On forced fast oscillations for delay differential equations on compact manifolds

We prove an existence result for forced oscillations of delay differential equations on compact manifolds with nonzero Euler-Poincaré characteristic. When the period is smaller than the delay we need the asymptotic fixed point index theory for C¹ maps due to Eells and Fournier, and Nussbaum.     

FBSDEs with time delayed generators: L-solutions, differentiability, representation formulas and path regularity

We extend the work of Delong and Imkeller (2010) [6] and [7] concerning backward stochastic differential equations with time delayed generators (delay BSDEs). We give moment and a priori estimates in general L^p-spaces and provide sufficient conditions for the solution of a delay BSDE to exist in L^p. We introduce decoupled systems of SDEs and delay BSDEs (delay FBSDEs) and give sufficient conditions for their variational differentiability. We connect these variational derivatives to the Malliavin derivatives of delay FBSDEs via the usual representation formulas. We conclude with several path regularity results, in particular we extend the classic L²-path regularity to delay FBSDEs.     

Stochastic functional differential equations with infinite delay

The stability and boundedness of the solution for stochastic functional differential equation with finite delay have been studied by several authors, but there is almost no work on the stability of the solutions for stochastic functional differential equations with infinite delay. The main aim of this paper is to close this gap. We establish criteria of pth moment ψγ(t)-bounded for neutral stochastic functional differential equations with infinite delay and exponentially stable criteria for stochastic functional differential equations with infinite delay, and we also illustrate the result with an example.     

Some results for the queues Mx/M/c and GIx/ G/c

We develop for the queue M^x/M/c an upper bound for the mean queue length and lower bounds for the delay probabilities (that of an arrival group and that of an arbitrary customer in the arrival group). An approximate formula is also developed for the general bulk-arrival queue GI^x/G/c. Preliminary numerical studies have indicated excellent performance of the results.     

Semigroup Properties and the Crandall Liggett Approximation for a Class of Differential Equations with State-Dependent Delays

We present an approach for the resolution of a class of differential equations with state-dependent delays by the theory of strongly continuous nonlinear semigroups. We show that this class determines a strongly continuous semigroup in a closed subset of C^0, 1. We characterize the infinitesimal generator of this semigroup through its domain. Finally, an approximation of the Crandall-Liggett type for the semigroup is obtained in a dense subset of (C, ‖·‖_∞). As far as we know this approach is new in the context of state-dependent delay equations while it is classical in the case of constant delay differential equations.     

A periodic solution of a differential equation with state-dependent delay

We study a differential equation for delayed negative feedback which models a situation where the delay depends on the present state and becomes effective in the future. The main result is existence of a periodic solution in case the equilibrium is linearly unstable. The proof employs the ejective fixed point principle on a compact convex set K₀⊂C([−h,0],R) of Lipschitz continuous functions and uses that the equation generates a smooth semiflow on an infinite-dimensional submanifold of the space C¹([−h,0],R).     

Global branches of periodic solutions for forced delay differential equations on compact manifolds

We prove a global bifurcation result for T-periodic solutions of the T-periodic delay differential equation x^′(t)=λf(t,x(t),x(t−1)) depending on a real parameter λ?0. The approach is based on the fixed point index theory for maps on ANRs.     

Stability and fixed points of point-dissipative systems

It is known that if T: X → X is completely continuous or if there exists an n₀ > 0 such that Tⁿ⁰ is completely continuous, then T point dissipative implies that there is a maximal compact invariant set which is uniformly asymptotically stable, attracts bounded sets, and has a fixed point (see Billotti and LaSalle [Bull. Amer. Math. Soc.6 1971]). The result is used, for example, in studying retarded functional differential equations, or parabolic partial differential equations. This result has been extended by Hale and Lopes [J. Differential Equations13 1973]. They get the result that if T is an α-contraction and compact dissipative then there is a maximal compact invariant set which is uniformly asymptotically stable, attracts neighborhoods of compact sets, and has a fixed point. The above result requires the stronger assumption of compact dissipative. The principal result of this paper is to get similar results under the weaker assumption of point dissipative. To do this we must make additional assumptions. We will show these assumptions are naturally satisfied by stable neutral functional differential equations and retarded functional differential equations with infinite delay. The result has applications to many other dynamical systems, of course.     

