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.     

Existence of positive periodic solutions for two kinds of neutral functional differential equations

In this work, we deal with a new existence theory for positive periodic solutions for two kinds of neutral functional differential equations by employing the Krasnoselskii fixed-point theorem. Applying our results to various mathematical models we improve some previous results.     

Existence of positive periodic solutions of impulsive functional differential equations with two parameters

In this paper, we employ a well-known fixed-point index theorem to study the existence and non-existence of positive periodic solutions for the periodic impulsive functional differential equations with two parameters. Several existence and non-existence results are established.     

Bifurcations of periodic solutions of delay differential equations

In this paper we develop Kaplan-Yorke's method and consider the existence of periodic solutions for some delay differential equations. We especially study Hopf and saddle-node bifurcations of periodic solutions with certain periods for these equations with parameters, and give conditions under which the bifurcations occur. We also give application examples and find that Hopf and saddle-node bifurcations often occur infinitely many times.     

Oscillation of fourth order sub-linear differential equations

In this paper, we deal with oscillatory and asymptotic properties of solutions of a fourth order sub-linear differential equation with the oscillatory operator. We establish conditions for the nonexistence of positive and bounded solutions and an oscillation criterion.     

Numbers of subnormal solutions for higher order periodic differential equations

In this paper, we estimate the number of subnormal solutions for higher order linear periodic differential equations, and estimate the growth of subnormal solutions and all other solutions. We also give a representation of subnormal solutions of a class of higher order linear periodic differential equations.     

Two periodic solutions of second-order neutral functional differential equations

In this paper, we consider a type of second-order neutral functional differential equations. We obtain some existence results of multiplicity and nonexistence of positive periodic solutions. Our approach is based on a fixed point theorem in cones.     

Nontrivial periodic solutions to a type of delayed resonant differential equations

In this paper, we consider a type of delayed resonant differential equations. We focus on the existence of periodic solutions. Employing the Clark dual, we provide two sets of criteria on the existence of at least one periodic solution. In fact, the periodic solutions are critical points minimizing the dual functional of the coupled Hamiltonian system on certain subspaces of a Banach space.     

Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations

By means of a monotone iterative technique, we establish the existence and uniqueness of the positive solutions for a class of higher conjugate-type fractional differential equation with one nonlocal term. In addition, the iterative sequences of solution and error estimation are also given. In particular, this model comes from economics, financial mathematics and other applied sciences, since the initial value of the iterative sequence can begin from an known function, this is simpler and helpful for computation.     

On the structure of positive solutions of a class of fourth order nonlinear differential equations

A detailed analysis is made of the structure of positive solutions of fourth-order differential equations of the form

Positive periodic solutions of first-order functional differential equations with parameter

We prove the existence and multiplicity of positive T$T$-periodic solution(s) for T$T$-periodic equation x^′(t)=h(t,x)−λb(t)f(x(t−τ(t)))$x^{\prime }\left(t\right)=h(t,x)-\lambda b\left(t\right)f\left(x(t-\tau \left(t\right))\right)$ by Krasnoselskii fixed point theorem, where f(x)$f\left(x\right)$ may be singular at x=0$x=0$. Our results improve some recent results in previous literature.     

On positive solutions of boundary value problems for second-order functional differential equations on infinite intervals

We study the existence of positive solutions for the following boundary value problem on infinite interval for second-order functional differential equations:

     

On oscillatory solutions of quasilinear differential equations

Necessary and sufficient conditions for the existence of at least one oscillatory solution of a second-order quasilinear differential equation are presented. These results yield also new conditions guaranteeing the coexistence of oscillatory and nonoscillatory solutions. Our approach is based on the asymptotic representation of solutions by means of a periodic function and of a suitable zero-counting function.     

Existence of multiple positive periodic solutions for functional differential equations

In our paper, by employing Krasnoselskii fixed point theorem, we investigate the existence of multiple positive periodic solutions for functional differential equations

     

Positive solutions to delayed differential equations of the second-order

A new criterion for the existence of positive solutions of the second-order delayed differential equation $\ddot{y}\left(t\right)=f(t,y_t,{\dot{y}}_t)$, $t\in [t_0,\infty )$ is given with applications to linear equations. Open problems for future research are formulated.     

Existence and uniqueness of periodic solutions for a kind of Rayleigh equation with finitely many deviating arguments

In this paper, we consider a kind of Rayleigh equation with finitely many deviating arguments of the form

     

Three periodic solutions for a nonlinear first order functional differential equation

This paper is concerned with the existence of three positive T-periodic solutions of the first order functional differential equations of the form

x^′(t)=a(t)x(t)-λb(t)f(t,x(h(t))),     

Positive periodic solutions of functional differential equations

We consider the existence, multiplicity and nonexistence of positive ω-periodic solutions for the periodic equation x′(t)=a(t)g(x)x(t)−λb(t)f(x(t−τ(t))), where are ω-periodic, , , f,g∈C([0,∞),[0,∞)), and f(u)>0 for u>0, g(x) is bounded, τ(t) is a continuous ω-periodic function. Define , , i₀=number of zeros in the set and i_∞=number of infinities in the set . We show that the equation has i₀ or i_∞ positive ω-periodic solution(s) for sufficiently large or small λ>0, respectively.