ABSTRACT The inclusion of kernels in Hardy‐type operators is a significant research direction. However, deriving results for general kernels remains an open problem. Consequently, current studies focus on Hardy‐type operators with kernels from specific classes satisfying particular conditions. This work investigates discrete Hardy‐type inequalities involving iterated operators and . The aim of the article is to obtain necessary and sufficient conditions for the inequality in the case of parameters and . For the case , the boundedness criterion for quasilinear operators and was established for a general class of matrices, and for the cases and , the criterion was obtained for matrices belonging to wider classes. These results allow us to study generalized bilinear operators. In particular, the results on quasilinear inequalities with matrices provide alternative proofs for estimates of bilinear inequalities. In addition, the importance of the obtained results lies in their application to the study of difference equations and the embedding theory of weighted Sobolev spaces.
ABSTRACT The standard class of fractional‐order neural networks with a single discrete‐time delay is examined in this study. By using the contraction mapping principle and Krasnoselskii fixed point result, we establish novel sufficient conditions to ensure the existence, uniqueness, and uniform stability of solutions. Under Lipschitz‐continuous activation functions, we derive a contraction condition guaranteeing the uniqueness of the solution on a finite time interval. Under bounded activation functions, we prove the existence of at least one solution via Krasnoselskii's theorem. Furthermore, if the initial histories of the solutions are close enough, uniform stability of the delayed fractional‐order neural networks is established in defined time intervals. Finally, the theoretical results are illustrated through examples. The work extends and generalizes previous studies on fractional‐order neural networks by incorporating time delays, which are biologically and physically significant in neural signal transmission.
ABSTRACT This study presents a combined analytical and dynamical investigation of the stochastic generalized nonlinear Schrödinger equation incorporating Kerr nonlinearity, higher‐order dispersion, and multiplicative stochastic perturbations. Using the modified Kudryashov method, explicit traveling‐wave solutions corresponding to bright, dark, singular, and composite soliton structures are derived. The dynamical behavior of the reduced system is investigated through phase portraits, Poincaré sections, sensitivity analysis, stroboscopic bifurcation diagrams, and largest Lyapunov exponents. The results reveal transitions among periodic, quasiperiodic, and chaotic regimes as the external forcing parameters are varied. In the stochastic solutions, the Wiener‐process contribution enters through the phase, producing random phase modulation while preserving the deterministic amplitude envelope. Additional visualizations of the real part and stochastic phase show that, within the considered parameter family, increasing the magnitude of the stochastic parameter enhances phase fluctuations and wavefront distortions. These findings demonstrate the combined influence of nonlinear, dispersive, forcing, and stochastic effects on the dynamics of the SGNLSE and provide a unified framework for exact soliton solutions and dynamical characterization in nonlinear optical media.
This article investigates a two‐point problem for a doubly singularly perturbed third‐order integro‐differential equation. The problem formulation is presented, which is then reduced to a parameterized problem using the parameterization method. The corresponding initial‐value problem is solved, and an algebraic equation system with respect to the introduced parameters is determined. The conditions for the existence and uniqueness of the solution are also established. The corresponding modified unperturbed problem is considered. Using the parameterization method, it becomes a parameter problem. The solution is obtained via the resolvent of a Fredholm integral equation of the second kind, and sufficient conditions for the uniqueness of the solution to the modified unperturbed problem are provided. The interrelation between singularly perturbed and modified unperturbed problems, an appropriate estimate is established, and provides an example illustrating the results obtained in the article.
We study interval‐valued Volterra dynamics under the ‐calculus, which combines the generalized Hukuhara difference with the general quantum operator on a continuous real interval; a thermal model with hereditary heat exchange is the main application. The coefficients and kernel are real‐valued, while the initial state and forcing may be interval‐valued. Under explicit boundedness and contraction assumptions, all bounded solutions share one scalar midpoint trajectory, and their radii solve a scalar ‐Volterra equation driven by a binary sign selection . We prove that every solution‐admissible selection generates a bounded solution and, conversely, that every such solution arises from at least one such selection. Under a sufficient uniform width condition, the constant positive selection yields an outer interval envelope. For the thermal model with nonnegative exchange coefficients, the type‐ band also encloses every admissible crisp trajectory, whereas the type‐ band need not contain them. Further results include an orbit form of the second fundamental theorem, fixed‐type well‐posedness and Ulam‐Hyers stability, a classical Jackson limit, and a truncated ‐orbit Picard method whose a priori error bound separates iteration and orbit‐truncation errors. Dimensionless experiments with constant and fading‐memory kernels illustrate the common midpoint, switching points and orbit‐induced jumps, the geometric truncation rate, and the convergence of fixed‐type solutions to the classical hereditary model.
