In the analysis of data on decays of the $\Upsilon$-meson family - $\Upsilon(2S)\to\Upsilon(1S)\pi\pi$, $\Upsilon(3S)\to\Upsilon(1S)\pi\pi$ and $\Upsilon(3S)\to\Upsilon(2S)\pi\pi$ - the contribution of multi-channel $\pi\pi$ scattering in the final-state interactions is considered. The analysis, which is aimed at studying the scalar mesons, is performed jointly considering the isoscalar S-wave processes $\pi\pi!\to!\pi\pi,K\overline{K},\eta\eta$, which are described in our model-independent approach based on analyticity and unitarity and using an uniformization procedure, and the charmonium decay processes $J/\psi\to\phi(\pi\pi, K\overline{K})$, $\psi(2S)\to J/\psi(\pi\pi)$. Results of the analysis confirm all our earlier conclusions on the scalar mesons. It is also shown that in the final states of the $\Upsilon$-meson family decays (except for the $\pi\pi$ scattering) the contribution of the coupled processes, e.g., $K\overline{K}\to\pi\pi$, is important even if these processes are energetically forbidden. This is in accordance with our previous conclusions on the wide resonances: If a wide resonance cannot decay into a channel which opens above its mass but the resonance is strongly connected with this channel (e.g. the $f_0$(500) and the $K\overline{K}$ channel), one should consider this resonance as a multi-channel state with allowing for the indicated channel taking into account the Riemann-surface sheets related to the threshold branch-point of this channel and performing the combined analysis of the considered and coupled channels.

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