On existence of the weak solution for non-linear partial differential equations of elliptic type. II.
The solvability of a class of monotone nonlinear variational inequality problems in a reflexive Banach space setting is presented.
We study the mappings of monotone type in Orlicz-Sobolev spaces. We introduce a new class as a generalization of and extend the definition of quasimonotone map. We also prove existence results for equations involving monotone-like mappings.
Notions as the numerical range and the spectrum of couple of homogeneous operators on a Banach space are used to derive theorems on solvability of the equation Conditions for the existence of eigenvalues of the couple are given.
The existence of positive solutions for a nonlocal boundary-value problem with vector-valued response is investigated. We develop duality and variational principles for this problem. Our variational approach enables us to approximate solutions and give a measure of a duality gap between the primal and dual functional for minimizing sequences.