The necessary and sufficient condition for the convergence of Ishikawa iteration on an arbitrary interval

As an important iteration, the Mann and Ishikawa iteration has extensive application in fixed point theory. In 1991, David Borwein and Jonathan Borwein proved the convergence of the Mann iteration on a closed bounded interval in their paper. In this paper, we will extend their result to an arbitrary interval and to the Ishikawa iteration, indicating the necessary and sufficient condition for the convergence of Ishikawa iteration of continuous functions on an arbitrary interval.     

A necessary and sufficient condition for input identifiability for linear time-invariant systems

A necessary and sufficient condition for input identifiability for linear autonomous systems is given. The result is based on a finite iterative process and its proof relies on elementary arguments involving matrices, finite dimensional linear spaces, Gronwall’s lemma, and linear differential systems. Our condition is equivalent to the classical condition involving the geometrical concept of controlled invariant [V. Basile, G. Marro, Controlled and Conditioned Invariants in Linear System Theory, Prentice Hall, Englewood Cliffs, NJ, 1992, p. 237] and the dimension reduction algorithm that we propose seems to be useful in designing deconvolution methods.     

Some sufficient conditions for oscillation of impulsive delay hyperbolic systems with Robin boundary conditions

This paper deals with the oscillation problems of delay hyperbolic systems with impulses. Some sufficient conditions for oscillations of impulsive delay hyperbolic systems with Robin boundary conditions are obtained and the criteria of oscillation of the systems are established.     

Development of stability research on Takagi-Sugeno fuzzy control systems and approximation of the necessary and sufficient conditions

The controller’s design and analysis based on Takagi-Sugeno (T-S) models play important roles in the research of fuzzy control. When fuzzy control is being widely applied, the stability research on T-S fuzzy control supports the applications on the theoretical standpoint. The development of the stability research on T-S fuzzy control is surveyed in this paper. For the Lyapunov functions, this paper considers common quadratic Lyapunov functions, piecewise quadratic Lyapunov functions, fuzzy Lyapunov functions, nonquadratic Lyapunov functions and homogenously polynomially parameterized Lyapunov functions. For the control laws, this paper considers parallelly distributed compensation, non-parallel distributed compensation laws and homogenously polynomially parameterized control laws. By extensively applying the Pólya’s theorem and the techniques for homogenous polynomials, the stability conditions are gradually developed towards necessary and sufficient. It is very important to recognize and master this development in order to further study fuzzy control and the related control theories.     

Necessary and sufficient conditions for distinguishability of linear control systems

Distinguishability plays a crucial rule in studying observability of hybrid system such as switched system. Recently, for two linear systems, Lou and Si gave a condition not only necessary but also sufficient to the distinguishability of linear systems. However, the condition is not easy enough to verify. This paper will give a new equivalent condition which is relatively easy to verify.     

Coefficient conditions for the asymptotic stability of solutions of systems of linear difference equations with continuous time and delay

We establish sufficient algebraic coefficient conditions for the asymptotic stability of solutions of systems of linear difference equations with continuous time and delay in the case of a rational correlation between delays. We use (n ² + m)-parameter Lyapunov functions (n is the dimension of the system of equations and m is the number of delays). Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 4, pp. 516–522, April, 1998. This work was partially supported by the Joint Foundation of the Ukrainian Government and the Soros International Science Foundation (grant No. K42199).     

Structural interpretation and derivation of necessary and sufficient conditions for delay moments in FIFO multiserver queues

Scheller-Wolf [12] established necessary and sufficient conditions for finite stationary delay moments in stable FIFO GI/GI/s queues that incorporate the interaction between service time distribution, traffic intensity (ρ) and the number of servers in the queue. These conditions can be used to show that when the service time has finite first but infinite αth moment, s slow servers can give lower delays than one fast server. In this paper, we derive an alternative derivation of these moment results: Both upper bounds, that serve as sufficient conditions, and lower bounds, that serve as necessary conditions are presented. In addition, we extend the class of service time distributions for which the necessary conditions are valid. Our new derivations provide a structural interpretation of the moment bounds, giving intuition into their origin: We show that FIFO GI/GI/s delay can be represented as the minimum of (s − k) i.i.d. GI/GI/1 delays, when ρ satisfies k < ρ < k+1. AMS Subject Classification 60K25     

Some necessary and sufficient conditions for Hypercyclicity Criterion

We give necessary and sufficient conditions for an operator on a separable Hilbert space to satisfy the hypercyclicity criterion. This paper is a part of the second author’s Doctoral thesis, written at Shiraz University under the direction of the first author.     

Necessary and sufficient conditions for the regularity and stability for some partial functional differential equations with infinite delay

In this work we give necessary and sufficient conditions for the regularity and stability of solutions for some partial functional differential equations with infinite delay. We establish also a new characterization of the infinitesimal generator of the solution semigroup.     

Algebraic criteria for stability of linear neutral systems with a single delay

The stability of linear neutral delay-differential systems with a single delay via Routh–Hurwitz and Schur–Cohn criteria is investigated. Some algebraic criteria for delay-independent stability are presented. These criteria may complement those reported in the literature. Finally, two examples illustrate the criteria.     

Conditions for the stability of an impulsive linear equation with pure delay

We establish necessary and sufficient conditions for the stability of one class of impulsive linear differential equations with delay.     

Delay independent stability of linear switching systems with time delay

We consider a switching system with time delay composed of a finite number of linear delay differential equations (DDEs). Each DDE consists of a sum of a linear ODE part and a linear DDE part. We study two particular cases: (a) all the ODE parts are stable and (b) all the ODE parts are unstable and determine conditions for delay independent stability. For case (a), we extend a standard result of linear DDEs via the multiple Lyapunov function and functional methods. For case (b) the standard DDE result is not directly applicable, however, we are able to obtain uniform asymptotic stability using the single Lyapunov function and functional methods.     

On near-optimal necessary and sufficient conditions for forward-backward stochastic systems with jumps,with applications to finance

We establish necessary and sufficient conditions of near-optimality for nonlinear systems governed by forward-backward stochastic differential equations with controlled jump processes (FBSDEJs in short). The set of controls under consideration is necessarily convex. The proof of our result is based on Ekeland’s variational principle and continuity in some sense of the state and adjoint processes with respect to the control variable. We prove that under an additional hypothesis, the near-maximum condition on the Hamiltonian function is a sufficient condition for near-optimality. At the end, as an application to finance, mean-variance portfolio selection mixed with a recursive utility optimization problem is given. Mokhtar Hafay     

Stability conditions for systems of linear nonautonomous delay differential equations

Systems of linear nonautonomous delay differential equations are considered which are of the form y′_i(t) = ∑_{k = 1}ⁿ ∝₀^T b_ik(t, s) y_k(t − s) dη_ik(s) − c_i(t) y_i(t), where I = 1,…, n. Sufficient conditions are derived for both the asymptotic stability and the instability of the zero solution. The main result is found by a monotone technique using elementary methods only. Moreover, additional criteria are obtained by using the method of Lyapunov functionals.     

A necessary and sufficient condition of positive solutions for nonlinear singular differential systems with four-point boundary conditions

In this paper, we concert with the existence of positive solution for the following nonlinear singular differential system with four-point boundary conditions