Existence, Uniqueness, and Exponential Stability of Uncertain Delayed Neural Networks with Inertial Term: Nonreduced Order Case

In this work, we mainly focus on uncertain delayed neural network system with inertial term. Here, the existence, uniqueness, and exponential stability of inertial neural networks are derived without shifting the second order differential system into first order through substituting variables. Initially, we construct a proper Lyapunov–Krasovskii functional to investigate the stability of novel uncertain delayed inertial neural networks, which is different from the classical Lyapunov functional approach. By utilizing the Kirchhoff’s matrix tree theorem, Cauchy–Schwartz inequality, homeomorphism theorem, and some inequality techniques, the necessary and sufficient conditions are derived for the designed framework. Subsequently, to exhibit the strength of this outcome, we framed a quantitative example.

Approximations of an Equilibrium Problem without Prior Knowledge of Lipschitz Constants in Hilbert Spaces with Applications

The objective of this paper is to introduce an iterative method with the addition of an inertial term to solve equilibrium problems in a real Hilbert space. The proposed iterative scheme is based on the Mann-type iterative scheme and the extragradient method. By imposing certain mild conditions on a bifunction, the corresponding theorem of strong convergence in real Hilbert space is well-established. The proposed method has the advantage of requiring no knowledge of Lipschitz-type constants. The applications of our results to solve particular classes of equilibrium problems is presented. Numerical results are established to validate the proposed method’s efficiency and to compare it to other methods in the literature.

Convergence analysis of a general inertial projection-type method for solving pseudomonotone equilibrium problems with applications

In this paper, we introduce a new algorithm by incorporating an inertial term with a subgradient extragradient algorithm to solve the equilibrium problems involving a pseudomonotone and Lipschitz-type continuous bifunction in real Hilbert spaces. A weak convergence theorem is well established under certain mild conditions for the bifunction and the control parameters involved. Some of the applications to solve variational inequalities and fixed point problems are considered. Finally, several numerical experiments are performed to demonstrate the numerical efficacy and superiority of the proposed algorithm over other well-known existing algorithms.

Regularity analysis for an abstract thermoelastic system with inertial term

In this paper, we provide a complete regularity analysis for an abstract system of coupled hyperbolic and parabolic equations. The asymptotic stability of this model was investigated in [6]. We are able to decompose the parameter region into three parts where the semigroup associated with the system is analytic, of Gevrey class of specific order, and non-smoothing, respectively. Moreover, by a detailed and creative spectral analysis,, we will show that the order of Gevrey class is sharp, under proper conditions. We also show that the orders of the polynomial stability obtained in [6] is optimal.

Pointwise Estimates of Solutions for the Viscous Cahn-Hilliard Equation with Inertial Term

In this paper, we study the pointwise estimates of solutions to the viscous Cahn-Hilliard equation with the inertial term in multidimensions. We use Green’s function method. Our approach is based on a detailed analysis on the Green’s function of the linear system. And we get the solution’s Lp convergence rate.

Free Vibrations of Rigid Kinematic Support of Yu.D. Cherepinsky

The author derives the equation of free vibrations for kinematical rolling support of Yu.D.Cherepinsky. Both support and the surface below are assumed rigid. It is shown that Lagrange equation is similar to the equation of motion for rotational oscillator where the rotation centre, rotational inertia and stiffness are changing every moment, depending on displacements. This equation can be further simplified to the linear form with the error proportional to the second degree of displacements. This equation looks somewhat like the equation for classical pendulum, but effective length in our case is controlled by curvature radii of support and of the rolling surface, as well as by the position of vertical load relative to the centre of the support. Non-linear characteristic is soft. The main no-linearity is in the inertial term, and not in the stiffness term.