A result on the singular perturbation theory for differential inclusions in Banach spaces
The following Cauchy problem for singularly perturbed systems of differential inclusions in infinite-dimensional space is considered \[ x'(t) \in Ax(t)+ f_1\bigl(t,x(t), y(t)\bigr),\;\varepsilon y'(t)\in By(t)+f_2 \bigl(t, x(t),y(t)\bigr), \tag{1} \] \[ x(0)=x_0,\;y(0)=y_0,\tag{2} \] where \(A\) and \(B\) are infinitesimal generators of \(C^0\)-semigroups of linear operators \(\exp(At)\) and \(\exp(Bt)\). Moreover, \(x\in E_1\), \(y\in E_2\), where \(E_1,E_2\) are infinite-dimensional Banach spaces. The nonlinear operators \(f_1,f_2\) are multivalued with nonempty, convex and compact values. Conditions ensuring that the solution set to (1)--(2) is upper semicontinuous with respect to the parameter \(\varepsilon\geq 0\) are investigated. The main tool used in the proof is a suitable measure of noncompactness being a combination of the Hausdorff measure (strong) noncompactness and the De Blasi measure of weak noncompactness. Apart from that the topological degree theory is also applied.
singularly perturbed systems of differential inclusions
scientific article; zbMATH DE number 1545383