Fractional order evolution equations. (English) Zbl 0839.34069

El-Sayed, A. M. A.

J. Fractional Calc. 7, 89-100 (1995).

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

Keywords:

fractional order; Cauchy problem; Banach space; existence; uniqueness; continuation; evolution equation; singular integro-differential equations