Global Hopf bifurcation for differential equations with state-dependent delay

We develop a global Hopf bifurcation theory for a system of functional differential equations with state-dependent delay. The theory is based on an application of the homotopy invariance of S¹-equivariant degree using the formal linearization of the system at a stationary state. Our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local Hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state-dependent delay.     

Center manifolds for infinite delay

We obtain a C¹ center manifold theorem for perturbations of delay difference equations in Banach spaces with infinite delay. Our results extend in several directions the existing center manifold theorems. Besides considering infinite delay equations, we consider perturbations of nonuniform exponential trichotomies and generalized trichotomies that may exhibit stable, unstable and central behaviors with respect to arbitrary asymptotic rates ecρ(n) for some diverging sequence ρ(n). This includes as a very special case the usual exponential behavior with ρ(n)=n.     

A Hamiltonian with periodic orbits having several delays

In 1974 Kaplan and Yorke introduced a certain delay equation with an arbitrary number of delays, x^′(t)=-f(x(t-1))-f(x(t-2))-?-f(x(t-n)), conjecturing that it has periodic solutions when f is an odd homeomorphism of the reals which is differentiable at the origin and infinity. By coupling the delay equation to a vector field on Rⁿ⁺¹ they were able to prove, when certain conditions on the derivative of f at 0 and ∞ are satisfied, one delay and two delay versions. We find that the closed orbits of the coupled vector field occur at the points of intersection of two hyperplane fields which are invariant under the flow and invariant under deformation to a linear vector field. By analyzing properties of the linear vector field we are able to give an elementary construction of periodic solutions to the general delay equation when conditions naturally extending those of Kaplan and Yorke are satisfied.     

Analysis of the dynamics of a delayed HIV pathogenesis model

In this paper, considering full Logistic proliferation of CD4⁺ T cells, we study an HIV pathogenesis model with antiretroviral therapy and HIV replication time. We first analyze the existence and stability of the equilibrium, and then investigate the effect of the time delay on the stability of the infected steady state. Sufficient conditions are given to ensure that the infected steady state is asymptotically stable for all delay. Furthermore, we apply the Nyquist criterion to estimate the length of delay for which stability continues to hold, and investigate the existence of Hopf bifurcation by using a delay τ as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the main results.     

A batch arrival retrial queue with general retrial times under Bernoulli vacation schedule for unreliable server and delaying repair

This paper deals with the steady state behaviour of an M^x/G/1 queue with general retrial time and Bernoulli vacation schedule for an unreliable server, which consists of a breakdown period and delay period. Here we assume that customers arrive according to compound Poisson processes. While the server is working with primary customers, it may breakdown at any instant and server will be down for short interval of time. Further concept of the delay time is also introduced. The primary customer finding the server busy, down or vacation are queued in the orbit in accordance with FCFS (first come first served) retrial policy. After the completion of a service, the server either goes for a vacation of random length with probability p or may continue to serve for the next customer, if any with probability (1 − p). We carry out an extensive analysis of this model. Finally, we obtain some important performance measures and reliability indices of this model.     

