Structure of solution sets to the nonlocal problems (Q2634921)

The authors consider the nonlocal problem \[ \dot{x}(t)+B(t,x(t))+Dx(t)\in F(t,x(t)),\text{ for a.e. }t\in I,\qquad x(0)=\tau (x), \] where \(I=[0,T]\), \(B:I\times V\rightarrow V^{\ast }\), \((V,H,V^{\ast })\) is an evolution triple of spaces, \(D:V\rightarrow V^{\ast }\) is a bounded linear self-adjoint operator, \(F:I\times H\rightarrow 2^{V^{\ast }}\) and \(\tau :C(I,H)\rightarrow H\) is continuous. Existence of weak solutions is guaranteed by [\textit{J. Zhang} et al., Bound. Value Probl. 2013, Article ID 15, 21 p. (2013; Zbl 1287.34050)]. Along with other conditions, the authors assume that \(B(t,x)\) is measurable in \(t\), satisfies a one-sided Lipschitz condition in \(x\) and satisfies a growth condition, and \(F\) is closed convex-valued, is graph measurable, satisfies a growth condition and for almost all \(t\in I\) \(F(t,\cdot )\) has a closed graph. It is then proven that the solution set \(S\) is an \(R_{\delta }\) set in \(C(I,H)\), i.e., the solution set is the intersection of a decreasing sequence of nonempty compact absolute retracts. It follows that the reachable set is compact and connected in \(H\). The proof uses results from [\textit{F. S. De Blasi} and \textit{J. Myjak}, Bull. Pol. Acad. Sci., Math. 33, 17--23 (1985; Zbl 0571.34008)]. Dropping the assumption that \(F\) is convex-valued and adding a Lipschitz-type assumption on \(F(t,\cdot )\), the authors then prove that \(S\) is path connected. Two examples are given to illustrate the application of these theorems.