In this study, we develop the concept of multivalued Suzuki-type θ-contractions via a gauge function and established two new related fixed point theorems on metric spaces. We also discuss an example to validate our results.

This paper aims to provide a comprehensive analysis of the advancements made in understanding Iterative Fixed-Point Schemes, which builds upon the concept of digital contraction mappings. Additionally, we introduce the notion of an Iterative Fixed-Point Schemes in digital metric spaces. In this study, we extend the idea of Iteration process Mann, Ishikawa, Agarwal, and Thakur based on the ϝ-Stable Iterative Scheme in digital metric space. We also design some fractal images, which frame the compression of Fixed-Point Iterative Schemes and contractive mappings. Furthermore, we present a concrete example that exemplifies the motivation behind our investigations. Moreover, we provide an application of the proposed Fractal image and Sierpinski triangle that compress the works by storing images as a collection of digital contractions, which addresses the issue of storing images with less storage memory in this paper.

In this study, inspired by the concept of B-metric-like space (BMLS), we introduce the concept of orthogonal B-metric-like space (OBMLS) via a hybrid pair of operators. Additionally, we establish the concept of orthogonal dynamic system (ODS) as a generalization of the dynamic system (DS), which improves the existing results for analysies such as those presented here. By applying this, some new refinements of the $F_{\perp }$-Suzuki-type ($F_{\perp }$-ST) fixed-point results are presented. These include some tangible instances, and applications in the field of nonlinear analysis are given to highlight the usability and validity of the theoretical results.

The manuscript is an extension of the investigations concerning θ-contractions which have been newly proposed in (Jleli and Samet in J. Inequal. Appl. 2014:38, 2014). The authors generalize the notion of θ-contractions to the case of nonlinear $\theta _{b} $θ b -contraction mappings and prove multi-valued fixed point theorems based on the b-Bianchini–Grandolfi gauge function in b-metric spaces. The manuscript consists of a tangible example which displays the motivation for such investigations. The work is compiled by the application of the proposed nonlinear $\theta _{b}$θ b -contractions to Liouville–Caputo fractional differential equations.

The focus of this paper is to conduct a comprehensive analysis of the advancements made in the understanding of Interpolative contraction, building upon the ideas initially introduced by Karapinar in 2018. In this paper, we develop the notion of Interpolative contraction mappings to the case of non-linear Kannan Interpolative, Riech Rus Ćirić interpolative and Hardy–Roger Interpolative contraction mappings based on controlled function, and prove some fixed point results in the context of controlled metric space, thereby enhancing the current understanding of this particular analysis. Furthermore, we provide a concrete example that illustrates the underlying drive for the investigations presented in this context. An application of the proposed non-linear Interpolative-contractions to the Liouville–Caputo fractional derivatives and fractional differential equations is provided in this paper.