## Fractional order evolution equations.(English)Zbl 0839.34069

Summary: The author is concerned with the Cauchy problem of the equation $$d^\alpha u(t)/dt^\alpha = Au(t)$$, $$t > 0$$, $$0 < \alpha \leq 2$$, where $$A$$ is a closed linear operator defined on the Banach space $$X$$. The existence, uniqueness and some other properties of the solution are proved. The continuation of the solution and its derivative (when $$\alpha \to 1)$$ to the solution and its derivative of the Cauchy problem of the evolution equation $$du (t)/dt = Au (t)$$ are established. As application of the results some different problems of singular integro-differential equations are given.

### MSC:

 34G10 Linear differential equations in abstract spaces 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 26A33 Fractional derivatives and integrals 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations