Rong Yuan

Critical type of Krasnosel’skii fixed point theorem

In this paper, by means of the technique of measures of noncompactness, we establish a generalized form of the fixed point theorem for the sum of T+S, where S is noncompact, I -T may not be injective, and T is not necessarily continuous. The obtained results unify and significantly extend a number of previously known generalizations of the Krasnoselskii fixed point theorem. The analysis presented here reveals the essential characteristics of the Krasnoselskii type fixed point theorem in strong topology setups. Further, the results are used to prove the existence of periodic solutions of a nonlinear neutral differential equation with delay in the critical case.

On the second-order differential equation with piecewise constant argument and almost periodic coefficients

In this paper, we study the existence of almost and quasi-periodic solutions to two classes of second-order differential equations. As a corollary, it is shown that periodic and unbounded solutions can coexist for the equation x(t) + ω2x(t) = bx([t]) + f(t), which is different from the case: b = 0. This phenomena is due to the piecewise constant argument and illustrates a crucial difference between ordinary differential equations and differential equations with piecewise constant argument. The results are extended to nonlinear equations.

Effect of harvesting, delay and diffusion in a generalist predator-prey model

In compared with specialist predators which feed almost exclusively on a specific species of prey, generalist predators feed on many types of species. Consequently, their dynamics is not coupled to the dynamics of a specific prey population, and the generalist predators has itself growth function which be extended a well-known logistic growth term. We develop a generalist predator-prey model with diffusion and study the effect of harvesting and delay under Neumann conditions. The stability of the equilibria is firstly investigated, and the existence of traveling wave solutions is then established by constructing a pair of upper-lower solutions and using the cross iteration method and Schauders fixed point theorem.

The impact of predation on the coexistence and competitive exclusion of pathogens in prey.

A two-strain epidemic model with saturating contact rate under a generalist predator is proposed. For a generalist predator which feeds on many types of prey, we assume that the predator can discriminate among susceptible and infected with each strain prey. First, mathematical analysis of the model with regard to invariance of nonnegativity, boundedness of solutions, nature of equilibria, persistence and global stability are analyzed. Second, the two strains will competitively exclude each other in the absence of predation with the strain with the larger reproduction number persisting. If predation is discriminate, then depending on the predation level, a dominant strain may occur. Thus, for some predation levels, the strain one may persist while for other predation levels strain two may persist. Furthermore, coexistence line and coexistent asymptotic-periodic solution are obtained when coexistence occur while heteroclinic is obtained when the two strains competitively exclude each other. Finally, the impact of predation is mentioned along with numerical results to provide some support to the analytical findings.

Homoclinic solutions for second order Hamiltonian systems with general potentials

Abstract In this paper we are concerned with the existence of infinitely many homoclinic solutions for the following second order non-autonomous Hamiltonian systems u ¨ t − L t u t + ∇ W t , u t = 0

Homoclinic solutions for a class of non-autonomous subquadratic second-order Hamiltonian systems

ddot uleft( t right) - Lleft( t right)uleft( t right) + nabla Wleft( {t,uleft( t right)} right) = 0

Psuedo-almost periodic solutions of second-order neutral delay differential equations with piecewise constant argument

(HS) where t ∈ ℝ, L ∈ C(ℝ, ℝn2) is a symmetric and positive definite matrix for all t ∈ ℝ, W ∈ C1(ℝ × ℝn, ℝ) and ∇W(t,u) is the gradient of W at u. The novelty of this paper is that, assuming that L meets some coercive condition and the potential W is of the form W(t, u) = W1(t, u) + W2(t, u), for the first time we show that (HS) possesses two different sequences of infinitely many homoclinic solutions via the Fountain theorem and the dual Fountain theorem such that the corresponding energy functional of (HS) goes to infinity and zero, respectively. Some recent results in the literature are generalized and significantly improved.

Homoclinic solutions of some second order non-autonomous systems

We consider the asymptotic stability and attractor bifurcation of the extended Fisher-Kolmogorov equation on the one-dimensional domain with Dirichlet or periodic boundary conditions. The novelty of this paper is that, based on a new method called attractor bifurcation, we investigate the existence of an attractor bifurcated from the trivial solution and give an explicit description of the bifurcated attractor. Moreover, the stability of the bifurcated branches is discussed.