A sufficient condition of viability for fractional differential equations with the Caputo derivative

AbstractIn this paper viability results for nonlinear fractional differential equations with the Caputo derivative are proved. We give the sufficient condition that guarantees fractional viability of a locally closed set with respect to nonlinear function. As an example we discuss positivity of solutions, particularly in linear case

Nabla derivatives associated with nonlinear control systems on homogeneous time scales

The backward shift and nabla derivative operators, defined by the control system on homogeneous time scale, in vector spaces of one-forms and vector fields are introduced and some of their properties are proven. In particular the formulas for components of the backward shift and nabla derivative of an arbitrary vector field are presented

The Z-Transform Method and Delta Type Fractional Difference Operators

The Caputo-, Riemann-Liouville-, and Grünwald-Letnikov-type difference initial value problems for linear fractional-order systems are discussed. We take under our consideration the possible solutions via the classical Z-transform method. We stress the formula for the image of the discrete Mittag-Leffler matrix function in the Z-transform. We also prove forms of images in the Z-transform of the expressed fractional difference summation and operators. Additionally, the stability problem of the considered systems is studied

Stability of nonlinear hh-difference systems with nn fractional orders

summary:In the paper we study the subject of stability of systems with $h$-differences of Caputo-, Riemann-Liouville- and Grünwald-Letnikov-type with $n$ fractional orders. The equivalent descriptions of fractional $h$-difference systems are presented. The sufficient conditions for asymptotic stability are given. Moreover, the Lyapunov direct method is used to analyze the stability of the considered systems with $n$-orders

On Solutions to Fractional Discrete Systems with Sequential h

We study the subject of a behaviour of the solutions of systems with sequential fractional h-differences. We give formulas for the unique solutions to initial value problems for systems in linear and semilinear cases. Moreover, the sufficient condition that guaranties the positivity of considered systems is presented