Second-order neutral delay-differential equations with piecewise constant time dependence

We find conditions for the existence and uniqueness of almost periodic solutions of second-order neutral delay-differential equations with almost periodic time dependence of the form (x(t)+px(t−1))″=qx([t])+f(t); here [·] is the greatest integer function, p and q are nonzero constants, and f(t) is Bohr almost periodic.     

Existence of almost periodic solutions to delay differential equations with Lipschitz nonlinearities

Differential equations whose nonlinearities depend upon both x(t) and x(t ? τ) arise in many settings. In this paper equations of this form subject to periodic and almost periodic forcing are studied: x′(t) + g(x(t), x(t ? τ)) = e(t), ?∞ < t < ∞. (E) A nonresonance type condition is found under which it is shown that (E) will have a unique Besicovitch almost periodic solution for any Besicovitch almost periodic forcing term e(t). These results are then generalized to systems of equations of the same form as (E). These results hold without any small parameter type restriction upon g.     

Almost periodic solutions of second-order neutral delay-differential equations with piecewise constant arguments

In this paper, we present an existence theorem of almost periodic solutions of second-order neutral delay-differential equations with piecewise constant arguments of the form (x(t)+x(t−1))″=qx([t])+f(t), where [·] denotes the greatest integer function, q is a nonzero constant, and f(t) is almost periodic.     

Global attractivity and positive almost periodic solution of a single species population model

3/2-criterion is built, which guarantees the global attractivity of positive solution for equation having the form x^′(t)=a(t)x(t)(1−L(t,xt)), where a(t)?0 and the linear functional L(t,⋅) is positive. Moreover, when the equation is almost periodic, the similar conditions can also guarantee the existence and uniqueness of almost periodic solution that is globally attractive. Our results improve those in literature.     

Block diagonal dominance and reducibility for linear differential systems

We consider the linear homogeneous differential equation x′(t) = A(t)x(t). Conditions are given on the entries of A(t) under which the equation is reducible. It is shown that if A(t) is almost periodic the reduction is via an almost periodic change of variables. The result on reducibility is used to give new conditions on A(t) under which the equation admits an exponential dichotomy and to prove a result on the reducibility of almost periodic perturbations of constant coefficient systems.     

A Massera type theorem for almost automorphic solutions of differential equations

In this note, we present a Massera type theorem for the existence of almost automorphic solutions of periodic linear evolution equations of the form x^′(t)=A(t)x(t)+f(t), where A(t) is unbounded linear operator depending on t periodically and generates a τ-periodic evolutionary process, f is almost automorphic. The main results are stated in terms of the almost automorphy of solutions and their Carleman spectra.     

On the optimal switching level for an M/G/1 queueing system

A cost function is studied for an M/G/1 queueing model for which the service rate of the virtual waiting time process U_t for U_t<K differs from that for U_t > K. The costs considered are costs for maintaining the service rate, costs for switching the service rate and costs proportional to the inventory U_t. The relevant cost factors for the system operating below level K differ from those when U_t > K. The cost function which is considered only for the stationary situation of the U_t-process expresses the average cost per unit time. The problem is to find that K for which the cost function reaches a minimum. Criteria for the possibly optimal cases are found; they have an interesting intuitive interpretation, and require the knowledge of only the first moment of the service time distribution.     

Cauchy matrix for linear almost periodic systems and some consequences

A novel method to construct the fundamental matrix for a linear almost periodic system is proposed, provided that the diagonal terms satisfy an average separation condition and the off-diagonal coefficients are L^∞-small. The idea is to transform the system in a set of Riccati type equations and use exponential dichotomy and its consequences. It is shown that the method yields easy computation procedures with simple and direct conditions depending on the coefficients. Finally, our result enables us to obtain: (i) explicit almost periodic matrices Q(t), Q⁻¹(t) and Q^′(t), which diagonalize the original system and (ii) sufficient conditions for the stability. Two illustrative examples are shown.     

The stochastic soliton-like solutions of stochastic KdV equations

By means of a generalized method and symbolic computation, we consider a stochastic KdV equation U_t + f(t)U ♢ U_x + g(t)U_xxx = W(t) ♢ R^♢(t, U, U_x, U_xxx). We construct new and more general formal solutions. At the same time, we recover all the solutions found by Xie [Phys. Lett. A 310 (2003) 161]. The solutions obtained include the nontravelling wave and coefficient function’s stochastic soliton-like solutions, singular stochastic soliton-like solutions, stochastic triangular functions solutions.     

On almost reducible systems with almost periodic coefficients

Let x=A(t)x be a system of two linear ordinary differential equations with almost periodic coefficients. Then there exists for any positive ε an almost reducible system of equations x=B(t)x with almost periodic coefficients and such that sup ∥A(t)?B (t)∥<ε.-∞相似文献   

Global attractivity in delayed differential equations with applications to “food-limited” population model

A new criterion is built, which guarantees the global attractivity of zero solution for equation having the form x^′(t)=(1+x(t))F(t,xt). The behavior of global solution on R is also addressed. As an application, we study the existence of globally attractive positive almost periodic solution for the “food-limited” single population model. Our results improve those in literature.     

