Strongly nonlinear multivalued systems involving singular \(\Phi \)-Laplacian operators
The paper studies the following problem \[ (k(t)\Phi (x'(t)))'\in Ax(t)+F(t,x(t),x'(t))\text{ a.e. on }[0,b],\quad x(0)=x(b),\tag{1} \] where \(k:[0,b]\to {\mathbb R}\) is a positive continuous function, \(\Phi :B\subset {\mathbb R}^n\to {\mathbb R}^n\) is a strictly monotone homeomorphism defined on a bounded domain \(B\), \(A:{\mathbb R}^n\to {\mathcal P}({\mathbb R}^n)\) is a maximal monotone operator and \(F:[0,b]\times {\mathbb R}^n\times {\mathbb R}^n\to {\mathcal P}({\mathbb R}^n)\) is a Carathéodory set-valued map with nonempty and compact values. Under certain technical hypotheses, the author proves the existence of solutions of problem (1) both in the cases when the set-valued map \(F\) has convex values. Afterwards, these results are used to obtain the existence of solutions for the more general problem \[ (a(t,x(t))\Phi (x'(t)))'\in Ax(t)+F(t,x(t),x'(t))\text{ a.e. on }[0,b],\quad x(0)=x(b), \] where \(a:[0,b]\times {\mathbb R}^n\to {\mathbb R}\) is a positive continuous function.
scientific article; zbMATH DE number 5790298