Maximal monotone differential inclusions with memory
Let \(b\), \(r>0\), \(\hat T=[-r,b]\), \(T_ 0=[-r,0]\), \(T=[0,b]\). The paper is dealing with the following differential inclusion with memory, in \(\mathbb{R}^ N\): \((*)\) \(-\dot x(t)\in Ax(t)+F(t,x_ t)\) a.e. on \(T\); \(x(v)=\varphi(v)\), \(v\in T_ 0\), where \(x_ t\in C(T_ 0;\mathbb{R}^ N)\), \(x_ t(s)=x(t+s)\), \(A:D(A)\subset\mathbb{R}^ N\to 2^{\mathbb{R}^ N}\) is maximal monotone and \(F\) is a multifunction defined on \(T\times C(T_ 0,\mathbb{R}^ N)\) which satisfies some Carathéodory type conditions and \(| F(t,y)|=\sup\{\| v\|:v\in F(t,y)\}\leq a(t)+b(t)\| y\|_ \infty\) a.e. \((a,b\in L^ 1_ +)\). The author proves two theorems on the existence of a solution \(x\in C(\hat T;\mathbb{R}^ N)\); one involving convex-valued mappings \(F\) and the other nonconvex-valued ones. Then he considers a parametrized family of problems, where \(A\), \(F\) and \(\varphi\) depend on a parameter \(r\) and the continuous dependence on \(r\) of the solution set is proved. The last result says that under reasonable hypotheses on \(F\), the solution set of the nonconvex problem \((*)\) is dense in that of the convexified problem \((*)_ c\) with \(\overline{\text{conv}} F\) instead of \(F\).
scientific article; zbMATH DE number 92806
scientific article