We have suggested a new modified iterative scheme for approximating a common fixed point of two $G$-nonexpansive mappings. Our approach was based on an iterative scheme in the context of Banach spaces via directed graphs. First, we proved a weak convergence theorem using the Opial's property of the underlying space. A weak convergence result without the Opial's property was also given. After this, we established several strong convergence theorems using various mild conditions. We also carried out some numerical simulations to examine the main techniques. Eventually, we obtained an application of our result to solve split feasibility problems (SFP) in the context of $G$-nonexpansive mappings.

We introduce a Ćirić type contraction principle in a vector-valued $b$-metric space that generalizes Perov's contraction principle. We investigate the possible conditions on the mappings $W,E:G\to G$ ( $G$ is a non-empty set), for which these mappings admit a unique common fixed point in $G$ subject to a nonlinear operator $\mathbf{F}:{\mathbb{P}}^m\to {\mathbb{R}}^m$. We illustrate the hypothesis of our findings with examples. We consider an infectious disease model represented by the system of delay integro-differential equations and apply the obtained fixed point theorem to show the existence of a solution to this model.

The target of this manuscript is to introduce new symmetric fractional $\alpha$- $\beta$- $\eta$- $\mathrm{{\rm Y}}$-contractions and prove some new fixed point results for such contractions in the setting of $M_b$-metric space. Moreover, we derive some results for said contractions on closed ball of mentioned space. The existence of the solution to a fractional-order differential equation with one boundary stipulation will be discussed.

Fixed-point theory, a major field of mathematics, analyzes outcomes that remain unchanged under particular operators, featuring multiple applications in mathematics, physics, engineering, computer science, and economics. This study presents the $D^{\ast }$ iteration technique, a robust and effective iterative scheme for approximating fixed points in Suzuki generalized nonexpansive mappings. Within the context of uniformly convex Banach spaces, the novel scheme's weak and strong convergence properties were carefully addressed. The efficiency of this approach was demonstrated through detailed theoretical, numerical, and graphical assessments. Additionally, the stability of the iterative process was established. The method is used to generalize and enhance previous findings by approximating solutions for a fractional differential problem.

Let $S$ be a polynomial ring over a field $K$ and $I$ be the edge ideal associated with the bristled graph of some four or five regular circulant graph. We discuss the depth, projective dimension, regularity and Stanley depth of $S/I$.