Existence of three periodic solutions for a nonlinear first order functional differential equation

In this paper, we use Leggett-Williams multiple fixed point theorem to obtain different sufficient conditions for the existence of at least three nonnegative periodic solutions of the first order functional differential equation of the form

y^′(t)=-a(t)y(t)+λf(t,y(h(t))).     

A Quantitative Stability Analysis of the Solutions to the Carson-Cambi Equation

A quantitative stability analysis of the Carson-Cambi equation [(1 + ε cos t)(d²y/dt²) + py = 0] is carried through, using a new, effective approach. The results are compared with a recent perturbation analysis, and show that this should not be used for ε0.4. In the present analysis we go up to ε = 0.8, and, in fact, with less effort than the perturbation analysis involves. Detailed stability diagrams are presented.     

Periodic solutions for p-Laplacian neutral Liénard equation with a sign-variable coefficient

Using Lu's continuation theorem, the extension one of Manásevich-Mawhin, we study the existence of periodic solutions for p-Laplacian neutral Liénard equation of the form

(?_p(x(t)-cx(t-σ))^′)^′+f(x(t))x^′(t)+β(t)g(x(t-τ(t))=e(t).     

Parameter identification of a class of time-varying systems via orthogonal shifted legendre polynomials

A method of using orthogonal shifted Legendre polynomials for identifying the parameters of a process whose behaviour can be modelled by a linear differential equation with time-varying coefficients in the form of finite-order polynomials is presented. It is based on the repeated integration of the differential equation and the representations of $\text{∫}\text{0}\text{t}$s(τ) dτ = Ps(t) and ts(t) = Rs(t), where P and R are constant matrices and s(t) is a shifted Legendre vector whose elements are shifted Legendre polynomials. The differential input-output equation is converted into a set of overdetermined linear algebraic equations for a least squares solution. The results of simulation studies are included to illustrate the applicability of the method.     

An experimental investigation of forced vibrations

The solution of the differential equation y″ + 2Ry′ + n²y = E cos pt is written in a new form which clearly exhibits many important facts thus far overlooked by theoretical and experimental investigators. Writing s = n ? p, and Δn = n ? √n² ? R², it is found: (a) When s ≠ Δn, there are “beats,” and the first “beat” maximum is greater than any later maximum while the first “beat” minimum is less than any later “beat” minimum. The “beat” frequency is $\text{(s ?}\text{Δ}\text{n)}\text{2π}$. (b) When n² ? p² = R², there are no “beats,” and the resultant amplitude grows monotonically from zero to the amplitude of the forced vibration, (c) At resonance, when n = p, we still have maxima which occur with a frequency $\text{Δ}\text{n}\text{2π}$ in a damped system. (d) The absence of “beats” is neither a sufficient nor a necessary condition for resonance in a damped system.In the experimental investigation the upper extremity of a simple pendulum was moved in simple harmonic motion and photographic records obtained of the motion of the pendulum bob. Different degrees of damping were used, ranging from very small to critical.The experimental results are in excellent agreement with theory.     

Growth dynamics of citations of cumulative papers of individual authors according to progressive nucleation mechanism: Concept of citation acceleration

Using data generated by progressive nucleation mechanism on the cumulative fraction of citations of individual papers published successively by a hypothetical author, an expression for the time dependence of the cumulative number L_sum(t) of citations of progressively published papers is proposed. It was found that, for all nonzero values of constant publication rate ΔN, the cumulative citations L_sum(t) of the cumulative N papers published by an author in his/her entire publication career spanning over T years may be represented in distinct regions: (1) in the region 0 < t < Θ₀ (where Θ₀ ≈ T/3), L_sum(t) slowly increases proportionally to the square of the citation time t, and (2) in the region t > Θ₀, L_sum(t) approaches a constant L_sum(max) at T. In the former region, the time dependence of L_sum(t) of an author is associated with three parameters, viz. the citability parameter λ₀, the publication rate ΔN and his/her publication career t. Based on the predicted dependence of L_sum(t) on t, a useful scientometric age-independent measure, defined as citation acceleration a = L_sum(t)/t², is suggested to analyze and compare the scientific activities of different authors. Confrontation of the time dependence of cumulative number L_sum(t) of citations of papers with the theoretical equation reveals one or more citation periods during the publication careers of different authors.     

Periodic solutions for a fourth-order p-Laplacian neutral functional differential equation

By means of Mawhin's continuation theorem, we study a kind of fourth-order p-Laplacian neutral functional differential equation with a deviating argument in the form:

(φ_p(x(t)−cx(t−δ))^″)^″=f(x(t))x^′(t)+g(t,x(t−τ(t,|x|_∞)))+e(t).     

