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Ulam-Hyers stability for operatorial equations. (English) Zbl 1265.54158

Let (X,d) be a metric space, 𝒫(X):={YX}, P(X):={Y𝒫(X):Y}, D d :P(X)×P(X) + the gap functional, given by

D d (A,B)=infd(a,b):aA,bB,

and let F:XP(X) be a multivalued operator. The fixed point inclusion


is said to be generalized Ulam-Hyers stable if and only if there exists an increasing function ψ: + + , continuous at 0 and with Ψ(0)=0 such that for each ϵ>0 and for each solution y * X of of the inequality

D d y,F(y)ϵ,

there exists a solution x * of the fixed point inclusion such that

d(x * ,y * )ψ(ϵ)·

In the paper under review, the authors establish generalized Ulam-Hyers stability results for fixed point problems as well as for coincidence point problems with multivalued operators.

54H25Fixed-point and coincidence theorems in topological spaces
54C60Set-valued maps (general topology)
54E40Special maps on metric spaces