Singular quadratic functionals of đť‘› dependent variables
E. C. Tomastik · Transactions of the American Mathematical Society · 1966
Introduction.In this paper singular quadratic functional« of n dependent variables are systematically studied.Necessary and sufficient conditions for the existence of a minimum are given, generalizing the results of Morse and Leighton [6].The study of principal quadratic functionals of n dependent variables is undertaken.A necessary condition is proved and sufficient conditions are obtained for certain cases.Finally, an oscillation theorem is given for systems of second-order linear differential equations.We shall make the usual conventions regarding notation.A repeated subscript indicates summation.Wc shall use the symbol (*) to indicate the transpose of a matrix or vector.In general capital letters indicate matrices, and we shall employ the notation A = || a¡¡ | for aimatrix when necessary.The determinant of a matrix A will be written det A. The notation | a | will be reserved to indicate the absolute value of a number a. I.The functional.Let f(x,y,y') -y*'(x)R(x)y'(x) + 2y*'(x)Q(x)y(x) (1.1) + y*(x)P(x)y(x), where R(x), Q'(x), and P(x) ate symmetric matrices continuous in the real variable x on [a, oo), and R(x) is positive definite for any fixed x on [a, oo).We consider the functional (1.2) J(y)\ba = f f(x,y,y')dx (a £ 6 < oo).J a Integrals employed throughout are Lebesgue integrals and their extensions.