Periodic solutions for third order p-Laplacian equation with a deviating argument

By means of Mawhin's continuation theorem, we study a third-order p-Laplacian differential equation

(?_p(u^″(t)))^′+f(t,u^′(t),u^″(t))+g(t,u(t-τ(t)))=e(t).     

The representation of radiation reaction in wave mechanics

It is well known that the wave mechanical ψ equation leads to the conclusion that the centroid of the wave mechanical electron should move according to the classical electrodynamic equation of motion in which, however, the terms representing what is commonly called radiation reaction are absent. If v is the velocity of the electron, the classical rate of change of momentum is $\text{md}\text{dt}\text{{}\text{v}\text{(}\text{I}\text{?}\text{v}{}^2\text{c}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{}}$. The equation of motion including radiation reaction terms may be regarded as obtainable by replacing this quantity by one obtained by operating upon it with the operator P^?1

$\text{P={I?}\text{α}{}_1\text{kd}\text{dt}\text{+}\text{α}{}_2\text{d}\text{dt}\text{(}\text{kd}\text{dt}\text{)?·}}{}^{\mathrm{?}}$

where α₁, α₂, etc., are constants and $\text{k = (}\text{I}\text{?}\text{v}{}^2\text{c}{}^2\text{)}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$. The main purpose of the paper is to show that if there be any relativistically invariant ψ equation which leads to the classical equation of motion without radiation reaction terms, then by replacing the vector and scalar potentials U and ? in that equation by P(U) and P(?), a relativistically invariant equation of motion will be obtained including the radiation reaction terms, provided that the $\text{d}\text{dt}$ in P be now regarded as $\text{?}\text{?t}$ + u · grad, where u is the velocity of the wave mechanical density distribution at a point. The purpose is to use the power to produce the equation of motion as a criterion for suggesting the proper modification of the ψ equation to apply in those cases where, on the classical theory, the electron would suffer great acceleration, as in ionization by rapidly moving corpuscles.     

Periodic solutions for a kind of Rayleigh equation with two deviating arguments

In this paper, we use the coincidence degree theory to establish new results on the existence of T-periodic solutions for the Rayleigh equation with two deviating arguments of the form

x^″+f(x^′(t))+g₁(t,x(t-τ₁(t)))+g₂(t,x(t-τ₂(t)))=p(t).     

Input-to-state stability of switched stochastic delayed systems with Lévy noise

This paper focuses on input-to-state stability of a class of switched stochastic delayed systems which are drived by Lévy noise. By multiple Lyapunov function and average dwell time approach, the sufficient conditions of the ψ^λ(t)-weighted input-to-state stability can be obtained if all the subsystems are input-to-state stable. Then utilizing comparison principle and the method of constant variation, the sufficient criteria of the e^λt-weighted input-to-state stability of the switched stochastic delayed systems containing both input-to-state stable subsystems and non-input-to-state stable subsystems can also be derived. Finally, an example is given to illustrate the effectiveness of the proposed results.     

Distributed consensus-based multi-agent convex optimization via gradient tracking technique

This paper considers solving a class of optimization problems over a network of agents, in which the cost function is expressed as the sum of individual objectives of the agents. The underlying communication graph is assumed to be undirected and connected. A distributed algorithm in which agents employ time-varying and heterogeneous step-sizes is proposed by combining consensus of multi-agent systems with gradient tracking technique. The algorithm not only drives the agents’ iterates to a global and consensual minimizer but also finds the optimal value of the cost function. When the individual objectives are convex and smooth, we prove that the algorithm converges at a rate of $O(1/\sqrt{t})$ if the homogeneous step-size does not exceed some upper bound, and it accelerates to O(1/t) if the homogeneous step-size is sufficiently small. When at least one of the individual objectives is strongly convex and all are smooth, we prove that the algorithm converges at a linear rate of O(λ^t) with 0?<?λ?<?1 even though the step-sizes are time-varying and heterogeneous. Two numerical examples are provided to demonstrate the efficiency of the proposed algorithm and to validate the theoretical findings.     

