By J. Aczél

Recently I taught brief classes on practical equations at numerous universities (Barcelona, Bern, Graz, Hamburg, Milan, Waterloo). My objective used to be to introduce crucial equations and techniques of resolution via genuine (not artifi­ cial) purposes which have been contemporary and with which I had anything to do. such a lot of them occurred to be concerning the social or behavioral sciences. All have been initially solutions to questions posed by way of experts within the respective utilized fields. the following I provide a a bit of prolonged model of those lectures, with more moderen effects and functions integrated. As past wisdom simply the elemental evidence of calculus and algebra are intended. components the place a little bit extra (measure conception) is required and sketches of lengthier calcula­ tions are set in tremendous print. i'm thankful to Drs. J. Baker (Waterloo, Ont.), W. Forg-Rob (Innsbruck, Austria) and C. Wagner (Knoxville, Tenn.) for serious feedback and to Mrs. Brenda legislation for care­ ful computer-typing of the manuscript (in numerous versions). A be aware on numbering of statements and references: The numbering of Lemmata, Propositions, Theorems, Corollaries and (separately) formulae begins anew in each one part. If quoted in one other part, the part quantity is extra, e.g. (2.10) or Theorem 1.2. References are quoted through the final names of the authors and the final digits of the yr, e.g. Daroczy-Losonczi [671. 1 1. An aggregation theorem for allocation difficulties. Cauchy equation for single-and multiplace services. extension theorems.

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Extra info for A Short Course on Functional Equations: Based Upon Recent Applications to the Social and Behavioral Sciences

Example text

Functions satisfying (13) are called multiplicative. Also this equation can be reduced to the Cauchy equation or to (8). It is clear that we will do the latter by taking logarithms of both sides of (13). For this we have to establish that a is everywhere positive. Actually, a(x) 0 is a solution of (13) but we exclude this (since, by (12) and (7), it would cause u to be constant). Then a is nowhere 0 because a(xo) = 0, would imply = x x x a(x) = a(-xo) = a(-) = a(-)a(xo) = 0 for all positive Xo Xo Xo x.

Keeping y constant and writing B(r) this becomes u(rx) = o:(r)u(x) + 3(r) = u(ry) - o:(r)u(y), (x,r EIRp) (5) the 'linear-affine functional equation' (so called, because x f-+ rx is a (homogeneous) linear and u f-+ 0: U + :3 an affine (inhomogeneous linear) transformation). Notice that (5) contains three unknown functions u. 5). It is remarkable that one functional equation can determine several unknown functions. We proceed by reducing the number of unknown functions. For this purpose, first put x = 1 into (5): u(r) = + f3(r) o:(r)u(l) and, by subtracting this from (5), get the functional equation v(rx) = o:(r)v(x) + v(r) (6) for the function v defined by v(x) = u(x) - u(l) (x EIR-,-+) .

Xn) = aA1(x) an A1 {Xi) n = i i=l with R{r) = A1(r) (x,r EJR:-,-) , (8) where A1:JR~+ -lR .... \1l, ... 'vfn :JR4-+ -lR-,-~ are arbitrary multiplicative functions and a is an arbitrary positive constant. These solutions are again too general. For (8), boundedness from above of u on a non degenerate n-dimensional interval is sufficient to have the same property for A1 (since a > 0) and boundedness from above on one-dimensional proper intervals for ;'vfi (by keeping Xj constant for j=l=i,i =l, ...

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