Topological degree theories for continuous perturbations of resolvent compact maximal monotone operators, existence theorems and applications
Teffera M. Asfaw · The Journal of Nonlinear Sciences and Applications · 2020
Let \(X\) be a real locally uniformly convex reflexive Banach space. Let \(T: X\supseteq D(T)\to 2^{X^*}\) and \(A:X\supseteq D(A)\to 2^{X^*}\) be maximal monotone operators such that \(T\) is of compact resolvents and \(A\) is strongly quasibounded, and \(C: X\supseteq D(C)\to X^*\) be a bounded and continuous operator with \(D(A)\subseteq D(C)\) or \(D(C)=\overline{U}\). The set \(U\) is a nonempty and open (possibly unbounded) subset of \(X\). New degree mappings are constructed for operators of the type \(T+A+C\). The operator \(C\) is neither pseudomonotone type nor defined everywhere. The theory for the case \(D(C)=\overline{U}\) presents a new degree mapping for possibly unbounded \(U\) and both of these theories are new even when \(A\) is identically zero. New existence theorems are derived. The existence theorems are applied to prove the existence of a solution for a nonlinear variational inequality problem.