An abstract topological point of view and a general averaging principle in the theory of differential inclusions
The authors deal with an abstract averaging principle for an operator equation \(x(t)=S F_\varepsilon(x)\) in a space \(C([0,d],E)\) of continuous functions with values in a Banach space \(E\) and with a measure of noncompactness \(\chi(\cdot)\); here \(S\) is a linear operator with properties similar to properties of a Volterra integral operator and \(F\) is a multifunction which tends in mean to a multifunction \(F_0\) as \(\varepsilon\to 0\). A fundamental example considered here is \[ \dot x(t)\in A(t)x(t)+ f\bigl(t,x(t) \bigr),\quad t\in[0,d],\;x(0)=x^0, \] where \(A(t)\) is a family of unbounded operators in \(E\), \(f(t,x)\) is a multifunction condensing in \(x\) with respect to \(x\); for this example, \(S\) is the operator \[ Sg(t)= U(t,0)x^0+ \int^t_0 U(t, \tau)g(\tau) d\tau, \] \(U(t,\tau)\) is the Cauchy operator for the family \(A(t)\). The constructions of the authors are based on the fixed-point theory for multivalued operators condensing with respect to a special measure of noncompactness in \(C([0,d],E)\) \[ \psi(\Omega) =\max_{D\in \Omega}\left( \sup_{t \in[0,d]} e^{-Lt}\chi \bigl(D(t) \bigr),\limsup_{\delta\to 0}\sup_{x\in D} \max_{|t-s|\leq \delta} \bigl\|x(t)-x(s)\bigr \|\right) \] generated by \(\chi(\cdot)\). The considerations of the authors are new even in the classical case when \(E\) is finite dimensional; in particular, their result does not demand the uniqueness for the solution to the averaging equation.
scientific article; zbMATH DE number 1526180
scientific article