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Nazim I. Mahmudov

Publications and source records attributed to Nazim I. Mahmudov.

At least 19 recordsLinked to original sources
arXiv2026

On a Diophantine Equation Involving Lucas Numbers

Let L_t denote the t-th Lucas number. We prove that the Diophantine equation L_m^{n+k} + L_m^n = L_r has no solutions in positive integers r, m, n, and k with m >= 2. In the case n = 1, the proof is based on a precise factorization formula for the difference of two Lucas numbers and the Carmichael Primitive Divisor Theorem. For n >= 2, we apply lower bounds for linear forms in logarithms due to Matveev, combined with Legendre's lemma, an exact divisibility property for powers of Lucas numbers, and computer-assisted computations to complete the proof.

math.NT↗
arXiv2026

Explicit representation of solutions to a linear wave equation with time delay

This paper develops an explicit spectral representation for solutions of a one-dimensional linear wave equation with a constant time delay. The model is considered on a bounded interval with non-homogeneous Dirichlet boundary data and a prescribed history function. To accommodate the loss of global smoothness in time caused by delay terms, solutions are understood in a \textit{stepwise classical sense}, allowing jump discontinuities in the second time derivative at multiples of the delay while maintaining continuity of the solution and its first time derivative. By combining separation of variables with Sturm-Liouville expansions, the delayed PDE is reduced to a family of scalar second-order delay differential equations. Using delay-dependent fundamental solutions, we derive closed-form representation formulas for the modal dynamics and reconstruct the PDE solution as a Fourier series. Convergence conditions guaranteeing uniform convergence and admissibility of termwise differentiation in space are established. A numerical example demonstrates the practical computation of truncated series solutions and their visualization.

math.AP↗
arXiv2026

Finite-Approximate Solvability of Linear Operator Equations

We introduce and study the finite-approximate solvability of operator equations \(Lu = h\) in a Hilbert space setting, where a bounded operator \(L \colon U \to H\) is paired with a finite-dimensional constraint operator \(π\colon H \to H_0\). The objective is to match exactly the prescribed component \(πh\) while approximating the remainder. We prove that the problem of finding \(u\) such that \(\|Lu - h\| < \varepsilon\) and \(π(Lu) = πh\) is solvable for all \(\varepsilon > 0\) if and only if \(αT_α^{-1}h \to 0\) as \(α\to 0^+\). We further show that dropping any of the structural assumptions on \(L\), \(Γ\), or \(π\) leads to a failure of the equivalence. When \(π\colon H \to H_0\) has an infinite-dimensional range that is compactly embedded in \(H\), the operator \(T_α\) may no longer be invertible. However, a Galerkin scheme \(π_n \to π\) recovers approximate solvability through the resolvents \((α(I - π_n) + Γ)^{-1}\).

math.DS↗
arXiv2026

Stochastic Maximum Principles and Linear-Quadratic Optimal Control Problems for Fractional Backward Stochastic Evolution Equations in Hilbert Spaces

This paper develops a comprehensive framework for optimal control of systems governed by fractional backward stochastic evolution equations (FBSEEs) in Hilbert spaces. We first establish a stochastic maximum principle (SMP) as a necessary condition for optimality. This is achieved by introducing spike variations, deriving precise estimates for the associated variational equations, and constructing an adjoint process tailored to the fractional dynamics. Subsequently, we apply this general principle to solve the linear-quadratic (LQ) optimal control problem explicitly. The resulting optimal control is characterized in closed form via the adjoint process and is shown to be governed by a system of coupled fractional forward-backward stochastic equations. Our work bridges fractional calculus with stochastic control theory, providing a rigorous foundation for controlling infinite-dimensional systems with memory and long-range dependencies.

math.OC↗
arXiv2025

Approximate Controllability of Linear Fractional Impulsive Evolution Equations in Hilbert Spaces

This paper investigates the approximate controllability of linear fractional impulsive evolution equations in Hilbert spaces. The system under consideration involves the Caputo fractional derivative of order $0<α\leq 1$, a closed linear operator generating a strongly continuous semigroup, and instantaneous state jumps governed by bounded linear impulse operators. We first derive an explicit representation of the mild solution by combining fractional solution operators with impulsive operators. Using this representation, we characterize the approximate controllability of the system through a necessary and sufficient condition expressed in terms of the convergence of an associated family of impulsive resolvent operators. This resolvent condition extends the classical criterion for approximate controllability to the fractional impulsive setting. To illustrate the applicability of our theoretical results, a concrete example is provided. The analysis presented here bridges the gap between the well-established theory for integer-order impulsive systems and the more complex fractional case, highlighting the distinct challenges and solutions arising from the interplay of fractional dynamics and impulsive effects.

