In the present work, we establish fixed-point theorems for rational and almost contractions in the framework of a θ_R-metric space. The concept of θ_R-metric space is based on the incorporation of a control function θ:[0,∞)³→[0,∞) into the quadrilateral inequality of the generalized metric space introduced by Branciari. To prove our results, we impose certain natural assumptions on the control function, namely normalization, monotonicity, continuity, and iterative contractive control. These conditions ensure the well-posedness of the θ_R-metric structure and play a crucial role in establishing the convergence of the associated iterative sequences. Then, we ensure the existence (and uniqueness) of a common fixed point for a pair of self-mappings, satisfying rational and almost contractive conditions within this framework. Moreover, a detailed comparison is carried out between the results obtained in this study and several well-known fixed-point and common fixed-point theorems established in rectangular metric spaces and other related generalized metric spaces. This comparison clearly demonstrates that our results significantly extend, generalize, and refine many existing theorems available in the literature. To illustrate the applicability and wide scope of the established theorems, appropriate and carefully constructed examples are presented. These examples substantiate the theoretical results and illustrate that the established theorems constitute proper extensions and meaningful improvements over previously known findings.

In this article, we examine the propagation of small-amplitude dust ion-acoustic (DIA) solitary waves in an unmagnetized three-component dusty plasma composed of negatively charged mobile dust, ions, and electrons obeying the Kaniadakis distribution. By applying the standard reductive perturbation method, the Korteweg–de Vries (KdV) and the modified KdV (mKdV) equations are derived to explore the effects of key plasma parameters, viz., the Kaniadakis deformation parameter (κ), the dust charge number (Z_d), and the initial streaming speed of ions (u_i0) and dusts (u_d0), on the evolution of DIA solitary waves. A wide range of notable features emerges, showing how the solitary profiles respond to variations in key plasma parameters. The analysis reveals that for certain values of initial ion streaming speed and dust charge number, a non-existence region of KdV-DIA solitons arises. Both compressive and rarefactive KdV-DIA solitons and only compressive mKdV-DIA solitons are observed under the same plasma conditions. For compressive structures, the soliton amplitude increases with increasing Z_d and decreasing κ. The results further demonstrate that the Kaniadakis deformation parameter (κ) significantly modifies the properties of mKdV-DIA solitons, while it does not influence the characteristics of KdV-DIA solitons. In addition, the dependence of the width of DIA solitons on various plasma parameters is also investigated.

Introduction.
Smoothness, roughness, and divisor structures are central topics in analytic and probabilistic number theory. In previous work, we introduced a ratio, R, associated with non-square integers, defined through a partition of their divisors, and conjectured that the condition R=2 is equivalent to P(n) being greater than the square root of n, where P(n) denotes the largest prime factor of n. Although this conjecture appears simple at first glance, its proof seems highly nontrivial. This difficulty motivates a broader investigation of the behaviour of R beyond the special case R=2.

Methods.
We analyse the general structure and distribution of the ratio R for non-square integers. Our approach combines asymptotic methods, heuristic arguments, and tools from probabilistic number theory, including the Dickman-de Bruijn function, to study how arithmetic properties such as smoothness and the size of the largest prime factor influence the possible values of R.

Results.
We identify several new patterns in the behaviour of R and provide evidence that its distribution reflects deeper structural properties of integer factorisation. In particular, we show that the case R=2 belongs to a broader family of phenomena, and we describe conditions under which other values of R arise.

Conclusions.
This work provides a more comprehensive perspective on the ratio R and highlights its potential as a tool for understanding smoothness and divisor structures. The results open several directions for future research, including a proof of the original conjecture and extensions to related arithmetic functions.

