Monotonicity And Approximation Properties Of Orlicz-Lorentz And Orlicz-Bochner Spaces

Since G.Birkhoff first introduced the uniform monotonicity, many studies laterhave shown that monotonicity properties are significant in the fixed point theory andapproximation etc; while approximative compactness is closely connected with thecontinuity of best approximation operator. Orlicz-Lorentz spaces are the connectionbetween some problems of harmonic analysis and interpolation theory, and Orlicz-Bochner spaces provide theoretical framework for evolution equations etc. Therefore inthis dissertation we discuss monotone points (monotonicity) in Orlicz-Lorentz spacesand Orlicz-Bochner spaces; and approximative compactness in Orlicz-Bochner functionspaces.Firstly we obtain the criteria for a point in Orlicz-Lorentz function spaces orsequence spaces being upper (lower) monotone and upper (lower) locally uniformlymonotone.Secondly we get su?cient and necessary conditions of upper locally uniformlymonotone point and upper monotone point under some conditions, the su?cient condi-tions or necessary conditions about a point in positive cone being lower monotone andlower locally uniformly monotone in Orlicz-Bochner function spaces. In Orlicz-Bochnersequence spaces we obtain the criteria for upper (lower) locally uniform monotonicityand some monotone points.Lastly we get the criteria for approximative compactness in the Orlicz-Bochnerfunction spaces.