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Hyers-Ulam stability of mean value points. (English) Zbl 1219.39013

The authors consider a few problems concerning the stability for Lagrange’s and Flett’s mean value points. The first result reads as follows.

Let f: be a continuously twice differentiable mapping and let η(a,b) be a unique Lagrange’s mean value point of f in (a,b) (i.e., f ' (η)=f(b)-f(a) b-a). It is proved that for each ε>0 there exists δ>0 such that for each differentiable function g: satisfying |f(x)-g(x)|δ, x[a,b] there exists a Lagrange’s mean value point ξ(a,b) of g such that |ξ-η|ε.

Other results are connected with approximate mean value points:

f ' (ξ)-f(b)-f(a) b-aε

and with the stability of the equation:

f ' (x)=f(x)-f(a) x-a·

MSC:
39B82Stability, separation, extension, and related topics
26A24Differentiation of functions of one real variable
39B22Functional equations for real functions