Dynamic Analysis of Software Systems with Aperiodic Impulse Rejuvenation

This paper aims to obtain the dynamical solution and instantaneous availability of software systems with aperiodic impulse rejuvenation. Firstly, we formulate the generic system with a group of coupled impulsive differential equations and transform it into an abstract Cauchy problem. Then we adopt a difference scheme and establish the convergence of this scheme by applying the Trotter–Kato theorem to obtain the system’s dynamical solution. Moreover, the instantaneous availability as an important evaluation index for software systems is derived, and its range is also estimated. At last, numerical examples are shown to illustrate the validity of theoretical results.

Existence of solutions for the semilinear abstract Cauchy problem of fractional order

In this paper we establish the existence of solutions for the nonlinear abstract Cauchy problem of order α ∈ (1, 2), where the fractional derivative is considered in the sense of Caputo. The autonomous and nonautonomous cases are studied. We assume the existence of an α-resolvent family for the homogeneous linear problem. By using this α-resolvent family and appropriate conditions on the forcing function, we study the existence of classical solutions of the nonhomogeneus semilinear problem. The non-autonomous problem is discussed as a perturbation of the autonomous case. We establish a variation of the constants formula for the nonautonomous and nonhomogeneous equation.

New interpolation spaces and strict Hölder regularity for fractional abstract Cauchy problem

We know that interpolation spaces in terms of analytic semigroup have a significant role into the study of strict Hölder regularity of solutions of classical abstract Cauchy problem (ACP). In this paper, we first construct interpolation spaces in terms of solution operators in fractional calculus and characterize these spaces. Then we establish strict Hölder regularity of mild solutions of fractional order ACP.

Normal periodic solutions for the fractional abstract Cauchy problem

We show that if A is a closed linear operator defined in a Banach space X and there exist $t_{0} \geq 0$ t 0 ≥ 0 and $M>0$ M > 0 such that $\{(im)^{\alpha }\}_{|m|> t_{0}} \subset \rho (A)$ { ( i m ) α } | m | > t 0 ⊂ ρ ( A ) , the resolvent set of A, and $$ \bigl\Vert (im)^{\alpha }\bigl(A+(im)^{\alpha }I \bigr)^{-1} \bigr\Vert \leq M \quad \text{ for all } \vert m \vert > t_{0}, m \in \mathbb{Z}, $$ ∥ ( i m ) α ( A + ( i m ) α I ) − 1 ∥ ≤ M for all | m | > t 0 , m ∈ Z , then, for each $\frac{1}{p}<\alpha \leq \frac{2}{p}$ 1 p < α ≤ 2 p and $1< p < 2$ 1 < p < 2 , the abstract Cauchy problem with periodic boundary conditions $$ \textstyle\begin{cases} _{GL}D^{\alpha }_{t} u(t) + Au(t) = f(t), & t \in (0,2\pi ); \\ u(0)=u(2\pi ), \end{cases} $$ { D t α G L u ( t ) + A u ( t ) = f ( t ) , t ∈ ( 0 , 2 π ) ; u ( 0 ) = u ( 2 π ) , where $_{GL}D^{\alpha }$ D α G L denotes the Grünwald–Letnikov derivative, admits a normal 2π-periodic solution for each $f\in L^{p}_{2\pi }(\mathbb{R}, X)$ f ∈ L 2 π p ( R , X ) that satisfies appropriate conditions. In particular, this happens if A is a sectorial operator with spectral angle $\phi _{A} \in (0, \alpha \pi /2)$ ϕ A ∈ ( 0 , α π / 2 ) and $\int _{0}^{2\pi } f(t)\,dt \in \operatorname{Ran}(A)$ ∫ 0 2 π f ( t ) d t ∈ Ran ( A ) .

Finite Rank Solution for Conformable Degenerate First-Order Abstract Cauchy Problem in Hilbert Spaces

In this paper, we find a solution of finite rank form of fractional Abstract Cauchy Problem. The fractional derivative used is the Conformable derivative. The main idea of the proofs are based on theory of tensor product of Banach spaces.

Reliability Analysis of Supply Chain System with the Multi-Suppliers and Single Demander

The time-dependent solution of a kind of supply chain system with the multi-suppliers and single demander is investigated in this paper. By choosing state space and defining operator of system, we transfer model into an abstract Cauchy problem. We are devoted to studying the unique existence of the system solution and its exponential stability by using the theory of C 0-semigroup. We prove that the system operator generates C 0-semigroup by the theory of cofinal operator and resolvent positive operator. We derive that the system has a unique nonnegative dynamic solution exponentially converging to its steady-state one which is the eigenfunction corresponding eigenvalue 0 of the system operator.

Stability and Bifurcation Analysis in a Predator–Prey Model with Age Structure and Two Delays

In this paper, a predator–prey model with age structure, Beddington–DeAngelis functional response and time delays is considered. Using a geometric method for studying transcendental equation with two delays, we conduct detailed analysis on the distribution of the roots for the characteristic equation of the model. Then, applying the integrated semigroup theory and the Hopf bifurcation theorem for an abstract Cauchy problem within a nondense domain, we proved the existence of Hopf bifurcation for the model. Stability switches can also occur, as the two time delays pass through a continuous curve in the parameter plane. To illustrate the theoretical results, numerical simulations are presented.

Stability of mild solutions of the fractional nonlinear abstract Cauchy problem

<abstract><p>Since the first work on Ulam-Hyers stabilities of differential equation solutions to date, many important and relevant papers have been published, both in the sense of integer order and fractional order differential equations. However, when we enter the field of fractional calculus, in particular, involving fractional differential equations, the path that is still long to be traveled, although there is a range of published works. In this sense, in this paper, we investigate the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of mild solutions for fractional nonlinear abstract Cauchy problem in the intervals $ [0, T] $ and $ [0, \infty) $ using Banach fixed point theorem.</p></abstract>