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In this paper, the solutions of a class of switch dynamical systems are investigated. The right-hand side of the underlying equations is discontinuous with respect to the state variable. The discontinuity is represented by jump discontinuous functions such as signum or Heaviside functions. In this paper, a novel approach of the solutions of this class of discontinuous equations is presented. The initial value problem is restated as a differential inclusion via Filippov’s regularization, after which, via the approximate selection results, the differential inclusion is transformed into a continuous, single-valued differential equation. Besides its existence, a sufficient uniqueness condition, the strengthened one-sided Lipschitz Condition, is also introduced. The important issue of the numerical integration of this class of equations is addressed, emphasizing by examples the errors that could appear if the discontinuity problem is neglected. The example of a mechanical system, a preloaded compliance system, is considered along with other examples.

Mathematical formulations play a crucial role in comprehending the dynamics of disease spread within a community. The objective of this research is to investigate the lumpy skin disease spread in cattle by taking control strategy as well as symptomatic and asymptomatic measures. The newly developed SVEI1I2R system is examined both qualitatively and quantitatively to ascertain its stable state. Local stability of the developed system is investigated for the validity of the developed system. The positiveness and existence of solution for the developed model are examined to ensure reliable findings, which are essential properties of epidemic models. The global derivative is illustrated to confirm the positivity with linear growth and Lipschitz conditions for the rate of effects in each sub-compartment on a global scale. The system is examined for global stability using Lyapunov first derivative functions to evaluate the collective impact of additional infections for acute as well as chronic stage. Simulations are conducted to observe the symptomatic and asymptomatic effects of the Lumpy skin disease acute as well as chronic infections worldwide in the cattle animals, providing insights into the actual behavior of the disease spread for both level acute as well as chronic. These investigations are valuable for comprehending the virus’s spread and effective control strategies at early detection or on acute stage infection from our justified outcomes.

In 2015, Mao (J. Comput. Appl. Math., 290, 370–384, 2015) proposed the truncated Euler-Maruyama (EM) method for stochastic differential equations (SDEs) under the local Lipschitz condition plus the Khasminskii-type condition. Adapting the truncation idea from Mao (J. Comput. Appl. Math., 290, 370–384, 2015) and Mao (Appl. Numer. Math., 296, 362–375, 2016), lots of modified truncated EM methods are proposed (see, e.g., Guo et al. (Appl. Numer. Math., 115, 235–251, 2017,) and Lan and Xia (J. Comput. Appl. Math., 334, 1–17, 2018) and Li et al. (IMA J. Numer. Anal., 39(2), 847–892, 2019) and the references therein). These truncated-type EM methods Mao (J. Comput. Appl. Math., 290, 370–384, 2015) and Mao (Appl. Numer. Math., 296, 362–375, 2016) and Guo et al. (Appl. Numer. Math., 115, 235–251, 2017,) and Lan and Xia (J. Comput. Appl. Math., 334, 1–17, 2018) and Li et al. (IMA J. Numer. Anal., 39(2), 847–892, 2019) construct the numerical solutions by defining an appropriate truncation projection, then applying the truncation projection to the numerical solutions before substituting them into the coefficients in each iteration. In this paper, we develop a new class of explicit schemes for superlinear stochastic differential equations with piecewise continuous arguments (SDEPCAs), which are defined by directly truncating the coefficients. Our method has a more simple structure and is easier to implement. We not only show the explicit schemes converge strongly to SDEPCAs but also demonstrate the convergence rate is optimal 1/2. A numerical example is provided to demonstrate the theoretical results.

