Abstract:
This paper investigates norm attainability and modular properties
in Orlicz spaces, which generalize Lp-spaces and are key in functional analysis
and nonlinear problems. It presents theorems on norm attainment, orthogonality,
weak compactness, and uniform convexity, and introduces a novel criterion connecting
the convexity of the Orlicz function with the smoothness and reflexivity
of the space. The research extends classical concepts such as the Δ2-condition
to ensure completeness and separability. The results have practical applications
in nonlinear optimization, variational analysis, machine learning, signal processing,
image reconstruction, and solving PDEs with nonlinear boundary conditions,
providing a strong foundation for future research in these areas.
