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

Let $\left(X,d\right)$ be a metric space, $𝒫\left(X\right):=\left\{Y\subset X\right\}$, $P\left(X\right):=\left\{Y\in 𝒫\left(X\right):Y\ne \varnothing \right\}$, ${D}_{d}:P\left(X\right)×P\left(X\right)\to {ℝ}_{+}$ the gap functional, given by

${D}_{d}\left(A,B\right)=inf\left\{d\left(a,b\right):a\in A,\phantom{\rule{0.166667em}{0ex}}b\in B\right\},$

and let $F:X\to P\left(X\right)$ be a multivalued operator. The fixed point inclusion

$x\in F\left(x\right),\phantom{\rule{0.166667em}{0ex}}x\in X$

is said to be generalized Ulam-Hyers stable if and only if there exists an increasing function $\psi :{ℝ}_{+}\to {ℝ}_{+}$, continuous at 0 and with ${\Psi }\left(0\right)=0$ such that for each $ϵ>0$ and for each solution ${y}^{*}\in X$ of of the inequality

${D}_{d}\left(y,F\left(y\right)\right)\le ϵ,$

there exists a solution ${x}^{*}$ of the fixed point inclusion such that

$d\left({x}^{*},{y}^{*}\right)\le \psi \left(ϵ\right)·$

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.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 54C60 Set-valued maps (general topology) 54E40 Special maps on metric spaces