We first introduce the concept of a triangular 2-α-η-admissible mapping which extends the notion of α-admissible mapping with respect to η to 2-metric spaces.Next, we introduce the concepts of modified weak and modified rational α-ψ-contractions and establish the existence and uniqueness of fixed points for such mappings in complete 2-metric spaces. As an application of the obtained results, we prove some fixed point results in partially ordered 2-metric spaces. The presented theorems generalize and improve certain existing results in the literature and provide main results in Dung and Hang (Fixed Point Theory Appl. 2013:161, 2013 as corollaries. Moreover, some examples and an application to integral equations are provided to illustrate the usability of the obtained results. MSC: 46N40; 47H10; 54H25; 46T99

We compare the rate of convergence for some iteration methods for contractions. We conclude that the coefficients involved in these methods have an important role to play in determining the speed of the convergence. By using Matlab software, we provide numerical examples to illustrate the results. Also, we compare mathematical and computer-calculating insights in the examples to explain the reason of the existence of the old difference between the points of view. MSC: 47H09; 47H10

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