Existence Theorems on Solvability of Constrained Inclusion Problems and Applications

Abstract

Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space X⁎. Let T:X⊇D(T)→2X⁎ be a maximal monotone operator and C:X⊇D(C)→X⁎ be bounded and continuous with D(T)⊆D(C). The paper provides new existence theorems concerning solvability of inclusion problems involving operators of the type T+C provided that C is compact or T is of compact resolvents under weak boundary condition. The Nagumo degree mapping and homotopy invariance results are employed. The paper presents existence results under the weakest coercivity condition on T+C. The operator C is neither required to be defined everywhere nor required to be pseudomonotone type. The results are applied to prove existence of solution for nonlinear variational inequality problems.