Vol. 74, No. 1, 1978

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On a theorem of S. Bernstein

Andrzej Granas, Ronald Bernard Guenther and John Walter Lee

Vol. 74 (1978), No. 1, 67–82

In 1912, S. Bernstein, in the first part of his memoir [2] devoted to the boundary value problems arising in calculus of variations, established sufficient conditions for the unique solvability of the Dirichlet problem for the equation y′′ = f(t,y,y). Our aim is to present a result which extends the scope of the Bernstein theorem and to show that the generalization obtained can be carried over (with only minor adjustments in the proof) to the case of all important boundary value problems which arise in applications.

Mathematical Subject Classification 2000
Primary: 34B15
Received: 3 June 1977
Published: 1 January 1978
Andrzej Granas
Ronald Bernard Guenther
John Walter Lee