Stability criteria for certain classes of nonlinear difference equations

The general mth order difference equation X(n+m)+a₁X(n+m?1)+…+a_mX(n) = F[n,X(n),…,X(n+m?1)] is considered. The stability properties of its solutions are studied using the discrete form of Liapunov's direct method. A quadratic form is selected as a possible Liapunov function V(n,X) and a scheme is developed for determining appropriate conditions on this function to insure that its total difference ΔV(n,X) is negative semi-definite or negative definite with respect to the difference equation. The approach is applied to the fourth-order difference equation in full detail to illustrate the method for determining the conditions which imply either uniform stability or uniform asymptotic stability and specific results are obtained. Several comments on, and extensions of, the work done by Puri and Drake for the cases m = 2 and m = 3 are presented.The results of the present approach in the homogeneous case where F[n,X(n),…,X(n+m?1)] = 0 are compared with the usual Schur-Cohn criteria and are shown to be at least as good.     

Stabilities and Settling Times of Nonlinear and Time-varying Feedback Systems

The bounded-input bounded-output stability, finite time stability and settling time of a single-loop feedback system consisting of a nonlinear time-varying gain followed by a linear time-invariant system are investigated via a nonlinear integral inequality. The gain has the form k₀+k₁(t)+k₂(t)g(bd) where g(bd) is a monotonic increasing function. The system is bounded-input bounded-output stable provided the time-varying gains are L¹(0, t8) functions and is finite time stable for bounded gains. The nonlinear integral inequality, which is used to obtain explicit and useful bounds on the output of the system, is also employed to determine the settling time.     

Fractional multiple birth-death processes with birth probabilities λi(Δt)+o((Δt))

The usual model for (Poissonian) linear birth-death processes is extended to multiple birth-death processes with fractional birth probabilities in the form λ_i(Δt)^α+o((Δt)^α, 0<α<1. The probability generating function for the time dependent population size is provided by a fractional partial differential equation. The solution of the latter is obtained and comparison with the usual model is made. The probability of ultimate extinction is obtained. One considers the special case of fractional Poissonian processes with individual arrivals only, and then one outlines basic results for continuous processes defined by fractional Poissonian noises. The key is the Taylor’s series of fractional order f(x+h)=E_α(h^αD_x^α)f(x), where E_α(·) is the Mittag-Leffler function, and D_x^α is the modified Riemann-Liouville fractional derivative, as previously introduced by the author.     

Uniform exponential stability of periodic discrete switched linear system

Let {Π_τ(m, n): m?≥?n?≥?0} be the family of periodic discrete transition matrices generated by bounded valued square matrices Λ_τ(n), where $\tau :[0,1,2,?)\to \Omega$ is an arbitrary switching signal. We prove that the family {Π_τ(m, n): m?≥?n?≥?0} of bounded linear operator is uniformly exponentially stable if and only if the sequence $n?{\sum }_{k=0}^n{e}^{i\alpha k}{\Pi }_{\tau }(n,k)w\left(k\right):{\mathbb{Z}}_{+}\to \mathbb{R}$ is bounded.     

Chain matrix decompositions,continued fractions and the optimal inhomogeneous ladder networks

Studies of two-element-kind ladder networks are well known in the classical literature, among them, the most celebrated ones are due to Cauer. Driving point immittance function synthesis by using continued fractions to obtain the series and shunt arm L-C element values is a standard and routine work. The idea of introducing a class of more general networks, the inhomogeneous ladder networks, was first developed by Lee and Brown, and subsequently the synthesis techniques of such a network were established.In this paper, new results are found such as: (1) the Iff. conditions of the existence of an inhomogeneous ladder network by a given chain matrix of the network satisfying: (a) determinant of the chain matrix is 1; (b) the zeros of A(s) and z^?1B(s) or A(s) and y^?1C(s) alternate with respect to [z(s)y(s), k] with an appropriate leading set of zeros of A(s); (c) the poles of A(s) and z^?1B(s) or A(s) and y^?1C(s) are the poles of z(s)y(s) of multiplicity of n and n?1, where n the number of sections of ladder networks; (2) the Iff. condition for the inhomogeneous ladder network to be optimal is that it be antimetrical, whereas for the extended class of inhomogeneous ladder networks it is symmetrical, where an optimal inhomogeneous ladder network is defined as the corresponding network with the minimum sum of immittance levels in the series and shunt arms; (3) algorithms of synthesis procedures were developed as the by-products of the Theorems.     

A new approach to the design of coupling networks

Mathematical results are derived, which enable one to find a vector of parameters k⁰ such that (P₁(s,k⁰)?H)∩(P₂(k⁰)=0), where P₁(s,k) is a polynomial in s and in the components of k,P₂(k) is a polynomial in the components of k, and H is the set of Hurwitz polynomials. The algorithm is based on an extension of the root locus technique to the multiparameter case. The design problem of coupling networks between a resistive generator and a passive load, under prescribed power gain characteristics, is translated into the above formulation. A numerical example is provided.