University of MaraghehSahand Communications in Mathematical Analysis2322-580713120190201Rational Geraghty Contractive Mappings and Fixed Point Theorems in Ordered $b_2$-metric Spaces1792122926310.22130/scma.2017.29263ENRoghaye Jalal ShahkoohiDepartment of Mathematics, Aliabad katoul Branch, Islamic Azad University, Aliabad katoul, Iran.Zohreh BagheriDepartment of Mathematics, Azadshahr Branch, Islamic Azad University, Azadshahr, Iran.Journal Article20170123In 2014, Zead Mustafa introduced $b_2$-metric spaces, as a generalization of both $2$-metric and $b$-metric spaces. Then new fixed point results for the classes of rational Geraghty contractive mappings of type I,II and III in the setup of $b_2$-metric spaces are investigated. Then, we prove some fixed point theorems under various contractive conditions in partially ordered $b_2$-metric spaces. These include Geraghty-type conditions, conditions that use comparison functions and almost generalized weakly contractive conditions. Berinde in [17-20] initiated the concept of almost contractions and obtained many interesting fixed point theorems. Results with similar conditions were obtained, textit{e.g.}, in [21] and [22]. In the last section of the paper, we define the notion of almost generalized $(psi ,varphi )_{s,a}$-contractive mappings and prove some new results. In particular, we extend Theorems 2.1, 2.2 and 2.3 of Ciric et.al. in [23] to the setting of $b_{2}$-metric spaces. Also, some examples are provided to illustrate the results presented herein and several interesting consequences of our theorems are also provided. The findings of the paper are based on generalization and modification of some recently reported theorems in the literature.http://scma.maragheh.ac.ir/article_29263_f6da7913f6ba4c54cf195c5e7a308a31.pdf