Oscillation Theorems for Second-Order Quasilinear Neutral Functional Differential Equations

New oscillation criteria are established for the second-order nonlinear neutral functional differential equations of the form (r(t)|z′(t)|α−1z′(t))’+f(t,x[σ(t)])=0, t≥t0, where z(t)=x(t)+p(t)x(τ(t)), p∈C1([t0,∞),[0,∞)), and α≥1. Our results improve and extend some known results in the literature. Some examples are also provided to show the importance of these results

Observer-based Leader-following Consensus for Positive Multi-agent Systems Over Time-varying Graphs

This paper addresses the leader-following consensus problem for discrete-time positive multi-agent systems over time-varying graphs. We assume that the followers may have mutually different positive dynamics which can also be different from the leader. Compared with most existing positive consensus works for homogeneous multi-agent systems, the formulated problem is more general and challenging due to the interplay between the positivity requirement and high-order heterogeneous dynamics. To solve the problem, we present an extended version of existing observer-based design for positive multi-agent systems. By virtue of the common quadratic Lyapunov function technique, we show the followers will maintain their state variables in the positive orthant and finally achieve an output consensus specified by the leader. A numerical example is used to verify the efficacy of our algorithms

Theory of fractional hybrid differential equations

AbstractIn this paper, we develop the theory of fractional hybrid differential equations involving Riemann–Liouville differential operators of order 0<q<1. An existence theorem for fractional hybrid differential equations is proved under mixed Lipschitz and Carathéodory conditions. Some fundamental fractional differential inequalities are also established which are utilized to prove the existence of extremal solutions. Necessary tools are considered and the comparison principle is proved which will be useful for further study of qualitative behavior of solutions