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Context. Dusty debris disks around pre- and main-sequence stars are potential signposts for the existence of planetesimals and exoplanets. Giant planet formation is therefore expected to play a key role in the evolution of the disk. This is indirectly confirmed by extant submillimeter near-infrared images of young protoplanetary and cool dusty debris disks around main-sequence stars that usually show substantial spatial structures. With two decades of direct imaging of exoplanets already studied, it is striking to note that a majority of recent discoveries of imaged giant planets have been obtained around young early-type stars hosting a circumstellar disk. Aims. Our aim was to create a direct imaging program designed to maximize our chances of giant planet discovery and target 22 young early-type stars. About half of them show indications of multi-belt architectures. Methods. Using the IRDIS dual-band imager and the IFS integral field spectrograph of SPHERE to acquire high-constrast coronagraphic differential near-infrared images, we conducted a systematic search in the close environment of these young, dusty, and early-type stars. We used a combination of angular and spectral differential imaging to reach the best detection performances down to the planetary mass regime. Results. We confirm that companions detected around HIP 34276, HIP 101800, and HIP 117452 are stationary background sources and binary companions. The companion candidates around HIP 8832, HIP 16095, and HIP 95619 are determined as background contaminations. Regarding the stars for which we infer the presence of debris belts, a theoretical minimum mass for planets required to clear the debris gaps can be calculated. The dynamical mass limit is at least 0.1 M-J and can exceed 1 M-J. Direct imaging data is typically sensitive to planets down to similar to 3.6 M-J at 1 '', and 1.7 M-J in the best case. These two limits tightly constrain the possible planetary systems present around each target. These systems will be probably detectable with the next generation of planet imagers. ; IDEXLyon project under University of Lyon ANR-16-IDEX-0005 Centre National de la Recherche Scientifique (CNRS) French National Research Agency (ANR) ANR-14-CE33-0018 European Union (EU) 823823 German Research Foundation (DFG) SPP 1992 KU 2849/7-1 Swiss National Science Foundation (SNSF) BSSGI0_155816 Swiss National Science Foundation (SNSF) ICM (Iniciativa Cientifica Milenio) via the Nucleo Milenio de Formacion Planetaria Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT) CONICYT FONDECYT 1190748 French National Research Agency (ANR) ANR10 LABX56