The Banach fixed-point theorem gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. By contrast, the Brouwer fixed-point theorem is a non- constructive result : it says that any continuous function from the closed unit ball in n -dimensional Euclidean space to itself must have a fixed point,  but it doesn't describe how to find the fixed point See also Sperner's lemma. The Lefschetz fixed-point theorem  and the Nielsen fixed-point theorem  from algebraic topology is notable because it gives, in some sense, a way to count fixed points. There are a number of generalisations to Banach fixed-point theorem and further; these are applied in PDE theory.
Fixed point (mathematics)
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In this paper we study and generalize some common fixed point theorems in compact and Hausdorff spaces for a pair of commuting mappings with new contraction conditions. The results presented in this paper include the generalization of some fixed point theorems of Fisher, Jungck, Mukherjee, Pachpatte and Sahu and Sharma. Fixed point theorey is a fascinating topic for research in modern analysis and topology. The study and research in fixed point theory began with the pioneering work of Banach 2 , who in presented his remarkable contraction mapping theorem popularly known as Banach contraction mapping principle. It has widespread applications in both pure and applied mathematics.
The special Issue is focused on latest achievements in fixed point theory and its applications. It reflects both state-of-the-art theoretical research and important recent advances in applications. This special issue will also focus on the iterative methods for finding the approximate solutions of various fixed point problems, equilibrium problems and optimization problems.