Asian Research Journal of Mathematics

Positive Periodic Solution to a Lienard Equation with Indefinite Weights

Fukun Li — 2026-08-18

This study establishes sufficient conditions for the existence of positive T-periodic solutions to a Lienard equation of the form \(x^{\prime \prime}+f(t, x) x^{\prime}+a(t) x=\frac{h(t)}{x^\lambda}+e(t),\) where a, h, and e are continuous T-periodic functions, h may change sign, λ > 0, and f(t, x) may depend explicitly on time and may be singular as x → 0+.

The analysis addresses the difficulty caused by the time-dependent resistance term, for which the integral cancellation available in the classical f(x)x′ setting is not generally valid, and also permits both a and h to change sign. By constructing appropriate cones in C and using positive Green functions, the periodic problem is reformulated as a fixedpoint problem. The Krasnoselski˘ı–Guo fixed-point theorem is then applied under explicit growth and sign conditions. Two regimes are considered according to the sign of the forcing term e. For positive forcing, the argument uses the Green function associated with x′′ +Nx; for negative forcing, it uses the Green function associated with −x′′ +a−(t)x. The resulting conditions cover both ρ > 1 and ρ = 1 in the estimate \(|f(t, x)| \leq \frac{m}{x^\rho} .\)

A concrete example illustrates the optimal choice of the splitting parameter in the first theorem. These results provide sufficient criteria for positive periodic solutions under indefinite weights and singular resistance. The findings remain confined to the stated sufficient hypotheses.

Oscillation Properties for First-Order Non-Linear Advanced Difference Equations

D. Palanisamy — 2026-08-21

This paper investigates the qualitative behaviour of the following class of first-order non-linear advanced difference equations featuring multiple non-monotone advanced arguments
\[\begin{equation*}
\Delta\omega(\vartheta)-\sum_{i=1}^{k}\rho_i(\vartheta)\omega^{\alpha}(\phi_i(\vartheta))=0;\qquad \vartheta\geq\vartheta_0,
\end{equation*}\]
where \(\alpha\) is a ratio of odd positive integers such that \(\alpha\ge1\), and k is a positive integer, {pi (\(\vartheta\))} are sequences of nonnegative real numbers for 1 ≤ i ≤ k and {\(\phi_i(\vartheta)\)} are sequences of positive integers such that \(\phi_i(\vartheta)\) ≥ \(\vartheta\) + 2. Specifically, we establish new sufficient conditions for the oscillation of all solutions to the difference equation in which the advanced arguments are not assumed to be monotonic. By constructing appropriate iterative sequences and analysing their convergence properties, distinct oscillation criteria are derived for both the linear case (\(\alpha\) = 1 ). The established criteria provide easily verifiable conditions that enhance the existing qualitative theory of non-linear difference equations. Finally, the theoretical findings are illustrated and validated using concrete examples.

Mathematical Framework for Finite-Element Fluid-structure Interaction: Governing Equations, Coupling Strategies, and Verification Protocols

Santosh Kumar — 2026-08-22

Fluid-structure interaction (FSI) couples fluid motion with structural deformation and is central to problems in aeroelasticity, marine and civil engineering, rotating machinery, and biomechanics. The mathematical difficulty arises because the fluid and solid occupy interacting domains, exchange traction and velocity at a moving interface, and may require a strongly coupled numerical solution when feedback between the two fields is significant. This paper develops a self-contained finite-element framework for an incompressible Newtonian fluid interacting with an elastic structure. The formulation is expressed in an arbitrary Lagrangian-Eulerian description for the fluid and a Lagrangian description for the solid. The governing Navier-Stokes and structural momentum equations, interface compatibility conditions, weak forms, nondimensional groups, mesh-motion requirements, and partitioned and monolithic coupling options are specified in a form suitable for implementation. Particular attention is given to numerical stability, including the added-mass difficulty in partitioned FSI, and to verification and validation procedures that separate discretisation error from agreement with physical or benchmark data. A benchmark matrix is proposed for time-dependent flow past a cylinder, vortex-induced vibration of an elastically mounted cylinder, and flexible-plate flutter. Error measures are defined without presenting ungenerated numerical results. The framework is intended as a reproducible methodological basis for subsequent computational studies and for extensions to turbulence, reduced-order modelling, and multiphysics FSI.

A Study of Fuzzy Bitopological Spaces Using α-γ Operators and Their Topological Properties

Anuradha Parmar — 2026-08-24

Fuzzy set theory provides a mathematical framework for representing uncertainty and imprecision through graded membership. This study examines fuzzy bitopological spaces using α-γ operators and considers the topological properties associated with two fuzzy topologies defined on a common underlying set. The analysis focuses on the interaction among fuzzy sets, α- and γ-level structures, and the two topological structures. Preliminary concepts concerning fuzzy sets, fuzzy topological spaces, level sets, bitopological spaces, and fuzzy bitopological spaces are reviewed to establish the framework used in the study. The α-γ interior and α-γ closure operators are then considered as generalised tools for describing openness and closedness in the fuzzy bitopological setting. The study establishes basic relationships between a fuzzy subset and its α-γ interior and closure, together with conditions characterising α-γ-open and α-γ-closed fuzzy sets. It also considers the repeated application of the α-γ interior operator and illustrates the relationships among selected membership levels through simple examples. The discussion shows how fuzzy membership, two topological structures, and α-γ operators can be considered within a common framework. The resulting formulation provides a focused basis for examining generalised topological behaviour in fuzzy bitopological spaces and for further investigation of related properties such as separation, continuity, and convergence.

