Decay of excited nuclei produced in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mmultiscripts><mml:mi mathvariant="normal">Kr</mml:mi><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>78</mml:mn><mml:mo>,</mml:mo><mml:mn>82</mml:mn></mml:mrow></mml:mmultiscripts></mml:math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mmultiscripts><mml:mi mathvariant="normal">Ca</mml:mi><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>40</mml:mn></mml:mrow></mml:mmultiscripts></mml:math>reactions at 5.5 MeV/nucleon

Decay modes of excited nuclei are investigated in $^{78,82}\mathrm{Kr}$$+$$^{40}\mathrm{Ca}$ reactions at 5.5 MeV/nucleon. Charged products were measured by means of the $4\ensuremath{\pi}$ INDRA array. Kinetic-energy spectra and angular distributions of fragments with atomic number 3 $\ensuremath{\leqslant}Z\ensuremath{\leqslant}$ 28 indicate a high degree of relaxation and are compatible with a fissionlike phenomenon. Persistence of structure effects is evidenced from elemental cross sections (${\ensuremath{\sigma}}_{Z}$) 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}\mathrm{Kr}$$+$$^{40}\mathrm{Ca}$ reaction is slightly higher than the one measured in the $^{82}\mathrm{Kr}$$+$$^{40}\mathrm{Ca}$ reaction. The fissionlike component is larger by $~$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 ${\ensuremath{\sigma}}_{Z}$. For both reactions, the transition-state formalism reasonably reproduces the $Z$ distribution of the fragments with charge 12 $\ensuremath{\leqslant}Z\ensuremath{\leqslant}$ 28. However, this model strongly overestimates the light-fragment cross sections and does not explain the o-e-s of the yields for 6 $\ensuremath{\leqslant}Z\ensuremath{\leqslant}$ 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 among 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 $\ensuremath{\leqslant}Z\ensuremath{\leqslant}$ 22 could signal a mechanism in addition to the capture process.