A central attraction field and a related nonuniqueness theorem Original Research Article

We study the attraction field defined by ordinary differential equations of the form View the MathML sourcex′(s)=v(x(s))−w(x(s))x(s)|x(s)|,x(0)=x0≠0 Turn MathJax on in a closed ball B(r)B(r) centered at the origin and of radius rr in ℜnℜn for n=2,3,…n=2,3,…. We prove that, if v(⋅)v(⋅) and w(⋅)w(⋅) are Lipschitz continuous, v(0)≠0v(0)≠0 and w(x)−|v(x)|≥c1w(x)−|v(x)|≥c1 for some c1>0c1>0, then the solution x(s)=x(s;x0)x(s)=x(s;x0) of the field equation converges to 0 and its derivative x′(s)x′(s) converges simultaneously to either (1−w(0)/|v(0)|)v(0)(1−w(0)/|v(0)|)v(0) or to (1+w(0)/|v(0)|)v(0)(1+w(0)/|v(0)|)v(0) but, under mild conditions, only to the former. Under further simple limitations, the convergence of x′(s;x0)x′(s;x0) is uniform for a set of initial values with a nonempty interior that ‘almost’ surrounds the origin. We then apply these results to establish a related nonuniqueness theorem for the initial value problem y′(t)=f(y(t)),y(0)=0y′(t)=f(y(t)),y(0)=0 in B(r)B(r), where f(⋅)f(⋅) is continuous, and which has a multiplicity of solutions closely related to the ‘backward’ solutions of the field equation.