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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.

MSC:
 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
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References:
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