Nonlinear evolution equations without convexity
The purpose of this article is to prove an existence result for the Cauchy problem \(x' \in - \partial^- f(x) + F(x)\), \(x(0) = x_0\), \(x_0 \in D (\partial^- f)\), where \(f : \Omega \subset \mathbb{R}^n \to \mathbb{R} \cup \{+ \infty\}\) is a function with \(\psi\)-monotone subdifferential of order 2 such that the mapping \(x \mapsto \text{grad}^- f(x)\) is locally bounded in \(x_0\) and \(F : U(x_0) \to 2^{\mathbb{R}^n}\) is an upper semicontinuous and cyclically monotone multivalued operator defined on some neighborhood of \(x_0\) and with nonempty compact values. This theorem extends a previous result of \textit{A. Cellina} and \textit{V. Staicu} [J. Differ. Equations 90, 71-80 (1991; Zbl 0719.34030)] and strictly contains a theorem due to \textit{A. Bressan}, \textit{A. Cellina} and \textit{G. Colombo} [Proc. Am. Math. Soc. 106, 771-775 (1989; Zbl 0698.34014)] for the problem \(x' \in F(x)\), \(x(0) = x_0\).
cyclically monotone multivalued operator
scientific article; zbMATH DE number 811180