Strongly nonlinear, multivalued, periodic problems with maximal monotone terms
The authors investigate solutions of the second-order differential inclusion for the function \(t \mapsto x(t)\) \[ \left (\alpha (x)\, \varphi (x')\right )' \in A(x) + F(t,x), \quad t \in [0,b], \qquad x(0) = x(b),\;x'(0) = x'(b), \] where \(\alpha \: \mathbb R^N \to \mathbb R\), \(\varphi \: \mathbb R^N \to \mathbb R^N\), \(\varphi (x) = | x| ^{p-2}x\), \(2 \leq p < \infty \), \(A\: D \subseteq \mathbb R^N \to 2^{\mathbb R^N}\) is a maximal monotone operator, and \(F\: [0,b] \times \mathbb R^N \to 2^{\mathbb R^N} \setminus \{0\}\) is a set-valued function (satisfying the Hartman condition). They extend results due to J. Mawhin. Making use of the theory of monotone operators, they find solutions \(x \in C^1 ([0,b], \mathbb R^N)\) of the corresponding equation with \(A\) replaced by its Yosida approximation \(A_{\lambda }\) and pass to the limit \(\lambda \to 0+\). In the particular case \(N=1 \) and \(p=2\), the nonlinearity \(F\) may depend more generally also on \(x'\). Another generalization is indicated if Dirichlet boundary conditions are considered instead of periodic ones and the operator \((\alpha (x)\, \varphi (x'))'\) is replaced by the more general operator \((\alpha (x,x'))'\).
scientific article; zbMATH DE number 5138413