Nonlocal \(m\)-dissipative evolution inclusions in general Banach spaces
Existence of solutions is proven for the nonlocal nonlinear evolution inclusion \[ \dot{x}\in Ax+F(t,x), \;t\in I\equiv \left[ t_{0},T\right], \] \[ x\left( t_{0}\right) =g\left( x\left( \cdot \right) \right) \] in a general Banach space \(X\), where \(A:D(A)\subseteq X\rightrightarrows X\) is an \(m\)-dissipative operator, the perturbation \(F:I\times X\rightrightarrows X\) has nonempty closed bounded values and \(g:C\left( I,X\right) \rightarrow \overline{D(A)}\). It is also assumed that \(F\left( \cdot ,x\right) \) has a strongly measurable selection for each \(x\in \overline{D(A)}\), \(F\) is Lebesgue integrably bounded and satisfies a weak Lipschitz type condition, and \(g\) is Lipschitz. The proof makes use of a variant of the Filippov-Pliss lemma, which concerns dependence on initial conditions. This result extends one found in [\textit{L. Zhu} et al., J. Math. Anal. Appl. 390, No. 2, 523--534 (2012; Zbl 1246.34059)]. A detailed example is also given.
scientific article; zbMATH DE number 6804311