In this paper, we prove some new fixed and common fixed point results in the framework of partially ordered quasi-metric spaces under linear and nonlinear contractions. Also we obtain some fixed point results in the framework of G-metric spaces.

The concepts of b-metric spaces and ω t -distance mappings play a key role in solving various kinds of equations through fixed point theory in mathematics and other science. In this article, we study some fixed point results through these concepts. We introduce a new kind of function namely, H -simulation function which is used in this manuscript together with the notion of ω t -distance mappings to furnish for new contractions. Many fixed point results are proved based on these new contractions as well as some examples are introduced. Moreover, we introduce an application on matrix equations to focus on the importance of our work.

Alegre and Marin [C. Alegre, J. Marin, Topol. Appl., 203 (2016), 32-41] introduced the concept of modified ω-distance mappings on a complete quasi metric space in which they studied some fixed point results. In this manuscript, we prove some fixed point results of nonlinear contraction conditions through modified ω-distance mapping on a complete quasi metric space in sense of Alegre and Marin.

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The ω -distance mapping is one of the important tools that can be used to get new contractions in fixed point theory. The aim of this paper is to use the concept of modified ω -distance mapping to introduce the notion of rational ( α , β ) φ - m ω contraction. We utilize our new notion to construct and formulate many fixed point results for a pair of two mappings defined on a nonempty set A. Our results modify many existing known results. In addition, we support our work by an example.