Edge-disjoint spanners in tori

A spanning subgraph S=(V,E^′) of a connected graph G=(V,E) is an (x+c)-spanner if for any pair of vertices u and v, d_S(u,v)≤d_G(u,v)+c where d_G and d_S are the usual distance functions in G and S, respectively. The parameter c is called the delay of the spanner. We study edge-disjoint spanners in graphs in multi-dimensional tori. We show that each two-dimensional torus has a set of two edge-disjoint spanners of delay approximately the size of the smaller dimension. Moreover, we show that this delay is close to the best possible. In three-dimensional tori, we find a set of three edge-disjoint spanners with delay approximately the sum of the sizes of the two smaller dimensions when all dimensions are of even size. Surprisingly, we also find a set of two edge-disjoint spanners in three-dimensional tori of constant delay. In d-dimensional tori, we show that for any k≤d/5, there is a set of k edge-disjoint spanners with delay depending only on k and the size of the smaller k dimensions.     

Negatively invariant sets of compact maps and an extension of a theorem of Cartwright

Let T be a C¹ map from an open subset of a separable Hilbert space into the Hilbert space, and Γ a negatively invariant compact set, that is T(Γ) ? Γ. Suppose the derivative of T for x?Γ is a uniform contraction on a subspace of finite codimension. Then the topological dimension of Γ is finite. This result may be used to show that for certain delay differential equations and partial differential equations, any almost periodic solution has only finitely many rationally independent frequencies, thus extending results of Cartwright for ODE's.     

Best constant in critical Sobolev inequalities of second-order in the presence of symmetries

Let (M,g) be a smooth compact Riemannian manifold. We first give the value of the best first constant for the critical embedding H²(M)?L^2?(M) for second-order Sobolev spaces of functions invariant by some subgroup of the isometry group of (M,g). We also prove that we can take ?=0 in the corresponding inequality under some geometric assumptions. As an application we give a sufficient condition for the existence of a smooth positive symmetric solution to a critical equation with a symmetric Paneitz-Branson-type operator. A sufficient condition for the existence of a nodal solution to such an equation is also derived. We eventually prove a multiplicity result for such an equation.     

Stochastic Lotka-Volterra system with infinite delay

This paper investigates a stochastic Lotka-Volterra system with infinite delay, whose initial data comes from an admissible Banach space C_r. We show that, under a simple hypothesis on the environmental noise, the stochastic Lotka-Volterra system with infinite delay has a unique global positive solution, and this positive solution will be asymptotic bounded. The asymptotic pathwise of the solution is also estimated by the exponential martingale inequality. Finally, two examples with their numerical simulations are provided to illustrate our result.     

Differentiability and growth bounds of solutions of delay equations

We consider the question of eventual differentiability of the delay semigroups associated with the retarded equation u^′(t)=Au(t)+Φut(t?0), where ut is the history function, A generates an immediately norm-continuous semigroup and Φ is bounded. We show that this is determined by the rate of decay of the resolvent of A along vertical lines.     

Circumference of 3-connected claw-free graphs and large Eulerian subgraphs of 3-edge-connected graphs

The circumference of a graph is the length of its longest cycles. Results of Jackson, and Jackson and Wormald, imply that the circumference of a 3-connected cubic n-vertex graph is Ω(n^0.694), and the circumference of a 3-connected claw-free graph is Ω(n^0.121). We generalize and improve the first result by showing that every 3-edge-connected graph with m edges has an Eulerian subgraph with Ω(m^0.753) edges. We use this result together with the Ryjá?ek closure operation to improve the lower bound on the circumference of a 3-connected claw-free graph to Ω(n^0.753). Our proofs imply polynomial time algorithms for finding large Eulerian subgraphs of 3-edge-connected graphs and long cycles in 3-connected claw-free graphs.     

The L p -L q analog of Morgan’s theorem on exponential solvable Lie groups

In this paper, we define an analog of the L ^p -L ^q Morgan’s uncertainty principle for any exponential solvable Lie group G (p, q ∈ [1,+∞]). When G is nilpotent and has a noncompact center, the proof of such an analog is given for p, q ∈ [2,+∞], extending the earlier settings ([2], [4] and [5]). Such a result is only known for some particular restrictive cases so far. We also prove the result for general exponential Lie groups with nontrivial center.