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Multiple Solutions for a Classical Problem in the Calculus of Variations

There are countless references for this problem. It was solved in full generality by Tonelli, who invented lower semi-continuity for that purpose. He was fortunate in working after Lebesgue had redefined the notion of integral, so that he could use Lebesgue's theory to prove that the integrals arising in the classical calculus of variations were lower semi-continuous. We refer to the treatise of Tonelli [To] for a history of the subject untill 1923, and to the recent book by Giusti [Gi] for a beautiful and concise description of the state of the art today (including its extension to several variables). Of course, there are many other books on the subject, some of which have been extremely popular: Bolza [Bo], Bliss [Bl], Caratheodory [C], Gelfand Fomin [G F]. The book by Bolza, in fact, became so popular that the fixed-endpoint problem we stated became known as the article no. 0162