The monotone iterative technique for impulsive hybrid set valued integro-differential equations

We develop the monotone method for impulsive hybrid set integro-differential equations in all its generality. Some interesting observations are presented.     

Stabilizing Trench''s algorithm to invert symmetric Toeplitz matrices

Presented here is a stable algorithm that uses Zohar's formulation of Trench's algorithm and computes the inverse of a symmetric Toeplitz matrix including those with vanishing or nearvanishing leading minors. The algorithm is based on a diagonal modification of the matrix, and exploits symmetry and persymmetry properties of the inverse matrix.     

Instability criteria for impulsive hybrid state dependent delay integrodifferential systems

In this paper, we present the sufficient conditions ensuring the instability of the zero solution of an impulsive state dependent delay integrodifferential system for different conditions on the delay function. We assume the instability of the associated linear impulsive system and apply the idea of dichotomies together with Schauder–Tychonoff theorem to establish the sufficient conditions for the unstable solution of the problem.     

Existence and approximation of solutions of forced duffing type integro-differential equations with three-point nonlinear boundary conditions

A generalized quasilinearization technique is developed to obtain an analytic approximation of the solutions of the forced Duffing type integro-differential equation with nonlinear three-point boundary conditions. Monotone sequences of approximate solutions converging uniformly and quadratically to a unique solution of the problem are presented.     

Multiple periodic solutions in impulsive hybrid neural networks with delays

The existence of multiple periodic solutions and their exponential stability are investigated for impulsive hybrid Hopfield-type neural networks with both time-dependent and distributed delays, using the Leray-Schauder fixed point theorem and Lyapunov functionals. The criteria given are easily verifiable, possess many adjustable parameters, and depend on impulses, providing flexibility for the analysis and design of delayed neural networks with impulse effects. Examples are given.     

Analytical and numerical methods for the stability analysis of linear fractional delay differential equations

In this paper, several analytical and numerical approaches are presented for the stability analysis of linear fractional-order delay differential equations. The main focus of interest is asymptotic stability, but bounded-input bounded-output (BIBO) stability is also discussed. The applicability of the Laplace transform method for stability analysis is first investigated, jointly with the corresponding characteristic equation, which is broadly used in BIBO stability analysis. Moreover, it is shown that a different characteristic equation, involving the one-parameter Mittag-Leffler function, may be obtained using the well-known method of steps, which provides a necessary condition for asymptotic stability. Stability criteria based on the Argument Principle are also obtained. The stability regions obtained using the two methods are evaluated numerically and comparison results are presented. Several key problems are highlighted.     

Extensions of the method of quasilinearization

Recently, the method of quasilinearization has been generalized, extended, and refined (Refs. 1–2). In this paper, various results are obtained which offer monotone sequences that provide lower and upper bounds for the solution and converge quadratically, when the function involved admits a decomposition of the difference of two convex functions.     

Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions

Using the Mellin transform approach, it is shown that, in contrast with integer-order derivatives, the fractional-order derivative of a periodic function cannot be a function with the same period. The three most widely used definitions of fractional-order derivatives are taken into account, namely, the Caputo, Riemann-Liouville and Grunwald-Letnikov definitions. As a consequence, the non-existence of exact periodic solutions in a wide class of fractional-order dynamical systems is obtained. As an application, it is emphasized that the limit cycle observed in numerical simulations of a simple fractional-order neural network cannot be an exact periodic solution of the system.