Transitivity and Imprimitivity of Product of Three Alternating Groups on Cartesian Product of Three Sets of Ordered Quadruples

Moses Khakame Maraka — 2026-08-27

In this paper, we determine the transitivity and primitivity of the product of three alternating groups acting on the cartesian product of three sets of ordered quadruples. When n ≥ 6 and using the Orbit-Stabilizer theorem, the action has been determined to be transitive and using the definition of primitivity and blocks, the action has been determined to be imprimitive.

A Continuous Chebyshevian Hybrid Method for Direct Solution of Second Order Differential Equations

F. O. Obarhua — 2026-08-28

This paper develops a hybrid linear multistep scheme based on Chebyshev polynomials for the direct solution of general second-order ordinary differential equations. The basis function is interpolated at grid points as well as selected off-grid points, and the resulting differential equation is collocated at every grid point. The derived method is shown to be continuous, consistent, symmetric, and zero-stable, and numerical experiments indicate that it produces more accurate results than several existing methods.

A New and Efficient Numerical Algorithm for Solving Fractional Boundary Value Problems

S. K. Talankar — 2026-08-28

This paper presents an efficient numerical approach based on Lagrange interpolation polynomials for solving higher-order nonlinear fractional boundary value problems. The proposed Lagrange Interpolation TransformMethod (LITM), formulated using the Caputo fractional derivative, effectively captures the memory effects inherent in fractional systems. An explicit computational framework is developed, in which the problem is reduced to a system of algebraic equations using Chebyshev collocation points and Lagrange basis functions. The applicability of the method is demonstrated through several nonlinear and higher-order test problems, and the results obtained are compared with existing techniques such as the ChebyshevWavelet Method (CWM) and the Optimal Homotopy Asymptotic Method (OHAM), with the comparisons showing improved accuracy and stable convergence. It is also observed that the fractional solutions converge to their corresponding integer-order solutions as the order approaches unity, confirming the consistency of the formulation. The method is simple, computationally efficient, and applicable to a wide class of fractional and classical boundary value problems.

(\(\vartheta\), \(\vartheta\))-Derivations and \(\vartheta\)-Centralizers of Prime Rings

Maysaa Zaki Salman — 2026-08-28

This paper examines the interaction between nonzero (\(\vartheta\), \(\vartheta\))-derivations and nonzero \(\vartheta\)-centralizers in prime rings, where \(\vartheta\) is an automorphism. The study considers a sequence of algebraic identities involving derivations, centralizers, commutators, and the centre of the ring, and establishes sufficient conditions under which either the derivation vanishes, the ring is commutative, or the derivation is commuting. After recalling the definitions of derivations, (\(\theta\), \(\varphi\))-derivations, centralizers, and \(\theta\)-centralizers, a preliminary lemma is used to support the subsequent arguments. The main results show that several commutator relations between a (\(\vartheta\), \(\vartheta\))-derivation and a \(\vartheta\)-centralizer force commutativity of the prime ring or triviality of the derivation. Additional conditions involving images of commutators, products of the derivation and centralizer, and central elements are shown to imply that the derivation is commuting. Further identities involving the \(\vartheta\)-centralizer alone also yield commutativity. The proofs rely on primeness, the automorphism property of \(\vartheta\), standard commutator identities, substitutions into the assumed relations, and repeated use of the preliminary lemma. Collectively, the results provide a unified set of sufficient conditions connecting (\(\vartheta\), \(\vartheta\))-derivations and \(\vartheta\)-centralizers with commutativity properties in prime rings, while remaining within the algebraic framework specified in the manuscript. The argument proceeds theorem by theorem, preserving the stated assumptions on the mappings and using the same algebraic setting throughou.

Range–kernel Orthogonality for Generalized Derivations Associated with Generalized Finite Pairs

Amenya Collins Sule — 2026-08-29

Let H be a complex Hilbert space, B(H) the algebra of bounded operators on H, and δ_(A,B)(X) = AX − XB the generalized derivation determined by A, B ∈ B(H). A pair (A, B) is generalized finite when ‖AX − XB − I‖ ≥ 1 for every X ∈ B(H). This paper develops a formulation that keeps generalized finiteness, which is a property of the pair (A, B), distinct from range–kernel orthogonality, which concerns the linear map δ_A,B. First, generalized finiteness is shown to be equivalent to the scalar family of estimates ‖δ_A,B(X) − λI‖ ≥ |λ| and hence to Birkhoff–James orthogonality of the identity operator to Ranδ_A,B. A Hahn–Banach separation argument yields a norm-one supporting functional annihilating Ranδ_A,B; this gives a controlled perturbation estimate for additional operators annihilated by the same functional. Second, for 1 < p < ∞, a Schatten-class duality criterion is established: if T ∈ C_p and its normalised duality element J_p(T) belongs to Kerδ_A^†,B^†, then ‖T + λδ _A,B(X)‖_p ≥ ‖T‖_p for all X ∈ C_p and λ ∈ ℂ. For normal A and B, the Fuglede–Putnam theorem ensures this criterion for every T ∈ Kerδ _A,B ∩ C_p , recovering the classical range–kernel orthogonality mechanism in a transparent norm-ideal form. A direct-sum compression corollary records the compatible block-operator setting. The resulting framework avoids assigning a linear kernel to the affine translate δ _A,B − I and provides a consistent basis for studying generalized finite pairs together with range–kernel orthogonality.