Operator equations and duality mappings in Sobolev spaces with variable exponents

After studying in a previous work the smoothness of the space U-Gamma 0={u epsilon W-1,(p(center dot)) (Omega);u=0 on Gamma 0 subset of Gamma=partial derivative Omega} where d Gamma - meas Gamma(0) > 0, with p(center dot) epsilon C ((Omega) over bar and p(x) > 1 for all x epsilon (Omega) over bar , the authors study in this paper the strict and uniform convexity as well as some special properties of duality mappings defined on the same space. The results obtained in this direction are used for proving existence results for operator equations having the form J(psi)u = N(f)u, where J(Psi) is a duality mapping on U-Gamma 0 corresponding to the gauge function Psi, and N (f) is the Nemytskij operator generated by a Carath,odory function f satisfying an appropriate growth condition ensuring that N (f) may be viewed as acting from U-Gamma 0 into its dual.