Let T be a time scale such that 0, T ∈ T. By means of the Schauder fixed point theorem and analysis method, we establish some existence criteria for positive solutions of the m-point boundary value problem on time scales where α ∈ Ctd（（O,T,[0,∞））,f∈ Ckd（[0,∞）×[0,∞））,β,γ ∈[0,∞）,ξi ∈（0,p（T）.b,ai∈ （0,∞） （for i = 1,..., m - 2） are given constants satisfying some suitable hypotheses. We show that this problem has at least one positive solution for sufficiently small b 〉 0 and no solution for sufficiently large b. Our results are new even for the corresponding differential equation （T = R） and difference equation （T = Z）.
The existence, uniqueness and global asymptotic stability for the equilibrium of Hopfield-type neural networks with diffusion effects are studied. When the activation functions are monotonously nondecreasing, differentiable, and the interconnected matrix is related to the Lyapunov diagonal stable matrix, the sufficient conditions guaranteeing the existence of the equilibrium of the system are obtained by applying the topological degree theory. By means of constructing the suitable average Lyapunov functions, the global asymptotic stability of the equilibrium of the system is also investigated. It is shown that the equilibrium （if it exists） is globally asymptotically stable and this implies that the equilibrium of the system is unique.
考虑了非线性项是变号的m-点奇异p-Laplacian动力方程(p(u~△(t)))~▽+ q(t)f(t,u(t))=0,t∈(0,T)_T,u(0)=0,_p(u~△(T))=sum from i=1 to m-2_i(u~△(ξ_i)),其中_p(s)= |s|^(p-2)s,p>1,ξ_i:R→R是连续的、不增的,0<ξ_1<ξ_2<…<ξ_(m-2)<ρ(T).利用Schauder不动点定理和上下解方法,证明了上述边值问题正解的一些存在性法则.这些结果在相应的微分方程(T=R)、差分方程(T=Z)以及通常的测度链上都是新的.特别是,如果非线性项容许变号,那么Sun和Li的结果[Appl.Math.Comput.,2006,182:478-491]仅仅是我们所得结果在相应微分方程(T=R)的一种特殊情形.作为应用,给出了一个例子验证了主要结果.
