This study investigates the dynamics of two oscillating rigid spheres moving through an infinite porous medium saturated with Stokes fluid flow, addressing the problem of how fluid properties, permeability, frequency, and slip length influence the system. The objective is to model the interactions between the spheres, which differ in size and velocity as they move along the axis connecting their centers while applying slip boundary conditions to their surfaces. We derive the governing field equations using a semi-analytical method and solve the resulting system of equations numerically through a collocation technique. Our novel quantitative results include insights into the drag force coefficients for both in-phase and out-of-phase oscillations of each hydrophobic sphere, considering parameters such as diameter ratio, permeability, frequency, velocity ratios, slip lengths, and the distances between the spheres. Notably, when the spheres are sufficiently far apart, the normalized drag force coefficients behave as if each sphere is moving independently. Additionally, we present streamlines that illustrate the interactions between the spheres across a range of parameters, highlighting the novelty of our findings. A purely viscous medium and no-slip conditions are used to validate the numerical approach and results.

The aim of this paper is to provide a new hybrid contraction that is a combination of Jaggi, Meir-Keeler and Zamfirescu-type contractions in the framework of a complete metric space. The existence and uniqueness conditions of a fixed point of such contractions are investigated. An example is constructed to demonstrate the validity of the obtained results. Furthermore, the presented result is utilized to formulate new criteria for solving a class of nonlinear Fredholm integral equation.

Let $\left({\mathfrak{H}}_{\mathsf{\Theta }},⟨·,·⟩\right)$ be a reproducing kernel Hilbert space over a non-empty set $\mathsf{\Theta }$, and let $\mathrm{A}$ be a non-zero positive operator on ${\mathfrak{H}}_{\mathsf{\Theta }}$. This operator induces a semi-inner product given by ${⟨\xi ,\eta ⟩}_{\mathrm{A}}=⟨\mathrm{A}\xi ,\eta ⟩$ for all $\xi ,\eta \in {\mathfrak{H}}_{\mathsf{\Theta }}$, with the associated seminorm ${\parallel \xi \parallel }_{\mathrm{A}}=\sqrt{{⟨\xi ,\xi ⟩}_{\mathrm{A}}}$. The $\mathrm{A}$-normalized Berezin number and the $\mathrm{A}$-normalized Berezin norm of an $\mathrm{A}$-bounded linear operator $\mathrm{C}$ on ${\mathfrak{H}}_{\mathsf{\Theta }}$ are defined by ${\displaystyle {\mathbf{b}}_{\mathrm{A}}\left(\mathrm{C}\right)=\underset{\gamma \in {\mathsf{\Theta }}_{\mathrm{A}}}{\sup }\left|{⟨\mathrm{C}{\widehat{\mathrm{x}}}_{\gamma }^{\mathrm{A}},{\widehat{\mathrm{x}}}_{\gamma }^{\mathrm{A}}⟩}_{\mathrm{A}}\right|}$ and ${\displaystyle {\parallel \mathrm{C}\parallel }_{{\mathbf{b}}_{\mathrm{A}}}=\underset{\gamma ,\delta \in {\mathsf{\Theta }}_{\mathrm{A}}}{\sup }\left|{⟨\mathrm{C}{\widehat{\mathrm{x}}}_{\gamma }^{\mathrm{A}},{\widehat{\mathrm{x}}}_{\delta }^{\mathrm{A}}⟩}_{\mathrm{A}}\right|}$, where ${\widehat{\mathrm{x}}}_{\gamma }^{\mathrm{A}}={\displaystyle \frac{{\mathrm{x}}_{\gamma }}{\parallel {\mathrm{x}}_{\gamma }{\parallel }_{\mathrm{A}}}}$ and ${\mathsf{\Theta }}_{\mathrm{A}}=\{\gamma \in \mathsf{\Theta }:\parallel {\mathrm{x}}_{\gamma }{\parallel }_{\mathrm{A}}\ne 0\}$. The primary aim of this paper is to establish new sharp bounds and inequalities involving these two quantities and related operator-theoretic notions. In doing so, we propose a novel method to address the challenges of operator bounds. Furthermore, we revisit recent results on generalized Berezin norms, in particular those of Huban’s work in 2022. We show that some of these results rely on the incorrect assumption that the $\mathrm{A}$-Berezin number coincides with the $\mathrm{A}$-Berezin norm for $\mathrm{A}$-selfadjoint operators. By providing corrected arguments and employing tools such as the $\mathrm{A}$-Cartesian decomposition and the generalized Buzano inequality, we develop a consistent and rigorous framework for the study of generalized Berezin symbols in semi-Hilbertian spaces.

This article has two objectives. The first objective is to utilize Darbo's fixed-point theorem ($ \mathbf{\mathcal{FPT}} $) and the measure of noncompactness ($ \mathbf{\mathcal{MNC}} $) technique to study the well-posedness (i.e., the existence and the uniqueness in addition to continuous dependence on the data) of a generalized fractional quadratic integral equation in Orlicz spaces $ L_\psi $. Our second objective is to prove and explain the novel properties of $ g $-fractional operators, such as boundedness, continuity, acting, and monotonicity within $ L_\psi $. As a result of our work, several fractional operators are generalized, unified, and extended, including Riemann-Liouville, Hadamard, and Erdélyi-Kober, as well as related classical and quadratic fractional problems. Our theories are supported by several examples.