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This paper deals with the second-order linear differential equation x ″ + a ( t ) x ′ + b ( t ) x = 0 , where a and b are periodic coefficients. The main purpose is to present new criteria which guarantee that all nontrivial solutions are nonoscillatory and that those are oscillatory. Our nonoscillation theorem and oscillation theorem are proved by using the Riccati technique. In our theorem, the composite function of an indefinite integral of b and a suitable multiple-valued continuously differentiable function are focused, and the composite function of them plays an important role. The results obtained here include a result by Kwong and Wong [15] and a result by Sugie and Matsumura [26] . An application to a equation of Whittaker–Hill type is given to show the usefulness of our results. Finally, simulations are also attached to illustrate that our oscillation criterion is sharp.

This paper studies the problem of stochastic stability and disturbance attenuation for a class of linear continuous-time uncertain systems with Markovian jumping parameters. The uncertainties are assumed to be nonlinear and state, control and external disturbance dependent. A sufficient condition is provided to solve the above problem. An H∞ controller is designed such that the resulting closed-loop system is stochastically stable and has a disturbance attenuation γ for all admissible uncertainties. It is shown that the control law is in terms of the solutions of a set of coupled Riccati inequalities. A numerical example is included to demonstrate the potential of the proposed technique.

For the delay differential equation x t + ∑ k = 1 m a k t x g k t = 0, g k t ≤ t, with impulsive conditions x τ j = A j x τ j − 0 , x τ j = B j x τ j − 0 , a connection between the following properties is established: nonoscillation of the differential equation and the corresponding differential inequality, positiveness of the fundamental function and the existence of a solution of a generalized Riccati inequality. Explicit conditions for nonoscillation and oscillation and comparison theorems are presented.

In this paper we establish several new results regarding the positivity and nonnegativity of discrete quadratic functionals F associated with discrete symplectic systems. In particular, we derive (i) the Riccati inequality for the positivity of F with separated endpoints, (ii) a characterization of the nonnegativity of F for the case of general (jointly varying) endpoints, and (iii) several perturbation-type inequalities regarding the nonnegativity of F with zero endpoints. Some of these results are new even for the special case of discrete Hamiltonian systems.