Asymptotic behavior of trajectories for some nonautonomous,almost periodic processes

In this paper, asymptotics are studied for some almost periodic processes on a complete metric space (X, d): (1) It is shown that any precompact positive trajectory of a contractive periodic process is asymptotically almost periodic as t → +∞. This property does not hold for general almost periodic contractive processes. (2) A compactness result is obtained for weakly almost periodic complete trajectories of some (possibly nonlinear) processes in a uniformly convex Banach space. (3) The existence of almost periodic trajectories is studied for “affine” processes in a uniformly convex Banach space. These results are applicable to some evolution equations of the form $\text{du}\text{dt}\text{+ A (t) u(t) ? f(t)}$, where $\text{?(t)}$ is almost periodic: R → V uniformly convex Banach space and A(t) is a periodic, time-dependent, m-accretive operator in V.     

The set of periodic scalar differential equations with cubic nonlinearities

We study the structure induced by the number of periodic solutions on the set of differential equations x^′=f(t,x) where f∈C³(R²) is T-periodic in t, fx³(t,x)<0 for every (t,x)∈R², and f(t,x)→?∞ as x→∞, uniformly on t. We find that the set of differential equations with a singular periodic solution is a codimension-one submanifold, which divides the space into two components: equations with one periodic solution and equations with three periodic solutions. Moreover, the set of differential equations with exactly one periodic singular solution and no other periodic solution is a codimension-two submanifold.     

On the existence and stability of periodic and almost periodic solutions of quasilinear equations with maxima

We study the problem of existence of periodic and almost periodic solutions of the scalar equation x′ (t) = − δx(t) + pmax _{u∈[t − h, t]} x(u) + f(t) where δ, p ∈ R, with a periodic (almost periodic) perturbation f(t). For these solutions, we establish conditions of global exponential stability and prove uniqueness theorems. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 747–754, June, 1998.     

Invariant almost Hermitian structures on flag manifolds

Let G be a complex semi-simple Lie group and form its maximal flag manifold where P is a minimal parabolic (Borel) subgroup, U a compact real form and T=U∩P a maximal torus of U. We study U-invariant almost Hermitian structures on . The (1,2)-symplectic (or quasi-Kähler) structures are naturally related to the affine Weyl groups. A special form for them, involving abelian ideals of a Borel subalgebra, is derived. From the (1,2)-symplectic structures a classification of the whole set of invariant structures is provided showing, in particular, that nearly Kähler invariant structures are Kähler, except in the A₂ case.     

Existence and uniqueness of positive periodic solutions of functional differential equations

In this paper we consider the existence and uniqueness of positive periodic solution for the periodic equation y′(t)=−a(t)y(t)+λh(t)f(y(t−τ(t))). By the eigenvalue problems of completely continuous operators and theory of α-concave or −α-convex operators and its eigenvalue, we establish some criteria for existence and uniqueness of positive periodic solution of above functional differential equations with parameter. In particular, the unique solution y_λ(t) of the above equation depends continuously on the parameter λ. Finally, as an application, we obtain sufficient condition for the existence of positive periodic solutions of the Nicholson blowflies model.     

Weighted pseudo almost periodic mild solutions of semilinear evolution equations with nonlocal conditions

In this paper, applying the theory of semigroups of operators to evolution families and Banach fixed point theorem, we prove the existence and uniqueness of the weighted pseudo almost periodic mild solution of the semilinear evolution equation x^′(t)=A(t)x(t)+f(t,x(t)) with nonlocal conditions x(0)=x₀+g(x) in Banach space X under some suitable hypotheses.     

Continuous families of vectors that are orbits of a unitary group

We characterize functions u from the real line into a Hilbert space that are the orbits of a unitary group {U(t)}_t∈R; that is, u(t)=U(t)u(0), for all real t. One of the characterizations is that u be the Fourier transform of a certain type of vector-valued measure Z; we then use our characterizations to construct Z from u.     

The linear model revisited

Following Gerber (1982), the annual gain of an insurance concern, G_t, is modeled as an autoregressive moving average process. The surplus process is defined as U_t = u + Σ^t_k=₁G_k. The distribution of U_t is obtained. It is proved that the traditional limit theorems, i.e., the central limit theorem, the strong law of large numbers and the law of the iterated logarithm, hold for U_t as t → ∞. Under certain conditions, bounds are provided for the probability of non-ruin in a finite time interval. It is proved that, if E[G_t] > 0 as t → ∞, the probability of non-ruin in an infinite time interval is positive.     

First Fundamental Theorem for Covariants of Classical Groups

Let U(G) be a maximal unipotent subgroup of one of the classical groups G=GL(V), O(V), Sp(V). Let W be a direct sum of copies of V and its dual V*. For the natural action U(G) : W, we describe a minimal system of homogeneous generators for the algebra of U(G)-invariant regular functions on W. For G=O(V), Sp(V), this result is connected with a construction for the irreducible representations of G due to H. Weyl.