In this paper, a low order asymptotic predictor is introduced for nonlinear systems based on measurable outputs. Predictors play basic role in the control of dead-time systems having no delay-free input. Generalized Lipschitz Condition is employed here to develop Sequential Sub-Predictors (SSP) for complex nonlinear systems. Compared with existing synthesis methods which involve Lipschitz Condition, Generalized Lipschitz condition relaxes the conservative of the stability results using some modified matrix inequalities. The proposed idea can be more attractive for unstable systems due to reduce significantly the order of the predictors. Also using the proposed method, the effect of the external disturbance can be minimized based on H∞ index. Afterward, a SSP-based controller is presented to illustrate the effectiveness of proposed method to stabilize nonlinear systems with input-delay. Finally, the predictor capability is investigated by simulation examples.

ABSTRACT In this paper, we consider general mean-field backward doubly stochastic differential equations with Poisson jumps (general mean-field BDSDEP in short), in which the coefficient f not only depends on the solution processes $(Y, Z, U)$ ( Y , Z , U ) , but also on their law $P_{(Y, Z, U)}$ P ( Y , Z , U ) , which describes the characteristic of the mean-field. In addition, the two independent Brownian motions are coupled with a independent Poisson random measure. First, under stochastic Lipschitz conditions, we establish through a Banach contraction theorem the existence and the uniqueness of solution of our general mean-field BDSDEP. Finally, we establish a comparison theorem of the solutions.

In this paper, we consider E the set of all infinitely differentiable functions with compact support included on the interval I = [ - 1 , 1 ] . We use the distributions in E , as a tool to prove the continuity of the Dunkl operator and the Dunkl translation. Some properties of the modulus of smoothness related to the Dunkl operator are verified. By means of generalized Dunkl–Lipschitz conditions on Dunkl–Sobolev spaces, a result of Younis on the torus, which is an analog of Titchmarsh’s theorem, is deduced as a special case. In addition, certain conditions and a characterization of the Dini–Lipschitz classes on I in terms of the behavior of their Fourier–Dunkl coefficients are derived.

This paper deals with the problem of iterative learning control algorithm for consensus of a class of multi-agent systems, and all the agents in the considered systems are governed by the nonlinear dynamics with quasi-one-sided Lipschitz condition. Based on the framework of network topologies, distributed consensus-based iterative learning control protocols are designed by using the nearest neighbor knowledge. Under the action of the iterative learning control law, consensus on the finite time interval along the iteration axis can be reached for all the directed communication graphs with spanning trees. A simulation example is finally used to illustrate the effectiveness of the proposed approach.

The strong convergence of numerical solutions is studied in this paper for stochastic Volterra integral differential equations (SVIDEs) with a Hölder diffusion coefficient using the truncated Euler–Maruyama method. Firstly, the numerical solutions of SVIDEs are obtained based on the Euler–Maruyama method. Then, the pth moment boundedness and strong convergence of truncated the Euler–Maruyama numerical solutions are proven under the local Lipschitz condition and the Khasminskii-type condition. Finally, the convergence rate of the truncated Euler–Maruyama method of the numerical solutions is also discussed under some suitable assumptions.

We deal with approximation of solutions of delay differential equations (DDEs) via the classical Euler algorithm. We investigate the pointwise error of the Euler scheme under nonstandard assumptions imposed on the right-hand side function f . Namely, we assume that f is globally of at most linear growth, satisfies globally one-side Lipschitz condition but it is only locally Hölder continuous. We provide a detailed error analysis of the Euler algorithm under such nonstandard regularity conditions. Moreover, we report results of numerical experiments.

In the present paper, we consider the semilocal convergence issue of two-step Newton method for solving nonlinear operator equation in Banach spaces. Under the assumption that the first derivative of the operator satisfies a generalized Lipschitz condition, a new semilocal convergence analysis for the two-step Newton method is presented. The Q-cubic convergence is obtained by an additional condition. This analysis also allows us to obtain three important spacial cases about the convergence results based on the premises of Kantorovich, Smale and Nesterov-Nemirovskii types. As applications of our convergence results, a nonsymmetric algebraic Riccati equation arising from transport theory and a two-dimensional nonlinear convection-diffusion equation are provided.