In this paper, we investigate the existence and uniqueness of a fixed point of almost contractions via simulation functions in metric spaces. Moreover, some examples and an application to integral equations are given to support availability of the obtained results.

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Orthogonal metric space is a considerable generalization of a usual metric space obtained by establishing a perpendicular relation on a set. Very recently, the notions of orthogonality of the set and orthogonality of the metric space are described and notable fixed point theorems are given in orthogonal metric spaces. Some fixed point theorems for the generalizations of contraction principle via altering distance functions on orthogonal metric spaces are presented and proved in this paper. Furthermore, an example is presented to clarify these theorems.

<abstract><p>In 2017, the concepts of orthogonal set and orthogonal metric spaces are presented. And an extension of Banach fixed point theorem is proved in this type metric spaces. Further in 2019, on orthogonal metric spaces, some fixed point theorems via altering distance functions are investigated. In this paper, presence and uniqueness of fixed points of the generalizations of contraction principle via auxiliary functions are investigated. And some consequences and an illustrative example are presented. On the other hand, homotopy theory constitute an important area of algebraic topology, but the application of fixed point results in orthogonal metric spaces to homotopy has not been done until now. As a different application in this field, the homotopy application of the one of the corollaries is given at the end of this paper.</p></abstract>

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Very recently, orthogonal cone metric spaces, orthogonal completeness and orthogonal continuity are described in [4]. Also some fixed point theorems and their conclusions are proved on orthogonal cone metric spaces in there. In this paper, some fixed point theorems for contractive mappings are presented on ordered orthogonal cone metric spaces. Also an illustrative example is given.

Some common fixed point results involving implicit contractions on soft quasi metric spaces are presented in this research article. Also, the well posedness property of the common fixed point problem of mappings is defined and a theorem is given about it. Finally, some fixed point results on soft G-metric spaces are indicated to be urgent outcomes of main theorems are given in this article .

Fixed-point theory and symmetry are major and vigorous tools to working nonlinear analysis and applications, specially nonlinear operator theory and applications. The subject of examining the presence and inimitableness of fixed points of self-mappings defined on orthogonal metric spaces has become very popular in the latest decade. As a result, many researchers reached more relevant conclusions. In this study, the notion of ϕ-Kannan orthogonal p-contractive conditions in orthogonal complete metric spaces is presented. W-distance mappings do not need to satisfy the symmetry condition, that is, such mappings can be symmetrical or asymmetrical. Self-distance does not need to be zero in w-distance mappings. The intent of this study is to enhance the recent development of fixed-point theory in orthogonal metric spaces and related nonlinear problems by using w-distance. On this basis, some fixed-point results are debated. Some explanatory examples are shown that indicate the currency of the hypotheses and grade of benefit of the suggested conclusions. Lastly, sufficient cases for the presence of a solution to nonlinear Fredholm integral equations are investigated through the main results in this study.