In nonlinear approximation theory, understanding the structure of approximation spaces and their interplay with greedy algorithms has been a central pursuit. In this paper, we study a generalization of the classical approximation spaces associated with a wide class of bases in separable, infinite-dimensional quasi-Banach and Banach spaces, including almost greedy bases. These approximation spaces, which quantify the decay of best $n$-term approximation errors relative to a basis, are deeply connected to structural properties of the basis and the efficiency of greedy selection procedures. Using weighted Lorentz sequence spaces as a tool, we provide a comprehensive characterization of these generalized approximation spaces in terms of embeddings into and from appropriate weighted Lorentz spaces. Our results extend and unify earlier findings for greedy and unconditional bases by identifying necessary and sufficient conditions under which the classical approximation spaces, greedy approximation classes defined via the Thresholding Greedy Algorithm, and Chebyshev–greedy classes coincide. In doing so, we relax several classical assumptions—such as unconditionality and democracy—replacing them with broader notions like truncation quasi-greediness, and extend the characterizations to encompass a larger family of weights. These embeddings and equivalences elucidate the intricate relationships between approximation error decay, greedy algorithm behavior, and the fine summability captured by Lorentz norms, offering new insights into approximation theory and its connections with nonlinear analysis.

In this study, the dynamics of plant-herbivore interactions is explored while incorporating the Allee effect on prey population. This study offers valuable insights into the dynamics of plant-herbivore systems by examining the role of the Allee effect and its influence on population dynamics. The Allee effect, which occurs when reduced population size negatively impacts individual fitness and population growth rate, is considered a critical factor in understanding these interactions. This investigation is focused on analyzing the stability of the system through the examination of fixed points, including a trivial steady-state, a predator-free steady-state, and a coexisting steady-state. Also their local stability criteria is examined with the help of the method of linearization. Bifurcation analysis is utilized to explore conditions leading to qualitative changes in the system’s behavior, including the emergence of complex dynamics. The system undergoes a transcritical bifurcation and Hopf bifurcation at positive steady-state. To address chaotic behavior, chaos control techniques is implemented to stabilize the system at the desired state including the emergence of chaotic attractors. Additionally, numerical simulations is conducted to validate the theoretical and mathematical analyses, offering a visual and qualitative comprehension of the system’s behavior across different parameter settings. For numerical simulations, MATHEMATICA software is used to illustrate the study.

Anisotropic diffusion phenomena arise naturally in several physical models, including crystalline media, non-Newtonian fluids, and nonlinear optics, where the material properties depend directly on direction. In the current work, we investigate a wide class of nonlinear anisotropic quasilinear equations in the whole space IRn, which is driven by a convex Finsler structure and involves a critical Sobolev growth. The critical fact of nonlinearity generates a loss of compactness, which leads to a fundamental difficulty in domains that are unbounded, and it is also closely related to the concentration phenomena.

To be able to model confinement effects and to restore compactness, we introduce an external potential that is assumed to be continuous, strictly positive, and coercive at infinity. This potential can play a role widely analogous to trapping mechanisms in a nonlinear Schrödinger-type model. Within this framework, we develop a method using a variational approach and analyze the associated energy functional. By combining compact embedding arguments with a careful comparison between the Mountain Pass level and the optimal anisotropic Sobolev constant, we prove the existence of at least one positive weak solution.

Our results provide a rigorous analytical framework for anisotropic critical models with external potentials and contribute to the mathematical understanding of direction-dependent nonlinear phenomena in unbounded media.

This paper investigates the occurence of a finite-time blow-up for a solutions of a problem of type stochastic quasilinear
viscoelastic wave equation posed in a given bounded domain. The studied model incorporates a memory
term that is governed by a decreasing relaxation kernel, a nonlinear damping mechanism, and a
logarithmic source nonlinearity, and it is also driven by an additive perturbation in form of a stochastic noise. In this problem we face the combinition of the influence of a viscoelastic memory and a nonlinear damping taht introduces the dissipative effects. Moreover, the combinition of
the logarithmic source term and the stochastic perturbation contributes to potential instability, which results in a delicate analytical balance.
Under appropriately choosen assumptions for each of our problem's component— the relaxation function, the damping exponent, and the
noise intensity—we construct a suitable energy functional and then derive refined energy estimates.
By introducing an appropriate Lyapunov functional and exploiting the non-convex structure
of the logarithmic potential, we establish sufficient conditions for finite-time blow-up with
positive probability. In particular, we prove that if the initial energy is below a critical
negative threshold depending explicitly on the noise intensity, then the corresponding solution
cannot exist globally in time. Moreover, an explicit upper bound for the blow-up time is
obtained in terms of the initial data and the system parameters.