math.OC↗
arXiv2025

Representation of solutions to continuous and discrete first-order linear matrix equations with delay

In this paper, we study continuous and discrete linear delay systems given respectively by \[ \dot{X}(ξ) = A_0 X(ξ) + X(ξ)A_1 + B_0 X(ξ-σ) + X(ξ-σ)B_1 + G(ξ), \] and its discrete analogue \[ X(u+1) = A_0 X(u) + X(u)A_1 + B_0 X(u-m) + X(u-m)B_1 + G(u), \] where \(A_0, A_1, B_0, B_1 \in \mathbb{R}^{d \times d}\) are constant noncommuting matrices, and \(σ>0\), \(m \in \mathbb{N}\) denote the delay parameters. The main objective is to generalize the classical results of \cite{diblik1, diblik2} and to provide explicit representations of the solutions. For this purpose, we present generalized delayed exponential-type systems for both continuous and discrete cases. This approach allows us to remove the restrictive commutativity conditions \(B_1G(ξ)=G(ξ)B_1\) and \(B_1Ψ(ξ)=Ψ(ξ)B_1\) imposed in \cite{diblik1, diblik2}, thus obtaining explicit solution formulas for more general classes of systems.

math.DS↗
arXiv2025

Necessary and sufficient conditions for relative controllability of discrete linear delay systems

In this paper, we investigate delayed linear difference systems and establish several fundamental results. We first provide a Kalman-type rank condition tailored for delayed linear difference systems. Furthermore, we construct the discrete Gramian matrix and prove its non-singularity, which is essential for analyzing system properties. Additionally, we obtain necessary and sufficient conditions ensuring the relative controllability of the system. Finally, we formulate the control function for effective system control and validate our theoretical findings with numerical examples. These examples illustrate the practical behavior of discrete linear delay systems.

math.OC↗
arXiv2025

Explicit solution of second-order delayed discrete equations

A system of inhomogeneous second-order difference equations with linear parts given by noncommutative matrix coefficients are considered. Closed form of its solution is derived by means of newly defined delayed matrix sine/cosine using the Z-transform and determining function.

math.DS↗
arXiv2025

Solvability and Optimal Controls of Impulsive Stochastic Evolution Equations in Hilbert Spaces

This paper investigates the solvability and optimal control of a class of impulsive stochastic differential equations (SDEs) within a Hilbert space setting. First, we establish the existence and uniqueness of mild solutions for the proposed impulsive stochastic system, leveraging fixed-point theorems and appropriate analytical techniques. Next, we identify and derive the necessary conditions for the existence of optimal control pairs, ensuring the feasibility and effectiveness of the control solutions. Finally, to validate and demonstrate the practical applicability of our theoretical findings, we provide a detailed example showcasing the utility of the results in real-world scenarios.

math.OC↗
arXiv2025

Remarks on finite-approximate controllability of impulsive evolution systems via resolvent-like operator in Hilbert spaces

In this manuscript, we examine impulsive evolution systems in Hilbert spaces. Using a resolvent-like operator, we first establish the finite-approximate controllability for linear systems. Subsequently, by applying the Schauder fixed-point theorem (SFPT), we prove the existence of a solution and demonstrate the finite-approximate controllability of semilinear impulsive systems in Hilbert spaces. Finally, we extend these results to a broader application, specifically to the heat equation.

math.OC↗
arXiv2024

On the Diophantine Equation $F_n = F_l^k (F_l^m-1)$

In this paper, we examine the Diophantine problem given by the equation $F_n = F_l^k (F_l^m - 1)$, where $n, l, m \geq 1$ and $k \geq 3$. Here, $\{ F_t \}_{t=0}^{\infty} $ denotes the Fibonacci numbers, defined by the recurrence relation $F_0 = 0$, $F_1 = 1$, and $F_t = F_{t-1} + F_{t-2}$ for $t \geq 2$. By applying Matveev's theorem, which provides lower bounds for linear forms in logarithms of algebraic numbers, along with a modified Baker-Davenport reduction method and a divisibility property of Fibonacci numbers, we show that $(n, l, k, m) = (6, 3, 3, 1)$ is the only positive integer quadruple that satisfies this equation.

math.NT↗
arXiv2024

Singular mean-field backward stochastic Volterra integral equations in infinite dimensional spaces

