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In this work, we investigate the axion-like particle (ALP)-photon oscillationeffect in the spectra of the blazar Markarian 421 (Mrk 421) using 4.5 years ofthe Astrophysical Radiation with Ground-based Observatory at YangBaJing(ARGO-YBJ) and Fermi Large Area Telescope (Fermi-LAT) data. These data arecollected during the common operation time, which cover ten activity phases ofMrk 421. No significant ALP-photon oscillation effect is confirmed. On theother hand, not all the observations of the ten phases can be individually usedto set the 95% confidence level ($\rm C.L.$) constraint on the ALP parameterspace. However, the constraint can be significantly improved if the analysesfor the ten phases are combined. We find that the upper limit at 95% $\rm C.L.$on the ALP-photon coupling $g_{a\gamma}$ set by the ARGO-YBJ and Fermi-LATobservations of Mrk 421 is within $\sim [2\times 10^{-11}, \ 6\times 10^{-11}]\rm \, GeV^{-1}$ for the ALP mass of $\sim 5\times 10^{-10}$ eV $\lesssim m_a\lesssim 5\times 10^{-7}\, \rm eV$.

Limits on axion-like particles from Mrk 421 with 4.5-year period observations by ARGO-YBJ and Fermi-LAT