A topological method for vector-valued and n th-order nonlinear boundary value problems (Q762665)

Topological methods are used to obtain the existence of solution to second order boundary value problems in \(R^ n\) of the form \(x''=f(t,x,x')\), \(g(x(c),x'(c))=0\), \(h(x(c),x'(c),x(d),x'(d))=0\) where f,g,h are continuous. The main techniques used include the Wazewski method, the maximum principle, and the Kneser-Hukuhara continuum theorem. A common ingredient in these approaches is the use of upper and lower solutions to obtain bounds on the solutions. The results generalize the scalar case as well as extend some results of others in \(R^ n\). Analytical conditions are imposed on f which ensure apriori bounds on the solution as well as on its derivative. Geometric conditions are given on the two sets \(\{\) (x,y): \(g(x,y)=0\}\) and \(\{\) (x,y,z,w): \(h(x,y,z,w)=0\}\) which allow one to apply the Wazewski method.