Topological degree theories for continuous perturbations of resolvent compact maximal monotone operators, existence theorems and applications

Asfaw, Teffera M. ¹

¹ Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA

Journal of nonlinear sciences and its applications, Tome 13 (2020) no. 5, p. 239-257.

Résumé

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.

Détail

DOI : 10.22436/jnsa.013.05.02

Classification : 47H11, 47H14, 47H07
Keywords: Compact resolvents, continuous operator, degree theory, variational inequality, homotopy invariance, maximal monotone

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