Geometric approach to fixed point

In this thesis, theorems that have been studied in the subject of Fixed Point Theory since 1922 (Banach, 1922) have been examined in more general metric space structures. Contraction transformations have enabled diversification of fixed point theorems and examples in these metric spaces. The mentioned examples are realized by re-giving the known definitions of shapes such as ellipses, circles, hyperbolas, known as fixed geometric shapes, from a new point of view in these spaces. It is explained which contraction transformations and under what conditions the examples that satisfy the conditions of the metric space in which each shape is located are provided. The relevant conditions are given using generalizations of Proinov-type E-contraction, Jleli-Samet-type and (θ,α,β) type contraction transformations on S-metric space and S_b-metric space, and the structure of these contractions is studied in the examples of circle, ellipse, hyperbola, Cassini curve, Apollonius curve, respectively.