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By Marek Kuczma

Marek Kuczma used to be born in 1935 in Katowice, Poland, and died there in 1991.

After completing highschool in his domestic city, he studied on the Jagiellonian collage in Kraków. He defended his doctoral dissertation less than the supervision of Stanislaw Golab. within the yr of his habilitation, in 1963, he got a place on the Katowice department of the Jagiellonian collage (now collage of Silesia, Katowice), and labored there until eventually his death.

Besides his numerous administrative positions and his impressive educating task, he entire first-class and wealthy medical paintings publishing 3 monographs and a hundred and eighty medical papers.

He is taken into account to be the founding father of the distinguished Polish institution of practical equations and inequalities.

"The moment half the identify of this e-book describes its contents effectively. most likely even the main committed expert do not have suggestion that approximately three hundred pages may be written almost about the Cauchy equation (and on a few heavily similar equations and inequalities). And the booklet is under no circumstances chatty, and doesn't even declare completeness. half I lists the necessary initial wisdom in set and degree idea, topology and algebra. half II offers info on ideas of the Cauchy equation and of the Jensen inequality [...], specifically on non-stop convex capabilities, Hamel bases, on inequalities following from the Jensen inequality [...]. half III bargains with comparable equations and inequalities (in specific, Pexider, Hosszú, and conditional equations, derivations, convex capabilities of upper order, subadditive features and balance theorems). It concludes with an day trip into the sector of extensions of homomorphisms in general." (Janos Aczel, Mathematical Reviews)

"This publication is a true vacation for the entire mathematicians independently in their strict speciality. you can actually think what deliciousness represents this e-book for useful equationists." (B. Crstici, Zentralblatt für Mathematik)

 

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Extra info for An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality

Example text

1. Let (X, +) be a topological group, and A, B ⊂ X two sets of the second category with the Baire property. Then int (A + B) = ∅ . 3 Added in the 2nd edition by K. Baron. 9. Theorem of S. Piccard 43 Proof. X itself is of the second category, since it contains sets of the second category. Then every non-empty open subset of X is of the second category. In fact, suppose that G ⊂ X is non-empty, open and of the first category. Fix an a ∈ G. 5) are homeomorphisms and x ∈ G − a + x. Thus X is of the first category at every point x ∈ X.

2) In fact, (H ∪ C) \ D = [H ∪ R ∪ (G \ H)] ∩ (P ∩ R ) = [H ∪ R ∪ (G ∩ H )] ∩ (P ∪ R) = (H ∪ R ∪ G ) ∩ (H ∪ R ∪ H ) ∩ (P ∪ R) . Now, H ∪ R ∪ H = X, and H ∪ R ∪ G = R ∪ G , since H = int G ⊂ G . Hence (H ∪C)\D = (R∪G )∩(P ∪R) = (R∩P )∪R∪(G ∩P )∪(G ∩R) = (G ∩P )∪R = A since R ∩ P ⊂ R and G ∩ R ⊂ R. 2). Now, the set H is open, D ⊂ P is of the first category, and G \ H is a closed set without inner points, and so it is nowhere dense, and hence of the first category. 2) that A has the Baire property. 2.

We do not presuppose any of those approaches. At any case the reader should be familiar with the following theorem, which we quote here without proof. 1. For any set A ⊂ RN the following conditions are equivalent: (i) A ∈ L. (ii) For every ε > 0 there exist open sets G and U such that A ⊂ G , G \ A ⊂ U , m(U ) < ε . (iii) There exist sets H ∈ Gδ and B ∈ L such that A ⊂ H , A = H \ B , m(B) = 0 . (iv) For every ε > 0 there exist a closed set F and an open set V such that F ⊂ A , A \ F ⊂ V , m(V ) < ε .

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