In this work, we study the production of strange quarkoniums, the $phi(2170)$, also named $Y(2175)$, and the $eta(2225)$, via a kaon induced reaction on a proton target in an effective Lagrangian approach. The total and differential cross sections of the reactions $K^{-}prightarrow phi (2170)Lambda $ and $K^{-}prightarrow eta (2225)Lambda $ are calculated by the Reggeized $t $-channel Born term under an assumption that the $phi(2170)$ and $eta(2225)$ are $Lambdabar{Lambda}$ molecular states. At the center of mass energies of about 4.2 GeV, the total cross section for the $phi(2170)$ production is predicted to be about 1 $mu $b. The numerical results indicate that it is feasible to produce the $phi (2170)$ via kaon beam scattering at the best energy window near 4.2 GeV. The total cross section for the $eta(2225)$ production is smaller than that for the $phi(2170)$ production and it may reach an order of the magnitude of 0.1 $mu$b. The differential cross sections for both reactions at different center of mass energies are also presented. It is found that the Reggeized $t$ channel gives a considerable contribution at forward angles. As the energy increases, the contribution from the $t$-channel almost concentrates at extreme forward angles. From these theoretical predictions, the relevant experimental research is suggested, which could provide important information to clarify the internal structure and production mechanism of these two strange quarkoniums.
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