Uniqueness and stability for generalized Caputo fractional differential quasi-variational inequalities

In this paper, we study a new kind of generalized Caputo fractional differential quasi-variational inequalities in Hilbert spaces. We prove the uniqueness and the stability of the abstract inequality by using generalized singular Gronwall’s lemma, projection operators, and contraction principle. Finally, an example is given to illustrate the abstract results.

Numerical algorithm for fractional order population dynamics model with delay

For a fractional-diffusion equation with nonlinearity in the differentiation operator and with the effect of functional delay, an implicit numerical method is constructed based on the approximation of the fractional derivative and the use of interpolation and extrapolation of discrete history. The source of this problem is a generalized model from population theory. Using a fractional discrete analogue of Gronwall's lemma, the convergence of the method is proved under certain conditions. The resulting system of nonlinear equations using Newton's method is reduced to a sequence of linear systems with tridiagonal matrices. Numerical results are given for a test example with distributed delay and a model example from the theory of population with concentrated constant delay.

On the Well-Posedness of a Fractional Model of HIV Infection

In this paper, we are concerned with the well-posedness of a fractional model of human immunodeficiency virus infection. Namely, using Grönwall’s lemma and Perov’s fixed point theorem, we obtain sufficient conditions for which the considered model admits a unique solution.

Stabilization of the Fractional-Order Chua Chaotic Circuit via the Caputo Derivative of a Single Input

A modified fractional-order Chua chaotic circuit is proposed in this paper, and the chaotic attractor is obtained forq=0.98. Based on the Mittag-Leffler function in two parameters and Gronwall’s Lemma, two control schemes are proposed to stabilize the modified fractional-order Chua chaotic system via the Caputo derivative of a single input. The numerical simulation shows the validity and feasibility of the control scheme.

Near viability for fully nonlinear differential inclusions

We consider the nonlinear differential inclusion x′(t) ∈ Ax(t) + F(x(t)), where A is an m-dissipative operator on a separable Banach space X and F is a multi-function. We establish a viability result under Lipschitz hypothesis on F, that consists in proving the existence of solutions of the differential inclusion above, starting from a given set, which remain arbitrarily close to that set, if a tangency condition holds. To this end, we establish a kind of set-valued Gronwall’s lemma and a compactness theorem, which are extensions to the nonlinear case of similar results for semilinear differential inclusions. As an application, we give an approximate null controllability result.

Uniqueness of the regular waiting-time type solution of the thin film equation

The main result of this paper is the proof of uniqueness of non-negative entropy solutions of the thin film equation ht + (|h|nhxxx)x = 0 for $\frac{7}{4}$ < n < 4. The uniqueness proved under assumptions that the initial data satisfy a finite β-entropy condition for some negative enough exponent β and that the solution is locally monotone at the touchdown point. The new dissipated functional recently constructed by Laugesen (Commun. Pure Appl. Anal., 4(3):613–634, 2005) is used to prove an auxiliary energy equality, and then Grönwall's lemma leads to uniqueness.

Stability Results for Switched Linear Systems with Constant Discrete Delays

This paper investigates the stability properties of switched systems possessing several parameterizations (or configurations) while being subject to internal constant point delays. Some of the stability results are formulated based on Gronwall's lemma for global exponential stability, and they are either dependent on or independent of the delay size but they depend on the switching law through the requirement of a minimum residence time. Another set of results concerned with the weaker property of global asymptotic stability is also obtained as being independent of the switching law, but still either dependent on or independent of the delay size, since they are based on the existence of a common Krasovsky-Lyapunov functional for all the above-mentioned configurations. Extensions to a class of polytopic systems and to a class of regular time-varying systems are also discussed.

Lyapunov stability solutions of fractional integrodifferential equations

Lyapunov stability and asymptotic stability conditions for the solutions of the fractional integrodiffrential equationsx(α)(t)=f(t,x(t))+∫t0tk(t,s,x(s))ds,0<α≤1, with the initial conditionx(α−1)(t0)=x0, have been investigated. Our methods are applications of Gronwall's lemma and Schwartz inequality.