We report searches for the processes $e^+e^-topi^+pi^-pi^0chi_{bJ}$ and $e^+e^-tophichi_{bJ}$ (J=1,2) based on data samples collected by the Belle experiment at the KEKB collider. We report the first observation of the process $e^+e^-to (pi^+pi^-pi^0)_{rm non-omega}chi_{b1}$ and first evidence for $e^+e^-toomegachi_{bJ}$ in the vicinity of the $Upsilon(11020)$ resonance, with center-of-mass energies from 10.96 to 11.05~GeV. The significances for $(pi^+pi^-pi^0)_{rm non-omega}chi_{b1}$ and $omegachi_{bJ}$ are greater than $5.3sigma$ and $4.0sigma$, respectively. We also investigate the energy dependence of the $e^+e^-topi^+pi^-pi^0chi_{bJ}$ cross section, but we cannot determine whether the contributions are from the $Upsilon(10860)$ and $Upsilon(11020)$ resonances or non-resonant continuum processes. The signals for $e^+e^-to phichi_{bJ}$ are not significant, and the upper limits of the Born cross sections at the 90% confidence level are $0.7$ and $1.0$~pb for $e^+e^-to phichi_{b1}$ and $phichi_{b2}$, respectively, for center-of-mass energies from 10.96 to 11.05~GeV.
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