The $Lambda$-hypernucleus production by $A(p, pK^+)_{Lambda}B$ reactions is investigated within the framework of the distorted wave impulse approximation(DWIA). The amplitude for the elementary process is evaluated in a fully covariant two-nucleon model based on the effective Lagrangian. The reaction cross sections for $Lambda$-hypernucleus productions on $^6Li$, $^{12}C$ and $^{16}O$ targets are calculated. It is found that the distortion effects tend to reduce the cross sections by a factor of 3$sim$10. Various differential cross sections (DCS) and double differential cross sections (DDCS) are presented. It is shown that for the $s_{Lambda}-$wave hypernucleus production, the DCS is decreased with increasing nuclear mass, and the DCS for the $p_{Lambda}-$wave hypernucleus production is normally higher than that for the $s_{Lambda}-$wave hypernucleus production. As a reference, the DDCS with respect to the momenta of the outgoing proton and kaon is also demonstrated. Finally, the missing mass spectra of the inclusive reaction $p+Ato p+ K^+ + X$ for $^6Li$, $^{12}C$ and $^{16}O$ targets are presented, from which the masses of hypernuclei can accurately be extracted. Thus, we conclude that the missing mass spectrum method is an alternative to study hypernuclear physics. And the study of hypernuclear physics can be carried out in COSY and CSR by the $A(p,pK^+)_{Lambda}B$ reaction due to the $mu$b-order reaction cross sections.
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