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 $T:X\supseteq D(T)\to {\mathrm{2}}^{X^{*}}$ be a maximal monotone operator and $C:X\supseteq D(C)\to X^{*}$ be bounded and continuous with $D(T)\subseteq D(C)$. 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.