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]).
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]).
To prove our main results, we need the following theorem (see [29]).
In Section 3, by using the symmetric mountain pass theorem (see [19]), we get the main result of this paper.
The main tool we use is the Leray-Schauder continuation theorem (see Mawhin [15, Theorem ]).
Finally, starting from (1.3) and using Schauder's fixed point theorem (see [21], Theorem 3.21), we get our main result.
A main tool of seeking the critical points of functional is the mountain pass theorem (see [1 3]).
Main result for self contractions on generalized metric spaces is Perov's fixed point theorem; see [1].
