Abstract
THIS paper is a direct continuation of the first three paperst in this series. The notation of the first three papers is assumed throughout and is used without further note. Citation of results from previous papers is made by equation number. The numbering of this paper follows that of the first three papers in sequence. IO. STATlONARIZATlON AND THE NON-LOCAL EULER-LAGRANGE OPERATOR The types of problems we now wish to study are those involving two or more dependent functions. In the case of only one dependent function, we found that it was sufficient to consider the linear spaces of functions of class C or of class Cl with the uniform convergence norm. We now consider the situation wherein we must use vector spaces of function spaces. Let {@*(xk)}, A = 1,. . ., N, denote an order collection of N functions of class C or of class Cl on D*. Such collections of functions form a linear space with the usual definitions of addition and scalar multiplication. If each of the {@*(I?)} is of class C on D * we define the norm by
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