In this paper, we obtain closed-form solutions of the following nonlinear fifth-order system of rational difference equations under four distinct cases: \[ \left\{ \begin{array}{l} S_{n+1} =\frac{I_{n-1}I_{n-4}}{S_{n-2}\left( \pm1 \pm I_{n-1}I_{n-4}\right) }, \\[10pt] I_{n+1} =\frac{S_{n-1}S_{n-4}}{I_{n-2}\left( \pm1 \pm S_{n-1}S_{n-4}\right) }, \end{array} \right. \] where \(n\) is a non-negative integer, and the initial conditions \(S_{-4},\ S_{-3}, S_{-2}, S_{-1}, S_{0}, I_{-4}, I_{-3}, I_{-2}, I_{-1}\), and \(I_{0}\) are arbitrary nonzero real numbers. We also explore the conditions under which such systems exhibit periodic solutions. An important part of the discussion is the contribution of initial conditions in deciding the overall behavior of such systems. To validate the theoretical results, numerical simulations were performed with different initial values.

In this paper, we introduce a newly proposed probability distribution, the Gamma Lindley \(q\)- distribution (\(q\)-GLD). We explore its structural properties and derive parameter estimation procedures. The proposed model is then applied in the context of quantum mechanics, where we demonstrate its suitability as a potential solution to the \(q\)-deformed Schrödinger equation. These findings reveal the distribution’s effectiveness in modeling complex physical systems within a \(q\)-deformed framework.

This note presents new criteria for the oscillation of all unbounded solutions of a second-order advanced differential equation \( y{''}(t)=p(t)y( \sigma(t))\). The results obtained are new and the progress is shown with help of Euler differential equation.

This study shows the conditions for local and global asymptotic stability of the equilibrium points in the nonlinear system of difference equations \[ Z_{\eta+1} =\beta_{1}Y_{\eta-1}+\frac{\delta_{1}Y_{\eta-1}Z_{\eta-4}% }{r+Y_{\eta-2}+Z_{-4}},\quad Y_{\eta+1} =\beta_{2}Z_{\eta-1}+\frac{\delta_{2}Z_{\eta-1}Y_{\eta-4}% }{r+Z_{\eta-2}\pm Y_{\eta-4}}.\] The boundedness of the positive solutions of the systems is examined. Additionally, the solutions of the systems are investigated. Numerical examples are presented to show the outcomes.

In this study, we address split variational inclusion problems in Hilbert spaces and introduce a new algorithm that combines multi-inertial extrapolation with a self-adaptive stepsize technique. The method is designed to enhance convergence performance while relaxing typical parameter constraints. Under suitable assumptions, we establish a weak convergence theorem. To validate the effectiveness of the approach, we apply it to real-world problems in image restoration and medical data classification. \begin{keyword}Split variational inclusion problems \sep multi-inertial extrapolation \sep weak convergence \sep image restoration \sep data classification. \MSC{47H09 \sep 47H10 \sep 47J05 \sep 47J25 \sep 49J40 \sep 35A15

The Spectral Tau method is a widely recognized numerical technique employed for solving fractional-order differential equations (FODEs). The central focus of the Spectral Tau method’s algorithm depends on the idea of operational matrices derived from the basis sets of orthogonal polynomials, which are utilized to approximate derivative terms in the problems. In this study, our main objective is to introduce a new numerical method within the class of spectral methods, but distinct in its formulation from the Spectral Tau method. While both approaches utilize operational matrices of orthogonal polynomials, the proposed method avoids the computation of residual functions, which is a key step in the Spectral Tau method. Another important feature of the proposed study is the construction of novel generalized integral operational matrices in the Riemann-Liouville sense, developed using a basis of orthogonal shifted Laguerre polynomials (OSLPs). This structure leads to simplified implementation, reduced computational cost, and enhanced spectral accuracy. To demonstrate the efficiency and practical applicability of the proposed method, we solve several test problems. Additionally, we compare the computational efficiency and the absolute errors obtained using our proposed method with those derived from the Spectral Tau method.

The aim of this paper is to propose Krasnosel'skii-Mann type iteration with double inertial steps for approximating fixed points of nonexpansive mappings in real Hilbert spaces. The weak convergence is proved under some suitable conditions of the parameters. Some applications to the problems of finding a common fixed point of a family of mappings are also given. Finally, several numerical experiments to show the efficiency and accuracy of our method in breast and cervical cancer diseases predictions are presented.

The exponential synchronization of the coronary artery chaos system (CACS) in complex dynamical networks (CDNs) with state and input time-varying delays is being studied for the first time. For the CACS in CDNs, feedback control was envisioned. To enable exponential synchronization of the CACS in CDNs with continuous differential time-varying delays, an appropriate Lyapunov-Krasovskii functional (LKF) was constructed. An extended reciprocally convex matrix inequality, Jensen inequality, and Wirtinger-based integral inequality, which further decreases conservativeness, were considered when establishing the synchronization criterion. The new linear matrix inequalities (LMIs) that are required for exponential synchronization have emerged. Numerical checks may be accomplished using MATLAB's LMI toolbox. To demonstrate the efficiency of the recommended strategies, numerical examples were provided.

This paper deals with the theoretical and numerical aspects for higher order Volterra-Fredholm fractional integro-differential equations (VF-IDEs) under \(\varphi\)-Caputo operator. Using Liptchiz conditions, Krasnoselskii's fixed point theorem, and Gronwall inequality with respect to the function \(\varphi\), existence and uniqueness of the solution are investigated. The stability of the solution is analyzed through the continuity of the parameters. Moreover, a new hybrid technique which is the combination of deep learning artificial neural network and finite difference method (FDL-ANN) is developed to approximate the solution of higher order VF-IDEs. This technique uses the Adaptive Moment Estimation Method (Adam) as an optimization algorithm with feed-forward deep learning to minimize the error function and training the model using five layers with different activation functions. The numerical analysis for the error bound and the computation complexity are provided for FDL-ANN. The numerical examples demonstrated the efficiency of the proposed method in solving the complicated higher order fractional problems of linear and non-linear terms.

The hyperharmonic numbers (and also harmonic numbers), which appear in analysis and combinatorial mathematics and play a significant role in the applications and theory of mathematics, are associated with the Daehee, Cauchy, and Stirling numbers with several extensions and have proven to be powerful tools in varied subjects in combinatorics and analysis. This work considers the generalized degenerate hyperharmonic numbers of order \(b\) and the generalized degenerate harmonic numbers of rank \(b\). Then, we investigate diverse combinatorial identities and summation formulas, including these numbers and some other special degenerate numbers such as Daehee numbers, type 2 Changhee numbers, derangement numbers, degenerate Bernoulli numbers of the second kind, and degenerate Stirling numbers utilizing series manipulation methods. Moreover, the first few values of the generalized degenerate hyperharmonic numbers and generalized degenerate harmonic numbers of rank \(b\) are provided, and some graphical representations are shown.