An approximation theory for focal points and focal intervals
John A. Gregory · Proceedings of the American Mathematical Society · 1972
The theory of focal points and conjugate points is an important part of the study of problems in the calculus of variations and control theory. In previous works we gave a theory of focal points and of focal intervals for an elliptic form J ( x ) J(x) on a Hilbert space A \mathcal {A} . These results were based upon inequalities dealing with the indices s ( σ ) s(\sigma ) and n ( σ ) n(\sigma ) of the elliptic form J ( x ; σ ) J(x;\sigma ) defined on the closed subspace A ( σ ) \mathcal {A}(\sigma ) of A \mathcal {A} , where σ \sigma belongs to the metric space ( Σ , ρ ) (\Sigma ,\rho ) . In this paper we give an approximation theory for focal point and focal interval problems. Our results are based upon inequalities dealing with the indices s ( μ ) s(\mu ) and u ( μ ) u(\mu ) , where μ