In this article, we prove some common fixed point theorems for generalized rational type contractions in bipolar metric spaces. These theorems also generalize and extend several interesting results of metric fixed point theory to the bipolar metric context. In addition, we provide some examples to illustrate our theorems, and applications are obtained in areas of homotopy theory and integral equations by using iterative methods for mathematical operators on a bipolar metric space.

We present convergence and common fixed point conclusions of the Krasnosel'skii iteration which is one of the iterative methods associated with $\alpha$-Krasnosel'skii mappings satisfying condition (E). Our conclusions extend, generalize and improve numerous conclusions existing in the literature. Examples are given to support our results.

Algal blooms pose a significant threat to the ecological integrity and biodiversity in aquatic ecosystems. In lakes, enriched with nutrients, these blooms result in overgrowth of periphyton, leading to biological clogging, oxygen depletion, and ultimately a decline in ecosystem's health and water quality. In this article, we presented a mathematical model centered around the role of aquatic species (specifically fish population) to alleviate algal blooms. The model analysis revealed significant shifts in dynamics, shedding light on the effectiveness of fish-mediated sustainability strategies to control algal proliferation. Notably, our study identified critical thresholds and regime transitions through the observation of saddle-node bifurcation within the proposed mathematical model. To validate our analytical findings, we have conducted numerical simulations, which provided robust evidence for the resilience of the ecosystem under different scenarios.

The paper deals with numerical analysis of solutions for state variables of a CoVID-19 model in integer and fractional order. The solution analysis for the fractional order model is done by the new generalized Caputo-type fractional derivative and Predictor-Corrector methodology, and that for the integer order model is carried out by Multi-step Differential Transformation Method. We have performed sensitivity analysis of the basic reproduction number with the help of a normalized forward sensitivity index. The Arzelá-Ascoli theorem and Fixed point theorems with other important properties are used to establish a mathematical analysis of the existence and uniqueness criteria for the solution of the fractional order. The obtained outcomes are depicted with the help of diagrams, narrating the nature of the state variables. According to the results, the Predictor-Corrector methodology is favorably unequivocal for the fractional model and very simple in administration for the system of equations that are non-linear. The research done in this manuscript can assure the execution and relevance of the new generalized Caputo-type fractional operator for mathematical physics.

Here we will investigate a retarded damped oscillator with double delays. We looked at the combined effect of retarded delay and feedback delay and found that the retarded delay plays a significant role in controlling the oscillation of the proposed system. Only the negative damping situation is considered in this research. At first, we will find conditions for which the origin of the proposed system becomes a Bogdanov-Takens (B-T) singularity. Also, we extract the second and the third-order normal forms of the Bogdanov-Takens bifurcation by using center manifold theory. At the end, an extensive numerical simulations have been presented to satisfy the theoretical results.

This study investigates set-valued contractions within the framework of $m_v^b-$metric spaces, extending classical contraction principles. By introducing and examining the Hausdorff $m_v^b-$metric, we establish a foundation for set-valued fixed point theorems, thereby contributing significantly to this area of research. Our findings generalize several well-known contraction concepts, including those of Banach, Sehgal, Wardowski, Altun, Bianchini, and Nadler, within the context of $m_v^b-$metric spaces. These advancements have practical implications, particularly in the study of nonlinear systems and the mathematical model of Fredholm integral inclusions. The results presented here emphasize the growing importance of set-valued fixed points and pave the way for further exploration and application across various scientific and engineering domains.

In this manuscript, we investigate the dynamics of higher-order polynomials with complex coefficients by employing the Jungck–Noor iteration scheme (one of the iterative methods) in conjunction with $s$-convexity, which controls the weighting of previous iterates versus current polynomial evaluations, tuning convergence speed, escape dynamics, and fractal density from compact, high-brightness patterns to intricate, detailed structures. This framework enables us to establish new escape criteria and to visualize nonclassical deviations of the celebrated Mandelbrot and Julia sets. The resulting fractal structures display intricate geometries that not only enrich the theoretical study of complex dynamics but also resemble patterns observed in natural systems. To highlight the novelty of our work, we provide both graphical and numerical illustrations that demonstrate how variations in polynomial parameters and iteration settings influence shape transformations, symmetry, color distributions, and computational complexity. A key observation is that each point in the Mandelbrot set encodes detailed information about the corresponding Julia fractal, reinforcing the deep interplay between the two families. Moreover, when real-valued parameters are considered in the polynomial map and iteration process, some fractals exhibit striking motifs suggestive of potential applications, for example, in pattern design within the textile industry. Our study also opens pathways for extending this framework to noise-perturbed systems and physical models, which will be explored in future work.

This paper introduces a novel numerical scheme for estimating option prices under the Black-Scholes (B-S) model. The proposed method utilizes a non-standard finite difference (NSFD) approach that incorporates the powerful techniques of methods of sub-equation and exact finite difference (EFD). The proposed technique exhibits several positive characteristics: It preserves positivity by design, works with large step sizes, ensures dynamic consistency, and enhances stability. Notably, its implicit scheme and construction ensures that the fundamental properties of the solution are accurately captured. Finally, some numerical simulations are provided to demonstrate the effectiveness of the proposed implicit NSFD scheme.

This article dealt with a class of coupled hybrid fractional differential system. It consisted of a mixed type of Caputo and Hilfer fractional derivatives with respect to two different kernel functions, ${\psi }_{{}_1}$ and ${\psi }_{{}_2}$, respectively, in addition to coupled boundary conditions. The existence of the solution of the system was investigated using the Dhage fixed point theorem. Finally, an illustration was presented to validate our findings.

This study investigated the boundary controllability of nonlinear impulsive integro-differential evolution systems (NIIESs) with time-varying delays within Banach spaces. Two classes of NIIESs were considered, and sufficient conditions for their controllability were established using fixed point theorems and semigroup theory. For the first class, Schaefer's fixed point theorem was employed in combination with compact semigroup theory, whereas for the second class, Schauder's fixed point theorem was utilized. The research defined essential hypotheses and mathematical structures to ensure the robustness and applicability of the results. Illustrative examples were provided to confirm the applicability and effectiveness of the developed theoretical framework. This work significantly contributes to the study of partial functional integro-differential equations in nonlinear systems, particularly systems influenced by impulsive effects and time delays, addressing gaps in the existing literature.