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∗. Let𝑇 : 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 underweak 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.