The goal of this study is to refine some numerical radius inequalities in a novel way. The new improvements and refinements purify some famous inequalities pertaining to Hilbert space operators numerical radii. The inequalities that have been demonstrated in this work are not only an improvement over old inequalities but also stronger than them. Several examples supporting the validity of our results are provided as well.

Interpolative Kannan contractions are a refinement of Kannan contraction, which is considered as one of the significant notions in fixed point theory. Gb-metric spaces is considered as a generalized concept of both concepts b-metric and G-metric spaces therefore, the significant fixed and common fixed point results of the contraction based on this concept is generalized results for both concepts. The purpose of this manuscript, is to take advantage to interpolative Kannan contraction together with the notion of Ωb which equipped with Gb-metric spaces and H simulation functions to formulate two new interpolative contractions namely, (H, Ωb)-interpolative contraction for self mapping f and generalized (H, Ωb)-interpolative contraction for pair of self mappings (f1, f2). We discuss new fixed and common fixed point theorems. Moreover, to demonstrate the applicability and novelty of our theorems, we formulate numerical examples and applications to illustrate the importance of fixed point theory in applied mathematics and other sciences.

The concepts of b-metric spaces and $\omega t$ -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, $\mathcal{H}$ -simulation function which is used in this manuscript together with the notion of $\omega t$ -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.

The conceptions of b-metric spaces and metric spaces play a remarkable role in proving many theorems of uniqueness and existence solution of such equations as integral or differential equations. The conception of extended b-metric spaces is considered as a generalization concept of b-metric spaces and metric spaces and this concept was employed to unify some new fixed point results in the literature. On the other hand, a new concept of simulation functions founded in 2020 in the name of ℋ-simulation functions and employed this class of functions to unify some fixed point results in the literature. The main objective of this manuscript, is to establish a new contraction namely, (γ, ϕ, θ) ℋ-contraction, this contraction based on the concepts of extended b-metric spaces and the class of functions (ℋ-simulation functions) and the class of functions Θ-functions and the class of functions Φ-functions, We utilize this new contraction to unify the uniqueness and existence of fixed point results in the literature. In addition, some illustrative examples and an interesting application were established to show the novelty of our work.

The main goal of this study is to formulate an effective iterative scheme, namely, an n^th− composite iterative scheme for approximating the fixed point of a self‐map with weak contraction property. We show that the n^th− composite iterative scheme is faster than the scheme obtained by Sintunavarat–Pitea’s iterative scheme. We present some examples using the MATLAB simulator to illustrate our results. Finally, we approximate the solution of some integral equations using our scheme and the Sintunavarat–Pitea scheme.

In this work, new refinements of some numerical radius inequalities are proved. Namely, new improvements and refinements purify the recent inequalities of some famous inequalities concerning the numerical radius of Hilbert space operators. The proven inequalities in this work are not only an improvement over old inequalities, but rather they are stronger than them. Several examples that support the validity of our results are established as well.