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In this work the Brouwer fixed point theorem for a triangle was proved by two methods based on the Sperner Lemma. One of the two proofs of Sperner’s Lemma given in the paper was carried out using the so-called index.

In this work it was proved Matkowski’s fixed point theorem. The consequences of this theorem are also presented.

The paper presents, among others, the golden number φas the limit of the quotient of neighboring terms of the Fibonacci and Fibonacci type sequence by means of a fixed point of a mapping f(x) = 1+1 x of a certain interval with the help of Edelstein's theorem. To demonstrate the equality limn → ∞ fn+1/fn = φ, where fn is n-th Fibonacci number also the formula from Corollary 1 has been applied. It was obtained using some relationships between Fibonacci and Lucas numbers, which were previously justified. [ABSTRACT FROM AUTHOR]