Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

This paper proposes an improved stability condition of discrete-time systems with variable delays. Based on some mathematical techniques, a series of new summation inequalities are obtained. These new inequalities are less conservative than the Jensen inequality. Based on these new summation inequalities and the reciprocally convex combination inequality, a novel sufficient criterion on asymptotical stability of discrete-time systems with variable delays is obtained by constructing a new Lyapunov-Krasovskii functional. The advantage of the proposed inequality in this paper is demonstrated by a classical example from the literature.

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Stability analysis of discrete-time systems with variable delays via some new summation inequalities

In this paper, we study some new class of nonlocal three-point fractional q-integral boundary value problems of a nonlinear fractional q-difference equation and a nonlinear fractional q-integrodifference equation. Our problems contain different numbers of order and q in derivatives and integrals. The existence and uniqueness results are based on Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem. In addition, some examples are presented to illustrate the importance of these results.

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On nonlocal fractional q-integral boundary value problems of fractional q-difference and fractional q-integrodifference equations involving different numbers of order and q

By applying Riccati substitution techniques triply, we establish some new oscillation and asymptotic nature of solutions to the third-order nonlinear differential equations with mixed neutral type. We present many theorems and related examples in order to illustrate and substantiate the main theory.

Some New Oscillatory Behavior of Certain Third-Order Nonlinear Neutral Differential Equations of Mixed Type

In this paper, state estimation for discrete-time BAM neural networks with time-varying delay is discussed. Under a weaker assumption on activation functions, by constructing a novel Lyapunov-Krasovskii functional (LKF), a set of sufficient conditions are derived in terms of linear matrix inequality (LMI) for the existence of state estimator such that the error system is global exponentially stable. Based on the delay partitioning method and the reciprocally convex approach, some less conservative stability criteria along with lower computational complexity are obtained. Finally, a numerical example is given to show the effectiveness of the derived result.

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New approach to state estimator for discrete-time BAM neural networks with time-varying delay

We consider a class of neutral stochastic partial differential equations driven by an α-stable process. We prove the existence and uniqueness of the mild solution to the equation by the Banach fixed-point theorem under some suitable assumptions. Sufficient conditions for the stability in the distribution of the mild solution are derived.

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Stability in distribution of neutral stochastic partial differential delay equations driven by α -stable process

This paper deals with a predator-prey model of Beddington-DeAngelis type functional response with Levy jumps. The proposed mathematical model consists of a system of two stochastic differential equations to stimulate the interactions between predator population and prey population. The dynamics of the system is discussed mainly from the point of view of persistence and extinction. To begin with, the global positivity, stochastically boundedness and other asymptotic properties have been derived. In addition, sufficient conditions for extinction, nonpersistence in the mean and weak persistence are obtained. It is proved that the variation of Levy jumps can affect the asymptotic property of the system.

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Analysis of a predator-prey model with Lévy jumps

This paper deals with stochastic nonautonomous Gompertz model with Levy jumps. To begin with, the existence of a global positive solution and an explicit solution have been derived. In addition, asymptotic moment properties are discussed. Besides, sufficient conditions for extinction, persistence in mean, and weak persistence are obtained. It is proved that the variability of Levy jumps can affect the asymptotic property of the system.

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Junping Li

Papers

Stability in distribution of neutral stochastic partial differential delay equations driven by α -stable process

Analysis of a predator-prey model with Lévy jumps

Stochastic nonautonomous Gompertz model with Lévy jumps