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An application of a fixed point theorem to a functional inequality. (English) Zbl 1184.39012

Let A be a unital C * -algebra with unitary group U(A). Assume that X,Y are left Banach modules over A and that Y is complete. Given f:XY, aA and x,y,zX let

D a f(x,y,z):=fax-ay 2+az+fay-az 2+ax+afz-x 2+y·

The authors prove the following theorem. Let f:XY be such that there is some ϕ:X 3 [0,) with

lim n 2 n ϕx 2 n ,y 2 n ,z 2 n =0andD a f(x,y,z)f(ax+ay+az)+ϕ(x,y,z)

for all x,y,zX and all aU(A). Assume furthermore that there is some L<1 such that the mapping Xxψ(x):=2ϕ(x/7,2x/7,-3x/7)+ϕ(4x/7,x/7,-5x/7) satisfies 2ψ(x)Lψ(2x) for all xX. Then there is a unique A-linear mapping T:XY such that

f(x)-T(x)1 1-Lψ(x)

for all xX.

By specializing the hypotheses, several corollaries are derived. The main tool for proving these results is a fixed point theorem in generalized metric spaces [cf. J. B. Diaz and B. Margolis, Bull. Am. Math. Soc. 74, 305–309 (1968; Zbl 0157.29904)].

MSC:
39B62Functional inequalities, including subadditivity, convexity, etc. (functional equations)
47H09Mappings defined by “shrinking” properties