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Semigroups and some nonlinear fractional differential equations. (English) Zbl 1046.34079
Summary: Equations of the form $\frac{d^\alpha u(t)} {dt^\alpha}= Au(t)+F\bigl(t,B_1(t), \dots,B_r(t)u(t)\bigr)$ are considered, where $$0<\alpha\leq 1$$, $$A$$ is a closed linear operator defined on a dense set in a Banach space $$E$$ into $$E$$, $$\{B_i(t)$$, $$i=1,\dots,r,\;t\geq 0\}$$ is a family of linear closed operators defined on dense sets in $$E$$ into $$E$$ and $$F$$ is a given abstract nonlinear function defined on $$[0,T]\times E^r$$ with values in $$E$$, $$T>0$$. It is assumed that $$A$$ generates an analytic semigroup. Under suitable conditions on the family of operators $$\{B_i(t):i=1, \dots,r,\;t\geq 0\}$$ and on $$F$$, we study the existence and uniqueness of the solution of the Cauchy problem for the considered equation. Some properties concerning the stability of solutions are obtained. We also give an application for nonlinear partial differential equations of fractional orders.

MSC:
 34G20 Nonlinear differential equations in abstract spaces
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References:
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