This paper investigates the well-posedness of singular mean-field backward stochastic Volterra integral equations (MF-BSVIEs) in infinite-dimensional spaces. We consider the equation: \[X(t) = Ψ(t) + \int_t^b P\big(t, s, X(s), \aleph(t, s), \aleph(s, t), \mathbb{E}[X(s)], \mathbb{E}[\aleph(t, s)], \mathbb{E}[\aleph(s, t)]\big) ds - \int_t^b \aleph(t, s) dB_s, \] where the focus lies on establishing the existence and uniqueness of adapted M-solutions under appropriate conditions. A key contribution of this work is the development of essential lemmas that provide a rigorous foundation for analyzing the well-posedness of these equations. In addition, we extend our analysis to singular mean-field forward stochastic Volterra integral equations (MF-FSVIEs) in infinite-dimensional spaces, demonstrating their solvability and unique adapted solutions. Finally, we strengthen our theoretical results by applying them to derive stochastic maximum principles, showcasing the practical relevance of the proposed framework. These findings contribute to the growing body of research on mean-field stochastic equations and their applications in control theory and mathematical finance.

math.PR↗
arXiv2024

Approximate controllability of impulsive semilinear evolution equations in Hilbert spaces

Several dynamical systems in fields such as engineering, chemistry, biology, and physics show impulsive behavior by reason of unexpected changes at specific times. These behaviors are described by differential systems under impulse effects. The current paper examines approximate controllability for semi-linear impulsive differential and neutral differential equations in Hilbert spaces. By applying a fixed-point method and semigroup theory, a new sufficient condition is provided for the ($\mathcal{A}$-controllability) approximate controllability of neutral and impulsive differential equations (IDEs). To demonstrate the value of the suggested consequences, three examples are presented, offering improvements over some recent findings.

math.OC↗
arXiv2024

Finite Time Stability Analysis for Fractional Stochastic Neutral Delay Differential Equations

In this manuscript, we investigate a fractional stochastic neutral differential equation with time delay, which includes both deterministic and stochastic components. Our primary objective is to rigorously prove the existence of a unique solution that satisfies given initial conditions. Furthermore, we extend our research to investigate the finite-time stability of the system by examining trajectory behavior over a given period. We employ advanced mathematical approaches to systematically prove finite-time stability, providing insights on convergence and stability within the stated interval. Using illustrative examples, we strengthen this all-encompassing examination into the complicated dynamics and stability features of fractionally ordered stochastic systems with time delays. The implications of our results extend to various fields, such as control theory, engineering, and financial mathematics, where understanding the stability of complex systems is crucial.

math.DS↗
arXiv2024

Euler-Maruyama approximation for stochastic fractional neutral integro-differential equations with weakly singular kernel

This manuscript examines the problem of nonlinear stochastic fractional neutral integro-differential equations with weakly singular kernels. Our focus is on obtaining precise estimates to cover all possible cases of Abel-type singular kernels. Initially, we establish the existence, uniqueness, and continuous dependence on the initial value of the true solution, assuming a local Lipschitz condition and linear growth condition. Additionally, we develop the Euler-Maruyama method for the numerical solution of the equation and prove its strong convergence under the same conditions as the well-posedness. Moreover, we determine the accurate convergence rate of this method under global Lipschitz conditions and linear growth conditions.

math.NA↗
arXiv2024

Delayed Gronwall inequality with weakly singular kernel

Delay Gronwall inequality with a weakly singular kernel has been a subject of interest in various mathematical studies. In this article, we will delve into the consideration of this inequality and its application in the study continuity of the state trajectory for a Volterra integral equation with delay. Using delay Gronwall inequality with a singular kernel, we investigate the behavior and properties of the state trajectory if there are delays. This analysis aims to improve our understanding of the dynamics associated in del Volterra integral equations with delay. In addition, we present a comprehensive example demonstrating the practical significance of our results.

math.DS↗
arXiv2024

A Pillai-Catalan-type problem involving Fibonacci numbers

This paper addresses A Pillai-Catalan-type problem assosiated with Fibonacci numbers. Let $F_{n}$ be the Fibonacci numbers defined by the recurrence relation $F_{1}=1$, $F_{2}=1$ and $F_{n}=F_{n-1}+F_{n-2}$ for all $n\geq 3$. We will find all positive integer solution to the equation $3^{x}-F_{n}2^{y}=1$ using properties of Fibonacci numbers, linear forms of logarithms, and the Baker-Davenport reduction method.

math.NT↗