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The fundamental solutions for fractional evolution equations of parabolic type. (English) Zbl 1081.34053
Let $$0<\alpha\leq 1$$, $$T> 0$$, $$E$$ a Banach space, $$\{A(t); t\in [0,T]\}$$ a family of linear closed operators defined on dense set $$D(A)$$ in $$E$$ with values in $$E$$, $$u: E\to E$$ a function, $$u_0\in D(A)$$, $$f: [0,T]\to E$$ a given function and $$B(E)$$ be the Banach space of linear bounded operators $$E\to E$$ endowed with the topology defined by the operator norm. Assume that $$D(A)$$ is independent of $$t$$. Suppose that the operator $(A(t)+\lambda I)^{-1}$ exists in $$B(E)$$ for any $$\lambda$$ with $$\text{Re\,}\lambda\geq 0$$ and \begin{aligned} &(\exists C> 0)(\forall t\in [0,T])(\forall\lambda\in \mathbb{C})(\|(A(t)+\lambda I)^{-1}\|\leq C(1+ |\lambda|)^{-1},\\ &(\exists C> 0)(\exists\gamma\in (0,1])(\forall(t_1, t_2,S)\in [0, T]^3)(\|(A(t_2)- A(t_1)(A^{-1}(s))\|\leq C|t_2- t_1|^\gamma),\\ &(\exists C> 0)(\exists\beta\in (0,1])(\forall(t_1, t_2)\in [0,T]^2)(\|(f(t_2)- f(t_1)\|\leq C|t_2- t_1|^\beta). \end{aligned} The author considers the fractional integral evolution equation $u(t)= u_0- (\Gamma(\alpha))^{-1} \int^t_0 (t-\theta)^{\alpha-1} (A(\theta) u(\theta)- f(\theta))\,d\theta,\tag{1}$ where $$\Gamma: (0,+\infty)\to \mathbb{C}$$ is the Gamma-function, constructs the fundamental solution of the homogeneous fractional differential equation ${d^\alpha v(t)\over dt^\alpha}+ A(t) v(t)= 0,\qquad t> 0,\tag{2}$ and proves the existence and uniqueness of the solution of (2) with the initial condition $$v(0)= u_0$$.
The author also proves the continuous dependence of the solutions of equation (1) on the element $$u_0$$ and the function $$f$$ and gives an application to a mixed problem of a parabolic partial differential equation of fractional order.
Reviewer: D. M. Bors (Iaşi)

##### MSC:
 34G10 Linear differential equations in abstract spaces 35K99 Parabolic equations and parabolic systems 45J05 Integro-ordinary differential equations 26A33 Fractional derivatives and integrals
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