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Some geometric properties in modular spaces and application to fixed point theory. (English) Zbl 1062.46011

The author considers several geometric properties in the setting of modular function spaces that lead to fixed point results for certain classes of mappings. For example, the author proves that, if \(\rho\) is a convex, additive function modular with the \(\Delta_2\)-type condition, then the modular space \(L_\rho\), endowed with the Luxemburg norm, has both the uniform Kadec-Klee property with respect to \(\rho\)-aė. convergence and the uniform Opial condition with respect to \(\rho\)-aė. convergence. With the same conditions on \(\rho\), the modular space \(L_\rho\), endowed with the Amemiya norm, has the uniform Opial condition with respect to \(\rho\)-aė. convergence. Using these results, the author proves that self-mappings of convex, bounded, \(\tau_\rho\)-compact subsets of some modular spaces \(L_\rho\) have fixed points if the mappings are of asymptotically nonexpansive type when \(L_\rho\) is endowed with the Luxemburg norm or if the mappings are asymptotically nonexpansive when \(L_\rho\) is endowed with the Amemiya norm.

MSC:

46A80 Modular spaces
46B20 Geometry and structure of normed linear spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47H10 Fixed-point theorems
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