Download A Short Course on Functional Equations: Based Upon Recent by J. Aczél PDF

By J. Aczél

Recently I taught brief classes on sensible equations at a number of universities (Barcelona, Bern, Graz, Hamburg, Milan, Waterloo). My objective used to be to introduce an important equations and techniques of resolution via genuine (not artifi­ cial) purposes which have been fresh and with which I had anything to do. so much of them occurred to be on the topic of the social or behavioral sciences. All have been initially solutions to questions posed through experts within the respective utilized fields. right here I supply a just a little prolonged model of those lectures, with newer effects and purposes incorporated. As prior wisdom simply the elemental evidence of calculus and algebra are meant. elements the place slightly extra (measure conception) is required and sketches of lengthier calcula­ tions are set in wonderful print. i'm thankful to Drs. J. Baker (Waterloo, Ont.), W. Forg-Rob (Innsbruck, Austria) and C. Wagner (Knoxville, Tenn.) for serious comments and to Mrs. Brenda legislations 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 every one part. If quoted in one other part, the part quantity is further, e.g. (2.10) or Theorem 1.2. References are quoted via 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 features. extension theorems.

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

Sample text

So we have the following. = Corollary 2. The general form of laws where all independent variables have the same ratio scale and the dependent variable also has a ratio case, that is, the general solution of (18) is given by 42 Section 3 X2 Xn Xl Xl . = M(xl)f(-, ... 1 -1R++ is an arbitrary function. ) Of course these laws, containing arbitrary multiplicative and logarithmic functions, are too general. 1 we can determine all solutions u bounded on an open n-dimensional interval I, which certainly is a reasonable supposition for the applications we have in mind (boundedness on a set of positive measure would be sufficient but does not sound very practical).

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-,-+) . The equation (6) contains only two unknown functions. solving (6) we ~istinguish two cases. (7) In Section 2 26 First, if o:{r) = 1 for all positive r, then (6) goes over into v{rx) = v{r) + v{x) (x,r E lR++) (8) Functions satisfying (8) are called logarithmic.

In the particular case x = (8, ... 8). We have called functions satisfying this equation 'logarithmic' functions. To remind us of this we will denote by L any function which satisfies L(rs) = L(r) + L(s) (r,s E 1R++) (6) So we have now v(r) = L(r). Notice that this is a statement about v(r) = v(r ,... ,r), not about v(x) = v(x1,. ,xn ). However, we can write (5) as v(rx) = v(x) + L{r) and determine v(x) immediately, for instance in the form 1 x2 xn xl Xl Xl v(x) = v(xl-x) = v(1,-, ... , - ) + L(xd If we take (4) into consideration and write we get (7) Conversely, every function of this form, with arbitrary L satisfying (6) and with arbitrary f, satisfies (3): X2 Xn xl Xl u(rx) = f(-, ...

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