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    Three Positive Periodic Solutions to Nonlinear Neutral Functional Differential Equations with Parameters on Variable Time Scales by Yongkun Li, Chao Wang

    Published 2012-01-01
    “…Using two successive reductions: B-equivalence of the system on a variable time scale to a system on a time scale and a reduction to an impulsive differential equation and by Leggett-Williams fixed point theorem, we investigate the existence of three positive periodic solutions to the nonlinear neutral functional differential equation on variable time scales with a transition condition between two consecutive parts of the scale (d/dt)(x(t)+c(t)x(t-α))=a(t)g(x(t))x(t)-∑j=1nλjfj(t,x(t-vj(t))), (t,x)∈T0(x),Δt|(t,x)∈S2i=Πi1(t,x)-t, Δx|(t,x)∈S2i=Πi2(t,x)-x, where Πi1(t,x)=t2i+1+τ2i+1(Πi2(t,x)) and Πi2(t,x)=Bix+Ji(x)+x,  i=1,2,….  λj   (j=1,2,…,n) are parameters, T0(x) is a variable time scale with (ω,p)-property, c(t),  a(t), vj(t), and fj(t,x)   (j=1,2,…,n) are ω-periodic functions of t, Bi+p=Bi,  Ji+p(x)=Ji(x) uniformly with respect to i∈Z.…”
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    Linear right ideal nearrings by Kenneth D. Magill

    Published 2001-01-01
    “…We determine, up to isomorphism, all those topological nearrings 𝒩n whose additive groups are the n-dimensional Euclidean groups, n>1, and which contain n one-dimensional linear subspaces {Ji}i=1n which are also right ideals of the nearring satisfying several additional properties. …”
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