=|A|), then (|D - A |=0).