Multiplicity of positive periodic solutions to a generalized delayed predator–prey system with stocking

In this paper, by using the continuation theorem of coincidence degree theory, the existence of multiple positive periodic solutions for a generalized delayed predator–prey system with stocking is established. When our result is applied to a delayed predator–prey system with nonmonotonic functional response and stocking, we establish the sufficient condition for the existence of multiple positive periodic solutions for the system.     

On stability of linear periodic differential difference equations

The paper considers the equation

where the operator-valued bounded functions a_j and b_j are 2π-periodic, and the operator-valued kernels m and n are 2π-periodic with respect to the first argument. The connection between the input-output stability of the equation and the invertibility of a family of operators acting on the space of periodic functions is investigated.     

Almost periodic solutions for an impulsive delay Nicholson’s blowflies model

By means of the contraction mapping principle and Gronwall-Bellman’s inequality, we prove the existence and exponential stability of positive almost periodic solution for an impulsive delay Nicholson’s blowflies model. The main results are illustrated by an example.     

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).     

Some results on second-order neutral functional differential equations with infinite distributed delay

In this paper, we consider two types of second-order neutral functional differential equations with infinite distributed delay. By choosing available operators and applying Krasnoselskii’s fixed-point theorem, we obtain sufficient conditions for the existence of periodic solutions to such equations.     

Existence and uniqueness of periodic solutions for a p-Laplacian Duffing equation with a deviating argument

By means of Mawhin’s continuation theorem, we study a class of p-Laplacian Duffing type differential equations of the form

(_φp(x^′(t)))^′=Cx^′(t)+g(t,x(t),x(t−τ(t)))+e(t).${\left({\phi }_p\left(x^{\prime }\left(t\right)\right)\right)}^{\prime }=C{x}^{\prime }\left(t\right)+g(t,x\left(t\right),x(t-\tau \left(t\right)))+e\left(t\right)\text{.}$

Analytical and numerical investigation of mixed-type functional differential equations

This paper is concerned with the approximate solution of a linear non-autonomous functional differential equation, with both advanced and delayed arguments. We search for a solution x(t), defined for t∈[−1,k], (k∈N), that satisfies this equation almost everywhere on [0,k−1] and assumes specified values on the intervals [−1,0] and (k−1,k]. We provide a discussion of existence and uniqueness theory for the problems under consideration and describe numerical algorithms for their solution, giving an analysis of their convergence.     

A regularization for discontinuous differential equations with application to state-dependent delay differential equations of neutral type

We consider a regularization for a class of discontinuous differential equations arising in the study of neutral delay differential equations with state dependent delays. For such equations the possible discontinuity in the derivative of the solution at the initial point may propagate along the integration interval giving rise to so-called “breaking points”, where the solution derivative is again discontinuous. Consequently, the problem of continuing the solution in a right neighborhood of a breaking point is equivalent to a Cauchy problem for an ode with a discontinuous right-hand side (see e.g. Bellen et al., 2009 [4]). Therefore a classical solution may cease to exist.The regularization is based on the replacement of the vector-field with its time average over an interval of length ε>0. The regularized solution converges as ε→⁰⁺ to the classical Filippov solution (Filippov, 1964, 1988 [13] and [14]). Several properties of the solutions corresponding to small ε>0 are presented.     

Existence of periodic solutions for a Liénard type p-Laplacian differential equation with a deviating argument

By means of Mawhin’s continuation theorem, a class of p-Laplacian type differential equation with a deviating argument of the form

(_φp(x^′(t)))^′+f(x(t))x^′(t)+β(t)g(t,x(t−τ(t,_|x|∞)))=e(t)${\left({\phi }_p\left(x^{\prime }\left(t\right)\right)\right)}^{\prime }+f\left(x\left(t\right)\right)x^{\prime }\left(t\right)+\beta \left(t\right)g(t,x(t-\tau (t,{\left|x\right|}_{\infty })))=e\left(t\right)$

Existence and multiplicity of wave trains in 2D lattices

We study the existence and branching patterns of wave trains in a two-dimensional lattice with linear and nonlinear coupling between nearest particles and a nonlinear substrate potential. The wave train equation of the corresponding discrete nonlinear equation is formulated as an advanced-delay differential equation which is reduced by a Lyapunov–Schmidt reduction to a finite-dimensional bifurcation equation with certain symmetries and an inherited Hamiltonian structure. By means of invariant theory and singularity theory, we obtain the small amplitude solutions in the Hamiltonian system near equilibria in non-resonance and p:q$p:q$ resonance, respectively. We show the impact of the direction θ of propagation and obtain the existence and branching patterns of wave trains in a one-dimensional lattice by investigating the existence of traveling waves of the original two-dimensional lattice in the direction θ of propagation satisfying tan θ is rational.     

Positive solutions of p-type retarded functional differential equations

For systems of retarded functional differential equations with unbounded delay and with finite memory sufficient and necessary conditions of existence of positive solutions on an interval of the form [_t0,∞)$[t_0,\infty )$ are derived. A general criterion is given together with corresponding applications (including a linear case, too). Examples are inserted to illustrate the results.     

Continuous transformations of differential equations with delays

The aim of this paper is to find the class of continuous pointwise transformations (as general as possible) in the framework of which Kummer's transformationz(t)=g(t)y(h(t)) represents the most general pointwise transformation converting every linear homogeneous differential equation of thenth order into an equation of the same type. Further, some forms of these equations having certain subspaces of solutions are considered.     

Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks

This paper addresses the local and global stability of n-dimensional Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks. Necessary and sufficient conditions for local stability independent of the choice of the delay functions are given, by imposing a weak nondelayed diagonal dominance which cancels the delayed competition effect. The global asymptotic stability of positive equilibria is established under conditions slightly stronger than the ones required for the linear stability. For the case of monotone interactions, however, sharper conditions are presented. This paper generalizes known results for discrete delays to systems with distributed delays. Several applications illustrate the results.     

Nonoscillation and positivity of solutions to first order state-dependent differential equations with impulses in variable moments

The state-dependent delay differential equation

     

Sharp conditions for global stability of Lotka-Volterra systems with distributed delays

We give a criterion for the global attractivity of a positive equilibrium of n-dimensional non-autonomous Lotka-Volterra systems with distributed delays. For a class of autonomous Lotka-Volterra systems, we show that such a criterion is sharp, in the sense that it provides necessary and sufficient conditions for the global asymptotic stability independently of the choice of the delay functions. The global attractivity of positive equilibria is established by imposing a diagonal dominance of the instantaneous negative feedback terms, and relies on auxiliary results showing the boundedness of all positive solutions. The paper improves and generalizes known results in the literature, namely by considering systems with distributed delays rather than discrete delays.     

Wave propagation in RTD-based cellular neural networks

This work investigates the existence of monotonic traveling wave and standing wave solutions of RTD-based cellular neural networks in the one-dimensional integer lattice . For nonzero wave speed c, applying the monotone iteration method with the aid of real roots of the corresponding characteristic function of the profile equation, we can partition the parameter space (γ,δ)-plane into four regions such that all the admissible monotonic traveling wave solutions connecting two neighboring equilibria can be classified completely. For the case of c=0, a discrete version of the monotone iteration scheme is established for proving the existence of monotonic standing wave solutions. Furthermore, if γ or δ is zero then the profile equation for the standing waves can be viewed as an one-dimensional iteration map and we then prove the multiplicity results of monotonic standing waves by using the techniques of dynamical systems for maps. Some numerical results of the monotone iteration scheme for traveling wave solutions are also presented.