## Download Functional Equations and Inequalities in Several Variables by Stefan Czerwik PDF

By Stefan Czerwik

This e-book outlines the trendy thought of practical equations and inequalities in numerous variables. It includes 3 components. the 1st is dedicated to additive and convex services outlined on linear areas with semilinear topologies. within the moment half, the issues of balance of sensible equations within the feel of Ulam-Hyers-Rassias and in a few functionality areas are thought of. within the final half, the sensible equations in set-valued capabilities are handled - for the 1st time within the mathematical literature. The publication comprises many clean effects referring to these difficulties.

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**Extra resources for Functional Equations and Inequalities in Several Variables**

**Sample text**

3, / is continuous which proves that T G A(X,T) and thus the proof of the theorem follows. H Note that in the space X = W1 convexity of / implies its continuity. However, in infinite-dimensional spaces, it is not any longer true. Taking this into account it is interesting to investigating the following set classes. A(X) := {T C X: every J-convex real-valued function / : A —>• R defined on an algebraically open and convex subset A of X containing T which is bounded above on T, is convex } , 32 Functional Equations and Inequalities in Several Variables B(X) : = { T C I : every additive function / : A -> E which is bounded above on T, is linear } .

Now we are going to present the main ideas and results of Gajda's paper [62]. ,an(x)), xeG, where (-|-) stands for the usual inner product in C™. 2) can be described in terms of unitary representations of the group G. In the sequel, we will need the following notions. By a unitary representation of a topological group G in a Hilbert space H we mean a homomorphism of G into the group of all unitary operators in H (see, e. g. [140]). A unitary representation U is said to be continuous iff for every vector ( e f f , the transformation G3x^ U(x)£ e H is continuous.

Isac and Th. M. Rassias [97] , Th. M. Rassias [176]). 3) f(xy) = f(x)f(y). 1) (see D. H. Hyers and Th. M. Rassias [95], as well as references [170], [164]). 1 Let X and Y be linear spaces. 1), then f (Ax) = A/(x) for every x G X and A € Q. 5) 36 Functional Equations and Inequalities in Several Variables Proof. 1) for x = y ~ 0 we get /(0) = 0. 1), we obtain 0 = f(x-x) = f(x) + f(-x), and therefore f(—x) = —f(x), which means that / is an odd function. Now let x\ = ... = xn = x, then f[xx + ... +xn) =/0&i) + f(x2) + f(x3 + ...