Some results of perturbed inclusions with compact-valued mapping (Q941951)

The problem \(x\in\Psi(x)+V\Phi(x)\) is studied in \(C([a,b],{\mathbb R}^n)\). Here, \(\Psi,\Phi\) are multivalued functions with values in the space of continuous or integrable functions, respectively, which are supposed to satisfy a contraction type hypothesis with respect to the Hausdorff distance, and \(V\) is an integral operator. Moreover, it is assumed that \(\Psi\) assumes compact values and \[ \Phi(x)=\{y:\text{\(y(t)\in\Delta(t,x)\) for a.a.\;\(t\)}\} \] for some function \(\Delta\) with values in the compact subsets of \({\mathbb R}^n\). The paper contains no proofs, but states a successive approximation type result including an a priori estimate. Also criteria are formulated when one may replace \(\Phi(x)\) by its closed convex hull or \(\Delta\) by the extreme points of its convex hull (``bang-bang principle'') without changing the set of quasi-solutions/solutions up to closure.