New reaction scheme are suggested for the initiated nonbranched-chain addition of free radicals to the multiple bond of the molecular oxygen. The scheme includes the reaction competing with chain propagation reactions through a reactive free radical. The chain evolution stage in this scheme involves a few of free radicals, one of which (tetraoxyl) is relatively low-reactive and inhibits the chain process by shortening of the kinetic chain length. Based on the proposed scheme rate equations (containing one to three parameters to be determined directly) are deduced using quasi-steady-state treatment. The kinetic description with use the obtained rate equations is applied to the ϒ-induced nonbranched-chain processes of the free-radical oxidation of liquid o-xylene at 373 K and hydrogen dissolved in water containing different amounts of oxygen at 296 K. In these processes the oxygen with the increase of its concentration begins to act as an oxidation autoingibitor (or an antioxidant). The energetics of the key radical-molecule reactions is considered.