Decay modes of excited nuclei are investigated in $^{78,82}$Kr + $^{40}$Ca reactions at 5.5 MeV/nucleon. Charged products were measured by means of the $4pi$ INDRA array. Kinetic-energy spectra and angular distributions of fragments with atomic number 3 $le Z le$ 28 indicate a high degree of relaxation and are compatible with a fission-like phenomenon. Persistence of structure effects is evidenced from elemental cross-sections ($sigma_{Z}$) as well as a strong odd-even-staggering (o-e-s) of the light-fragment yields. The magnitude of the staggering does not significantly depend on the neutron content of the emitting system. Fragment-particle coincidences suggest that the light partners in very asymmetric fission are emitted either cold or at excitation energies below the particle emission thresholds. The evaporation residue cross-section of the $^{78}$Kr + $^{40}$Ca reaction is slightly higher than the one measured in $^{82}$Kr + $^{40}$Ca reaction. The fission-like component is larger by $sim$ 25% for the reaction having the lowest neutron-to-proton ratio. These experimental features are confronted to the predictions of theoretical models. The Hauser-Feshbach approach including the emission of fragments up to $Z$ = 14 in their ground states as well as excited states does not account for the main features of $sigma_{Z}$. For both reactions, the transition-state formalism reasonably reproduces the $Z$-distribution of the fragments with charge 12 $le Z le$ 28. However, this model strongly overestimates the light-fragment cross-sections and does not explain the o-e-s of the yields for 6 $le Z le$ 10. The shape of the whole $Z$-distribution and the o-e-s of the light-fragment yields are satisfactorily reproduced within the dinuclear system framework which treats the competition between evaporation, fusion-fission and quasifission processes. The model suggests that heavy fragments come mainly from quasifission while light fragments are predominantly populated by fusion. An underestimation of the cross sections for 16 $le Z le$ 22 could signal a mechanism in addition to the capture process.
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