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We discuss here the boundedness of the fractional integral operator I α and its generalized version on generalized nonhomogeneous Morrey spaces. To prove the boundedness of I α , we employ the boundedness of the so-called maximal fractional integral operator I * a,κ . In addition, we prove an Olsentype inequality, which is analogous to that in the case of homogeneous type. 2000 Mathematics subject classification: primary 42B20; secondary 42B35, 47G10, 31B10, 26A33.doi:10.1017/s0004972709000343 fatcat:kwhkuny27jha7bygkzs25mwgae