Periodic problems for strongly nonlinear second-order differential inclusions
The authors study the following nonlinear, second-order periodic problem in \(\mathbb{R}^N\) \[ a(x'(t))'\in Ax(t)+F(t,x(t),x'(t))\text{ a.e. on }T,\quad x(0)=x(b),\;x'(0)=x'(b), \] where \(a:\mathbb{R}^N\rightarrow\mathbb{R}^N\) and \(A:\mathbb{R}^N\rightarrow 2^{\mathbb{R}^N}\) are maximal monotone maps and \(F\) is a multifunction. The hypotheses they use on the nonlinear differential operator do not assume any homogeneity nor any growth condition. Moreover, they allow the operator \(A\) to be either defined everywhere on \(\mathbb{R}^N\) or not. The methods used come from multivalued analysis and from the theory of operators of monotone type. The existence of solutions for both the ``convex'' and the ``nonconvex'' problem is proved.
scientific article; zbMATH DE number 1836797
scientific article