The Variation of Constants Formula in Lebesgue Spaces with Variable Exponents

This study looks closely into the analysis of the variation of constants formula given by Phi(t)=S(t)Phi(0)+integral(t)(0) S(t - sigma)F(sigma,Phi(sigma))d sigma, for t is an element of[0, T], T >0, within the context of modular function spaces L-rho. Additionally, this research explores practical applications of the variation of constants formula in variable exponent Lebesgue spaces L-P(.). Specifically, the study examines these spaces under certain conditions applied to the exponent function p(.) and the functions F as well as the semigroup S(t), utilizing the symmetry properties of the algebraic semigroup. This investigation sheds light on the intricate interplay between parameters and functions within these mathematical frameworks, offering valuable insights into their behavior and properties in L-P(.).