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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 some theorems about fixed points of nonexpansive mappings of certain convex subsets of Banach spaces, as well as examples of applications of these theorems, among others, to justify the correctness of the square roots approximation process in Heron's algorithm (from the 1st century AD). [ABSTRACT FROM AUTHOR]

The paper contains several theorems about the Browder type contraction fixed points and some of their applications. [ABSTRACT FROM AUTHOR]

This work is devoted to the application of selected fixed point theorems in the problems of convergence of certain sequences to the golden number. It contains the theorem about the fixed point of so-called ψ-contraction specified on the closed interval < a, b > and the local version of Banach Contraction Principle as a conclusion. It will also be used to approximate the golden number. [ABSTRACT FROM AUTHOR]

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]