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An abstract topological point of view and a general averaging principle in the theory of differential inclusions. (English) Zbl 0972.34049
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.

34G20 Nonlinear differential equations in abstract spaces
34C29 Averaging method for ordinary differential equations
47H11 Degree theory for nonlinear operators
34G25 Evolution inclusions
34A60 Ordinary differential inclusions
Full Text: DOI
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