Geometric properties of analytic functions provide deep insights into the structure of the domains on which they act. In particular, 2-degree Blaschke products exhibit a remarkable geometric feature in the unit disc: any two distinct points with equal images lie on a straight line passing through a fixed point. This striking collinearity property reveals a strong connection between algebraic expressions and geometric configurations. The present study aims to generalize this property beyond the unit disc, first to the upper half-plane and then to the compact Riemann surface associated with the multi-valued function √ z, thereby establishing a unified geometric framework.

The investigation proceeds in three stages. First, the classical geometric structure of 2-degree Blaschke products is analyzed using algebraic manipulation and geometric interpretation. Second, the unit disc is mapped conformally to the upper half-plane, where rational Nevanlinna functions are introduced as natural analogues. Their geometric behavior is studied to determine how their collinearity property transforms under this change of domain. Finally, the nonlinear mapping z = w ² and stereographic projection are employed to lift planar curves to the compactified Riemann surface of √ z, allowing a global geometric interpretation on the Riemann sphere.

It is shown that the collinearity property in the unit disc becomes a concyclicity property in the upper half-plane: points with equal function values lie on a circle passing through a fixed base point. Under the squaring map, chords of the unit disc transform into parabolas, which lift to spherical curves on the compact Riemann surface, all intersecting at infinity.

This study demonstrates that geometric properties of analytic functions are preserved and meaningfully transformed under conformal equivalence and compactification. The results highlight the role of Riemann surfaces in connecting algebraic structure, analytic behavior, and global geometry within a coherent framework.

Blaschke products are a class of inner functions that map the unit disk onto itself and the unit circle onto itself, preserving the boundary modulus equal to one, and they play a fundamental role in complex analysis and geometric properties.
The quotients of finite Blaschke products do not map the unit disk onto itself, since they may have poles inside the unit disk, but they preserve the unit circle, maintaining unimodular boundary values almost everywhere on it.
The present study systematically investigates the critical points of quotients of Blaschke products of degrees two and three,
$B(z) = z \left(\displaystyle\frac{z-a}{1-\bar{a}z}\right), |a|>1$,
and there are two distinct critical points on the unit circle given by the intersection of the circles $|z-a|=\sqrt{|a|^{2}-1} ~\text{and}~ |z|=1$. For the case of degree three,
$B(z)=z\left( \displaystyle\frac{z-a}{1-\bar{a}z} \right)\left( \displaystyle\frac{z-b}{1-\bar{b}z} \right)$,
where $a,b > 1$, the critical points are determined by a degree four polynomial. This coincidence shows that the quotient of Blaschke products always has at least one critical point on the unit circle.
These results establish a connection between the critical points and the locus of points, satisfying the pre-image of the unit circle under finite Blaschke products of degrees two and three, as well as under quotients of Blaschke products.

We study the homogeneous Dirichlet problem for the double-phase evolution equation
z = (x, t) ∈ Q_T = Ω × (0, T ), Ω ⊂ R^N , N ≥ 2.
The non-differentiable coefficients a(z), b(z) and the variable exponents p(z), q(z) are given functions. The coefficients a, b are nonnegative and bounded, with and such that in Q_{T, .}
It is shown that if u₀ ∈ W₀^1,r (Ω) with r ≥ max{2, max p(z), max q(z)}, f ∈ L^{N +2}(Q_T ), max |p(z) − q(z)| < , then the problem admits a unique solution with the following properties:
• The solution preserves the initial regularity, for ;
• The gradient acquires higher integrability, |∇u|^{min{p(z),q(z)}+s+r} ∈ L1(Q_T ) for any ;• The solution possesses the second-order regularity,.
The regularity properties remain true in the case f ∈ L^σ (Q_T ) with σ ∈ (2, N + 2), but the admissible values of r are in a bounded interval depending on N and σ.
This is a joint project [1, 2] with Dr. Rakesh Arora from the Indian Institute of Technology at Varanasi, India.
[1] R.Arora, S.Shmarev “Global gradient estimates for solutions of parabolic equations with nonstandard growth”. J. Math. Anal. Appl. 549 (2), 129582, 36 pp., (2025). DOI 10.1016/j.jmaa.2025.129582
[2] R.Arora, S.Shmarev “Irregular double-phase evolution problem: Existence and global regularity” ArXiv, 41 pp., 2025, https://arxiv.org/abs/2507.04924