This paper presents a new generalized Nicholson’s blowflies system with patch structure and nonlinear density-dependent mortality terms. Under appropriate conditions, we establish some criteria to guarantee the exponential extinction of this system. Moreover, we give two examples and numerical simulations to demonstrate our main results.

For nonautonomous linear equations $x^{\prime }=A(t)x$, we give a complete characterization of general nonuniform contractions in terms of Lyapunov functions. We consider the general case of nonuniform contractions, which corresponds to the existence of what we call nonuniform $(D,\mu )$-contractions. As an application, we establish the robustness of the nonuniform contraction under sufficiently small linear perturbations. Moreover, we show that the stability of a nonuniform contraction persists under sufficiently small nonlinear perturbations.

The relation between the notions of nonuniform asymptotic stability and admissibility is considered. Using appropriate Lyapunov norms, it is showed that if any of their associated ${ℒ}^p$ spaces, with $p\in (1,\infty ]$, is admissible for a given evolution process, then this process is a nonuniform $(\mu ,\nu )$ contraction and dichotomy. A collection of admissible Banach spaces for any given nonuniform $(\mu ,\nu )$ contraction and dichotomy is provided.

Analytical properties like existence, uniqueness, and asymptotic behavior of solutions are studied for the following singular initial value problem: $g_i(t)y_i^{\prime }(t)=a_i{y}_i(t)(1+f_i(t,\mathbf{y}(t),{\int }_{0^{+}}^t{K}_i(t,s,\mathbf{y}(t),\mathbf{y}(s))ds))$, $y_i(0^{+})=0$, $t\in (0,\hspace{0.17em}{t}_0]$, where $\mathbf{y}=(y_1,\hspace{0.17em}\dots ,\hspace{0.17em}{y}_n)$, $a_i>0$, $i=1,\hspace{0.17em}\dots ,\hspace{0.17em}n$ are constants and $t_0>0$. An approach which combines topological method of T. Ważewski and Schauder's fixed point theorem is used. Particular attention is paid to construction of asymptotic expansions of solutions for certain classes of systems of integrodifferential equations in a right-hand neighbourhood of a singular point.

Three-point boundary value problems of second-order differential equation with a p-Laplacian on finite and infinite intervals are investigated in this paper. By using a new continuation theorem, sufficient conditions are given, under the resonance conditions, to guarantee the existence of solutions to such boundary value problems with the nonlinear term involving in the first-order derivative explicitly.

We consider the existence of multiple solutions of the singular elliptic problem $-\mathrm{div}({|x|}^{-ap}{|\nabla u|}^{p-2}\nabla u)+{|u|}^{p-2}u/{|x|}^{(a+1)p}=f{|u|}^{r-2}u+h{|u|}^{s-2}u+{|x|}^{-b{p}^{*}}{|u|}^{p^{*}-2}u$, $u(x)\to 0$ as $|x|\to +\infty$, where $x\in {ℝ}^N$, , , $a\le b\le a+1,r$, $s>1$, $p^{*}=Np/(N-pd)$, $d=a+1-b$. By the variational method and the theory of genus, we prove that the above-mentioned problem has infinitely many solutions when some conditions are satisfied.

We study the existence of periodic solutions of some second-order Hamiltonian systems with impulses. We obtain some new existence theorems by variational methods.

The existence and multiplicity of solutions for second-order differential equations with a parameter are discussed in this paper. We are mainly concerned with the semipositone case. The analysis relies on the nonlinear alternative principle of Leray-Schauder and Krasnosel'skii's fixed point theorem in cones.

Let $L$ denote the operator generated in ${\ell }_2(\mathbb{N},{ℂ}^2)$ by $a_{n+1}{y}_{n+1}^{(2)}+b_n{y}_n^{(2)}+p_n{y}_n^{(1)}=\lambda y_n^{(1)}$, $a_{n-1}{y}_{n-1}^{(1)}+b_n{y}_n^{(1)}+q_n{y}_n^{(2)}=\lambda y_n^{(2)}$, $n\in \mathbb{N}$, and the boundary condition $({\gamma }_0+{\gamma }_1\lambda )y_1^{(2)}+({\beta }_0+{\beta }_1\lambda )y_0^{(1)}=\mathbf{0}$, where $(a_n)$, $(b_n)$, $(p_n)$, and $(q_n)$, $n\in \mathbb{N}$ are complex sequences, ${\gamma }_i,{\beta }_i\in ℂ$, $i=\mathrm{0,1}$, and $\lambda$ is an eigenparameter. In this paper we investigated the principal functions corresponding to the eigenvalues and the spectral singularities of $L$.

We consider a predator-prey model in which the preys disperse among $n$ patches ($n\ge 2$) with stochastic perturbation. We show that there is a unique positive solution and find out the sufficient conditions for the extinction to the system with any given positive initial value. In addition, we investigate that there exists a stationary distribution for the system and it has ergodic property. Finally, we illustrate the dynamic behavior of the system with $n=2$ via numerical simulation.

This paper deals with the periodic solutions of a class of fourth-order superlinear differential equations. By using the classical variational techniques and symmetric mountain pass lemma, the periodic solutions of a single equation in literature are extended to that of equations, and also, the cubic growth of nonlinear term is extended to a general form of superlinear growth.

We investigate the existence of multiple solutions for a class of nonhomogeneous Neumann problem with a perturbed term. By using variational methods and three critical point theorems of B. Ricceri, we establish some new sufficient conditions under which such a problem possesses three solutions in an appropriate Orlicz-Sobolev space.

This paper gives an application of Mawhin’s coincidence degree and matrix spectral theory to a predator-prey model with M-predators and N-preys. The method is different from that used in the previous work. Some new sufficient conditions are obtained for the existence and global asymptotic stability of the periodic solution. The existence and stability conditions are given in terms of spectral radius of explicit matrices which are much different from the conditions given by the algebraic inequalities. Finally, an example is given to show the feasibility of our results.